现代分析化学（英文）：(ebook-pdf) - Science - Modern Analytic Chemistry

Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei Toronto Chemistry Modern Analytical Chemistry David Harvey DePauw University 1400-Fm 9/9/99 7:37 AM Page i MODERN ANALYTICAL CHEMISTRY Copyright ? 2000 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 KGP/KGP 0 9 8 7 6 5 4 3 2 1 0 ISBN 0–07–237547–7 Vice president and editorial director: Kevin T. Kane Publisher: James M. Smith Sponsoring editor: Kent A. Peterson Editorial assistant: Jennifer L. Bensink Developmental editor: Shirley R. Oberbroeckling Senior marketing manager: Martin J. Lange Senior project manager: Jayne Klein Production supervisor: Laura Fuller Coordinator of freelance design: Michelle D. Whitaker Senior photo research coordinator: Lori Hancock Senior supplement coordinator: Audrey A. Reiter Compositor: Shepherd, Inc. Typeface: 10/12 Minion Printer: Quebecor Printing Book Group/Kingsport Freelance cover/interior designer: Elise Lansdon Cover image: ? George Diebold/The Stock Market Photo research: Roberta Spieckerman Associates Colorplates: Colorplates 1–6, 8, 10: ? David Harvey/Marilyn E. Culler, photographer; Colorplate 7: Richard Megna/Fundamental Photographs; Colorplate 9: ? Alfred Pasieka/Science Photo Library/Photo Researchers, Inc.; Colorplate 11: From H. Black, Environ. Sci. Technol., 1996, 30, 124A. Photos courtesy D. Pesiri and W. Tumas, Los Alamos National Laboratory; Colorplate 12: Courtesy of Hewlett-Packard Company; Colorplate 13: ? David Harvey. Library of Congress Cataloging-in-Publication Data Harvey, David, 1956– Modern analytical chemistry / David Harvey. — 1st ed. p. cm. Includes bibliographical references and index. ISBN 0–07–237547–7 1. Chemistry, Analytic. I. Title. QD75.2.H374 2000 543—dc21 99–15120 CIP INTERNATIONAL EDITION ISBN 0–07–116953–9 Copyright ? 2000. Exclusive rights by The McGraw-Hill Companies, Inc. for manufacture and export. This book cannot be re-exported from the country to which it is consigned by McGraw-Hill. The International Edition is not available in North America. www.mhhe.com McGraw-Hill Higher Education A Division of The McGraw-Hill Companies 1400-Fm 9/9/99 7:37 AM Page ii iii ContentsContents Preface xii Chapter 1 Introduction 1 1A What is Analytical Chemistry? 2 1B The Analytical Perspective 5 1C Common Analytical Problems 8 1D Key Terms 9 1E Summary 9 1F Problems 9 1G Suggested Readings 10 1H References 10 Chapter 2 Basic Tools of Analytical Chemistry 11 2A Numbers in Analytical Chemistry 12 2A.1 Fundamental Units of Measure 12 2A.2 Significant Figures 13 2B Units for Expressing Concentration 15 2B.1 Molarity and Formality 15 2B.2 Normality 16 2B.3 Molality 18 2B.4 Weight, Volume, and Weight-to-Volume Ratios 18 2B.5 Converting Between Concentration Units 18 2B.6 p-Functions 19 2C Stoichiometric Calculations 20 2C.1 Conservation of Mass 22 2C.2 Conservation of Charge 22 2C.3 Conservation of Protons 22 2C.4 Conservation of Electron Pairs 23 2C.5 Conservation of Electrons 23 2C.6 Using Conservation Principles in Stoichiometry Problems 23 2D Basic Equipment and Instrumentation 25 2D.1 Instrumentation for Measuring Mass 25 2D.2 Equipment for Measuring Volume 26 2D.3 Equipment for Drying Samples 29 2E Preparing Solutions 30 2E.1 Preparing Stock Solutions 30 2E.2 Preparing Solutions by Dilution 31 2F The Laboratory Notebook 32 2G Key Terms 32 2H Summary 33 2I Problems 33 2J Suggested Readings 34 2K References 34 Chapter 3 The Language of Analytical Chemistry 35 3A Analysis, Determination, and Measurement 36 3B Techniques, Methods, Procedures, and Protocols 36 3C Classifying Analytical Techniques 37 3D Selecting an Analytical Method 38 3D.1 Accuracy 38 3D.2 Precision 39 3D.3 Sensitivity 39 3D.4 Selectivity 40 3D.5 Robustness and Ruggedness 42 3D.6 Scale of Operation 42 3D.7 Equipment, Time, and Cost 44 3D.8 Making the Final Choice 44 1400-Fm 9/9/99 7:37 AM Page iii iv Modern Analytical Chemistry 4E.4 Errors in Significance Testing 84 4F Statistical Methods for Normal Distributions 85 4F.1 Comparing – X to m 85 4F.2 Comparing s 2 to s 2 87 4F.3 Comparing Two Sample Variances 88 4F.4 Comparing Two Sample Means 88 4F.5 Outliers 93 4G Detection Limits 95 4H Key Terms 96 4I Summary 96 4J Suggested Experiments 97 4K Problems 98 4L Suggested Readings 102 4M References 102 Chapter 5 Calibrations, Standardizations, and Blank Corrections 104 5A Calibrating Signals 105 5B Standardizing Methods 106 5B.1 Reagents Used as Standards 106 5B.2 Single-Point versus Multiple-Point Standardizations 108 5B.3 External Standards 109 5B.4 Standard Additions 110 5B.5 Internal Standards 115 5C Linear Regression and Calibration Curves 117 5C.1 Linear Regression of Straight-Line Calibration Curves 118 5C.2 Unweighted Linear Regression with Errors in y 119 5C.3 Weighted Linear Regression with Errors in y 124 5C.4 Weighted Linear Regression with Errors in Both x and y 127 5C.5 Curvilinear and Multivariate Regression 127 5D Blank Corrections 128 5E Key Terms 130 5F Summary 130 5G Suggested Experiments 130 5H Problems 131 5I Suggested Readings 133 5J References 134 3E Developing the Procedure 45 3E.1 Compensating for Interferences 45 3E.2 Calibration and Standardization 47 3E.3 Sampling 47 3E.4 Validation 47 3F Protocols 48 3G The Importance of Analytical Methodology 48 3H Key Terms 50 3I Summary 50 3J Problems 51 3K Suggested Readings 52 3L References 52 Chapter 4 Evaluating Analytical Data 53 4A Characterizing Measurements and Results 54 4A.1 Measures of Central Tendency 54 4A.2 Measures of Spread 55 4B Characterizing Experimental Errors 57 4B.1 Accuracy 57 4B.2 Precision 62 4B.3 Error and Uncertainty 64 4C Propagation of Uncertainty 64 4C.1 A Few Symbols 65 4C.2 Uncertainty When Adding or Subtracting 65 4C.3 Uncertainty When Multiplying or Dividing 66 4C.4 Uncertainty for Mixed Operations 66 4C.5 Uncertainty for Other Mathematical Functions 67 4C.6 Is Calculating Uncertainty Actually Useful? 68 4D The Distribution of Measurements and Results 70 4D.1 Populations and Samples 71 4D.2 Probability Distributions for Populations 71 4D.3 Confidence Intervals for Populations 75 4D.4 Probability Distributions for Samples 77 4D.5 Confidence Intervals for Samples 80 4D.6 A Cautionary Statement 81 4E Statistical Analysis of Data 82 4E.1 Significance Testing 82 4E.2 Constructing a Significance Test 83 4E.3 One-Tailed and Two-Tailed Significance Tests 84 1400-Fm 9/9/99 7:37 AM Page iv Contents v Chapter 7 Obtaining and Preparing Samples for Analysis 179 7A The Importance of Sampling 180 7B Designing a Sampling Plan 182 7B.1 Where to Sample the Target Population 182 7B.2 What Type of Sample to Collect 185 7B.3 How Much Sample to Collect 187 7B.4 How Many Samples to Collect 191 7B.5 Minimizing the Overall Variance 192 7C Implementing the Sampling Plan 193 7C.1 Solutions 193 7C.2 Gases 195 7C.3 Solids 196 7D Separating the Analyte from Interferents 201 7E General Theory of Separation Efficiency 202 7F Classifying Separation Techniques 205 7F.1 Separations Based on Size 205 7F.2 Separations Based on Mass or Density 206 7F.3 Separations Based on Complexation Reactions (Masking) 207 7F.4 Separations Based on a Change of State 209 7F.5 Separations Based on a Partitioning Between Phases 211 7G Liquid–Liquid Extractions 215 7G.1 Partition Coefficients and Distribution Ratios 216 7G.2 Liquid–Liquid Extraction with No Secondary Reactions 216 7G.3 Liquid–Liquid Extractions Involving Acid–Base Equilibria 219 7G.4 Liquid–Liquid Extractions Involving Metal Chelators 221 7H Separation versus Preconcentration 223 7I Key Terms 224 7J Summary 224 7K Suggested Experiments 225 7L Problems 226 7M Suggested Readings 230 7N References 231 Chapter 6 Equilibrium Chemistry 135 6A Reversible Reactions and Chemical Equilibria 136 6B Thermodynamics and Equilibrium Chemistry 136 6C Manipulating Equilibrium Constants 138 6D Equilibrium Constants for Chemical Reactions 139 6D.1 Precipitation Reactions 139 6D.2 Acid–Base Reactions 140 6D.3 Complexation Reactions 144 6D.4 Oxidation–Reduction Reactions 145 6E Le Chatelier’s Principle 148 6F Ladder Diagrams 150 6F.1 Ladder Diagrams for Acid–Base Equilibria 150 6F.2 Ladder Diagrams for Complexation Equilibria 153 6F.3 Ladder Diagrams for Oxidation–Reduction Equilibria 155 6G Solving Equilibrium Problems 156 6G.1 A Simple Problem: Solubility of Pb(IO 3 ) 2 in Water 156 6G.2 A More Complex Problem: The Common Ion Effect 157 6G.3 Systematic Approach to Solving Equilibrium Problems 159 6G.4 pH of a Monoprotic Weak Acid 160 6G.5 pH of a Polyprotic Acid or Base 163 6G.6 Effect of Complexation on Solubility 165 6H Buffer Solutions 167 6H.1 Systematic Solution to Buffer Problems 168 6H.2 Representing Buffer Solutions with Ladder Diagrams 170 6I Activity Effects 171 6J Two Final Thoughts About Equilibrium Chemistry 175 6K Key Terms 175 6L Summary 175 6M Suggested Experiments 176 6N Problems 176 6O Suggested Readings 178 6P References 178 1400-Fm 9/9/99 7:38 AM Page v vi Modern Analytical Chemistry Chapter 8 Gravimetric Methods of Analysis 232 8A Overview of Gravimetry 233 8A.1 Using Mass as a Signal 233 8A.2 Types of Gravimetric Methods 234 8A.3 Conservation of Mass 234 8A.4 Why Gravimetry Is Important 235 8B Precipitation Gravimetry 235 8B.1 Theory and Practice 235 8B.2 Quantitative Applications 247 8B.3 Qualitative Applications 254 8B.4 Evaluating Precipitation Gravimetry 254 8C Volatilization Gravimetry 255 8C.1 Theory and Practice 255 8C.2 Quantitative Applications 259 8C.3 Evaluating Volatilization Gravimetry 262 8D Particulate Gravimetry 262 8D.1 Theory and Practice 263 8D.2 Quantitative Applications 264 8D.3 Evaluating Precipitation Gravimetry 265 8E Key Terms 265 8F Summary 266 8G Suggested Experiments 266 8H Problems 267 8I Suggested Readings 271 8J References 272 Chapter 9 Titrimetric Methods of Analysis 273 9A Overview of Titrimetry 274 9A.1 Equivalence Points and End Points 274 9A.2 Volume as a Signal 274 9A.3 Titration Curves 275 9A.4 The Buret 277 9B Titrations Based on Acid–Base Reactions 278 9B.1 Acid–Base Titration Curves 279 9B.2 Selecting and Evaluating the End Point 287 9B.3 Titrations in Nonaqueous Solvents 295 9B.4 Representative Method 296 9B.5 Quantitative Applications 298 9B.6 Qualitative Applications 308 9B.7 Characterization Applications 309 9B.8 Evaluation of Acid–Base Titrimetry 311 9C Titrations Based on Complexation Reactions 314 9C.1 Chemistry and Properties of EDTA 315 9C.2 Complexometric EDTA Titration Curves 317 9C.3 Selecting and Evaluating the End Point 322 9C.4 Representative Method 324 9C.5 Quantitative Applications 327 9C.6 Evaluation of Complexation Titrimetry 331 9D Titrations Based on Redox Reactions 331 9D.1 Redox Titration Curves 332 9D.2 Selecting and Evaluating the End Point 337 9D.3 Representative Method 340 9D.4 Quantitative Applications 341 9D.5 Evaluation of Redox Titrimetry 350 9E Precipitation Titrations 350 9E.1 Titration Curves 350 9E.2 Selecting and Evaluating the End Point 354 9E.3 Quantitative Applications 354 9E.4 Evaluation of Precipitation Titrimetry 357 9F Key Terms 357 9G Summary 357 9H Suggested Experiments 358 9I Problems 360 9J Suggested Readings 366 9K References 367 Chapter 10 Spectroscopic Methods of Analysis 368 10A Overview of Spectroscopy 369 10A.1 What Is Electromagnetic Radiation 369 10A.2 Measuring Photons as a Signal 372 10B Basic Components of Spectroscopic Instrumentation 374 10B.1 Sources of Energy 375 10B.2 Wavelength Selection 376 10B.3 Detectors 379 10B.4 Signal Processors 380 10C Spectroscopy Based on Absorption 380 10C.1 Absorbance of Electromagnetic Radiation 380 10C.2 Transmittance and Absorbance 384 10C.3 Absorbance and Concentration: Beer’s Law 385 1400-Fm 9/9/99 7:38 AM Page vi Contents vii 11B Potentiometric Methods of Analysis 465 11B.1 Potentiometric Measurements 466 11B.2 Reference Electrodes 471 11B.3 Metallic Indicator Electrodes 473 11B.4 Membrane Electrodes 475 11B.5 Quantitative Applications 485 11B.6 Evaluation 494 11C Coulometric Methods of Analysis 496 11C.1 Controlled-Potential Coulometry 497 11C.2 Controlled-Current Coulometry 499 11C.3 Quantitative Applications 501 11C.4 Characterization Applications 506 11C.5 Evaluation 507 11D Voltammetric Methods of Analysis 508 11D.1 Voltammetric Measurements 509 11D.2 Current in Voltammetry 510 11D.3 Shape of Voltammograms 513 11D.4 Quantitative and Qualitative Aspects of Voltammetry 514 11D.5 Voltammetric Techniques 515 11D.6 Quantitative Applications 520 11D.7 Characterization Applications 527 11D.8 Evaluation 531 11E Key Terms 532 11F Summary 532 11G Suggested Experiments 533 11H Problems 535 11I Suggested Readings 540 11J References 541 Chapter 12 Chromatographic and Electrophoretic Methods 543 12A Overview of Analytical Separations 544 12A.1 The Problem with Simple Separations 544 12A.2 A Better Way to Separate Mixtures 544 12A.3 Classifying Analytical Separations 546 12B General Theory of Column Chromatography 547 12B.1 Chromatographic Resolution 549 12B.2 Capacity Factor 550 12B.3 Column Selectivity 552 12B.4 Column Efficiency 552 10C.4 Beer’s Law and Multicomponent Samples 386 10C.5 Limitations to Beer’s Law 386 10D Ultraviolet-Visible and Infrared Spectrophotometry 388 10D.1 Instrumentation 388 10D.2 Quantitative Applications 394 10D.3 Qualitative Applications 402 10D.4 Characterization Applications 403 10D.5 Evaluation 409 10E Atomic Absorption Spectroscopy 412 10E.1 Instrumentation 412 10E.2 Quantitative Applications 415 10E.3 Evaluation 422 10F Spectroscopy Based on Emission 423 10G Molecular Photoluminescence Spectroscopy 423 10G.1 Molecular Fluorescence and Phosphorescence Spectra 424 10G.2 Instrumentation 427 10G.3 Quantitative Applications Using Molecular Luminescence 429 10G.4 Evaluation 432 10H Atomic Emission Spectroscopy 434 10H.1 Atomic Emission Spectra 434 10H.2 Equipment 435 10H.3 Quantitative Applications 437 10H.4 Evaluation 440 10I Spectroscopy Based on Scattering 441 10I.1 Origin of Scattering 441 10I.2 Turbidimetry and Nephelometry 441 10J Key Terms 446 10K Summary 446 10L Suggested Experiments 447 10M Problems 450 10N Suggested Readings 458 10O References 459 Chapter 11 Electrochemical Methods of Analysis 461 11A Classification of Electrochemical Methods 462 11A.1 Interfacial Electrochemical Methods 462 11A.2 Controlling and Measuring Current and Potential 462 1400-Fm 9/9/99 7:38 AM Page vii 12B.5 Peak Capacity 554 12B.6 Nonideal Behavior 555 12C Optimizing Chromatographic Separations 556 12C.1 Using the Capacity Factor to Optimize Resolution 556 12C.2 Using Column Selectivity to Optimize Resolution 558 12C.3 Using Column Efficiency to Optimize Resolution 559 12D Gas Chromatography 563 12D.1 Mobile Phase 563 12D.2 Chromatographic Columns 564 12D.3 Stationary Phases 565 12D.4 Sample Introduction 567 12D.5 Temperature Control 568 12D.6 Detectors for Gas Chromatography 569 12D.7 Quantitative Applications 571 12D.8 Qualitative Applications 575 12D.9 Representative Method 576 12D.10 Evaluation 577 12E High-Performance Liquid Chromatography 578 12E.1 HPLC Columns 578 12E.2 Stationary Phases 579 12E.3 Mobile Phases 580 12E.4 HPLC Plumbing 583 12E.5 Sample Introduction 584 12E.6 Detectors for HPLC 584 12E.7 Quantitative Applications 586 12E.8 Representative Method 588 12E.9 Evaluation 589 12F Liquid–Solid Adsorption Chromatography 590 12G Ion-Exchange Chromatography 590 12H Size-Exclusion Chromatography 593 12I Supercritical Fluid Chromatography 596 12J Electrophoresis 597 12J.1 Theory of Capillary Electrophoresis 598 12J.2 Instrumentation 601 12J.3 Capillary Electrophoresis Methods 604 12J.4 Representative Method 607 12J.5 Evaluation 609 12K Key Terms 609 12L Summary 610 12M Suggested Experiments 610 12N Problems 615 viii Modern Analytical Chemistry 12O Suggested Readings 620 12P References 620 Chapter 1 3 Kinetic Methods of Analysis 622 13A Methods Based on Chemical Kinetics 623 13A.1 Theory and Practice 624 13A.2 Instrumentation 634 13A.3 Quantitative Applications 636 13A.4 Characterization Applications 638 13A.5 Evaluation of Chemical Kinetic Methods 639 13B Radiochemical Methods of Analysis 642 13B.1 Theory and Practice 643 13B.2 Instrumentation 643 13B.3 Quantitative Applications 644 13B.4 Characterization Applications 647 13B.5 Evaluation 648 13C Flow Injection Analysis 649 13C.1 Theory and Practice 649 13C.2 Instrumentation 651 13C.3 Quantitative Applications 655 13C.4 Evaluation 658 13D Key Terms 658 13E Summary 659 13F Suggested Experiments 659 13G Problems 661 13H Suggested Readings 664 13I References 665 Chapter 1 4 Developing a Standard Method 666 14A Optimizing the Experimental Procedure 667 14A.1 Response Surfaces 667 14A.2 Searching Algorithms for Response Surfaces 668 14A.3 Mathematical Models of Response Surfaces 674 14B Verifying the Method 683 14B.1 Single-Operator Characteristics 683 14B.2 Blind Analysis of Standard Samples 683 14B.3 Ruggedness Testing 684 14B.4 Equivalency Testing 687 1400-Fm 9/9/99 7:38 AM Page viii Contents ix 15D Key Terms 721 15E Summary 722 15F Suggested Experiments 722 15G Problems 722 15H Suggested Readings 724 15I References 724 Appendixes Appendix 1A Single-Sided Normal Distribution 725 Appendix 1B t-Table 726 Appendix 1C F-Table 727 Appendix 1D Critical Values for Q-Test 728 Appendix 1E Random Number Table 728 Appendix 2 Recommended Reagents for Preparing Primary Standards 729 Appendix 3A Solubility Products 731 Appendix 3B Acid Dissociation Constants 732 Appendix 3C Metal–Ligand Formation Constants 739 Appendix 3D Standard Reduction Potentials 743 Appendix 3E Selected Polarographic Half-Wave Potentials 747 Appendix 4 Balancing Redox Reactions 748 Appendix 5 Review of Chemical Kinetics 750 Appendix 6 Countercurrent Separations 755 Appendix 7 Answers to Selected Problems 762 Glossary 769 Index 781 14C Validating the Method as a Standard Method 687 14C.1 Two-Sample Collaborative Testing 688 14C.2 Collaborative Testing and Analysis of Variance 693 14C.3 What Is a Reasonable Result for a Collaborative Study? 698 14D Key Terms 699 14E Summary 699 14F Suggested Experiments 699 14G Problems 700 14H Suggested Readings 704 14I References 704 Chapter 1 5 Quality Assurance 705 15A Quality Control 706 15B Quality Assessment 708 15B.1 Internal Methods of Quality Assessment 708 15B.2 External Methods of Quality Assessment 711 15C Evaluating Quality Assurance Data 712 15C.1 Prescriptive Approach 712 15C.2 Performance-Based Approach 714 1400-Fm 9/9/99 7:38 AM Page ix x Modern Analytical Chemistry A Guide to Using This Text . . . in Chapter Representative Methods Annotated methods of typical analytical procedures link theory with practice. The format encourages students to think about the design of the procedure and why it works. 246 Modern Analytical Chemistry Representative Methods An additional problem is encountered when the isolated solid is non- stoichiometric. For example, precipitating Mn 2+ as Mn(OH)2, followed by heating to produce the oxide, frequently produces a solid with a stoichiometry of MnOx, where x varies between 1 and 2. In this case the nonstoichiometric product results from the formation of a mixture of several oxides that differ in the oxidation state of manganese. Other nonstoichiometric compounds form as a result of lattice de- fects in the crystal structure. 6 Representative Method The best way to appreciate the importance of the theoreti- cal and practical details discussed in the previous section is to carefully examine the procedure for a typical precipitation gravimetric method. Although each method has its own unique considerations, the determination of Mg 2+ in water and waste- water by precipitating MgNH4PO4 6H2O and isolating Mg2P2O7 provides an in- structive example of a typical procedure. Method 8.1 Determination of Mg 2+ in Water and Wastewater 7 Description of Method. Magnesium is precipitated as MgNH4PO4 6H2O using (NH4)2HPO4 as the precipitant. The precipitate’s solubility in neutral solutions (0.0065 g/100 mL in pure water at 10 °C) is relatively high, but it is much less soluble in the presence of dilute ammonia (0.0003 g/100 mL in 0.6 M NH3). The precipitant is not very selective, so a preliminary separation of Mg 2+ from potential interferents is necessary. Calcium, which is the most significant interferent, is usually removed by its prior precipitation as the oxalate. The presence of excess ammonium salts from the precipitant or the addition of too much ammonia can lead to the formation of Mg(NH4)4(PO4)2, which is subsequently isolated as Mg(PO3)2 after drying. The precipitate is isolated by filtration using a rinse solution of dilute ammonia. After filtering, the precipitate is converted to Mg2P2O7 and weighed. Procedure. Transfer a sample containing no more than 60 mg of Mg 2+ into a 600-mL beaker. Add 2–3 drops of methyl red indicator, and, if necessary, adjust the volume to 150 mL. Acidify the solution with 6 M HCl, and add 10 mL of 30% w/v (NH4)2HPO4. After cooling, add concentrated NH3 dropwise, and while constantly stirring, until the methyl red indicator turns yellow (pH > 6.3). After stirring for 5 min, add 5 mL of concentrated NH3, and continue stirring for an additional 10 min. Allow the resulting solution and precipitate to stand overnight. Isolate the precipitate by filtration, rinsing with 5% v/v NH3. Dissolve the precipitate in 50 mL of 10% v/v HCl, and precipitate a second time following the same procedure. After filtering, carefully remove the filter paper by charring. Heat the precipitate at 500 °C until the residue is white, and then bring the precipitate to constant weight at 1100 °C. Questions 1. Why does the procedure call for a sample containing no more than 60 mg of qy There is a serious limitation, however, to an external standardization. The relationship between Sstand and CS in equation 5.3 is determined when the ana- lyte is present in the external standard’s matrix. In using an external standardiza- tion, we assume that any difference between the matrix of the standards and the sample’s matrix has no effect on the value of k. A proportional determinate error is introduced when differences between the two matrices cannot be ignored. This is shown in Figure 5.4, where the relationship between the signal and the amount of analyte is shown for both the sample’s matrix and the standard’s matrix. In this example, using a normal calibration curve results in a negative determinate error. When matrix problems are expected, an effort is made to match the matrix of the standards to that of the sample. This is known as matrix matching. When the sample’s matrix is unknown, the matrix effect must be shown to be negligi- ble, or an alternative method of standardization must be used. Both approaches are discussed in the following sections. 5 B. 4 Standard Additions The complication of matching the matrix of the standards to that of the sample can be avoided by conducting the standardization in the sample. This is known as the method of standard additions. The simplest version of a standard addi- tion is shown in Figure 5.5. A volume, Vo, of sample is diluted to a final volume, Vf, and the signal, Ssamp is measured. A second identical aliquot of sample is matrix matching Adjusting the matrix of an external standard so that it is the same as the matrix of the samples to be analyzed. method of standard additions A standardization in which aliquots of a standard solution are added to the sample. Examples of Typical Problems Each example problem includes a detailed solution that helps students in applying the chapter’s material to practical problems. Margin Notes Margin notes direct students to colorplates located toward the middle of the book Bold-faced Key Terms with Margin Definitions Key words appear in boldface when they are introduced within the text. The term and its definition appear in the margin for quick review by the student. All key words are also defined in the glossary. 110 Modern Analytical Chemistry either case, the calibration curve provides a means for relating Ssamp to the ana- lyte’s concentration. EXAMPLE 5 . 3 A second spectrophotometric method for the quantitative determination of Pb 2+ levels in blood gives a linear normal calibration curve for which Sstand = (0.296 ppb –1 ) · CS + 0.003 What is the Pb 2+ level (in ppb) in a sample of blood if Ssamp is 0.397? SOLUTION To determine the concentration of Pb 2+ in the sample of blood, we replace Sstand in the calibration equation with Ssamp and solve for CA It is worth noting that the calibration equation in this problem includes an extra term that is not in equation 5.3. Ideally, we expect the calibration curve to give a signal of zero when CS is zero. This is the purpose of using a reagent blank to correct the measured signal. The extra term of +0.003 in our calibration equation results from uncertainty in measuring the signal for the reagent blank and the standards. An external standardization allows a related series of samples to be analyzed using a single calibration curve. This is an important advantage in laboratories where many samples are to be analyzed or when the need for a rapid throughput of l i iti l t i i l f th t l t d C S A samp ppb === –. . .–. . . – 0 003 0 296 0 397 0 003 0 296 133 1 ppb ppb –1 Color plate 1 shows an example of a set of external standards and their corresponding normal calibration curve. x 1400-Fm 9/9/99 7:38 AM Page x List of Key Terms The key terms introduced within the chapter are listed at the end of each chapter. Page references direct the student to the definitions in the text. Summary The summary provides the student with a brief review of the important concepts within the chapter. Suggested Experiments An annotated list of representative experiments is provided from the Journal of Chemical Education. . . . End of Chapter yy 5 E KEY TERMS aliquot (p. 111) external standard (p. 109) internal standard (p. 116) linear regression (p. 118) matrix matching (p. 110) method of standard additions (p. 110) multiple-point standardization (p. 109) normal calibration curve (p. 109) primary reagent (p. 106) reagent grade (p. 107) residual error (p. 118) secondary reagent (p. 107) single-point standardization (p. 108) standard deviation about the regression (p. 121) total Youden blank (p. 129) In a quantitative analysis, we measure a signal and calculate the amount of analyte using one of the following equations. Smeas = knA + Sreag Smeas = kCA + Sreag To obtain accurate results we must eliminate determinate errors affecting the measured signal, Smeas, the method’s sensitivity, k, and any signal due to the reagents, Sreag. To ensure that Smeas is determined accurately, we calibrate the equipment or instrument used to obtain the signal. Balances are calibrated using standard weights. When necessary, we can also correct for the buoyancy of air. Volumetric glassware can be calibrated by measuring the mass of water contained or de- livered and using the density of water to calculate the true vol- ume. Most instruments have calibration standards suggested by the manufacturer. An analytical method is standardized by determining its sensi- tivity. There are several approaches to standardization, including the use of external standards, the method of standard addition, and the use of an internal standard. The most desirable standard- ization strategy is an external standardization. The method of standard additions, in which known amounts of analyte are added to the sample, is used when the sample’s matrix complicates the analysis. An internal standard, which is a species (not analyte) added to all samples and standards, is used when the procedure does not allow for the reproducible handling of samples and standards. Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple- point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regres- sion analysis can provide an equation for the standardization. A reagent blank corrects the measured signal for signals due to reagents other than the sample that are used in an analysis. The most common reagent blank is prepared by omitting the sample. When a simple reagent blank does not compensate for all constant sources of determinate error, other types of blanks, such as the total Youden blank, can be used. 5 F SUMMARY Calibration—Volumetric glassware (burets, pipets, and volumetric flasks) can be calibrated in the manner described in Example 5.1. Most instruments have a calibration sample that can be prepared to verify the instrument’s accuracy and precision. For example, as described in this chapter, a solution of 60.06 ppm K2Cr2O7 in 0.0050 M H2SO4 should give an absorbance of 0.640 ± 0.010 at a wavelength of 350.0 nm when using 0.0050 M H2SO4 as a reagent blank. These exercises also provide practice with using volumetric glassware, weighing samples, and preparing solutions. Standardization—External standards, standard additions, and internal standards are a common feature of many quantitative analyses. Suggested experiments using these standardization methods are found in later chapters. A good project experiment for introducing external standardization, standard additions, and the importance of the sample’s matrix is to explore the effect of pH on the quantitative analysis of an acid–base indicator. Using bromothymol blue as an example, external standards can be prepared in a pH 9 buffer and used to analyze samples buffered to different pHs in the range of 6–10. Results can be compared with those obtained using a standard addition. 5 G Suggested EXPERIMENTS The following exercises and experiments help connect the material in this chapter to the analytical laboratory. Experiments 1. When working with a solid sample, it often is necessary to bring the analyte into solution by dissolving the sample in a suitable solvent. Any solid impurities that remain are removed by filtration before continuing with the analysis. In a typical total analysis method, the procedure might read After dissolving the sample in a beaker, remove any solid impurities by passing the solution containing the analyte through filter paper, collecting the solution in a clean Erlenmeyer flask. Rinse the beaker with several small portions of solvent, passing these rinsings through the filter paper, and collecting them in the same Erlenmeyer flask. Finally, rinse the filter paper with several portions of solvent, collecting the rinsings in the same Erlenmeyer flask. For a typical concentration method, however, the procedure might state 4. A sample was analyzed to determine the concentration of an analyte. Under the conditions of the analysis, the sensitivity is 17.2 ppm –1 . What is the analyte’s concentration if Smeas is 35.2 and Sreag is 0.6? 5. A method for the analysis of Ca 2+ in water suffers from an interference in the presence of Zn 2+ . When the concentration of Ca 2+ is 50 times greater than that of Zn 2+ , an analysis for Ca 2+ gives a relative error of –2.0%. What is the value of the selectivity coefficient for this method? 6. The quantitative analysis for reduced glutathione in blood is complicated by the presence of many potential interferents. In one study, when analyzing a solution of 10-ppb glutathione and 1.5-ppb ascorbic acid, the signal was 5.43 times greater than that obtained for the analysis of 10-ppb glutathione. 12 What is the selectivity coefficient for this analysis? The same study found that when analyzing a solution of 350-ppb methionine and 10-ppb glutathione the signal was 0 906 times less than that obtained for the analysis 3J PROBLEMS yy The role of analytical chemistry within the broader discipline of chemistry has been discussed by many prominent analytical chemists. Several notable examples follow. Baiulescu, G. E.; Patroescu, C.; Chalmers, R. A. Education and Teaching in Analytical Chemistry. Ellis Horwood: Chichester, 1982. Hieftje, G. M. “The Two Sides of Analytical Chemistry,” Anal. Chem. 1985, 57, 256A–267A. Kissinger, P. T. “Analytical Chemistry—What is It? Who Needs It? Why Teach It?” Trends Anal. Chem. 1992, 11, 54–57. Laitinen, H. A. “Analytical Chemistry in a Changing World,” Anal. Chem. 1980, 52, 605A–609A. Laitinen, H. A. “History of Analytical Chemistry in the U.S.A.,” Talanta 1989, 36, 1–9. Laitinen, H. A.; Ewing, G. (eds). A History of Analytical Chemistry. The Division of Analytical Chemistry of the American Chemical Society: Washington, D.C., 1972. McLafferty, F. W. “Analytical Chemistry: Historic and Modern,” Acc. Chem. Res. 1990, 23, 63–64. 1G SUGGESTED READINGS 1. Ravey, M. Spectroscopy 1990, 5(7), 11. 2. de Haseth, J. Spectroscopy 1990, 5(7), 11. 3. Fresenius, C. R. A System of Instruction in Quantitative Chemical Analysis. John Wiley and Sons: New York, 1881. 4. Hillebrand, W. F.; Lundell, G. E. F. Applied Inorganic Analysis, John Wiley and Sons: New York, 1953. 5. Van Loon, J. C. Analytical Atomic Absorption Spectroscopy. Academic Press: New York, 1980. 6. Murray, R. W. Anal. Chem. 1991, 63, 271A. 7. For several different viewpoints see (a) Beilby, A. L. J. Chem. Educ. 1970, 47, 237–238; (b) Lucchesi, C. A. Am. Lab. 1980, October, 113–119; (c) Atkinson, G. F. J. Chem. Educ. 1982, 59, 201–202; (d) Pardue, H. L.; Woo, J. J. Chem. Educ. 1984, 61, 409–412; (e) Guarnieri, M. J. Chem. Educ. 1988, 65, 201–203; (f) de Haseth, J. Spectroscopy 1990, 5, 20–21; (g) Strobel, H. A. Am. Lab. 1990, October, 17–24. 8. Hieftje, G. M. Am. Lab. 1993, October, 53–61. 9. See, for example, the following laboratory texts: (a) Sorum, C. H.; Lagowski, J. J. Introduction to Semimicro Qualitative Analysis, 5th ed. Prentice-Hall: Englewood Cliffs, NJ, 1977.; (b) Shriner, R. L.; Fuson, R. C.; Curtin, D. Y. The Systematic Identification of Organic Compounds, 5th ed. John Wiley and Sons: New York, 1964. 1H REFERENCES Problems A variety of problems, many based on data from the analytical literature, provide the student with practical examples of current research. Suggested Readings Suggested readings give the student access to more comprehensive discussion of the topics introduced within the chapter. References The references cited in the chapter are provided so the student can access them for further information. xi 1400-Fm 9/9/99 7:38 AM Page xi As currently taught, the introductory course in analytical chemistry emphasizes quantitative (and sometimes qualitative) methods of analysis coupled with a heavy dose of equilibrium chemistry. Analytical chemistry, however, is more than equilib- rium chemistry and a collection of analytical methods; it is an approach to solving chemical problems. Although discussing different methods is important, that dis- cussion should not come at the expense of other equally important topics. The intro- ductory analytical course is the ideal place in the chemistry curriculum to explore topics such as experimental design, sampling, calibration strategies, standardization, optimization, statistics, and the validation of experimental results. These topics are important in developing good experimental protocols, and in interpreting experi- mental results. If chemistry is truly an experimental science, then it is essential that all chemistry students understand how these topics relate to the experiments they conduct in other chemistry courses. Currently available textbooks do a good job of covering the diverse range of wet and instrumental analysis techniques available to chemists. Although there is some disagreement about the proper balance between wet analytical techniques, such as gravimetry and titrimetry, and instrumental analysis techniques, such as spec- trophotometry, all currently available textbooks cover a reasonable variety of tech- niques. These textbooks, however, neglect, or give only brief consideration to, obtaining representative samples, handling interferents, optimizing methods, ana- lyzing data, validating data, and ensuring that data are collected under a state of sta- tistical control. In preparing this textbook, I have tried to find a more appropriate balance between theory and practice, between “classical” and “modern” methods of analysis, between analyzing samples and collecting and preparing samples for analysis, and between analytical methods and data analysis. Clearly, the amount of material in this textbook exceeds what can be covered in a single semester; it’s my hope, however, that the diversity of topics will meet the needs of different instructors, while, per- haps, suggesting some new topics to cover. The anticipated audience for this textbook includes students majoring in chem- istry, and students majoring in other science disciplines (biology, biochemistry, environmental science, engineering, and geology, to name a few), interested in obtaining a stronger background in chemical analysis. It is particularly appropriate for chemistry majors who are not planning to attend graduate school, and who often do not enroll in those advanced courses in analytical chemistry that require physical chemistry as a pre-requisite. Prior coursework of a year of general chemistry is assumed. Competence in algebra is essential; calculus is used on occasion, however, its presence is not essential to the material’s treatment. xii Preface Preface 1400-Fm 9/9/99 7:38 AM Page xii Preface xiii Key Features of This Textbook Key features set this textbook apart from others currently available. ? A stronger emphasis on the evaluation of data. Methods for characterizing chemical measurements, results, and errors (including the propagation of errors) are included. Both the binomial distribution and normal distribution are presented, and the idea of a confidence interval is developed. Statistical methods for evaluating data include the t-test (both for paired and unpaired data), the F-test, and the treatment of outliers. Detection limits also are discussed from a statistical perspective. Other statistical methods, such as ANOVA and ruggedness testing, are presented in later chapters. ? Standardizations and calibrations are treated in a single chapter. Selecting the most appropriate calibration method is important and, for this reason, the methods of external standards, standard additions, and internal standards are gathered together in a single chapter. A discussion of curve-fitting, including the statistical basis for linear regression (with and without weighting) also is included in this chapter. ? More attention to selecting and obtaining a representative sample. The design of a statistically based sampling plan and its implementation are discussed earlier, and in more detail than in other textbooks. Topics that are covered include how to obtain a representative sample, how much sample to collect, how many samples to collect, how to minimize the overall variance for an analytical method, tools for collecting samples, and sample preservation. ? The importance of minimizing interferents is emphasized. Commonly used methods for separating interferents from analytes, such as distillation, masking, and solvent extraction, are gathered together in a single chapter. ? Balanced coverage of analytical techniques. The six areas of analytical techniques—gravimetry, titrimetry, spectroscopy, electrochemistry, chromatography, and kinetics—receive roughly equivalent coverage, meeting the needs of instructors wishing to emphasize wet methods and those emphasizing instrumental methods. Related methods are gathered together in a single chapter encouraging students to see the similarities between methods, rather than focusing on their differences. ? An emphasis on practical applications. Throughout the text applications from organic chemistry, inorganic chemistry, environmental chemistry, clinical chemistry, and biochemistry are used in worked examples, representative methods, and end-of-chapter problems. ? Representative methods link theory with practice. An important feature of this text is the presentation of representative methods. These boxed features present typical analytical procedures in a format that encourages students to think about why the procedure is designed as it is. ? Separate chapters on developing a standard method and quality assurance. Two chapters provide coverage of methods used in developing a standard method of analysis, and quality assurance. The chapter on developing a standard method includes topics such as optimizing experimental conditions using response surfaces, verifying the method through the blind analysis of standard samples and ruggedness testing, and collaborative testing using Youden’s two-sample approach and ANOVA. The chapter on quality assurance covers quality control and internal and external techniques for quality assessment, including the use of duplicate samples, blanks, spike recoveries, and control charts. 1400-Fm 9/9/99 7:38 AM Page xiii ? Problems adapted from the literature. Many of the in-chapter examples and end- of-chapter problems are based on data from the analytical literature, providing students with practical examples of current research in analytical chemistry. ? An emphasis on critical thinking. Critical thinking is encouraged through problems in which students are asked to explain why certain steps in an analytical procedure are included, or to determine the effect of an experimental error on the results of an analysis. ? Suggested experiments from the Journal of Chemical Education. Rather than including a short collection of experiments emphasizing the analysis of standard unknowns, an annotated list of representative experiments from the Journal of Chemical Education is included at the conclusion of most chapters. These experiments may serve as stand alone experiments, or as starting points for individual or group projects. The Role of Equilibrium Chemistry in Analytical Chemistry Equilibrium chemistry often receives a significant emphasis in the introductory ana- lytical chemistry course. While an important topic, its overemphasis can cause stu- dents to confuse analytical chemistry with equilibrium chemistry. Although atten- tion to solving equilibrium problems is important, it is equally important for stu- dents to recognize when such calculations are impractical, or when a simpler, more qualitative approach is all that is needed. For example, in discussing the gravimetric analysis of Ag + as AgCl, there is little point in calculating the equilibrium solubility of AgCl since the concentration of Cl – at equilibrium is rarely known. It is impor- tant, however, to qualitatively understand that a large excess of Cl – increases the sol- ubility of AgCl due to the formation of soluble silver-chloro complexes. Balancing the presentation of a rigorous approach to solving equilibrium problems, this text also introduces the use of ladder diagrams as a means for providing a qualitative pic- ture of a system at equilibrium. Students are encouraged to use the approach best suited to the problem at hand. Computer Software Many of the topics covered in analytical chemistry benefit from the availability of appropriate computer software. In preparing this text, however, I made a conscious decision to avoid a presentation tied to a single computer platform or software pack- age. Students and faculty are increasingly experienced in the use of computers, spreadsheets, and data analysis software; their use is, I think, best left to the person- al choice of each student and instructor. Organization The textbook’s organization can be divided into four parts. Chapters 1–3 serve as an introduction, providing an overview of analytical chemistry (Chapter 1); a review of the basic tools of analytical chemistry, including significant figures, units, and stoi- chiometry (Chapter 2); and an introduction to the terminology used by analytical chemists (Chapter 3). Familiarity with the material in these chapters is assumed throughout the remainder of the text. Chapters 4–7 cover a number of topics that are important in understanding how a particular analytical method works. Later chapters are mostly independent of the material in these chapters. Instructors may pick and choose from among the topics xiv Preface 1400-Fm 9/9/99 7:38 AM Page xiv Preface xv of these chapters, as needed, to support individual course goals. The statistical analy- sis of data is covered in Chapter 4 at a level that is more complete than that found in other introductory analytical textbooks. Methods for calibrating equipment, stan- dardizing methods, and linear regression are gathered together in Chapter 5. Chapter 6 provides an introduction to equilibrium chemistry, stressing both the rigorous solution to equilibrium problems, and the use of semi-quantitative approaches, such as ladder diagrams. The importance of collecting the right sample, and methods for separating analytes and interferents are covered in Chapter 7. Chapters 8–13 cover the major areas of analysis, including gravimetry (Chapter 8), titrimetry (Chapter 9), spectroscopy (Chapter 10), electrochemistry (Chapter 11), chromatography and electrophoresis (Chapter 12), and kinetic meth- ods (Chapter 13). Related techniques, such as acid–base titrimetry and redox titrimetry, or potentiometry and voltammetry, are gathered together in single chap- ters. Combining related techniques together encourages students to see the similar- ities between methods, rather than focusing on their differences. The first technique presented in each chapter is generally that which is most commonly covered in the introductory course. Finally, the textbook concludes with two chapters discussing the design and maintenance of analytical methods, two topics of importance to analytical chemists. Chapter 14 considers the development of an analytical method, including its opti- mization, verification, and validation. Quality control and quality assessment are discussed in Chapter 15. Acknowledgments Before beginning an academic career I was, of course, a student. My interest in chemistry and teaching was nurtured by many fine teachers at Westtown Friends School, Knox College, and the University of North Carolina at Chapel Hill; their col- lective influence continues to bear fruit. In particular, I wish to recognize David MacInnes, Alan Hiebert, Robert Kooser, and Richard Linton. I have been fortunate to work with many fine colleagues during my nearly 17 years of teaching undergraduate chemistry at Stockton State College and DePauw University. I am particularly grateful for the friendship and guidance provided by Jon Griffiths and Ed Paul during my four years at Stockton State College. At DePauw University, Jim George and Bryan Hanson have willingly shared their ideas about teaching, while patiently listening to mine. Approximately 300 students have joined me in thinking and learning about ana- lytical chemistry; their questions and comments helped guide the development of this textbook. I realize that working without a formal textbook has been frustrating and awkward; all the more reason why I appreciate their effort and hard work. The following individuals reviewed portions of this textbook at various stages during its development. David Ballantine Northern Illinois University John E. Bauer Illinois State University Ali Bazzi University of Michigan–Dearborn Steven D. Brown University of Delaware Wendy Clevenger University of Tennessee–Chattanooga Cathy Cobb Augusta State University Paul Flowers University of North Carolina–Pembroke Nancy Gordon University of Southern Maine 1400-Fm 9/9/99 7:38 AM Page xv Virginia M. Indivero Swarthmore College Michael Janusa Nicholls State University J. David Jenkins Georgia Southern University Richard S. Mitchell Arkansas State University George A. Pearse, Jr. Le Moyne College Gary Rayson New Mexico State University David Redfield NW Nazarene University I am particularly grateful for their detailed written comments and suggestions for improving the manuscript. Much of what is good in the final manuscript is the result of their interest and ideas. George Foy (York College of Pennsylvania), John McBride (Hofstra University), and David Karpovich (Saginaw Valley State University) checked the accuracy of problems in the textbook. Gary Kinsel (University of Texas at Arlington) reviewed the page proofs and provided additional suggestions. This project began in the summer of 1992 with the support of a course develop- ment grant from DePauw University’s Faculty Development Fund. Additional finan- cial support from DePauw University’s Presidential Discretionary Fund also is acknowledged. Portions of the first draft were written during a sabbatical leave in the Fall semester of the 1993/94 academic year. A Fisher Fellowship provided release time during the Fall 1995 semester to complete the manuscript’s second draft. Alltech and Associates (Deerfield, IL) graciously provided permission to use the chromatograms in Chapter 12; the assistance of Jim Anderson, Vice-President, and Julia Poncher, Publications Director, is greatly appreciated. Fred Soster and Marilyn Culler, both of DePauw University, provided assistance with some of the photographs. The editorial staff at McGraw-Hill has helped guide a novice through the process of developing this text. I am particularly thankful for the encouragement and confidence shown by Jim Smith, Publisher for Chemistry, and Kent Peterson, Sponsoring Editor for Chemistry. Shirley Oberbroeckling, Developmental Editor for Chemistry, and Jayne Klein, Senior Project Manager, patiently answered my ques- tions and successfully guided me through the publishing process. Finally, I would be remiss if I did not recognize the importance of my family’s support and encouragement, particularly that of my parents. A very special thanks to my daughter, Devon, for gifts too numerous to detail. How to Contact the Author Writing this textbook has been an interesting (and exhausting) challenge. Despite my efforts, I am sure there are a few glitches, better examples, more interesting end- of-chapter problems, and better ways to think about some of the topics. I welcome your comments, suggestions, and data for interesting problems, which may be addressed to me at DePauw University, 602 S. College St., Greencastle, IN 46135, or electronically at harvey@depauw.edu. xvi Preface Vincent Remcho West Virginia University Jeanette K. Rice Georgia Southern University Martin W. Rowe Texas A&M University Alexander Scheeline University of Illinois James D. Stuart University of Connecticut Thomas J. Wenzel Bates College David Zax Cornell University 1400-Fm 9/9/99 7:38 AM Page xvi Chapter 1 1 Introduction Chemistry is the study of matter, including its composition, structure, physical properties, and reactivity. There are many approaches to studying chemistry, but, for convenience, we traditionally divide it into five fields: organic, inorganic, physical, biochemical, and analytical. Although this division is historical and arbitrary, as witnessed by the current interest in interdisciplinary areas such as bioanalytical and organometallic chemistry, these five fields remain the simplest division spanning the discipline of chemistry. Training in each of these fields provides a unique perspective to the study of chemistry. Undergraduate chemistry courses and textbooks are more than a collection of facts; they are a kind of apprenticeship. In keeping with this spirit, this text introduces the field of analytical chemistry and the unique perspectives that analytical chemists bring to the study of chemistry. 1400-CH01 9/9/99 2:20 PM Page 1 2 Modern Analytical Chemistry *Attributed to C. N. Reilley (1925–1981) on receipt of the 1965 Fisher Award in Analytical Chemistry. Reilley, who was a professor of chemistry at the University of North Carolina at Chapel Hill, was one of the most influential analytical chemists of the last half of the twentieth century. 1A What Is Analytical Chemistry? “Analytical chemistry is what analytical chemists do.”* We begin this section with a deceptively simple question. What is analytical chem- istry? Like all fields of chemistry, analytical chemistry is too broad and active a disci- pline for us to easily or completely define in an introductory textbook. Instead, we will try to say a little about what analytical chemistry is, as well as a little about what analytical chemistry is not. Analytical chemistry is often described as the area of chemistry responsible for characterizing the composition of matter, both qualitatively (what is present) and quantitatively (how much is present). This description is misleading. After all, al- most all chemists routinely make qualitative or quantitative measurements. The ar- gument has been made that analytical chemistry is not a separate branch of chem- istry, but simply the application of chemical knowledge. 1 In fact, you probably have performed quantitative and qualitative analyses in other chemistry courses. For ex- ample, many introductory courses in chemistry include qualitative schemes for identifying inorganic ions and quantitative analyses involving titrations. Unfortunately, this description ignores the unique perspective that analytical chemists bring to the study of chemistry. The craft of analytical chemistry is not in performing a routine analysis on a routine sample (which is more appropriately called chemical analysis), but in improving established methods, extending existing methods to new types of samples, and developing new methods for measuring chemical phenomena. 2 Here’s one example of this distinction between analytical chemistry and chemi- cal analysis. Mining engineers evaluate the economic feasibility of extracting an ore by comparing the cost of removing the ore with the value of its contents. To esti- mate its value they analyze a sample of the ore. The challenge of developing and val- idating the method providing this information is the analytical chemist’s responsi- bility. Once developed, the routine, daily application of the method becomes the job of the chemical analyst. Another distinction between analytical chemistry and chemical analysis is that analytical chemists work to improve established methods. For example, sev- eral factors complicate the quantitative analysis of Ni 2+ in ores, including the presence of a complex heterogeneous mixture of silicates and oxides, the low con- centration of Ni 2+ in ores, and the presence of other metals that may interfere in the analysis. Figure 1.1 is a schematic outline of one standard method in use dur- ing the late nineteenth century. 3 After dissolving a sample of the ore in a mixture of H 2 SO 4 and HNO 3 , trace metals that interfere with the analysis, such as Pb 2+ , Cu 2+ and Fe 3+ , are removed by precipitation. Any cobalt and nickel in the sample are reduced to Co and Ni, isolated by filtration and weighed (point A). After dissolving the mixed solid, Co is isolated and weighed (point B). The amount of nickel in the ore sample is determined from the difference in the masses at points A and B. %Ni = mass point A – mass point B mass sample × 100 1400-CH01 9/9/99 2:20 PM Page 2 Chapter 1 Introduction 3 Original Sample PbSO 4 Sand Basic ferric acetate CuS 1:3 H 2 SO 4 /HNO 3 100°C (8–10 h) dilute w/H 2 O, digest 2–4 h Cu 2+ , Fe 3+ Co 2+ , Ni 2+ Fe 3+ , Co 2+ , Ni 2+ Fe(OH) 3 CoS, NiS CuS, PbS Co(OH) 2 , Ni(OH) 2 CoO, NiO cool, add NH 3 digest 50°–70°, 30 min Co 2+ , Ni 2+ Fe 3+ Waste Waste Co 2+ , Ni 2+ aqua regia heat, add HCl until strongly acidic bubble H 2 S (g) WasteCo 2+ Solid Key Solution H 2 O, HCl heat add Na 2 CO 3 until alkaline NaOH K 3 Co(NO 3 ) 5 Ni 2+ neutralize w/ NH 3 Na 2 CO 3 , CH 3 COOH slightly acidify w/ HCl heat, bubble H 2 S (g) HCl heat Co as above Co, Ni heat, H 2 (g) HNO 3 K 2 CO 3 , KNO 3 CH 3 COOH digest 24 h dilute bubble H 2 S(g) A B Figure 1.1 Analytical scheme outlined by Fresenius 3 for the gravimetric analysis of Ni in ores. 1400-CH01 9/9/99 2:20 PM Page 3 The combination of determining the mass of Ni 2+ by difference, coupled with the need for many reactions and filtrations makes this procedure both time-consuming and difficult to perform accurately. The development, in 1905, of dimethylgloxime (DMG), a reagent that selec- tively precipitates Ni 2+ and Pd 2+ , led to an improved analytical method for deter- mining Ni 2+ in ores. 4 As shown in Figure 1.2, the mass of Ni 2+ is measured directly, requiring fewer manipulations and less time. By the 1970s, the standard method for the analysis of Ni 2+ in ores progressed from precipitating Ni(DMG) 2 to flame atomic absorption spectrophotometry, 5 resulting in an even more rapid analysis. Current interest is directed toward using inductively coupled plasmas for determin- ing trace metals in ores. In summary, a more appropriate description of analytical chemistry is “. . . the science of inventing and applying the concepts, principles, and . . . strategies for measuring the characteristics of chemical systems and species.” 6 Analytical chemists typically operate at the extreme edges of analysis, extending and improving the abil- ity of all chemists to make meaningful measurements on smaller samples, on more complex samples, on shorter time scales, and on species present at lower concentra- tions. Throughout its history, analytical chemistry has provided many of the tools and methods necessary for research in the other four traditional areas of chemistry, as well as fostering multidisciplinary research in, to name a few, medicinal chem- istry, clinical chemistry, toxicology, forensic chemistry, material science, geochem- istry, and environmental chemistry. 4 Modern Analytical Chemistry Original sample Residue Ni(DMG) 2 (s) HNO 3 , HCl, heat Solution Solid Key Solution 20% NH 4 Cl 10% tartaric acid take alkaline with 1:1 NH 3 Ye s No A take acid with HCl 1% alcoholic DMG take alkaline with 1:1 NH 3 take acid with HCl 10% tartaric acid take alkaline with 1:1 NH 3 Is solid present? %Ni = · 100 mass A · 0.2031 g sample Figure 1.2 Analytical scheme outlined by Hillebrand and Lundell 4 for the gravimetric analysis of Ni in ores (DMG = dimethylgloxime). The factor of 0.2031 in the equation for %Ni accounts for the difference in the formula weights of Ni(DMG) 2 and Ni; see Chapter 8 for more details. 1400-CH01 9/9/99 2:20 PM Page 4 Chapter 1 Introduction 5 You will come across numerous examples of qualitative and quantitative meth- ods in this text, most of which are routine examples of chemical analysis. It is im- portant to remember, however, that nonroutine problems prompted analytical chemists to develop these methods. Whenever possible, we will try to place these methods in their appropriate historical context. In addition, examples of current re- search problems in analytical chemistry are scattered throughout the text. The next time you are in the library, look through a recent issue of an analyti- cally oriented journal, such as Analytical Chemistry. Focus on the titles and abstracts of the research articles. Although you will not recognize all the terms and methods, you will begin to answer for yourself the question “What is analytical chemistry”? 1B The Analytical Perspective Having noted that each field of chemistry brings a unique perspective to the study of chemistry, we now ask a second deceptively simple question. What is the “analyt- ical perspective”? Many analytical chemists describe this perspective as an analytical approach to solving problems. 7 Although there are probably as many descriptions of the analytical approach as there are analytical chemists, it is convenient for our purposes to treat it as a five-step process: 1. Identify and define the problem. 2. Design the experimental procedure. 3. Conduct an experiment, and gather data. 4. Analyze the experimental data. 5. Propose a solution to the problem. Figure 1.3 shows an outline of the analytical approach along with some im- portant considerations at each step. Three general features of this approach de- serve attention. First, steps 1 and 5 provide opportunities for analytical chemists to collaborate with individuals outside the realm of analytical chemistry. In fact, many problems on which analytical chemists work originate in other fields. Sec- ond, the analytical approach is not linear, but incorporates a “feedback loop” consisting of steps 2, 3, and 4, in which the outcome of one step may cause a reevaluation of the other two steps. Finally, the solution to one problem often suggests a new problem. Analytical chemistry begins with a problem, examples of which include evalu- ating the amount of dust and soil ingested by children as an indicator of environ- mental exposure to particulate based pollutants, resolving contradictory evidence regarding the toxicity of perfluoro polymers during combustion, or developing rapid and sensitive detectors for chemical warfare agents.* At this point the analyti- cal approach involves a collaboration between the analytical chemist and the indi- viduals responsible for the problem. Together they decide what information is needed. It is also necessary for the analytical chemist to understand how the prob- lem relates to broader research goals. The type of information needed and the prob- lem’s context are essential to designing an appropriate experimental procedure. Designing an experimental procedure involves selecting an appropriate method of analysis based on established criteria, such as accuracy, precision, sensitivity, and detection limit; the urgency with which results are needed; the cost of a single analy- sis; the number of samples to be analyzed; and the amount of sample available for *These examples are taken from a series of articles, entitled the “Analytical Approach,” which has appeared as a regular feature in the journal Analytical Chemistry since 1974. 1400-CH01 9/9/99 2:20 PM Page 5 Figure 1.3 Flow diagram for the analytical approach to solving problems; modified after Atkinson. 7c analysis. Finding an appropriate balance between these parameters is frequently complicated by their interdependence. For example, improving the precision of an analysis may require a larger sample. Consideration is also given to collecting, stor- ing, and preparing samples, and to whether chemical or physical interferences will affect the analysis. Finally, a good experimental procedure may still yield useless in- formation if there is no method for validating the results. The most visible part of the analytical approach occurs in the laboratory. As part of the validation process, appropriate chemical or physical standards are used to calibrate any equipment being used and any solutions whose concentrations must be known. The selected samples are then analyzed and the raw data recorded. The raw data collected during the experiment are then analyzed. Frequently the data must be reduced or transformed to a more readily analyzable form. A statistical treatment of the data is used to evaluate the accuracy and precision of the analysis and to validate the procedure. These results are compared with the criteria estab- lished during the design of the experiment, and then the design is reconsidered, ad- ditional experimental trials are run, or a solution to the problem is proposed. When a solution is proposed, the results are subject to an external evaluation that may re- sult in a new problem and the beginning of a new analytical cycle. 6 Modern Analytical Chemistry 1. Identify the problem Determine type of information needed (qualitative, quantitative, characterization, or fundamental) Identify context of the problem 2. Design the experimental procedure Establish design criteria (accuracy, precision, scale of operation, sensitivity, selectivity, cost, speed) Identify interferents Select method Establish validation criteria Establish sampling strategy Feedback loop 3. Conduct an experiment Calibrate instruments and equipment Standardize reagents Gather data 4. Analyze the experimental data Reduce or transform data Analyze statistics Verify results Interpret results 5. Propose a solution Conduct external evaluation 1400-CH01 9/9/99 2:20 PM Page 6 As an exercise, let’s adapt this model of the analytical approach to a real prob- lem. For our example, we will use the determination of the sources of airborne pol- lutant particles. A description of the problem can be found in the following article: “Tracing Aerosol Pollutants with Rare Earth Isotopes” by Ondov, J. M.; Kelly, W. R. Anal. Chem. 1991, 63, 691A–697A. Before continuing, take some time to read the article, locating the discussions per- taining to each of the five steps outlined in Figure 1.3. In addition, consider the fol- lowing questions: 1. What is the analytical problem? 2. What type of information is needed to solve the problem? 3. How will the solution to this problem be used? 4. What criteria were considered in designing the experimental procedure? 5. Were there any potential interferences that had to be eliminated? If so, how were they treated? 6. Is there a plan for validating the experimental method? 7. How were the samples collected? 8. Is there evidence that steps 2, 3, and 4 of the analytical approach are repeated more than once? 9. Was there a successful conclusion to the problem? According to our model, the analytical approach begins with a problem. The motivation for this research was to develop a method for monitoring the transport of solid aerosol particulates following their release from a high-temperature com- bustion source. Because these particulates contain significant concentrations of toxic heavy metals and carcinogenic organic compounds, they represent a signifi- cant environmental hazard. An aerosol is a suspension of either a solid or a liquid in a gas. Fog, for exam- ple, is a suspension of small liquid water droplets in air, and smoke is a suspension of small solid particulates in combustion gases. In both cases the liquid or solid par- ticulates must be small enough to remain suspended in the gas for an extended time. Solid aerosol particulates, which are the focus of this problem, usually have micrometer or submicrometer diameters. Over time, solid particulates settle out from the gas, falling to the Earth’s surface as dry deposition. Existing methods for monitoring the transport of gases were inadequate for studying aerosols. To solve the problem, qualitative and quantitative information were needed to determine the sources of pollutants and their net contribution to the total dry deposition at a given location. Eventually the methods developed in this study could be used to evaluate models that estimate the contributions of point sources of pollution to the level of pollution at designated locations. Following the movement of airborne pollutants requires a natural or artificial tracer (a species specific to the source of the airborne pollutants) that can be exper- imentally measured at sites distant from the source. Limitations placed on the tracer, therefore, governed the design of the experimental procedure. These limita- tions included cost, the need to detect small quantities of the tracer, and the ab- sence of the tracer from other natural sources. In addition, aerosols are emitted from high-temperature combustion sources that produce an abundance of very re- active species. The tracer, therefore, had to be both thermally and chemically stable. On the basis of these criteria, rare earth isotopes, such as those of Nd, were selected as tracers. The choice of tracer, in turn, dictated the analytical method (thermal ionization mass spectrometry, or TIMS) for measuring the isotopic abundances of Chapter 1 Introduction 7 1400-CH01 9/9/99 2:20 PM Page 7 8 Modern Analytical Chemistry qualitative analysis An analysis in which we determine the identity of the constituent species in a sample. Nd in samples. Unfortunately, mass spectrometry is not a selective technique. A mass spectrum provides information about the abundance of ions with a given mass. It cannot distinguish, however, between different ions with the same mass. Consequently, the choice of TIMS required developing a procedure for separating the tracer from the aerosol particulates. Validating the final experimental protocol was accomplished by running a model study in which 148 Nd was released into the atmosphere from a 100-MW coal utility boiler. Samples were collected at 13 locations, all of which were 20 km from the source. Experimental results were compared with predictions determined by the rate at which the tracer was released and the known dispersion of the emissions. Finally, the development of this procedure did not occur in a single, linear pass through the analytical approach. As research progressed, problems were encoun- tered and modifications made, representing a cycle through steps 2, 3, and 4 of the analytical approach. Others have pointed out, with justification, that the analytical approach out- lined here is not unique to analytical chemistry, but is common to any aspect of sci- ence involving analysis. 8 Here, again, it helps to distinguish between a chemical analysis and analytical chemistry. For other analytically oriented scientists, such as physical chemists and physical organic chemists, the primary emphasis is on the problem, with the results of an analysis supporting larger research goals involving fundamental studies of chemical or physical processes. The essence of analytical chemistry, however, is in the second, third, and fourth steps of the analytical ap- proach. Besides supporting broader research goals by developing and validating an- alytical methods, these methods also define the type and quality of information available to other research scientists. In some cases, the success of an analytical method may even suggest new research problems. 1C Common Analytical Problems In Section 1A we indicated that analytical chemistry is more than a collection of qualitative and quantitative methods of analysis. Nevertheless, many problems on which analytical chemists work ultimately involve either a qualitative or quantita- tive measurement. Other problems may involve characterizing a sample’s chemical or physical properties. Finally, many analytical chemists engage in fundamental studies of analytical methods. In this section we briefly discuss each of these four areas of analysis. Many problems in analytical chemistry begin with the need to identify what is present in a sample. This is the scope of a qualitative analysis, examples of which include identifying the products of a chemical reaction, screening an athlete’s urine for the presence of a performance-enhancing drug, or determining the spatial dis- tribution of Pb on the surface of an airborne particulate. Much of the early work in analytical chemistry involved the development of simple chemical tests to identify the presence of inorganic ions and organic functional groups. The classical labora- tory courses in inorganic and organic qualitative analysis, 9 still taught at some schools, are based on this work. Currently, most qualitative analyses use methods such as infrared spectroscopy, nuclear magnetic resonance, and mass spectrometry. These qualitative applications of identifying organic and inorganic compounds are covered adequately elsewhere in the undergraduate curriculum and, so, will receive no further consideration in this text. 1400-CH01 9/9/99 2:20 PM Page 8 Perhaps the most common type of problem encountered in the analytical lab is a quantitative analysis. Examples of typical quantitative analyses include the ele- mental analysis of a newly synthesized compound, measuring the concentration of glucose in blood, or determining the difference between the bulk and surface con- centrations of Cr in steel. Much of the analytical work in clinical, pharmaceutical, environmental, and industrial labs involves developing new methods for determin- ing the concentration of targeted species in complex samples. Most of the examples in this text come from the area of quantitative analysis. Another important area of analytical chemistry, which receives some attention in this text, is the development of new methods for characterizing physical and chemical properties. Determinations of chemical structure, equilibrium constants, particle size, and surface structure are examples of a characterization analysis. The purpose of a qualitative, quantitative, and characterization analysis is to solve a problem associated with a sample. A fundamental analysis, on the other hand, is directed toward improving the experimental methods used in the other areas of analytical chemistry. Extending and improving the theory on which a method is based, studying a method’s limitations, and designing new and modify- ing old methods are examples of fundamental studies in analytical chemistry. Chapter 1 Introduction 9 characterization analysis An analysis in which we evaluate a sample’s chemical or physical properties. fundamental analysis An analysis whose purpose is to improve an analytical method’s capabilities. quantitative analysis An analysis in which we determine how much of a constituent species is present in a sample. 1D KEY TERMS characterization analysis (p. 9) fundamental analysis (p. 9) qualitative analysis (p. 8) quantitative analysis (p. 9) Analytical chemists work to improve the ability of all chemists to make meaningful measurements. Chemists working in medicinal chemistry, clinical chemistry, forensic chemistry, and environ- mental chemistry, as well as the more traditional areas of chem- istry, need better tools for analyzing materials. The need to work with smaller quantities of material, with more complex materi- als, with processes occurring on shorter time scales, and with species present at lower concentrations challenges analytical chemists to improve existing analytical methods and to develop new analytical techniques. Typical problems on which analytical chemists work include qualitative analyses (what is present?), quantitative analyses (how much is present?), characterization analyses (what are the material’s chemical and physical properties?), and funda- mental analyses (how does this method work and how can it be improved?). 1E SUMMARY 1. For each of the following problems indicate whether its solution requires a qualitative, quantitative, characterization, or fundamental study. More than one type of analysis may be appropriate for some problems. a. A hazardous-waste disposal site is believed to be leaking contaminants into the local groundwater. b. An art museum is concerned that a recent acquisition is a forgery. c. A more reliable method is needed by airport security for detecting the presence of explosive materials in luggage. d. The structure of a newly discovered virus needs to be determined. e. A new visual indicator is needed for an acid–base titration. f. A new law requires a method for evaluating whether automobiles are emitting too much carbon monoxide. 2. Read a recent article from the column “Analytical Approach,” published in Analytical Chemistry, or an article assigned by your instructor, and write an essay summarizing the nature of the problem and how it was solved. As a guide, refer back to Figure 1.3 for one model of the analytical approach. 1F PROBLEMS 1400-CH01 9/9/99 2:20 PM Page 9 10 Modern Analytical Chemistry The role of analytical chemistry within the broader discipline of chemistry has been discussed by many prominent analytical chemists. Several notable examples follow. Baiulescu, G. E.; Patroescu, C.; Chalmers, R. A. Education and Teaching in Analytical Chemistry. Ellis Horwood: Chichester, 1982. Hieftje, G. M. “The Two Sides of Analytical Chemistry,” Anal. Chem. 1985, 57, 256A–267A. Kissinger, P. T. “Analytical Chemistry—What is It? Who Needs It? Why Teach It?” Trends Anal. Chem. 1992, 11, 54–57. Laitinen, H. A. “Analytical Chemistry in a Changing World,” Anal. Chem. 1980, 52, 605A–609A. Laitinen, H. A. “History of Analytical Chemistry in the U.S.A.,” Talanta 1989, 36, 1–9. Laitinen, H. A.; Ewing, G. (eds). A History of Analytical Chemistry. The Division of Analytical Chemistry of the American Chemical Society: Washington, D.C., 1972. McLafferty, F. W. “Analytical Chemistry: Historic and Modern,” Acc. Chem. Res. 1990, 23, 63–64. Mottola, H. A. “The Interdisciplinary and Multidisciplinary Nature of Contemporary Analytical Chemistry and Its Core Components,” Anal. Chim. Acta 1991, 242, 1–3. Tyson, J. Analysis: What Analytical Chemists Do. Royal Society of Chemistry: Cambridge, England, 1988. Several journals are dedicated to publishing broadly in the field of analytical chemistry, including Analytical Chemistry, Analytica Chimica Acta, Analyst, and Talanta. Other journals, too numerous to list, are dedicated to single areas of analytical chemistry. Current research in the areas of quantitative analysis, qualitative analysis, and characterization analysis are reviewed biannually (odd-numbered years) in Analytical Chemistry’s “Application Reviews.” Current research on fundamental developments in analytical chemistry are reviewed biannually (even-numbered years) in Analytical Chemistry’s “Fundamental Reviews.” 1G SUGGESTED READINGS 1. Ravey, M. Spectroscopy 1990, 5(7), 11. 2. de Haseth, J. Spectroscopy 1990, 5(7), 11. 3. Fresenius, C. R. A System of Instruction in Quantitative Chemical Analysis. John Wiley and Sons: New York, 1881. 4. Hillebrand, W. F.; Lundell, G. E. F. Applied Inorganic Analysis, John Wiley and Sons: New York, 1953. 5. Van Loon, J. C. Analytical Atomic Absorption Spectroscopy. Academic Press: New York, 1980. 6. Murray, R. W. Anal. Chem. 1991, 63, 271A. 7. For several different viewpoints see (a) Beilby, A. L. J. Chem. Educ. 1970, 47, 237–238; (b) Lucchesi, C. A. Am. Lab. 1980, October, 113–119; (c) Atkinson, G. F. J. Chem. Educ. 1982, 59, 201–202; (d) Pardue, H. L.; Woo, J. J. Chem. Educ. 1984, 61, 409–412; (e) Guarnieri, M. J. Chem. Educ. 1988, 65, 201–203; (f) de Haseth, J. Spectroscopy 1990, 5, 20–21; (g) Strobel, H. A. Am. Lab. 1990, October, 17–24. 8. Hieftje, G. M. Am. Lab. 1993, October, 53–61. 9. See, for example, the following laboratory texts: (a) Sorum, C. H.; Lagowski, J. J. Introduction to Semimicro Qualitative Analysis, 5th ed. Prentice-Hall: Englewood Cliffs, NJ, 1977.; (b) Shriner, R. L.; Fuson, R. C.; Curtin, D. Y. The Systematic Identification of Organic Compounds, 5th ed. John Wiley and Sons: New York, 1964. 1H REFERENCES 1400-CH01 9/9/99 2:20 PM Page 10 Chapter 2 11 Basic Tools of Analytical Chemistry In the chapters that follow we will learn about the specifics of analytical chemistry. In the process we will ask and answer questions such as How do we treat experimental data? How do we ensure that our results are accurate? How do we obtain a representative sample? and How do we select an appropriate analytical technique? Before we look more closely at these and other questions, we will first review some basic numerical and experimental tools of importance to analytical chemists. 1400-CH02 9/8/99 3:47 PM Page 11 12 Modern Analytical Chemistry 2A Numbers in Analytical Chemistry Analytical chemistry is inherently a quantitative science. Whether determining the concentration of a species in a solution, evaluating an equilibrium constant, mea- suring a reaction rate, or drawing a correlation between a compound’s structure and its reactivity, analytical chemists make measurements and perform calculations. In this section we briefly review several important topics involving the use of num- bers in analytical chemistry. 2A.1 Fundamental Units of Measure Imagine that you find the following instructions in a laboratory procedure: “Trans- fer 1.5 of your sample to a 100 volumetric flask, and dilute to volume.” How do you do this? Clearly these instructions are incomplete since the units of measurement are not stated. Compare this with a complete instruction: “Transfer 1.5 g of your sample to a 100-mL volumetric flask, and dilute to volume.” This is an instruction that you can easily follow. Measurements usually consist of a unit and a number expressing the quantity of that unit. Unfortunately, many different units may be used to express the same physical measurement. For example, the mass of a sample weighing 1.5 g also may be expressed as 0.0033 lb or 0.053 oz. For consistency, and to avoid confusion, sci- entists use a common set of fundamental units, several of which are listed in Table 2.1. These units are called SI units after the Système International d’Unités. Other measurements are defined using these fundamental SI units. For example, we mea- sure the quantity of heat produced during a chemical reaction in joules, (J), where Table 2.2 provides a list of other important derived SI units, as well as a few com- monly used non-SI units. Chemists frequently work with measurements that are very large or very small. A mole, for example, contains 602,213,670,000,000,000,000,000 particles, and some analytical techniques can detect as little as 0.000000000000001 g of a compound. For simplicity, we express these measurements using scientific notation; thus, a mole contains 6.0221367 · 10 23 particles, and the stated mass is 1 · 10 –15 g. Some- times it is preferable to express measurements without the exponential term, replac- ing it with a prefix. A mass of 1 · 10 –15 g is the same as 1 femtogram. Table 2.3 lists other common prefixes. 1 J = 1 mkg 2 s 2 Table 2.1 Fundamental SI Units Measurement Unit Symbol mass kilogram kg volume liter L distance meter m temperature kelvin K time second s current ampere A amount of substance mole mol scientific notation A shorthand method for expressing very large or very small numbers by indicating powers of ten; for example, 1000 is 1 · 10 3 . SI units Stands for Système International d’Unités. These are the internationally agreed on units for measurements. 1400-CH02 9/8/99 3:47 PM Page 12 2A.2 Significant Figures Recording a measurement provides information about both its magnitude and un- certainty. For example, if we weigh a sample on a balance and record its mass as 1.2637 g, we assume that all digits, except the last, are known exactly. We assume that the last digit has an uncertainty of at least ±1, giving an absolute uncertainty of at least ±0.0001 g, or a relative uncertainty of at least Significant figures are a reflection of a measurement’s uncertainty. The num- ber of significant figures is equal to the number of digits in the measurement, with the exception that a zero (0) used to fix the location of a decimal point is not con- sidered significant. This definition can be ambiguous. For example, how many sig- nificant figures are in the number 100? If measured to the nearest hundred, then there is one significant figure. If measured to the nearest ten, however, then two ± ×=± 0 0001 1 2637 100 0 0079 . . .% g g Chapter 2 Basic Tools of Analytical Chemistry 13 Table 2.2 Other SI and Non-SI Units Measurement Unit Symbol Equivalent SI units length angstrom ? 1 ? = 1 · 10 –10 m force newton N 1 N = 1 m kg/s 2 pressure pascal Pa 1 Pa = 1 N/m 2 = 1 kg/(m s 2 ) atmosphere atm 1 atm = 101,325 Pa energy, work, heat joule J 1 J = 1 N m = 1 m 2 kg/s 2 power watt W 1 W = 1 J/s = 1 m 2 kg/s 3 charge coulomb C 1 C = 1 A s potential volt V 1 V = 1 W/A = 1 m 2 kg/(s 3 A) temperature degree Celsius °C °C = K – 273.15 degree Fahrenheit °F °F = 1.8(K – 273.15) + 32 Table 2. 3 Common Prefixes for Exponential Notation Exponential Prefix Symbol 10 12 tera T 10 9 giga G 10 6 mega M 10 3 kilo k 10 –1 deci d 10 –2 centi c 10 –3 milli m 10 –6 micro m 10 –9 nano n 10 –12 pico p 10 –15 femto f 10 –18 atto a significant figures The digits in a measured quantity, including all digits known exactly and one digit (the last) whose quantity is uncertain. 1400-CH02 9/8/99 3:47 PM Page 13 significant figures are included. To avoid ambiguity we use scientific notation. Thus, 1 · 10 2 has one significant figure, whereas 1.0 · 10 2 has two significant figures. For measurements using logarithms, such as pH, the number of significant figures is equal to the number of digits to the right of the decimal, including all zeros. Digits to the left of the decimal are not included as significant figures since they only indicate the power of 10. A pH of 2.45, therefore, contains two signifi- cant figures. Exact numbers, such as the stoichiometric coefficients in a chemical formula or reaction, and unit conversion factors, have an infinite number of significant figures. A mole of CaCl 2 , for example, contains exactly two moles of chloride and one mole of calcium. In the equality 1000 mL = 1 L both numbers have an infinite number of significant figures. Recording a measurement to the correct number of significant figures is im- portant because it tells others about how precisely you made your measurement. For example, suppose you weigh an object on a balance capable of measuring mass to the nearest ±0.1 mg, but record its mass as 1.762 g instead of 1.7620 g. By failing to record the trailing zero, which is a significant figure, you suggest to others that the mass was determined using a balance capable of weighing to only the nearest ±1 mg. Similarly, a buret with scale markings every 0.1 mL can be read to the nearest ±0.01 mL. The digit in the hundredth’s place is the least sig- nificant figure since we must estimate its value. Reporting a volume of 12.241 mL implies that your buret’s scale is more precise than it actually is, with divi- sions every 0.01 mL. Significant figures are also important because they guide us in reporting the re- sult of an analysis. When using a measurement in a calculation, the result of that calculation can never be more certain than that measurement’s uncertainty. Simply put, the result of an analysis can never be more certain than the least certain mea- surement included in the analysis. As a general rule, mathematical operations involving addition and subtraction are carried out to the last digit that is significant for all numbers included in the cal- culation. Thus, the sum of 135.621, 0.33, and 21.2163 is 157.17 since the last digit that is significant for all three numbers is in the hundredth’s place. 135.621 + 0.33 + 21.2163 = 157.1673 = 157.17 When multiplying and dividing, the general rule is that the answer contains the same number of significant figures as that number in the calculation having the fewest significant figures. Thus, It is important to remember, however, that these rules are generalizations. What is conserved is not the number of significant figures, but absolute uncertainty when adding or subtracting, and relative uncertainty when multiplying or dividing. For example, the following calculation reports the answer to the correct number of significant figures, even though it violates the general rules outlined earlier. 101 99 102= . 22 91 0 152 16 302 0 21361 0 214 .. . .. × == 14 Modern Analytical Chemistry 1400-CH02 9/8/99 3:48 PM Page 14 Chapter 2 Basic Tools of Analytical Chemistry 15 Since the relative uncertainty in both measurements is roughly 1% (101 ±1, 99 ±1), the relative uncertainty in the final answer also must be roughly 1%. Reporting the answer to only two significant figures (1.0), as required by the general rules, implies a relative uncertainty of 10%. The correct answer, with three significant figures, yields the expected relative uncertainty. Chapter 4 presents a more thorough treat- ment of uncertainty and its importance in reporting the results of an analysis. Finally, to avoid “round-off ” errors in calculations, it is a good idea to retain at least one extra significant figure throughout the calculation. This is the practice adopted in this textbook. Better yet, invest in a good scientific calculator that allows you to perform lengthy calculations without recording intermediate values. When the calculation is complete, the final answer can be rounded to the correct number of significant figures using the following simple rules. 1. Retain the least significant figure if it and the digits that follow are less than halfway to the next higher digit; thus, rounding 12.442 to the nearest tenth gives 12.4 since 0.442 is less than halfway between 0.400 and 0.500. 2. Increase the least significant figure by 1 if it and the digits that follow are more than halfway to the next higher digit; thus, rounding 12.476 to the nearest tenth gives 12.5 since 0.476 is more than halfway between 0.400 and 0.500. 3. If the least significant figure and the digits that follow are exactly halfway to the next higher digit, then round the least significant figure to the nearest even number; thus, rounding 12.450 to the nearest tenth gives 12.4, but rounding 12.550 to the nearest tenth gives 12.6. Rounding in this manner prevents us from introducing a bias by always rounding up or down. 2B Units for Expressing Concentration Concentration is a general measurement unit stating the amount of solute present in a known amount of solution 2.1 Although the terms “solute” and “solution” are often associated with liquid sam- ples, they can be extended to gas-phase and solid-phase samples as well. The actual units for reporting concentration depend on how the amounts of solute and solu- tion are measured. Table 2.4 lists the most common units of concentration. 2B.1 Molarity and Formality Both molarity and formality express concentration as moles of solute per liter of solu- tion. There is, however, a subtle difference between molarity and formality. Molarity is the concentration of a particular chemical species in solution. Formality, on the other hand, is a substance’s total concentration in solution without regard to its spe- cific chemical form. There is no difference between a substance’s molarity and for- mality if it dissolves without dissociating into ions. The molar concentration of a so- lution of glucose, for example, is the same as its formality. For substances that ionize in solution, such as NaCl, molarity and formality are different. For example, dissolving 0.1 mol of NaCl in 1 L of water gives a solution containing 0.1 mol of Na + and 0.1 mol of Cl – . The molarity of NaCl, therefore, is zero since there is essentially no undissociated NaCl in solution. The solution, Concentration amount of solute amount of solution = molarity The number of moles of solute per liter of solution (M). formality The number of moles of solute, regardless of chemical form, per liter of solution (F). concentration An expression stating the relative amount of solute per unit volume or unit mass of solution. 1400-CH02 9/8/99 3:48 PM Page 15 instead, is 0.1 M in Na + and 0.1 M in Cl – . The formality of NaCl, however, is 0.1 F because it represents the total amount of NaCl in solution. The rigorous definition of molarity, for better or worse, is largely ignored in the current literature, as it is in this text. When we state that a solution is 0.1 M NaCl we understand it to consist of Na + and Cl – ions. The unit of formality is used only when it provides a clearer de- scription of solution chemistry. Molar concentrations are used so frequently that a symbolic notation is often used to simplify its expression in equations and writing. The use of square brackets around a species indicates that we are referring to that species’ molar concentration. Thus, [Na + ] is read as the “molar concentration of sodium ions.” 2B.2 Normality Normality is an older unit of concentration that, although once commonly used, is frequently ignored in today’s laboratories. Normality is still used in some hand- books of analytical methods, and, for this reason, it is helpful to understand its meaning. For example, normality is the concentration unit used in Standard Meth- ods for the Examination of Water and Wastewater, 1 a commonly used source of ana- lytical methods for environmental laboratories. Normality makes use of the chemical equivalent, which is the amount of one chemical species reacting stoichiometrically with another chemical species. Note that this definition makes an equivalent, and thus normality, a function of the chemical reaction in which the species participates. Although a solution of H 2 SO 4 has a fixed molarity, its normality depends on how it reacts. 16 Modern Analytical Chemistry Table 2.4 Common Units for Reporting Concentration Name Units a Symbol molarity M formality F normality N molality m weight % % w/w volume % % v/v weight-to-volume % % w/v parts per million ppm parts per billion ppb a FW = formula weight; EW = equivalent weight. moles solute liters solution number F solute liters solution Ws number E solute liters solution Ws m solute k solvent oles g g solute solution g 100 m solute solution L mL 100 g solute solution mL 100 g solute solution g 10 6 g solute g solution 10 9 normality The number of equivalents of solute per liter of solution (N). 1400-CH02 9/8/99 3:48 PM Page 16 The number of equivalents, n, is based on a reaction unit, which is that part of a chemical species involved in a reaction. In a precipitation reaction, for example, the reaction unit is the charge of the cation or anion involved in the reaction; thus for the reaction Pb 2+ (aq) + 2I – (aq) t PbI 2 (s) n = 2 for Pb 2+ and n = 1 for I – . In an acid–base reaction, the reaction unit is the number of H + ions donated by an acid or accepted by a base. For the reaction be- tween sulfuric acid and ammonia H 2 SO 4 (aq) + 2NH 3 (aq) t 2NH 4 + (aq) + SO 4 2– (aq) we find that n = 2 for H 2 SO 4 and n = 1 for NH 3 . For a complexation reaction, the reaction unit is the number of electron pairs that can be accepted by the metal or donated by the ligand. In the reaction between Ag + and NH 3 Ag + (aq) + 2NH 3 (aq) t Ag(NH 3 ) 2 + (aq) the value of n for Ag + is 2 and that for NH 3 is 1. Finally, in an oxidation–reduction reaction the reaction unit is the number of electrons released by the reducing agent or accepted by the oxidizing agent; thus, for the reaction 2Fe 3+ (aq) + Sn 2+ (aq) t Sn 4+ (aq) + 2Fe 2+ (aq) n = 1 for Fe 3+ and n = 2 for Sn 2+ . Clearly, determining the number of equivalents for a chemical species requires an understanding of how it reacts. Normality is the number of equivalent weights (EW) per unit volume and, like formality, is independent of speciation. An equivalent weight is defined as the ratio of a chemical species’ formula weight (FW) to the number of its equivalents Consequently, the following simple relationship exists between normality and molarity. N = n · M Example 2.1 illustrates the relationship among chemical reactivity, equivalent weight, and normality. EXAMPLE 2.1 Calculate the equivalent weight and normality for a solution of 6.0 M H 3 PO 4 given the following reactions: (a) H 3 PO 4 (aq) + 3OH – (aq) t PO 4 3– (aq) + 3H 2 O(l) (b) H 3 PO 4 (aq) + 2NH 3 (aq) t HPO 4 2– (aq) + 2NH 4 + (aq) (c) H 3 PO 4 (aq) + F – (aq) t H 2 PO 4 – (aq) + HF(aq) SOLUTION For phosphoric acid, the number of equivalents is the number of H + ions donated to the base. For the reactions in (a), (b), and (c) the number of equivalents are 3, 2, and 1, respectively. Thus, the calculated equivalent weights and normalities are EW = FW n Chapter 2 Basic Tools of Analytical Chemistry 17 equivalent The moles of a species that can donate one reaction unit. equivalent weight The mass of a compound containing one equivalent (EW). formula weight The mass of a compound containing one mole (FW). 1400-CH02 9/8/99 3:48 PM Page 17 18 Modern Analytical Chemistry 2B. 3 Molality Molality is used in thermodynamic calculations where a temperature independent unit of concentration is needed. Molarity, formality and normality are based on the volume of solution in which the solute is dissolved. Since density is a temperature de- pendent property a solution’s volume, and thus its molar, formal and normal concen- trations, will change as a function of its temperature. By using the solvent’s mass in place of its volume, the resulting concentration becomes independent of temperature. 2B. 4 Weight, Volume, and Weight-to-Volume Ratios Weight percent (% w/w), volume percent (% v/v) and weight-to-volume percent (% w/v) express concentration as units of solute per 100 units of sample. A solution in which a solute has a concentration of 23% w/v contains 23 g of solute per 100 mL of solution. Parts per million (ppm) and parts per billion (ppb) are mass ratios of grams of solute to one million or one billion grams of sample, respectively. For example, a steel that is 450 ppm in Mn contains 450 m g of Mn for every gram of steel. If we approxi- mate the density of an aqueous solution as 1.00 g/mL, then solution concentrations can be expressed in parts per million or parts per billion using the following relationships. For gases a part per million usually is a volume ratio. Thus, a helium concentration of 6.3 ppm means that one liter of air contains 6.3 m L of He. 2B. 5 Converting Between Concentration Units The units of concentration most frequently encountered in analytical chemistry are molarity, weight percent, volume percent, weight-to-volume percent, parts per mil- lion, and parts per billion. By recognizing the general definition of concentration given in equation 2.1, it is easy to convert between concentration units. EXAMPLE 2.2 A concentrated solution of aqueous ammonia is 28.0% w/w NH 3 and has a density of 0.899 g/mL. What is the molar concentration of NH 3 in this solution? SOLUTION 28 0 100 0 899 1 17 04 1000 14 8 33 3 .. . . L gNH g solution g solution m solution mole NH gNH mL liter M×××= ppm = mg liter ppb = g liter = = μ μ g mL ng mL (a) EW = FW = 97.994 3 = 32.665 N = M = 3 6.0 = 18 N (b) EW = FW = 97.994 2 = 48.997 N = M = 2 6.0 = 12 N (c) EW = = 97.994 1 = 97.994 N = M = 1 6.0 = 6.0 N n n n n n n ×× ×× ×× FW molality The number of moles of solute per kilogram of solvent (m). weight percent Grams of solute per 100 g of solution. (% w/w). volume percent Milliliters of solute per 100 mL of solution (% v/v). weight-to-volume percent Grams of solute per 100 mL of solution (% w/v). parts per million Micrograms of solute per gram of solution; for aqueous solutions the units are often expressed as milligrams of solute per liter of solution (ppm). parts per billion Nanograms of solute per gram of solution; for aqueous solutions the units are often expressed as micrograms of solute per liter of solution (ppb). 1400-CH02 9/8/99 3:48 PM Page 18 Chapter 2 Basic Tools of Analytical Chemistry 19 EXAMPLE 2. 3 The maximum allowed concentration of chloride in a municipal drinking water supply is 2.50 · 10 2 ppm Cl – . When the supply of water exceeds this limit, it often has a distinctive salty taste. What is this concentration in moles Cl – /liter? SOLUTION 2B.6 p-Functions Sometimes it is inconvenient to use the concentration units in Table 2.4. For exam- ple, during a reaction a reactant’s concentration may change by many orders of mag- nitude. If we are interested in viewing the progress of the reaction graphically, we might wish to plot the reactant’s concentration as a function of time or as a function of the volume of a reagent being added to the reaction. Such is the case in Figure 2.1, where the molar concentration of H + is plotted (y-axis on left side of figure) as a function of the volume of NaOH added to a solution of HCl. The initial [H + ] is 0.10 M, and its concentration after adding 75 mL of NaOH is 5.0 · 10 –13 M. We can easily follow changes in the [H + ] over the first 14 additions of NaOH. For the last ten addi- tions of NaOH, however, changes in the [H + ] are too small to be seen. When working with concentrations that span many orders of magnitude, it is often more convenient to express the concentration as a p-function. The p-func- tion of a number X is written as pX and is defined as pX = –log(X) Thus, the pH of a solution that is 0.10 M H + is pH = –log[H + ] = –log(0.10) = 1.00 and the pH of 5.0 · 10 –13 M H + is pH = –log[H + ] = –log(5.0 · 10 –13 ) = 12.30 Figure 2.1 shows how plotting pH in place of [H + ] provides more detail about how the concentration of H + changes following the addition of NaOH. EXAMPLE 2. 4 What is pNa for a solution of 1.76 · 10 –3 M Na 3 PO 4 ? SOLUTION Since each mole of Na 3 PO 4 contains three moles of Na + , the concentration of Na + is and pNa is pNa = –log[Na + ] = –log(5.28 · 10 –3 ) = 2.277 [] . . –– Na mol Na mol Na PO MM + + =××=× 3 1 76 10 5 28 10 34 33 250 10 1 1 1 35 453 705 10 2 3 . . . –– – – × ×× =× 000 g mg Cl L g m mole Cl gCl M p-function A function of the form pX, where pX = -log(X). 1400-CH02 9/8/99 3:48 PM Page 19 20 Modern Analytical Chemistry 0.12 0.10 0.08 0.06 0.04 0.02 0.00 [H + ] (M) Volume NaOH (mL) 0 204060 [H + ] pH 80 pH 14 12 10 8 6 4 2 0 Figure 2.1 Graph of [H + ] versus volume of NaOH and pH versus volume of NaOH for the reaction of 0.10 M HCl with 0.10 M NaOH. EXAMPLE 2. 5 What is the [H + ] in a solution that has a pH of 5.16? SOLUTION The concentration of H + is pH = –log[H + ] = 5.16 log[H + ] = –5.16 [H + ] = antilog(–5.16) = 10 –5.16 = 6.9 · 10 –6 M 2C Stoichiometric Calculations A balanced chemical reaction indicates the quantitative relationships between the moles of reactants and products. These stoichiometric relationships provide the basis for many analytical calculations. Consider, for example, the problem of deter- mining the amount of oxalic acid, H 2 C 2 O 4 , in rhubarb. One method for this analy- sis uses the following reaction in which we oxidize oxalic acid to CO 2 . 2Fe 3+ (aq) + H 2 C 2 O 4 (aq) + 2H 2 O(l) fi 2Fe 2+ (aq) + 2CO 2 (g) + 2H 3 O + (aq) 2.2 The balanced chemical reaction provides the stoichiometric relationship between the moles of Fe 3+ used and the moles of oxalic acid in the sample being analyzed— specifically, one mole of oxalic acid reacts with two moles of Fe 3+ . As shown in Ex- ample 2.6, the balanced chemical reaction can be used to determine the amount of oxalic acid in a sample, provided that information about the number of moles of Fe 3+ is known. 1400-CH02 9/8/99 3:48 PM Page 20 Chapter 2 Basic Tools of Analytical Chemistry 21 EXAMPLE 2.6 The amount of oxalic acid in a sample of rhubarb was determined by reacting with Fe 3+ as outlined in reaction 2.2. In a typical analysis, the oxalic acid in 10.62 g of rhubarb was extracted with a suitable solvent. The complete oxidation of the oxalic acid to CO 2 required 36.44 mL of 0.0130 M Fe 3+ . What is the weight percent of oxalic acid in the sample of rhubarb? SOLUTION We begin by calculating the moles of Fe 3+ used in the reaction The moles of oxalic acid reacting with the Fe 3+ , therefore, is Converting moles of oxalic acid to grams of oxalic acid and converting to weight percent gives the concentration of oxalic acid in the sample of rhubarb as In the analysis described in Example 2.6 oxalic acid already was present in the desired form. In many analytical methods the compound to be determined must be converted to another form prior to analysis. For example, one method for the quan- titative analysis of tetraethylthiuram disulfide (C 10 H 20 N 2 S 4 ), the active ingredient in the drug Antabuse (disulfiram), requires oxidizing the S to SO 2 , bubbling the SO 2 through H 2 O 2 to produce H 2 SO 4 , followed by an acid–base titration of the H 2 SO 4 with NaOH. Although we can write and balance chemical reactions for each of these steps, it often is easier to apply the principle of the conservation of reaction units. A reaction unit is that part of a chemical species involved in a reaction. Con- sider, for example, the general unbalanced chemical reaction A + B fi Products Conservation of reaction units requires that the number of reaction units associated with the reactant A equal the number of reaction units associated with the reactant B. Translating the previous statement into mathematical form gives Number of reaction units per A · moles A = number of reaction units per B · moles B 2.3 If we know the moles of A and the number of reaction units associated with A and B, then we can calculate the moles of B. Note that a conservation of reaction units, as defined by equation 2.3, can only be applied between two species. There are five important principles involving a conservation of reaction units: mass, charge, pro- tons, electron pairs, and electrons. 2.132 10 g C C –2 2 2 × ×= HO g rhubarb ww H O 24 24 10 62 100 0 201 . .%/ 2 10 mol C 90.03 g C mol C = 2.132 10 oxalic acid –4 2 2 2 –2 .369 24 24 24 ×× ×HO HO HO g 4.737 10 mol Fe 1 mol C mol Fe = 2.369 10 mol C –4 3+ 2 3+ –4 2 ×× × HO HO 24 24 2 0 0130 0 03644 . . mol Fe L L = 4.737 10 Fe 3+ –4 3+ ××mol 1400-CH02 9/8/99 3:48 PM Page 21 22 Modern Analytical Chemistry 2C.1 Conservation of Mass The easiest principle to appreciate is conservation of mass. Except for nuclear reac- tions, an element’s total mass at the end of a reaction must be the same as that pres- ent at the beginning of the reaction; thus, an element serves as the most fundamen- tal reaction unit. Consider, for example, the combustion of butane to produce CO 2 and H 2 O, for which the unbalanced reaction is C 4 H 10 (g) + O 2 (g) fi CO 2 (g) + H 2 O(g) All the carbon in CO 2 comes from the butane, thus we can select carbon as a reac- tion unit. Since there are four carbon atoms in butane, and one carbon atom in CO 2 , we write 4 · moles C 4 H 10 = 1 · moles CO 2 Hydrogen also can be selected as a reaction unit since all the hydrogen in butane ends up in the H 2 O produced during combustion. Thus, we can write 10 · moles C 4 H 10 = 2 · moles H 2 O Although the mass of oxygen is conserved during the reaction, we cannot apply equation 2.3 because the O 2 used during combustion does not end up in a single product. Conservation of mass also can, with care, be applied to groups of atoms. For example, the ammonium ion, NH 4 + , can be precipitated as Fe(NH 4 ) 2 (SO 4 ) 2 6H 2 O. Selecting NH 4 + as the reaction unit gives 2 · moles Fe(NH 4 ) 2 (SO 4 ) 2 · 6H 2 O = 1 · moles NH 4 + 2C.2 Conservation of Charge The stoichiometry between two reactants in a precipitation reaction is governed by a conservation of charge, requiring that the total cation charge and the total anion charge in the precipitate be equal. The reaction units in a precipitation reaction, therefore, are the absolute values of the charges on the cation and anion that make up the precipitate. Applying equation 2.3 to a precipitate of Ca 3 (PO 4 ) 2 formed from the reaction of Ca 2+ and PO 4 3– , we write 2 · moles Ca 2+ = 3 · moles PO 4 3– 2C. 3 Conservation of Protons In an acid–base reaction, the reaction unit is the proton. For an acid, the num- ber of reaction units is given by the number of protons that can be donated to the base; and for a base, the number of reaction units is the number of protons that the base can accept from the acid. In the reaction between H 3 PO 4 and NaOH, for example, the weak acid H 3 PO 4 can donate all three of its pro- tons to NaOH, whereas the strong base NaOH can accept one proton. Thus, we write 3 · moles H 3 PO 4 = 1 · moles NaOH Care must be exercised in determining the number of reaction units associ- ated with the acid and base. The number of reaction units for an acid, for in- stance, depends not on how many acidic protons are present, but on how many 1400-CH02 9/8/99 3:48 PM Page 22 Chapter 2 Basic Tools of Analytical Chemistry 23 of the protons are capable of reacting with the chosen base. In the reaction be- tween H 3 PO 4 and NH 3 H 3 PO 4 (aq) + 2NH 3 (aq) t HPO 4 – (aq) + 2NH 4 + (aq) a conservation of protons requires that 2 · moles H 3 PO 4 = moles of NH 3 2C. 4 Conservation of Electron Pairs In a complexation reaction, the reaction unit is an electron pair. For the metal, the number of reaction units is the number of coordination sites available for binding ligands. For the ligand, the number of reaction units is equivalent to the number of electron pairs that can be donated to the metal. One of the most important analyti- cal complexation reactions is that between the ligand ethylenediaminetetracetic acid (EDTA), which can donate 6 electron pairs and 6 coordinate metal ions, such as Cu 2+ ; thus 6 · mole Cu 2+ = 6 · moles EDTA 2C. 5 Conservation of Electrons In a redox reaction, the reaction unit is an electron transferred from a reducing agent to an oxidizing agent. The number of reaction units for a reducing agent is equal to the number of electrons released during its oxidation. For an oxidizing agent, the number of reaction units is given by the number of electrons needed to cause its reduction. In the reaction between Fe 3+ and oxalic acid (reaction 2.2), for example, Fe 3+ undergoes a 1-electron reduction. Each carbon atom in oxalic acid is initially present in a +3 oxidation state, whereas the carbon atom in CO 2 is in a +4 oxidation state. Thus, we can write 1 · moles Fe 3+ = 2 · moles of H 2 C 2 O 4 Note that the moles of oxalic acid are multiplied by 2 since there are two carbon atoms, each of which undergoes a 1-electron oxidation. 2C.6 Using Conservation Principles in Stoichiometry Problems As shown in the following examples, the application of conservation principles sim- plifies stoichiometric calculations. EXAMPLE 2. 7 Rework Example 2.6 using conservation principles. SOLUTION Conservation of electrons for this redox reaction requires that moles Fe 3+ = 2 · moles H 2 C 2 O 4 which can be transformed by writing moles as the product of molarity and volume or as grams per formula weight. M gCO FW C O Fe Fe 33 2 24 24 ++×= × V H H 2 2 1400-CH02 9/8/99 3:48 PM Page 23 24 Modern Analytical Chemistry Solving for g H 2 C 2 O 4 gives and the weight percent oxalic acid is EXAMPLE 2.8 One quantitative analytical method for tetraethylthiuram disulfide, C 10 H 20 N 2 S 4 (Antabuse), requires oxidizing the sulfur to SO 2 , and bubbling the resulting SO 2 through H 2 O 2 to produce H 2 SO 4 . The H 2 SO 4 is then reacted with NaOH according to the reaction H 2 SO 4 (aq) + 2NaOH(aq) fi Na 2 SO 4 (aq) + 2H 2 O(l) Using appropriate conservation principles, derive an equation relating the moles of C 10 H 20 N 2 S 4 to the moles of NaOH. What is the weight percent C 10 H 20 N 2 S 4 in a sample of Antabuse if the H 2 SO 4 produced from a 0.4613-g portion reacts with 34.85 mL of 0.02500 M NaOH? SOLUTION The unbalanced reactions converting C 10 H 20 N 2 S 4 to H 2 SO 4 are C 10 H 20 N 2 S 4 fi SO 2 SO 2 fi H 2 SO 4 Using a conservation of mass we have 4 · moles C 10 H 20 N 2 S 4 = moles SO 2 = moles H 2 SO 4 A conservation of protons for the reaction of H 2 SO 4 with NaOH gives 2 · moles H 2 SO 4 = moles of NaOH Combining the two conservation equations gives the following stoichiometric equation between C 10 H 20 N 2 S 4 and NaOH 8 · moles C 10 H 20 N 2 S 4 = moles NaOH Now we are ready to finish the problem. Making appropriate substitutions for moles of C 10 H 20 N 2 S 4 and moles of NaOH gives Solving for g C 10 H 20 N 2 S 4 gives 1 8 0 02500 20 2 4 ( . M)(0.03485 L)(296.54 g/mol) = 0.032295 g C 10 HNS g C W C 10 10 HNS M F HNS NaOH NaOH20 2 4 20 2 4 1 8 =× × ×V 8 20 2 4 20 2 4 × =× g C W C 10 10 HNS FHNS M NaOH NaOH V 2 132 10 10 62 100 0 201 2 24 24 . . .%/ – × ×= g C C 2 2 HO g rhubarb ww H O MFWCO CO Fe Fe 33 24 2 24 2 0 0130 2 2 132 10 ++×× = =× V H M)(0.03644 L)(90.03 g/mole) g H 2 2 (. . – 1400-CH02 9/8/99 3:48 PM Page 24 Chapter 2 Basic Tools of Analytical Chemistry 25 The weight percent C 10 H 20 N 2 S 4 in the sample, therefore, is 2D Basic Equipment and Instrumentation Measurements are made using appropriate equipment or instruments. The array of equipment and instrumentation used in analytical chemistry is impressive, ranging from the simple and inexpensive, to the complex and costly. With two exceptions, we will postpone the discussion of equipment and instrumentation to those chap- ters where they are used. The instrumentation used to measure mass and much of the equipment used to measure volume are important to all analytical techniques and are therefore discussed in this section. 2D.1 Instrumentation for Measuring Mass An object’s mass is measured using a balance. The most common type of balance is an electronic balance in which the balance pan is placed over an electromagnet (Figure 2.2). The sample to be weighed is placed on the sample pan, displacing the pan downward by a force equal to the product of the sample’s mass and the acceler- ation due to gravity. The balance detects this downward movement and generates a counterbalancing force using an electromagnet. The current needed to produce this force is proportional to the object’s mass. A typical electronic balance has a capacity of 100–200 g and can measure mass to the nearest ±0.01 to ±1 mg. Another type of balance is the single-pan, unequal arm balance (Figure 2.3). In this mechanical balance the balance pan and a set of removable standard weights on one side of a beam are balanced against a fixed counterweight on the beam’s other side. The beam itself is balanced on a fulcrum consisting of a sharp knife edge. Adding a sample to the balance pan tilts the beam away from its balance point. Selected stan- dard weights are then removed until the beam is brought back into balance. The com- bined mass of the removed weights equals the sample’s mass. The capacities and mea- surement limits of these balances are comparable to an electronic balance. 0 32295 0 100 7 001 20 2 4 20 2 4 . .% g C .4613 sample w/w C 10 10 HNS g HNS×= Detector Light source N SS Balance pan Electromagnetic servomotor Control circuitry Figure 2.2 (a) Photo of a typical electronic balance. (b) Schematic diagram of electronic balance; adding a sample moves the balance pan down, allowing more light to reach the detector. The control circuitry directs the electromagnetic servomotor to generate an opposing force, raising the sample up until the original intensity of light at the detector is restored. Photo courtesy of Fisher Scientific. (a) (b) balance An apparatus used to measure mass. 1400-CH02 9/8/99 3:48 PM Page 25 The mass of a sample is determined by difference. If the material being weighed is not moisture-sensitive, a clean and dry container is placed on the balance. The mass of this container is called the tare. Most balances allow the tare to be automat- ically adjusted to read a mass of zero. The sample is then transferred to the con- tainer, the new mass is measured and the sample’s mass determined by subtracting the tare. Samples that absorb moisture from the air are weighed differently. The sample is placed in a covered weighing bottle and their combined mass is deter- mined. A portion of the sample is removed, and the weighing bottle and remaining sample are reweighed. The difference between the two masses gives the mass of the transferred sample. Several important precautions help to minimize errors in measuring an object’s mass. Balances should be placed on heavy surfaces to minimize the effect of vibra- tions in the surrounding environment and should be maintained in a level position. Analytical balances are sensitive enough that they can measure the mass of a finger- print. For this reason, materials placed on a balance should normally be handled using tongs or laboratory tissues. Volatile liquid samples should be weighed in a covered container to avoid the loss of sample by evaporation. Air currents can sig- nificantly affect a sample’s mass. To avoid air currents, the balance’s glass doors should be closed, or the balance’s wind shield should be in place. A sample that is cooler or warmer than the surrounding air will create convective air currents that adversely affect the measurement of its mass. Finally, samples dried in an oven should be stored in a desiccator to prevent them from reabsorbing moisture from the atmosphere. 2D.2 Equipment for Measuring Volume Analytical chemists use a variety of glassware to measure volume, several examples of which are shown in Figure 2.4. The type of glassware used depends on how exact the volume needs to be. Beakers, dropping pipets, and graduated cylinders are used to measure volumes approximately, typically with errors of several percent. Pipets and volumetric flasks provide a more accurate means for measuring vol- ume. When filled to its calibration mark, a volumetric flask is designed to contain a specified volume of solution at a stated temperature, usually 20 °C. The actual vol- 26 Modern Analytical Chemistry Fulcrum Balance beam Fulcrum Removable weights Balance pan Counterweight Figure 2.3 Schematic diagram of single-arm mechanical balance. volumetric flask Glassware designed to contain a specific volume of solution when filled to its calibration mark. 1400-CH02 9/8/99 3:48 PM Page 26 ume contained by the volumetric flask is usually within 0.03–0.2% of the stated value. Volumetric flasks containing less than 100 mL generally measure volumes to the hundredth of a milliliter, whereas larger volumetric flasks measure volumes to the tenth of a milliliter. For example, a 10-mL volumetric flask contains 10.00 mL, but a 250-mL volumetric flask holds 250.0 mL (this is important when keeping track of significant figures). Because a volumetric flask contains a solution, it is useful in preparing solu- tions with exact concentrations. The reagent is transferred to the volumetric flask, and enough solvent is added to dissolve the reagent. After the reagent is dissolved, additional solvent is added in several portions, mixing the solution after each addi- tion. The final adjustment of volume to the flask’s calibration mark is made using a dropping pipet. To complete the mixing process, the volumetric flask should be in- verted at least ten times. A pipet is used to deliver a specified volume of solution. Several different styles of pipets are available (Figure 2.5). Transfer pipets provide the most accurate means for delivering a known volume of solution; their volume error is similar to that from an equivalent volumetric flask. A 250-mL transfer pipet, for instance, will deliver 250.0 mL. To fill a transfer pipet, suction from a rubber bulb is used to pull the liquid up past the calibration mark (never use your mouth to suck a solu- tion into a pipet). After replacing the bulb with your finger, the liquid’s level is ad- justed to the calibration mark, and the outside of the pipet is wiped dry. The pipet’s contents are allowed to drain into the receiving container with the tip of the pipet touching the container walls. A small portion of the liquid remains in the pipet’s tip and should not be blown out. Measuring pipets are used to deliver vari- able volumes, but with less accuracy than transfer pipets. With some measuring Chapter 2 Basic Tools of Analytical Chemistry 27 Figure 2.4 Common examples of glassware used to measure volume: (a) beaker; (b) graduated cylinder; (c) volumetric flask; (d) pipet; (e) dropping pipet. Photos courtesy of Fisher Scientific. (b) (c) (d) (e) (a) pipet Glassware designed to deliver a specific volume of solution when filled to its calibration mark. 1400-CH02 9/8/99 3:48 PM Page 27 pipets, delivery of the calibrated volume requires that any solution remaining in the tip be blown out. Digital pipets and syringes can be used to deliver volumes as small as a microliter. Three important precautions are needed when working with pipets and volumetric flasks. First, the volume delivered by a pipet or contained by a volu- metric flask assumes that the glassware is clean. Dirt and grease on the inner glass surface prevents liquids from draining evenly, leaving droplets of the liquid on the container’s walls. For a pipet this means that the delivered volume is less than the calibrated volume, whereas drops of liquid above the calibration mark mean that a volumetric flask contains more than its calibrated volume. Commercially available cleaning solutions can be used to clean pipets and volu- metric flasks. 28 Modern Analytical Chemistry (a) (b) (c) (d) Figure 2.5 Common types of pipets and syringes: (a) transfer pipet; (b) measuring pipet; (c) digital pipet; (d) syringe. Photos courtesy of Fisher Scientific. Meniscus Calibration mark Figure 2.6 Proper means of reading the meniscus on a volumetric flask or pipet. 1400-CH02 9/8/99 3:48 PM Page 28 Chapter 2 Basic Tools of Analytical Chemistry 29 Figure 2.7 Conventional laboratory oven used for drying materials. Figure 2.8 Example of a muffle furnace used for heating samples to maximum temperatures of 1100–1700 °C. Courtesy of Fisher Scientific. (a) (b) Second, when filling a pipet or volumetric flask, set the liquid’s level exactly at the calibration mark. The liquid’s top surface is curved into a meniscus, the bottom of which should be exactly even with the glassware’s calibration mark (Figure 2.6). The meniscus should be adjusted with the calibration mark at eye level to avoid parallax errors. If your eye level is above the calibration mark the pipet or volumetric flask will be overfilled. The pipet or volumetric flask will be underfilled if your eye level is below the calibration mark. Finally, before using a pipet or volumetric flask you should rinse it with several small portions of the solution whose volume is being measured. This ensures that any residual liquid remaining in the pipet or volumetric flask is removed. 2D. 3 Equipment for Drying Samples Many materials need to be dried prior to their analysis to remove residual moisture. Depending on the material, heating to a temperature of 110–140 °C is usually suffi- cient. Other materials need to be heated to much higher temperatures to initiate thermal decomposition. Both processes can be accomplished using a laboratory oven capable of providing the required temperature. Commercial laboratory ovens (Figure 2.7) are used when the maximum de- sired temperature is 160–325 °C (depending on the model). Some ovens include the ability to circulate heated air, allowing for a more efficient removal of moisture and shorter drying times. Other ovens provide a tight seal for the door, allowing the oven to be evacuated. In some situations a conventional laboratory oven can be re- placed with a microwave oven. Higher temperatures, up to 1700° C, can be achieved using a muffle furnace (Figure 2.8). After drying or decomposing a sample, it should be cooled to room tempera- ture in a desiccator to avoid the readsorption of moisture. A desiccator (Figure 2.9) is a closed container that isolates the sample from the atmosphere. A drying agent, called a desiccant, is placed in the bottom of the container. Typical desiccants in- clude calcium chloride and silica gel. A perforated plate sits above the desiccant, providing a shelf for storing samples. Some desiccators are equipped with stopcocks that allow them to be evacuated. Figure 2.9 (a) Desiccator. (b) Desiccator with stopcock for evacuating the desiccator. Photos courtesy of Fisher Scientific. meniscus The curved surface of a liquid contained in a tube. desiccator A closed container containing a desiccant; used to store samples in a moisture-free environment. desiccant A drying agent. 1400-CH02 9/8/99 3:48 PM Page 29 30 Modern Analytical Chemistry 2E Preparing Solutions Preparing a solution of known concentration is perhaps the most common activity in any analytical lab. The method for measuring out the solute and solvent depend on the desired concentration units, and how exact the solution’s concentration needs to be known. Pipets and volumetric flasks are used when a solution’s concen- tration must be exact; graduated cylinders, beakers, and reagent bottles suffice when concentrations need only be approximate. Two methods for preparing solutions are described in this section. 2E.1 Preparing Stock Solutions A stock solution is prepared by weighing out an appropriate portion of a pure solid or by measuring out an appropriate volume of a pure liquid and diluting to a known volume. Exactly how this is done depends on the required concentration units. For example, to prepare a solution with a desired molarity you would weigh out an appropriate mass of the reagent, dissolve it in a portion of solvent, and bring to the desired volume. To prepare a solution where the solute’s concentration is given as a volume percent, you would measure out an appropriate volume of solute and add sufficient solvent to obtain the desired total volume. EXAMPLE 2. 9 Describe how you would prepare the following three solutions: (a) 500 mL of approximately 0.20 M NaOH using solid NaOH; (b) 1 L of 150.0 ppm Cu 2+ using Cu metal; and (c) 2 L of 4% v/v acetic acid using concentrated glacial acetic acid. SOLUTION (a) Since the concentration only needs to be known to two significant figures, the mass of NaOH and volume of solution do not need to be measured exactly. The desired mass of NaOH is To prepare the solution we place 4.0 g of NaOH, weighed to the nearest tenth of a gram, in a bottle or beaker and add approximately 500 mL of water. (b) Since the concentration of Cu 2+ needs to be exact, the mass of Cu metal and the final solution volume must be measured exactly. The desired mass of Cu metal is To prepare the solution we measure out exactly 0.1500 g of Cu into a small beaker. To dissolve the Cu we add a small portion of concentrated HNO 3 and gently heat until it completely dissolves. The resulting solution is poured into a 1-L volumetric flask. The beaker is rinsed repeatedly with small portions of water, which are added to the volumetric flask. This process, which is called a quantitative transfer, ensures that the Cu 2+ is completely transferred to the volumetric flask. Finally, additional water is added to the volumetric flask’s calibration mark. 1 1 000 50.0 mg L L = 150.0 mg = 0.1500 g× . 020 400 050 .. . mol L g mol L = 4.0 g×× stock solution A solution of known concentration from which other solutions are prepared. quantitative transfer The process of moving a sample from one container to another in a manner that ensures all material is transferred. 1400-CH02 9/8/99 3:48 PM Page 30 Chapter 2 Basic Tools of Analytical Chemistry 31 (c) The concentration of this solution is only approximate, so volumes do not need to be measured exactly. The necessary volume of glacial acetic acid is To prepare the solution we use a graduated cylinder to transfer 80 mL of glacial acetic acid to a container that holds approximately 2 L, and we then add sufficient water to bring the solution to the desired volume. 2E.2 Preparing Solutions by Dilution Solutions with small concentrations are often prepared by diluting a more concen- trated stock solution. A known volume of the stock solution is transferred to a new container and brought to a new volume. Since the total amount of solute is the same before and after dilution, we know that C o · V o = C d · V d 2.4 where C o is the concentration of the stock solution, V o is the volume of the stock solution being diluted, C d is the concentration of the dilute solution, and V d is the volume of the dilute solution. Again, the type of glassware used to measure V o and V d depends on how exact the solution’s concentration must be known. EXAMPLE 2.10 A laboratory procedure calls for 250 mL of an approximately 0.10 M solution of NH 3 . Describe how you would prepare this solution using a stock solution of concentrated NH 3 (14.8 M). SOLUTION Substituting known volumes in equation 2.4 14.8 M · V o = 0.10 M · 0.25 L and solving for V o gives 1.69 · 10 –3 L, or 1.7 mL. Since we are trying to make a solution that is approximately 0.10 M NH 3 , we can measure the appropriate amount of concentrated NH 3 using a graduated cylinder, transfer the NH 3 to a beaker, and add sufficient water to bring the total solution volume to approximately 250 mL. As shown in the following example, equation 2.4 also can be used to calculate a solution’s original concentration using its known concentration after dilution. EXAMPLE 2.11 A sample of an ore was analyzed for Cu 2+ as follows. A 1.25-g sample of the ore was dissolved in acid and diluted to volume in a 250-mL volumetric flask. A 20-mL portion of the resulting solution was transferred by pipet to a 50-mL volumetric flask and diluted to volume. An analysis showed that the concentration of Cu 2+ in the final solution was 4.62 ppm. What is the weight percent of Cu in the original ore? 4 2000 mL CH 100 mL mL = 80 mL CH COOH 3 3 COOH × dilution The process of preparing a less concentrated solution from a more concentrated solution. 1400-CH02 9/8/99 3:48 PM Page 31 32 Modern Analytical Chemistry SOLUTION Substituting known volumes (with significant figures appropriate for pipets and volumetric flasks) into equation 2.4 (ppm Cu 2+ ) o · 20.00 mL = 4.62 ppm · 50.00 mL and solving for (ppm Cu 2+ ) o gives the original solution concentration as 11.55 ppm. To calculate the grams of Cu 2+ we multiply this concentration by the total volume The weight percent Cu is then given by 2F The Laboratory Notebook Finally, we cannot end a chapter on the basic tools of analytical chemistry without mentioning the laboratory notebook. Your laboratory notebook is your most im- portant tool when working in the lab, providing a complete record of all your work. If kept properly, you should be able to look back at your laboratory notebook sev- eral years from now and reconstruct the experiments on which you worked. Your instructor will probably provide you with detailed instructions on how he or she wants you to maintain your notebook. Of course, you should expect to bring your notebook to the lab. Everything you do, measure, or observe while working in the lab should be recorded in your notebook as it takes place. Preparing data tables to organize your data will help ensure that you record the data you need and that you can find the data when it is time to calculate and analyze your results. Writing a narrative to accompany your data will help you remember what you did, why you did it, and why you thought it was significant. Reserve space for your calculations, for analyzing your data, and for interpreting your results. Take your notebook with you when you do research in the library. Maintaining a laboratory notebook may seem like a great deal of effort, but if you do it well you have a permanent record of your work. Scientists working in aca- demic, industrial, and governmental research labs rely on their notebooks to pro- vide a written record of their work. Questions about research carried out at some time in the past can be answered by finding the appropriate pages in the laboratory notebook. A laboratory notebook is also a legal document that helps establish patent rights and proof of discovery. 2.888 10 g Cu 1.25 g sample Cu –3 2+ × ×=100 0 231.%/ww 11 55 250 0 . . Cu mL mL 1 g 10 = 2.888 10 g Cu 2+ 6 –3 2+ μ μ g g ×× × 2G KEY TERMS balance (p. 25) concentration (p. 15) desiccant (p. 29) desiccator (p. 29) dilution (p. 31) equivalent (p. 17) equivalent weight (p. 17) formality (p. 15) formula weight (p. 17) meniscus (p. 29) molality (p. 18) molarity (p. 15) 1400-CH02 9/8/99 3:48 PM Page 32 Chapter 2 Basic Tools of Analytical Chemistry 33 normality (p. 16) parts per billion (p. 18) parts per million (p. 18) p-function (p. 19) pipet (p. 27) quantitative transfer (p. 30) scientific notation (p. 12) significant figures (p. 13) SI units (p. 12) stock solution (p. 30) volume percent (p. 18) volumetric flask (p. 26) weight percent (p. 18) weight-to-volume percent (p. 18) There are a few basic numerical and experimental tools with which you must be familiar. Fundamental measurements in ana- lytical chemistry, such as mass and volume, use base SI units, such as the kilogram (kg) and the liter (L). Other units, such as power, are defined in terms of these base units. When reporting measure- ments, we must be careful to include only those digits that are sig- nificant and to maintain the uncertainty implied by these signifi- cant figures when transforming measurements into results. The relative amount of a constituent in a sample is expressed as its concentration. There are many ways to express concentration, the most common of which are molarity, weight percent, volume percent, weight-to-volume percent, parts per million, and parts per billion. Concentrations also can be expressed using p-functions. Stoichiometric relationships and calculations are important in many quantitative analyses. The stoichiometry between the reac- tants and products of a chemical reaction is given by the coeffi- cients of a balanced chemical reaction. When it is inconvenient to balance reactions, conservation principles can be used to establish the stoichiometric relationships. Balances, volumetric flasks, pipets, and ovens are standard pieces of laboratory instrumentation and equipment that are routinely used in almost all analytical work. You should be fa- miliar with the proper use of this equipment. You also should be familiar with how to prepare a stock solution of known con- centration, and how to prepare a dilute solution from a stock solution. 2H SUMMARY 1. Indicate how many significant figures are in each of the following numbers. a. 903 b. 0.903 c. 1.0903 d. 0.0903 e. 0.09030 f. 9.03 · 10 2 2. Round each of the following to three significant figures. a. 0.89377 b. 0.89328 c. 0.89350 d. 0.8997 e. 0.08907 3. Round each of the following to the stated number of significant figures. a. The atomic weight of carbon to four significant figures b. The atomic weight of oxygen to three significant figures c. Avogadro’s number to four significant figures d. Faraday’s constant to three significant figures 4. Report results for the following calculations to the correct number of significant figures. a. 4.591 + 0.2309 + 67.1 = b. 313 – 273.15 = c. 712 · 8.6 = d. 1.43/0.026 = e. (8.314 · 298)/96485 = f. log(6.53 · 10 –5 ) = g. 10 –7.14 = h. (6.51 · 10 –5 ) (8.14 · 10 –9 ) = 5. A 12.1374-g sample of an ore containing Ni and Co was carried through Fresenius’ analytical scheme shown in Figure 1.1. At point A the combined mass of Ni and Co was found to be 0.2306 g, and at point B the mass of Co was found to be 0.0813 g. Report the weight percent Ni in the ore to the correct number of significant figures. 6. Hillebrand and Lundell’s analytical scheme (see Figure 1.2) for the analysis of Ni in ores involves precipitating Ni 2+ using dimethylgloxime. The formula for the precipitate is Ni(C 4 H 7 N 2 O 2 ) 2 . Calculate the precipitate’s formula weight to the correct number of significant figures. 7. An analyst wishes to add 256 mg of Cl – to a reaction mixture. How many milliliters of 0.217 M BaCl 2 should be added? 8. A solution of 0.10 M SO 4 2– is available. What is the normality of this solution when used in the following reactions? a. Pb 2+ (aq) + SO 4 2– (aq) t PbSO 4 (s) b. HCl(aq) + SO 4 2– (aq) t HSO 4 – (aq) + Cl – (aq) c. SO 4 2– + 4H 3 O + (aq) + 2e – t H 2 SO 3 (aq) + 5H 2 O(l) 9. The concentration of lead in an industrial waste stream is 0.28 ppm. What is its molar concentration? 10. Commercially available concentrated hydrochloric acid is 37.0% w/w HCl. Its density is 1.18 g/mL. Using this information calculate (a) the molarity of concentrated HCl, and (b) the mass and volume (in milliliters) of solution containing 0.315 mol of HCl. 11. The density of concentrated ammonia, which is 28.0% w/w NH 3 , is 0.899 g/mL. What volume of this reagent should be diluted to 1.0 · 10 3 mL to make a solution that is 0.036 M in NH 3 ? 2I PROBLEMS 1400-CH02 9/8/99 3:49 PM Page 33 34 Modern Analytical Chemistry 12. A 250.0-mL aqueous solution contains 45.1 m g of a pesticide. Express the pesticide’s concentration in weight percent, parts per million, and parts per billion. 13. A city’s water supply is fluoridated by adding NaF. The desired concentration of F – is 1.6 ppm. How many milligrams of NaF should be added per gallon of treated water if the water supply already is 0.2 ppm in F – ? 14. What is the pH of a solution for which the concentration of H + is 6.92 · 10 –6 M? What is the [H + ] in a solution whose pH is 8.923? 15. Using conservation principles, write stoichiometric relationships for the following a. The precipitation of Mg 2+ as Mg 2 P 2 O 7 b. The acid–base reaction between CaCO 3 and HCl in which H 2 CO 3 is formed c. The reaction between AgCl and NH 3 to form Ag(NH 3 ) 2 + d. The redox reaction between Cr 2 O 7 2– and Fe 2+ to form Cr 3+ and Fe 3+ 16. Calculate the molarity of a potassium dichromate solution prepared by placing 9.67 g of K 2 Cr 2 O 7 in a 100-mL volumetric flask, dissolving, and diluting to the calibration mark. 17. For each of the following, explain how you would prepare 1.0 L of a solution that is 0.10 M in K + . Repeat for concentrations of 1.0 · 10 2 ppm K + and 1.0% w/v K + . a. KCl b. K 2 SO 4 c. K 3 Fe(CN) 6 18. A series of dilute NaCl solutions is prepared, starting with an initial stock solution of 0.100 M NaCl. Solution A is prepared by pipeting 10 mL of the stock solution into a 250-mL volumetric flask and diluting to volume. Solution B is prepared by pipeting 25 mL of solution A into a 100-mL volumetric flask and diluting to volume. Solution C is prepared by pipeting 20 mL of solution B into a 500-mL volumetric flask and diluting to volume. What is the molar concentration of NaCl in solutions A, B, and C? 19. Calculate the molar concentration of NaCl, to the correct number of significant figures, if 1.917 g of NaCl is placed in a beaker and dissolved in 50 mL of water measured with a graduated cylinder. This solution is quantitatively transferred to a 250-mL volumetric flask and diluted to volume. Calculate the concentration of this second solution to the correct number of significant figures. 20. What is the molar concentration of NO 3 – in a solution prepared by mixing 50.0 mL of 0.050 M KNO 3 with 40.0 mL of 0.075 M NaNO 3 ? What is pNO 3 for the mixture? 21. What is the molar concentration of Cl – in a solution prepared by mixing 25.0 mL of 0.025 M NaCl with 35.0 mL of 0.050 M BaCl 2 ? What is pCl for the mixture? 22. To determine the concentration of ethanol in cognac a 5.00-mL sample of cognac is diluted to 0.500 L. Analysis of the diluted cognac gives an ethanol concentration of 0.0844 M. What is the molar concentration of ethanol in the undiluted cognac? Two useful articles providing additional information on topics covered in this chapter are MacCarthy, P. “A Novel Classification of Concentration Units,” J. Chem. Educ. 1983, 60, 187–189. Schwartz, L. M. “Propagation of Significant Figures,” J. Chem. Educ. 1985, 62, 693–697. A useful resource for information on maintaining a useful laboratory notebook is Kanare, H. M. Writing the Laboratory Notebook, American Chemical Society: Washington, DC; 1985. 2 J SUGGESTED READINGS 1. American Public Health Association. Standard Methods for the Analysis of Waters and Wastewaters, 19th ed., Washington, DC. 1995. 2K REFERENCES 1400-CH02 9/8/99 3:49 PM Page 34 Chapter 3 35 The Language of Analytical Chemistry Analytical chemists converse using terminology that conveys specific meaning to other analytical chemists. To discuss and learn analytical chemistry you must first understand its language. You are probably already familiar with some analytical terms, such as accuracy and precision, but you may not have placed them in their appropriate analytical context. Other terms, such as analyte and matrix, may be less familiar. This chapter introduces many important terms routinely used by analytical chemists. Becoming comfortable with these terms will make the material in the chapters that follow easier to read and understand. 1400-CH03 9/8/99 3:51 PM Page 35 36 Modern Analytical Chemistry analysis A process that provides chemical or physical information about the constituents in the sample or the sample itself. analytes The constituents of interest in a sample. matrix All other constituents in a sample except for the analytes. determination An analysis of a sample to find the identity, concentration, or properties of the analyte. measurement An experimental determination of an analyte’s chemical or physical properties. 3 A Analysis, Determination, and Measurement The first important distinction we will make is among the terms “analysis,” “deter- mination,” and “measurement.” An analysis provides chemical or physical infor- mation about a sample. The components of interest in the sample are called ana- lytes, and the remainder of the sample is the matrix. In an analysis we determine the identity, concentration, or properties of the analytes. To make this determina- tion we measure one or more of the analyte’s chemical or physical properties. An example helps clarify the differences among an analysis, a determination, and a measurement. In 1974, the federal government enacted the Safe Drinking Water Act to ensure the safety of public drinking water supplies. To comply with this act municipalities regularly monitor their drinking water supply for potentially harmful substances. One such substance is coliform bacteria. Municipal water de- partments collect and analyze samples from their water supply. To determine the concentration of coliform bacteria, a portion of water is passed through a mem- brane filter. The filter is placed in a dish containing a nutrient broth and incu- bated. At the end of the incubation period the number of coliform bacterial colonies in the dish is measured by counting (Figure 3.1). Thus, municipal water departments analyze samples of water to determine the concentration of coliform bacteria by measuring the number of bacterial colonies that form during a speci- fied period of incubation. 3 B Techniques, Methods, Procedures, and Protocols Suppose you are asked to develop a way to determine the concentration of lead in drinking water. How would you approach this problem? To answer this question it helps to distinguish among four levels of analytical methodology: techniques, meth- ods, procedures, and protocols. 1 A technique is any chemical or physical principle that can be used to study an analyte. Many techniques have been used to determine lead levels. 2 For example, in graphite furnace atomic absorption spectroscopy lead is atomized, and the ability of the free atoms to absorb light is measured; thus, both a chemical principle (atom- ization) and a physical principle (absorption of light) are used in this technique. Chapters 8–13 of this text cover techniques commonly used to analyze samples. A method is the application of a technique for the determination of a specific analyte in a specific matrix. As shown in Figure 3.2, the graphite furnace atomic ab- sorption spectroscopic method for determining lead levels in water is different from that for the determination of lead in soil or blood. Choosing a method for deter- mining lead in water depends on how the information is to be used and the estab- lished design criteria (Figure 3.3). For some analytical problems the best method might use graphite furnace atomic absorption spectroscopy, whereas other prob- lems might be more easily solved by using another technique, such as anodic strip- ping voltammetry or potentiometry with a lead ion-selective electrode. A procedure is a set of written directions detailing how to apply a method to a particular sample, including information on proper sampling, handling of interfer- ents, and validating results. A method does not necessarily lead to a single proce- dure, as different analysts or agencies will adapt the method to their specific needs. As shown in Figure 3.2, the American Public Health Agency and the American Soci- ety for Testing Materials publish separate procedures for the determination of lead levels in water. technique A chemical or physical principle that can be used to analyze a sample. method A means for analyzing a sample for a specific analyte in a specific matrix. procedure Written directions outlining how to analyze a sample. 1400-CH03 9/8/99 3:51 PM Page 36 Chapter 3 The Language of Analytical Chemistry 37 Figure 3.1 Membrane filter showing colonies of coliform bacteria. The number of colonies are counted and reported as colonies/100 mL of sample. PourRite? is a trademark of Hach Company/photo courtesy of Hach Company. Graphite furnace atomic absorption spectroscopy Pb in Soil Procedures Techniques Methods Protocols Pb in Blood Pb in Water EPA APHA ASTM 1. Identify the problem Determine type of information needed (qualitative, quantitative, or characterization) Identify context of the problem 2. Design the experimental procedure Establish design criteria (accuracy, precision, scale of operation, sensitivity, selectivity, cost, speed) Identify interferents Select method Establish validation criteria Establish sampling strategy Figure 3.2 Chart showing hierarchical relationship among a technique, methods using that technique, and procedures and protocols for one method. (Abbreviations: APHA = American Public Health Association, ASTM = American Society for Testing Materials, EPA = Environmental Protection Agency) Figure 3.3 Subsection of the analytical approach to problem solving (see Figure 1.3), of relevance to the selection of a method and the design of an analytical procedure. protocol A set of written guidelines for analyzing a sample specified by an agency. signal An experimental measurement that is proportional to the amount of analyte (S). Finally, a protocol is a set of stringent written guidelines detailing the proce- dure that must be followed if the agency specifying the protocol is to accept the re- sults of the analysis. Protocols are commonly encountered when analytical chem- istry is used to support or define public policy. For purposes of determining lead levels in water under the Safe Drinking Water Act, labs follow a protocol specified by the Environmental Protection Agency. There is an obvious order to these four facets of analytical methodology. Ide- ally, a protocol uses a previously validated procedure. Before developing and vali- dating a procedure, a method of analysis must be selected. This requires, in turn, an initial screening of available techniques to determine those that have the potential for monitoring the analyte. We begin by considering a useful way to classify analyti- cal techniques. 3 C Classifying Analytical Techniques Analyzing a sample generates a chemical or physical signal whose magnitude is pro- portional to the amount of analyte in the sample. The signal may be anything we can measure; common examples are mass, volume, and absorbance. For our pur- poses it is convenient to divide analytical techniques into two general classes based on whether this signal is proportional to an absolute amount of analyte or a relative amount of analyte. Consider two graduated cylinders, each containing 0.01 M Cu(NO 3 ) 2 (Fig- ure 3.4). Cylinder 1 contains 10 mL, or 0.0001 mol, of Cu 2+ ; cylinder 2 contains 20 mL, or 0.0002 mol, of Cu 2+ . If a technique responds to the absolute amount of analyte in the sample, then the signal due to the analyte, S A , can be expressed as S A = kn A 3.1 where n A is the moles or grams of analyte in the sample, and k is a proportionality constant. Since cylinder 2 contains twice as many moles of Cu 2+ as cylinder 1, an- alyzing the contents of cylinder 2 gives a signal that is twice that of cylinder 1. 1400-CH03 9/8/99 3:51 PM Page 37 38 Modern Analytical Chemistry total analysis techniques A technique in which the signal is proportional to the absolute amount of analyte; also called “classical” techniques. concentration techniques A technique in which the signal is proportional to the analyte’s concentration; also called “instrumental” techniques. Figure 3.4 Graduated cylinders containing 0.01 M Cu(NO 3 ) 2 . (a) Cylinder 1 contains 10 mL, or 0.0001 mol, of Cu 2+ . (b) Cylinder 2 contains 20 mL, or 0.0002 mol, of Cu 2+ . ? David Harvey/Marilyn Culler, photographer. (a) (b) accuracy A measure of the agreement between an experimental result and its expected value. A second class of analytical techniques are those that respond to the relative amount of analyte; thus S A = kC A 3.2 where C A is the concentration of analyte in the sample. Since the solutions in both cylinders have the same concentration of Cu 2+ , their analysis yields identical signals. Techniques responding to the absolute amount of analyte are called total analysis techniques. Historically, most early analytical methods used total analysis techniques, hence they are often referred to as “classical” techniques. Mass, volume, and charge are the most common signals for total analysis techniques, and the cor- responding techniques are gravimetry (Chapter 8), titrimetry (Chapter 9), and coulometry (Chapter 11). With a few exceptions, the signal in a total analysis tech- nique results from one or more chemical reactions involving the analyte. These re- actions may involve any combination of precipitation, acid–base, complexation, or redox chemistry. The stoichiometry of each reaction, however, must be known to solve equation 3.1 for the moles of analyte. Techniques, such as spectroscopy (Chapter 10), potentiometry (Chapter 11), and voltammetry (Chapter 11), in which the signal is proportional to the relative amount of analyte in a sample are called concentration techniques. Since most concentration techniques rely on measuring an optical or electrical signal, they also are known as “instrumental” techniques. For a concentration technique, the rela- tionship between the signal and the analyte is a theoretical function that depends on experimental conditions and the instrumentation used to measure the signal. For this reason the value of k in equation 3.2 must be determined experimentally. 3 D Selecting an Analytical Method A method is the application of a technique to a specific analyte in a specific matrix. Methods for determining the concentration of lead in drinking water can be devel- oped using any of the techniques mentioned in the previous section. Insoluble lead salts such as PbSO 4 and PbCrO 4 can form the basis for a gravimetric method. Lead forms several soluble complexes that can be used in a complexation titrimetric method or, if the complexes are highly absorbing, in a spectrophotometric method. Lead in the gaseous free-atom state can be measured by an atomic ab- sorption spectroscopic method. Finally, the availability of multiple oxidation states (Pb, Pb 2+ , Pb 4+ ) makes coulometric, potentiometric, and voltammetric methods feasible. The requirements of the analysis determine the best method. In choosing a method, consideration is given to some or all the following design criteria: accuracy, precision, sensitivity, selectivity, robustness, ruggedness, scale of operation, analysis time, availability of equipment, and cost. Each of these criteria is considered in more detail in the following sections. 3 D.1 Accuracy Accuracy is a measure of how closely the result of an experiment agrees with the ex- pected result. The difference between the obtained result and the expected result is usually divided by the expected result and reported as a percent relative error % Error = obtained result – expected result expected result × 100 1400-CH03 9/8/99 3:51 PM Page 38 Figure 3.5 Two determinations of the concentration of K + in serum, showing the effect of precision. The data in (a) are less scattered and, therefore, more precise than the data in (b). Analytical methods may be divided into three groups based on the magnitude of their relative errors. 3 When an experimental result is within 1% of the correct result, the analytical method is highly ac- curate. Methods resulting in relative errors between 1% and 5% are moderately accurate, but methods of low accuracy produce rel- ative errors greater than 5%. The magnitude of a method’s relative error depends on how accurately the signal is measured, how accurately the value of k in equations 3.1 or 3.2 is known, and the ease of handling the sample without loss or contamination. In general, total analysis methods produce results of high accuracy, and concentration methods range from high to low accuracy. A more detailed discussion of accuracy is presented in Chapter 4. 3 D.2 Precision When a sample is analyzed several times, the individual results are rarely the same. Instead, the results are randomly scattered. Precision is a measure of this variability. The closer the agreement between individual analyses, the more precise the results. For example, in determining the concentration of K + in serum, the results shown in Figure 3.5(a) are more precise than those in Figure 3.5(b). It is important to realize that precision does not imply accuracy. That the data in Figure 3.5(a) are more pre- cise does not mean that the first set of results is more accurate. In fact, both sets of results may be very inaccurate. As with accuracy, precision depends on those factors affecting the relationship between the signal and the analyte (equations 3.1 and 3.2). Of particular impor- tance are the uncertainty in measuring the signal and the ease of handling samples reproducibly. In most cases the signal for a total analysis method can be measured with a higher precision than the corresponding signal for a concentration method. Precision is covered in more detail in Chapter 4. 3 D. 3 Sensitivity The ability to demonstrate that two samples have different amounts of analyte is an essential part of many analyses. A method’s sensitivity is a measure of its ability to establish that such differences are significant. Sensitivity is often confused with a method’s detection limit. 4 The detection limit is the smallest amount of analyte that can be determined with confidence. The detection limit, therefore, is a statisti- cal parameter and is discussed in Chapter 4. Sensitivity is the change in signal per unit change in the amount of analyte and is equivalent to the proportionality constant, k, in equations 3.1 and 3.2. If D S A is the smallest increment in signal that can be measured, then the smallest difference in the amount of analyte that can be detected is Suppose that for a particular total analysis method the signal is a measurement of mass using a balance whose smallest increment is ±0.0001 g. If the method’s ? ? ? ? n S k C S k A A A A = = () () total analysis method concentration method Chapter 3 The Language of Analytical Chemistry 39 5.8 (a) (b) 5.9 6.0 ppm K + 6.1 6.2 5.8 5.9 6.0 ppm K + 6.1 6.2 precision An indication of the reproducibility of a measurement or result. sensitivity A measure of a method’s ability to distinguish between two samples; reported as the change in signal per unit change in the amount of analyte (k). detection limit A statistical statement about the smallest amount of analyte that can be determined with confidence. 1400-CH03 9/8/99 3:51 PM Page 39 40 Modern Analytical Chemistry selectivity A measure of a method’s freedom from interferences as defined by the method’s selectivity coefficient. selectivity coefficient A measure of a method’s sensitivity for an interferent relative to that for the analyte (K A,I ). sensitivity is 0.200, then the method can conceivably detect a difference of as little as in the absolute amount of analyte in two samples. For methods with the same D S A , the method with the greatest sensitivity is best able to discriminate among smaller amounts of analyte. 3 D. 4 Selectivity An analytical method is selective if its signal is a function of only the amount of an- alyte present in the sample. In the presence of an interferent, equations 3.1 and 3.2 can be expanded to include a term corresponding to the interferent’s contribution to the signal, S I , S samp = S A + S I = k A n A + k I n I (total analysis method) 3.3 S samp = S A + S I = k A C A + k I C I (concentration method) 3.4 where S samp is the total signal due to constituents in the sample; k A and k I are the sensitivities for the analyte and the interferent, respectively; and n I and C I are the moles (or grams) and concentration of the interferent in the sample. The selectivity of the method for the interferent relative to the analyte is de- fined by a selectivity coefficient, K A,I 3.5 which may be positive or negative depending on whether the interferent’s effect on the signal is opposite that of the analyte.* A selectivity coefficient greater than +1 or less than –1 indicates that the method is more selective for the interferent than for the analyte. Solving equation 3.5 for k I k I = K A,I · k A 3.6 substituting into equations 3.3 and 3.4, and simplifying gives S samp = k A (n A + K A,I · n I ) (total analysis method) 3.7 S samp = k A (C A + K A,I · C I ) (concentration method) 3.8 The selectivity coefficient is easy to calculate if k A and k I can be independently determined. It is also possible to calculate K A,I by measuring S samp in the presence and absence of known amounts of analyte and interferent. EXAMPLE 3 .1 A method for the analysis of Ca 2+ in water suffers from an interference in the presence of Zn 2+ . When the concentration of Ca 2+ is 100 times greater than that of Zn 2+ , an analysis for Ca 2+ gives a relative error of +0.5%. What is the selectivity coefficient for this method? K k k A,I A = I ?n A = 0.0001 g 0.200 0.0005 g ± =± *Although k A and k I are usually positive, they also may be negative. For example, some analytical methods work by measuring the concentration of a species that reacts with the analyte. As the analyte’s concentration increases, the concentration of the species producing the signal decreases, and the signal becomes smaller. If the signal in the absence of analyte is assigned a value of zero, then the subsequent signals are negative. 1400-CH03 9/8/99 3:51 PM Page 40 Chapter 3 The Language of Analytical Chemistry 41 SOLUTION Since only relative concentrations are reported, we can arbitrarily assign absolute concentrations. To make the calculations easy, let C Ca = 100 (arbitrary units) and C Zn = 1. A relative error of +0.5% means that the signal in the presence of Zn 2+ is 0.5% greater than the signal in the absence of zinc. Again, we can assign values to make the calculation easier. If the signal in the absence of zinc is 100 (arbitrary units), then the signal in the presence of zinc is 100.5. The value of k Ca is determined using equation 3.2 In the presence of zinc the signal is S samp = 100.5 = k Ca C Ca + k Zn C Zn = (1)(100) + k Zn (1) Solving for k Zn gives a value of 0.5. The selectivity coefficient, therefore, is Knowing the selectivity coefficient provides a useful way to evaluate an inter- ferent’s potential effect on an analysis. An interferent will not pose a problem as long as the term K A,I · n I in equation 3.7 is significantly smaller than n A , or K A,I · C I in equation 3.8 is significantly smaller than C A. EXAMPLE 3 .2 Barnett and colleagues 5 developed a new method for determining the concentration of codeine during its extraction from poppy plants. As part of their study they determined the method’s response to codeine relative to that for several potential interferents. For example, the authors found that the method’s signal for 6-methoxycodeine was 6 (arbitrary units) when that for an equimolar solution of codeine was 40. (a) What is the value for the selectivity coefficient K A,I when 6-methoxycodeine is the interferent and codeine is the analyte? (b) If the concentration of codeine is to be determined with an accuracy of ±0.50%, what is the maximum relative concentration of 6-methoxycodeine (i.e., [6-methoxycodeine]/[codeine]) that can be present? SOLUTION (a) The signals due to the analyte, S A, and the interferent, S I , are S A = k A C A S I = k I C I Solving these two expressions for k A and k I and substituting into equation 3.6 gives K SC SC AI II AA , / / = K k k Ca Zn/ == . = 0.5 Zn Ca 05 1 k S C Ca Ca Ca = == 100 100 1 1400-CH03 9/8/99 3:51 PM Page 41 Since equimolar concentrations of analyte and interferent were used (C A = C I ), we have (b) To achieve an accuracy of better than ±0.50% the term K A,I · C I in equation 3.8 must be less than 0.50% of C A; thus 0.0050 · C A ? K A,I · C I Solving this inequality for the ratio C I /C A and substituting the value for K A,I determined in part (a) gives Therefore, the concentration of 6-methoxycodeine cannot exceed 3.3% of codeine’s concentration. Not surprisingly, methods whose signals depend on chemical reactivity are often less selective and, therefore, more susceptible to interferences. Problems with selec- tivity become even greater when the analyte is present at a very low concentration. 6 3 D. 5 Robustness and Ruggedness For a method to be useful it must provide reliable results. Unfortunately, methods are subject to a variety of chemical and physical interferences that contribute uncer- tainty to the analysis. When a method is relatively free from chemical interferences, it can be applied to the determination of analytes in a wide variety of sample matri- ces. Such methods are considered robust. Random variations in experimental conditions also introduce uncertainty. If a method’s sensitivity is highly dependent on experimental conditions, such as tem- perature, acidity, or reaction time, then slight changes in those conditions may lead to significantly different results. A rugged method is relatively insensitive to changes in experimental conditions. 3 D.6 Scale of Operation Another way to narrow the choice of methods is to consider the scale on which the analysis must be conducted. Three limitations of particular importance are the amount of sample available for the analysis, the concentration of analyte in the sample, and the absolute amount of analyte needed to obtain a measurable signal. The first and second limitations define the scale of operations shown in Figure 3.6; the last limitation positions a method within the scale of operations. 7 The scale of operations in Figure 3.6 shows the analyte’s concentration in weight percent on the y-axis and the sample’s size on the x-axis. For convenience, we divide analytes into major (>1% w/w), minor (0.01% w/w – 1% w/w), trace (10 –7 % w/w – 0.01% w/w) and ultratrace (<10 –7 % w/w) components, and we divide samples into macro (>0.1 g), meso (10 mg – 100 mg), micro (0.1 mg – 10 mg) and ultramicro (<0.1 mg) sample sizes. Note that both the x-axis and the y-axis use a logarithmic scale. The analyte’s concentration and the amount of C CK I AAI ≤== 0 0050 0 0050 015 0 033 .. . . , K S S AI I A , .=== 6 40 015 42 Modern Analytical Chemistry rugged A method that is insensitive to changes in experimental conditions is considered rugged. robust A method that can be applied to analytes in a wide variety of matrices is considered robust. 1400-CH03 9/8/99 3:51 PM Page 42 Figure 3.6 Scale of operation for analytical methods. Adapted from references 7a and 7b. sample used provide a characteristic description for an analysis. For example, samples in a macro–major analysis weigh more than 0.1 g and contain more than 1% analyte. Diagonal lines connecting the two axes show combinations of sample size and concentration of analyte containing the same absolute amount of analyte. As shown in Figure 3.6, for example, a 1-g sample containing 1% analyte has the same amount of analyte (0.010 g) as a 100-mg sample containing 10% analyte or a 10-mg sample containing 100% analyte. Since total analysis methods respond to the absolute amount of analyte in a sample, the diagonal lines provide an easy way to define their limitations. Consider, for example, a hypothetical total analysis method for which the minimum de- tectable signal requires 100 mg of analyte. Using Figure 3.6, the diagonal line repre- senting 100 mg suggests that this method is best suited for macro samples and major analytes. Applying the method to a minor analyte with a concentration of 0.1% w/w requires a sample of at least 100 g. Working with a sample of this size is rarely practical, however, due to the complications of carrying such a large amount of material through the analysis. Alternatively, the minimum amount of required analyte can be decreased by improving the limitations associated with measuring the signal. For example, if the signal is a measurement of mass, a decrease in the minimum amount of analyte can be accomplished by switching from a con- ventional analytical balance, which weighs samples to ±0.1 mg, to a semimicro (±0.01 mg) or microbalance (±0.001 mg). Chapter 3 The Language of Analytical Chemistry 43 Ultratrace Trace Minor g mg μg ng 1 g sample, 1% analyte ppm ppb 100 mg 10 mg 1 mg 100 μ g 10 μ g 1 μ g 0.1 g sample, 10% analyte 0.01 g sample, 100% analyte Macro Micro Meso Ultramicro Major 10 –10 % 10 –9 % 10 –8 % 10 –7 % 10 –6 % 10 –5 % 10 –4 % 10 –3 % 10 –2 % 0.1% 1% 10% 100% –log(% anal yte as %w/w) –log(Weight of sample) 1 0.1 0.01 0.1 0.01 0.1 0.01 100 10 1 100 100 10 10 1 1 1400-CH03 9/8/99 3:51 PM Page 43 Concentration methods frequently have both lower and upper limits for the amount of analyte that can be determined. The lower limit is dictated by the small- est concentration of analyte producing a useful signal and typically is in the parts per million or parts per billion concentration range. Upper concentration limits exist when the sensitivity of the analysis decreases at higher concentrations. An upper concentration level is important because it determines how a sam- ple with a high concentration of analyte must be treated before the analysis. Con- sider, for example, a method with an upper concentration limit of 1 ppm (micro- grams per milliliter). If the method requires a sample of 1 mL, then the upper limit on the amount of analyte that can be handled is 1 m g. Using Figure 3.6, and following the diagonal line for 1 m g of analyte, we find that the analysis of an ana- lyte present at a concentration of 10% w/w requires a sample of only 10 m g! Ex- tending such an analysis to a major analyte, therefore, requires the ability to ob- tain and work with very small samples or the ability to dilute the original sample accurately. Using this example, analyzing a sample for an analyte whose concen- tration is 10% w/w requires a 10,000-fold dilution. Not surprisingly, concentra- tion methods are most commonly used for minor, trace, and ultratrace analytes, in macro and meso samples. 3 D. 7 Equipment, Time, and Cost Finally, analytical methods can be compared in terms of their need for equipment, the time required to complete an analysis, and the cost per sample. Methods relying on instrumentation are equipment-intensive and may require significant operator training. For example, the graphite furnace atomic absorption spectroscopic method for determining lead levels in water requires a significant capital investment in the instrument and an experienced operator to obtain reliable results. Other methods, such as titrimetry, require only simple equipment and reagents and can be learned quickly. The time needed to complete an analysis for a single sample is often fairly simi- lar from method to method. This is somewhat misleading, however, because much of this time is spent preparing the solutions and equipment needed for the analysis. Once the solutions and equipment are in place, the number of samples that can be analyzed per hour differs substantially from method to method. This is a significant factor in selecting a method for laboratories that handle a high volume of samples. The cost of an analysis is determined by many factors, including the cost of necessary equipment and reagents, the cost of hiring analysts, and the number of samples that can be processed per hour. In general, methods relying on instruments cost more per sample than other methods. 3 D.8 Making the Final Choice Unfortunately, the design criteria discussed earlier are not mutually independent. 8 Working with smaller amounts of analyte or sample, or improving selectivity, often comes at the expense of precision. Attempts to minimize cost and analysis time may decrease accuracy. Selecting a specific method requires a careful balance among these design criteria. Usually, the most important design criterion is accuracy, and the best method is that capable of producing the most accurate results. When the need for results is urgent, as is often the case in clinical labs, analysis time may be- come the critical factor. The best method is often dictated by the sample’s properties. Analyzing a sam- ple with a complex matrix may require a method with excellent selectivity to avoid 44 Modern Analytical Chemistry 1400-CH03 9/8/99 3:51 PM Page 44 Chapter 3 The Language of Analytical Chemistry 45 interferences. Samples in which the analyte is present at a trace or ultratrace con- centration usually must be analyzed by a concentration method. If the quantity of sample is limited, then the method must not require large amounts of sample. Determining the concentration of lead in drinking water requires a method that can detect lead at the parts per billion concentrations. Selectivity is also im- portant because other metal ions are present at significantly higher concentrations. Graphite furnace atomic absorption spectroscopy is a commonly used method for determining lead levels in drinking water because it meets these specifications. The same method is also used in determining lead levels in blood, where its ability to detect low concentrations of lead using a few microliters of sample are important considerations. 3 E Developing the Procedure After selecting a method, it is necessary to develop a procedure that will accomplish the goals of the analysis. In developing the procedure, attention is given to compen- sating for interferences, selecting and calibrating equipment, standardizing the method, acquiring a representative sample, and validating the method. 3 E.1 Compensating for Interferences The accuracy of a method depends on its selectivity for the analyte. Even the best methods, however, may not be free from interferents that contribute to the mea- sured signal. Potential interferents may be present in the sample itself or the reagents used during the analysis. In this section we will briefly look at how to mini- mize these two sources of interference. In the absence of an interferent, the total signal measured during an analy- sis, S meas , is a sum of the signal due to the analyte, and the signal due to the rea- gents, S reag S meas = S A + S reag = kn A + S reag (total analysis method) 3.9 S meas = S A + S reag = kC A + S reag (concentration method) 3.10 Without an independent determination of S reag , equation 3.9 or 3.10 cannot be solved for the moles or concentration of analyte. The contribution of S reag is deter- mined by measuring the signal for a reagent or method blank that does not contain the sample. Consider, for example, a procedure in which a 0.1-g sample is dissolved in 100 mL of solvent. After dissolving the sample, several reagents are added, and the signal is measured. The reagent blank is prepared by omitting the sample and adding the reagents to 100 mL of solvent. When the sample is a liquid, or is in solu- tion, an equivalent volume of an inert solvent is substituted for the sample. Once S reag is known, it is easy to correct S meas for the reagent’s contribution to the overall signal. Compensating for an interference in the sample’s matrix is more difficult. If the identity and concentration of the interferent are known, then it can be added to the reagent blank. In most analyses, however, the identity or concentration of matrix interferents is not known, and their contribution to S meas is not included in S reag . In- stead, the signal from the interferent is included as an additional term S meas = k A n A + k I n I + S reag (total analysis method) 3.11 S meas = k A C A + k I C I + S reag (concentration method) 3.12 method blank A sample that contains all components of the matrix except the analyte. 1400-CH03 9/8/99 3:51 PM Page 45 46 Modern Analytical Chemistry Solving either equation 3.11 or 3.12 for the amount of analyte can be accomplished by separating the analyte and interferent before the analysis, thus eliminating the term for the interferent. Methods for effecting this separation are discussed in Chapter 7. Alternatively, equations 3.11 or 3.12 can be solved for the amounts of both the analyte and the interferent. To do so, however, we must obtain two independent values for S meas . Using a concentration method as an example, gives two equations S meas,1 = k A,1 C A + k I,1 C I + S reag,1 S meas,2 = k A,2 C A + k I,2 C I + S reag,2 that can be solved simultaneously for C A and C I . This treatment is general. The composition of a solution with a total of n analytes and interferents can be deter- mined by measuring n independent signals, and solving n independent simultane- ous equations of the general form of equation 3.11 or 3.12. EXAMPLE 3 . 3 A sample was analyzed for the concentration of two analytes, A and B, under two sets of conditions. Under condition 1, the calibration sensitivities are k A,1 = 76 ppm –1 k B,1 = 186 ppm –1 and for condition 2 k A,2 = 33 ppm –1 k B,2 = 243 ppm –1 The signals under the two sets of conditions are S meas,1 = 33.4 S meas,2 = 29.7 Determine the concentration of A and B. You may assume that S reag is zero under both conditions. SOLUTION Using equation 3.12, we write the following simultaneous equations 33.4 = (76 ppm –1 )C A + (186 ppm –1 )C B 29.7 = (33 ppm –1 )C A + (243 ppm –1 )C B Multiplying the first equation by the ratio 33/76 gives the two equations as 14.5 = (33 ppm –1 )C A + (80.8 ppm –1 )C B 29.7 = (33 ppm –1 )C A + (243 ppm –1 )C B Subtracting the first equation from the second gives 15.2 = (162.2 ppm –1 )C B Solving for C B gives the concentration of B as 0.094 ppm. Substituting this concentration back into either of the two original equations gives the concentration of A, C A , as 0.21 ppm. 1400-CH03 9/8/99 3:51 PM Page 46 3 E.2 Calibration and Standardization Analytical chemists make a distinction between calibration and standardization. 9 Calibration ensures that the equipment or instrument used to measure the signal is operating correctly by using a standard known to produce an exact signal. Balances, for example, are calibrated using a standard weight whose mass can be traced to the internationally accepted platinum–iridium prototype kilogram. Standardization is the process of experimentally determining the relation- ship between the signal and the amount of analyte (the value of k in equations 3.1 and 3.2). For a total analysis method, standardization is usually defined by the stoichiometry of the chemical reactions responsible for the signal. For a con- centration method, however, the relationship between the signal and the ana- lyte’s concentration is a theoretical function that cannot be calculated without experimental measurements. To standardize a method, the value of k is deter- mined by measuring the signal for one or more standards, each containing a known concentration of analyte. When several standards with different concen- trations of analyte are used, the result is best viewed visually by plotting S meas versus the concentration of analyte in the standards. Such a plot is known as a calibration curve. A more detailed discussion of calibration and standardization is found in Chapter 5. 3 E. 3 Sampling Selecting an appropriate method helps ensure that an analysis is accurate. It does not guarantee, however, that the result of the analysis will be sufficient to solve the problem under investigation or that a proposed answer will be correct. These latter concerns are addressed by carefully collecting the samples to be analyzed. A proper sampling strategy ensures that samples are representative of the mate- rial from which they are taken. Biased or nonrepresentative sampling and contami- nation of samples during or after their collection are two sources of sampling error that can lead to significant errors. It is important to realize that sampling errors are completely independent of analysis errors. As a result, sampling errors cannot be corrected by evaluating a reagent blank. A more detailed discussion of sampling is found in Chapter 7. 3 E. 4 Validation Before a procedure can provide useful analytical information, it is necessary to demonstrate that it is capable of providing acceptable results. Validation is an eval- uation of whether the precision and accuracy obtained by following the procedure are appropriate for the problem. In addition, validation ensures that the written procedure has sufficient detail so that different analysts or laboratories following the same procedure obtain comparable results. Ideally, validation uses a standard sam- ple whose composition closely matches the samples for which the procedure was developed. The comparison of replicate analyses can be used to evaluate the proce- dure’s precision and accuracy. Intralaboratory and interlaboratory differences in the procedure also can be evaluated. In the absence of appropriate standards, accuracy can be evaluated by comparing results obtained with a new method to those ob- tained using a method of known accuracy. Chapter 14 provides a more detailed dis- cussion of validation techniques. Chapter 3 The Language of Analytical Chemistry 47 standardization The process of establishing the relationship between the amount of analtye and a method’s signal. calibration curve The result of a standardization showing graphically how a method’s signal changes with respect to the amount of analyte. validation The process of verifying that a procedure yields acceptable results. calibration The process of ensuring that the signal measured by a piece of equipment or an instrument is correct. 1400-CH03 9/8/99 3:51 PM Page 47 48 Modern Analytical Chemistry 3 F Protocols Earlier we noted that a protocol is a set of stringent written guidelines, specifying an exact procedure that must be followed if results are to be accepted by the agency specifying the protocol. Besides all the considerations taken into account when de- signing the procedure, a protocol also contains very explicit instructions regarding internal and external quality assurance and quality control (QA/QC) procedures. 10 Internal QA/QC includes steps taken to ensure that the analytical work in a given laboratory is both accurate and precise. External QA/QC usually involves a process in which the laboratory is certified by an external agency. As an example, we will briefly outline some of the requirements in the Envi- ronmental Protection Agency’s Contract Laboratory Program (CLP) protocol for the analysis of trace metals in aqueous samples by graphite furnace atomic ab- sorption spectrophotometry. The CLP protocol (Figure 3.7) calls for daily stan- dardization with a reagent blank and three standards, one of which is at the labo- ratory’s contract required detection limit. The resulting calibration curve is then verified by analyzing initial calibration verification (ICV) and initial calibration blank (ICB) samples. The reported concentration of the ICV sample must fall within ±10% of the expected concentration. If the concentration falls outside this limit, the analysis must be stopped and the problem identified and corrected be- fore continuing. After a successful analysis of the ICV and ICB samples, standardization is rever- ified by analyzing a continuing calibration verification (CCV) sample and a contin- uing calibration blank (CCB). Results for the CCV also must be within ±10% of the expected concentration. Again, if the concentration of the CCV falls outside the es- tablished limits, the analysis must be stopped, the problem identified and corrected, and the system standardized as described earlier. The CCV and the CCB are ana- lyzed before the first and after the last sample, and after every set of ten samples. Whenever the CCV or the CCB is unacceptable, the results for the most recent set of ten samples are discarded, the system is standardized, and the samples are reana- lyzed. By following this protocol, every result is bound by successful checks on the standardization. Although not shown in Figure 3.7, the CLP also contains detailed instructions regarding the analysis of duplicate or split samples and the use of spike testing for accuracy. 3 G The Importance of Analytical Methodology The importance of analytical methodology is evident when examining the results of environmental monitoring programs. The purpose of a monitoring program is to determine the present status of an environmental system and to assess long- term trends in the quality of the system. These are broad and poorly defined goals. In many cases, such studies are initiated with little thought to the ques- tions the data will be used to answer. This is not surprising since it can be hard to formulate questions in the absence of initial information about the system. Without careful planning, however, a poor experimental design may result in data that has little value. quality assurance and quality control Those steps taken to ensure that the work conducted in an analytical lab is capable of producing acceptable results; also known as QA/QC. 1400-CH03 9/8/99 3:51 PM Page 48 Chapter 3 The Language of Analytical Chemistry 49 Figure 3.7 Schematic diagram of a portion of the Contract Laboratory Program protocol for the analysis of trace metals by graphite furnace atomic spectrophotometry, as specified by the Environmental Protection Agency. (Abbreviations: ICV = initial calibration verification; ICB = initial calibration blank, CCV = continuing calibration verification, CCB = continuing calibration blank) Standardization Run 10 samples Discard results for last set of samples Identify and correct problem Ye s No Ye s Ye s Ye s No No No ICV, ICB OK? CCV, CCB OK? CCV, CCB OK? More samples? Start End 1400-CH03 9/8/99 3:51 PM Page 49 50 Modern Analytical Chemistry Every discipline has its own terminology. Your success in studying analytical chemistry will improve if you master the language used by analytical chemists. Be sure that you understand the difference between an analyte and its matrix, a technique and a method, a procedure and a protocol, and a total analysis technique and a concentration technique. An analytical method is selected on the basis of criteria such as accuracy, precision, sensitivity, selectivity, robustness, ruggedness, the amount of available sample, the amount of analyte in the sam- ple, time, cost, and the availability of equipment. These criteria are not mutually independent, and it often is necessary to find an ac- ceptable balance among them. In developing a procedure or protocol, consideration is given to compensating for interferences, calibrating equipment and standardizing the method, obtaining an appropriate sample, and validating the analysis. Poorly designed procedures and protocols produce results that are insufficient to meet the needs of the analysis. 3 I SUMMARY 3 H KEY TERMS accuracy (p. 38) analysis (p. 36) analytes (p. 36) calibration (p. 47) calibration curve (p. 47) concentration techniques (p. 38) detection limit (p. 39) determination (p. 36) matrix (p. 36) measurement (p. 36) method (p. 36) method blank (p. 45) precision (p. 39) procedure (p. 36) protocol (p. 37) quality assurance and quality control (p. 48) robust (p. 42) rugged (p. 42) selectivity (p. 40) selectivity coefficient (p. 40) sensitivity (p. 39) signal (p. 37) standardization (p. 47) technique (p. 36) total analysis techniques (p. 38) validation (p. 47) These concerns are illustrated by the Chesapeake Bay monitoring program. This research program, designed to study nutrients and toxic pollutants in the Chesapeake Bay, was initiated in 1984 as a cooperative venture between the fed- eral government, the state governments of Maryland, Virginia, and Pennsylvania, and the District of Columbia. A 1989 review of some of the problems with this program highlights the difficulties common to many monitoring programs. 11 At the beginning of the Chesapeake Bay monitoring program, little attention was given to the proper choice of analytical methods, in large part because the intended uses of the monitoring data were not specified. The analytical methods initially chosen were those standard methods already approved by the EPA. In many cases these methods proved to be of little value for this monitoring project. Most of the EPA-approved methods were designed to detect pollutants at their legally mandated maximum allowed concentrations. The concentrations of these contaminants in natural waters, however, are often well below the detection limit of the EPA methods. For example, the EPA-approved standard method for phos- phate had a detection limit of 7.5 ppb. Since actual phosphate concentrations in Chesapeake Bay usually were below the EPA detection limit, the EPA method provided no useful information. On the other hand, a nonapproved variant of the EPA method commonly used in chemical oceanography had a detection limit of 0.06 ppb. In other cases, such as the elemental analysis for particulate forms of carbon, nitrogen, and phosphorus, EPA-approved procedures provided poorer reproducibility than nonapproved methods. 1400-CH03 9/8/99 3:51 PM Page 50 Chapter 3 The Language of Analytical Chemistry 51 1. When working with a solid sample, it often is necessary to bring the analyte into solution by dissolving the sample in a suitable solvent. Any solid impurities that remain are removed by filtration before continuing with the analysis. In a typical total analysis method, the procedure might read After dissolving the sample in a beaker, remove any solid impurities by passing the solution containing the analyte through filter paper, collecting the solution in a clean Erlenmeyer flask. Rinse the beaker with several small portions of solvent, passing these rinsings through the filter paper, and collecting them in the same Erlenmeyer flask. Finally, rinse the filter paper with several portions of solvent, collecting the rinsings in the same Erlenmeyer flask. For a typical concentration method, however, the procedure might state After dissolving the sample in a beaker, remove any solid impurities by filtering a portion of the solution containing the analyte. Collect and discard the first several milliliters of solution before collecting a sample of approximately 5 mL for further analysis. Explain why these two procedures are different. 2. A certain concentration method works best when the analyte’s concentration is approximately 10 ppb. a. If the sampling volume for the method is 0.5 mL, about what mass of analyte is being measured? b. If the analyte is present at 10% w/v, how would you prepare the sample for analysis? c. Repeat for the case in which the analyte is present at 10% w/w. d. Based on your results, comment on the suitability of this method for the analysis of a major analyte. 3. An analyst needs to evaluate the potential effect of an interferent, I, on the quantitative analysis for an analyte, A. She begins by measuring the signal for a sample in which the interferent is absent and the analyte is present with a concentration of 15 ppm, obtaining an average signal of 23.3 (arbitrary units). When analyzing a sample in which the analyte is absent and the interferent is present with a concentration of 25 ppm, she obtains an average signal of 13.7. a. What is the analyte’s sensitivity? b. What is the interferent’s sensitivity? c. What is the value of the selectivity coefficient? d. Is the method more selective for the analyte or the interferent? e. What is the maximum concentration of interferent relative to that of the analyte (i.e., [interferent]/[analyte]), if the error in the analysis is to be less than 1%? 4. A sample was analyzed to determine the concentration of an analyte. Under the conditions of the analysis, the sensitivity is 17.2 ppm –1 . What is the analyte’s concentration if S meas is 35.2 and S reag is 0.6? 5. A method for the analysis of Ca 2+ in water suffers from an interference in the presence of Zn 2+ . When the concentration of Ca 2+ is 50 times greater than that of Zn 2+ , an analysis for Ca 2+ gives a relative error of –2.0%. What is the value of the selectivity coefficient for this method? 6. The quantitative analysis for reduced glutathione in blood is complicated by the presence of many potential interferents. In one study, when analyzing a solution of 10-ppb glutathione and 1.5-ppb ascorbic acid, the signal was 5.43 times greater than that obtained for the analysis of 10-ppb glutathione. 12 What is the selectivity coefficient for this analysis? The same study found that when analyzing a solution of 350-ppb methionine and 10-ppb glutathione the signal was 0.906 times less than that obtained for the analysis of 10 ppb-glutathione. What is the selectivity coefficient for this analysis? In what way do these interferents behave differently? 7. Oungpipat and Alexander described a new method for determining the concentration of glycolic acid (GA) in a variety of samples, including physiological fluids such as urine. 13 In the presence of only GA, the signal is given as S samp,1 = k GA C GA and in the presence of both glycolic acid and ascorbic acid (AA), the signal is S samp,2 = k GA C GA + k AA C AA When the concentration of glycolic acid is 1.0 · 10 –4 M and the concentration of ascorbic acid is 1.0 · 10 –5 M, the ratio of the two signals was found to be a. Using the ratio of the two signals, determine the value of the selectivity ratio b. Is the method more selective toward glycolic acid or ascorbic acid? c. If the concentration of ascorbic acid is 1.0 · 10 –5 M, what is the smallest concentration of glycolic acid that can be determined such that the error introduced by failing to account for the signal from ascorbic acid is less than 1%? K k k GA AA AA GA , = S S samp samp , , 2 1 144=. 3J PROBLEMS 1400-CH03 9/8/99 3:51 PM Page 51 8. Ibrahim and co-workers developed a new method for the quantitative analysis of hypoxanthine, a natural compound of some nucleic acids. 14 As part of their study they evaluated the method’s selectivity for hypoxanthine in the presence of several possible interferents, including ascorbic acid. a. When analyzing a solution of 1.12 · 10 –6 M hypoxanthine, the authors obtained a signal of 7.45 · 10 –5 amperes (A). What is the sensitivity for hypoxanthine? You may assume that the signal has been corrected for the method blank. b. When a solution containing 1.12 · 10 –6 M hypoxanthine and 6.5 · 10 –5 M ascorbic acid was analyzed a signal of 4.04 · 10 –5 A was obtained. What is the selectivity coefficient for this method? c. Is the method more selective for hypoxanthine or for ascorbic acid? d. What is the largest concentration of ascorbic acid that may be present if a concentration of 1.12 · 10 –6 M hypoxanthine is to be determined within ±1%? 52 Modern Analytical Chemistry The following papers provide alternative schemes for classifying analytical methods Booksh, K. S.; Kowalski, B. R. “Theory of Analytical Chemistry,” Anal. Chem. 1994, 66, 782A–791A. Phillips, J. B. “Classification of Analytical Methods,” Anal. Chem. 1981, 53, 1463A–1470A. Valcárcel, M.; Luque de Castro, M. D. “A Hierarchical Approach to Analytical Chemistry,” Trends Anal. Chem., 1995, 14, 242–250. Further details on evaluating analytical methods may be found in Wilson, A. L. “The Performance-Characteristics of Analytical Methods,” Part I-Talanta, 1970, 17, 21–29; Part II-Talanta, 1970, 17, 31–44; Part III-Talanta, 1973, 20, 725–732; Part IV- Talanta, 1974, 21, 1109–1121. Several texts provide numerous examples of analytical procedures for specific analytes in well-defined matrices. Basset, J.; Denney, R. C.; Jeffery, G. H.; et al. Vogel’s Textbook of Quantitative Inorganic Analysis, 4th ed. Longman: London, 1981. Csuros, M. Environmental Sampling and Analysis for Technicians, Lewis: Boca Raton, 1994. Keith, L. H., ed. Compilation of EPA’s Sampling and Analysis Methods, Lewis: Boca Raton, 1996. Rump, H. H.; Krist, H. Laboratory Methods for the Examination of Water, Wastewater and Soil. VCH Publishers: New York, 1988. Standard Methods for the Analysis of Waters and Wastewaters, 19th ed. American Public Health Association: Washington, DC, 1995. 3 K SUGGESTED READINGS 1. Taylor, J. K. Anal. Chem. 1983, 55, 600A–608A. 2. Fitch, A.; Wang, Y.; Mellican, S.; et al. Anal. Chem. 1996, 68, 727A–731A. 3. Basset, J.; Denney, R. C.; Jeffery, G. H.; et al. Vogel’s Textbook of Quantitative Inorganic Analysis, 4th ed. Longman: London, 1981, p. 8. 4. Ingle, J. D.; Crouch, S. R. Spectrochemical Analysis. Prentice-Hall: Englewood, NJ, 1988, pp. 171–172. 5. Barnett, N. W.; Bowser, T. A.; Gerardi, R. D.; et al. Anal. Chim. Acta 1996, 318, 309–317. 6. Rogers, L. B. J. Chem. Ed. 1986, 63, 3–6. 7. (a) Sandell, E. B.; Elving, P. J. In Kolthoff, I. M.; Elving, P. J., eds. Treatise on Analytical Chemistry, Interscience: New York; Part 1, Vol. 1, Chapter 1, pp. 3–6; (b) Potts, L. W. Quantitative Analysis—Theory and Practice. Harper and Row: New York, 1987, p. 12. 8. Valcárcel, M.; Ríos, A. Anal. Chem. 1993, 65, 781A–787A. 9. Valcárcel, M.; Ríos, A. Analyst, 1995, 120, 2291–2297. 10. (a) Amore, F. Anal. Chem. 1979, 51, 1105A–1110A; (b) Taylor, J. K. Anal. Chem. 1981, 53, 1588A–1593A. 11. D’Elia, C. F.; Sanders, J. G.; Capone, D. G. Environ. Sci. Technol. 1989, 23, 768–774. 12. Jiménez-Prieto, R.; Velasco, A.; Silva, M.; et al. Anal. Chim. Acta 1992, 269, 273–279. 13. Oungpipat, W.; Alexander, P. W. Anal. Chim. Acta 1994, 295, 36–46. 14. Ibrahim, M. S.; Ahmad, M. E.; Temerk, Y. M.; et al. Anal. Chim. Acta 1996, 328, 47–52. 3 L REFERENCES 9. A sample was analyzed for the concentration of two analytes, C and D, under two sets of conditions. Under condition 1 the calibration sensitivities are k C,1 = 23 ppm –1 k D,1 = 415 ppm –1 and for condition 2 k C,2 = 115 ppm –1 k D,2 = 45 ppm –1 The signals under the two sets of conditions are S meas,1 = 78.6 S meas,2 = 47.9 Determine the concentration of C and D. You may assume that S reag is zero under both conditions. 10. Examine a procedure from Standard Methods for the Analysis of Waters and Wastewaters (or another manual of standard analytical methods), and identify the steps taken to compensate for interferences, to calibrate equipment and instruments, to standardize the method, and to acquire a representative sample. 1400-CH03 9/8/99 3:51 PM Page 52 Chapter 4 53 Evaluating Analytical Data Aproblem dictates the requirements we place on our measurements and results. Regulatory agencies, for example, place stringent requirements on the reliability of measurements and results reported to them. This is the rationale for creating a protocol for regulatory problems. Screening the products of an organic synthesis, on the other hand, places fewer demands on the reliability of measurements, allowing chemists to customize their procedures. When designing and evaluating an analytical method, we usually make three separate considerations of experimental error. 1 First, before beginning an analysis, errors associated with each measurement are evaluated to ensure that their cumulative effect will not limit the utility of the analysis. Errors known or believed to affect the result can then be minimized. Second, during the analysis the measurement process is monitored, ensuring that it remains under control. Finally, at the end of the analysis the quality of the measurements and the result are evaluated and compared with the original design criteria. This chapter is an introduction to the sources and evaluation of errors in analytical measurements, the effect of measurement error on the result of an analysis, and the statistical analysis of data. 1400-CH04 9/8/99 3:53 PM Page 53 54 Modern Analytical Chemistry mean The average value of a set of data ( – X). 4 A Characterizing Measurements and Results Let’s begin by choosing a simple quantitative problem requiring a single measure- ment. The question to be answered is—What is the mass of a penny? If you think about how we might answer this question experimentally, you will realize that this problem is too broad. Are we interested in the mass of United State pennies or Cana- dian pennies, or is the difference in country of importance? Since the composition of a penny probably differs from country to country, let’s limit our problem to pennies minted in the United States. There are other considerations. Pennies are minted at several locations in the United States (this is the meaning of the letter, or absence of a letter, below the date stamped on the lower right corner of the face of the coin). Since there is no reason to expect a difference between where the penny was minted, we will choose to ignore this consideration. Is there a reason to expect a difference between a newly minted penny not yet in circulation, and a penny that has been in circulation? The answer to this is not obvious. Let’s simplify the problem by narrow- ing the question to—What is the mass of an average United States penny in circula- tion? This is a problem that we might expect to be able to answer experimentally. A good way to begin the analysis is to acquire some preliminary data. Table 4.1 shows experimentally measured masses for seven pennies from my change jar at home. Looking at these data, it is immediately apparent that our question has no simple answer. That is, we cannot use the mass of a single penny to draw a specific conclusion about the mass of any other penny (although we might conclude that all pennies weigh at least 3 g). We can, however, characterize these data by providing a measure of the spread of the individual measurements around a central value. 4 A.1 Measures of Central Tendency One way to characterize the data in Table 4.1 is to assume that the masses of indi- vidual pennies are scattered around a central value that provides the best estimate of a penny’s true mass. Two common ways to report this estimate of central tendency are the mean and the median. Mean The mean, – X, is the numerical average obtained by dividing the sum of the individual measurements by the number of measurements where X i is the i th measurement, and n is the number of independent measurements. X X n i i n = = ∑ 1 Table 4 .1 Masses of Seven United States Pennies in Circulation Penny Mass (g) 1 3.080 2 3.094 3 3.107 4 3.056 5 3.112 6 3.174 7 3.198 1400-CH04 9/8/99 3:53 PM Page 54 Chapter 4 Evaluating Analytical Data 55 EXAMPLE 4 .1 What is the mean for the data in Table 4.1? SOLUTION To calculate the mean, we add the results for all measurements 3.080 + 3.094 + 3.107 + 3.056 + 3.112 + 3.174 + 3.198 = 21.821 and divide by the number of measurements The mean is the most common estimator of central tendency. It is not consid- ered a robust estimator, however, because extreme measurements, those much larger or smaller than the remainder of the data, strongly influence the mean’s value. 2 For example, mistakenly recording the mass of the fourth penny as 31.07 g instead of 3.107 g, changes the mean from 3.117 g to 7.112 g! Median The median, X med , is the middle value when data are ordered from the smallest to the largest value. When the data include an odd number of measure- ments, the median is the middle value. For an even number of measurements, the median is the average of the n/2 and the (n/2) + 1 measurements, where n is the number of measurements. EXAMPLE 4 .2 What is the median for the data in Table 4.1? SOLUTION To determine the median, we order the data from the smallest to the largest value 3.056 3.080 3.094 3.107 3.112 3.174 3.198 Since there is a total of seven measurements, the median is the fourth value in the ordered data set; thus, the median is 3.107. As shown by Examples 4.1 and 4.2, the mean and median provide similar esti- mates of central tendency when all data are similar in magnitude. The median, however, provides a more robust estimate of central tendency since it is less sensi- tive to measurements with extreme values. For example, introducing the transcrip- tion error discussed earlier for the mean only changes the median’s value from 3.107 g to 3.112 g. 4 A.2 Measures of Spread If the mean or median provides an estimate of a penny’s true mass, then the spread of the individual measurements must provide an estimate of the variability in the masses of individual pennies. Although spread is often defined relative to a specific measure of central tendency, its magnitude is independent of the central value. Changing all X == 21 821 7 3 117 . . g median That value for a set of ordered data, for which half of the data is larger in value and half is smaller in value ( – X med ). 1400-CH04 9/8/99 3:53 PM Page 55 56 Modern Analytical Chemistry measurements in the same direction, by adding or subtracting a constant value, changes the mean or median, but will not change the magnitude of the spread. Three common measures of spread are range, standard deviation, and variance. Range The range, w, is the difference between the largest and smallest values in the data set. Range = w = X largest – X smallest The range provides information about the total variability in the data set, but does not provide any information about the distribution of individual measurements. The range for the data in Table 4.1 is the difference between 3.198 g and 3.056 g; thus w = 3.198 g – 3.056 g = 0.142 g Standard Deviation The absolute standard deviation, s, describes the spread of individual measurements about the mean and is given as 4.1 where X i is one of n individual measurements, and – X is the mean. Frequently, the relative standard deviation, s r , is reported. The percent relative standard deviation is obtained by multiplying s r by 100%. EXAMPLE 4 . 3 What are the standard deviation, the relative standard deviation, and the percent relative standard deviation for the data in Table 4.1? SOLUTION To calculate the standard deviation, we obtain the difference between the mean value (3.117; see Example 4.1) and each measurement, square the resulting differences, and add them to determine the sum of the squares (the numerator of equation 4.1) (3.080 – 3.117) 2 = (–0.037) 2 = 0.00137 (3.094 – 3.117) 2 = (–0.023) 2 = 0.00053 (3.107 – 3.117) 2 = (–0.010) 2 = 0.00010 (3.056 – 3.117) 2 = (–0.061) 2 = 0.00372 (3.112 – 3.117) 2 = (–0.005) 2 = 0.00003 (3.174 – 3.117) 2 = (+0.057) 2 = 0.00325 (3.198 – 3.117) 2 = (+0.081) 2 = 0.00656 0.01556 The standard deviation is calculated by dividing the sum of the squares by n – 1, where n is the number of measurements, and taking the square root. s . = ? = 0 01556 71 0 051. s s X r = s XX n i i n = ? = ∑ (–) 1 2 1 standard deviation A statistical measure of the “average” deviation of data from the data’s mean value (s). range The numerical difference between the largest and smallest values in a data set (w). 1400-CH04 9/8/99 3:53 PM Page 56 Chapter 4 Evaluating Analytical Data 57 s r == 0 051 3 117 0 016 . . . The relative standard deviation and percent relative standard deviation are s r (%) = 0.016 · 100% = 1.6% It is much easier to determine the standard deviation using a scientific calculator with built-in statistical functions.* Variance Another common measure of spread is the square of the standard devia- tion, or the variance. The standard deviation, rather than the variance, is usually re- ported because the units for standard deviation are the same as that for the mean value. EXAMPLE 4 . 4 What is the variance for the data in Table 4.1? SOLUTION The variance is just the square of the absolute standard deviation. Using the standard deviation found in Example 4.3 gives the variance as Variance = s 2 = (0.051) 2 = 0.0026 4 B Characterizing Experimental Errors Realizing that our data for the mass of a penny can be characterized by a measure of central tendency and a measure of spread suggests two questions. First, does our measure of central tendency agree with the true, or expected value? Second, why are our data scattered around the central value? Errors associated with central tendency reflect the accuracy of the analysis, but the precision of the analysis is determined by those errors associated with the spread. 4 B.1 Accuracy Accuracy is a measure of how close a measure of central tendency is to the true, or expected value, m . ? Accuracy is usually expressed as either an absolute error E = – X – m 4.2 or a percent relative error, E r . 4.3E X r = ? × μ μ 100 *Many scientific calculators include two keys for calculating the standard deviation, only one of which corresponds to equation 4.3. Your calculator’s manual will help you determine the appropriate key to use. ?The standard convention for representing experimental parameters is to use a Roman letter for a value calculated from experimental data, and a Greek letter for the corresponding true value. For example, the experimentally determined mean is – X, and its underlying true value is m . Likewise, the standard deviation by experiment is given the symbol s, and its underlying true value is identified as s . variance The square of the standard deviation (s 2 ). 1400-CH04 9/8/99 3:53 PM Page 57 Although the mean is used as the measure of central tendency in equations 4.2 and 4.3, the median could also be used. Errors affecting the accuracy of an analysis are called determinate and are char- acterized by a systematic deviation from the true value; that is, all the individual measurements are either too large or too small. A positive determinate error results in a central value that is larger than the true value, and a negative determinate error leads to a central value that is smaller than the true value. Both positive and nega- tive determinate errors may affect the result of an analysis, with their cumulative ef- fect leading to a net positive or negative determinate error. It is possible, although not likely, that positive and negative determinate errors may be equal, resulting in a central value with no net determinate error. Determinate errors may be divided into four categories: sampling errors, method errors, measurement errors, and personal errors. Sampling Errors We introduce determinate sampling errors when our sampling strategy fails to provide a representative sample. This is especially important when sampling heterogeneous materials. For example, determining the environmental quality of a lake by sampling a single location near a point source of pollution, such as an outlet for industrial effluent, gives misleading results. In determining the mass of a U.S. penny, the strategy for selecting pennies must ensure that pennies from other countries are not inadvertently included in the sample. Determinate errors as- sociated with selecting a sample can be minimized with a proper sampling strategy, a topic that is considered in more detail in Chapter 7. Method Errors Determinate method errors are introduced when assumptions about the relationship between the signal and the analyte are invalid. In terms of the general relationships between the measured signal and the amount of analyte S meas = kn A+ S reag (total analysis method) 4.4 S meas = kC A+ S reag (concentration method) 4.5 method errors exist when the sensitivity, k, and the signal due to the reagent blank, S reag , are incorrectly determined. For example, methods in which S meas is the mass of a precipitate containing the analyte (gravimetric method) assume that the sensitiv- ity is defined by a pure precipitate of known stoichiometry. When this assumption fails, a determinate error will exist. Method errors involving sensitivity are mini- mized by standardizing the method, whereas method errors due to interferents present in reagents are minimized by using a proper reagent blank. Both are dis- cussed in more detail in Chapter 5. Method errors due to interferents in the sample cannot be minimized by a reagent blank. Instead, such interferents must be sepa- rated from the analyte or their concentrations determined independently. Measurement Errors Analytical instruments and equipment, such as glassware and balances, are usually supplied by the manufacturer with a statement of the item’s maximum measurement error, or tolerance. For example, a 25-mL volumetric flask might have a maximum error of ±0.03 mL, meaning that the actual volume contained by the flask lies within the range of 24.97–25.03 mL. Although expressed as a range, the error is determinate; thus, the flask’s true volume is a fixed value within the stated range. A summary of typical measurement errors for a variety of analytical equipment is given in Tables 4.2–4.4. 58 Modern Analytical Chemistry sampling error An error introduced during the process of collecting a sample for analysis. heterogeneous Not uniform in composition. method error An error due to limitations in the analytical method used to analyze a sample. determinate error Any systematic error that causes a measurement or result to always be too high or too small; can be traced to an identifiable source. measurement error An error due to limitations in the equipment and instruments used to make measurements. tolerance The maximum determinate measurement error for equipment or instrument as reported by the manufacturer. 1400-CH04 9/8/99 3:53 PM Page 58 Chapter 4 Evaluating Analytical Data 59 Table 4 .2 Measurement Errors for Selected Glassware a Measurement Errors for Volume Class A Glassware Class B Glassware Glassware (mL) (±mL) (±mL) Transfer Pipets 1 0.006 0.012 2 0.006 0.012 5 0.01 0.02 10 0.02 0.04 20 0.03 0.06 25 0.03 0.06 50 0.05 0.10 Volumetric Flasks 5 0.02 0.04 10 0.02 0.04 25 0.03 0.06 50 0.05 0.10 100 0.08 0.16 250 0.12 0.24 500 0.20 0.40 1000 0.30 0.60 2000 0.50 1.0 Burets 10 0.02 0.04 25 0.03 0.06 50 0.05 0.10 a Specifications for class A and class B glassware are taken from American Society for Testing and Materials E288, E542 and E694 standards. Table 4 . 4 Measurement Errors for Selected Digital Pipets Volume Measurement Error Pipet Range (mL or m L) a (±%) 10–100 m L b 10 1.0 50 0.6 100 0.6 200–1000 m L c 200 1.5 1000 0.8 1–10 mL d 1 0.6 5 0.4 10 0.3 a Units for volume same as for pipet range. b Data for Eppendorf Digital Pipet 4710. c Data for Oxford Benchmate. d Data for Eppendorf Maxipetter 4720 with Maxitip P. Table 4 . 3 Measurement Errors for Selected Balances Capacity Measurement Balance (g) Error Precisa 160M 160 ±1 mg A & D ER 120M 120 ±0.1 mg Metler H54 160 ±0.01 mg 1400-CH04 9/8/99 3:53 PM Page 59 Volumetric glassware is categorized by class. Class A glassware is manufactured to comply with tolerances specified by agencies such as the National Institute of Standards and Technology. Tolerance levels for class A glassware are small enough that such glassware normally can be used without calibration. The tolerance levels for class B glassware are usually twice those for class A glassware. Other types of vol- umetric glassware, such as beakers and graduated cylinders, are unsuitable for accu- rately measuring volumes. Determinate measurement errors can be minimized by calibration. A pipet can be calibrated, for example, by determining the mass of water that it delivers and using the density of water to calculate the actual volume delivered by the pipet. Al- though glassware and instrumentation can be calibrated, it is never safe to assume that the calibration will remain unchanged during an analysis. Many instruments, in particular, drift out of calibration over time. This complication can be minimized by frequent recalibration. Personal Errors Finally, analytical work is always subject to a variety of personal errors, which can include the ability to see a change in the color of an indicator used to signal the end point of a titration; biases, such as consistently overestimat- ing or underestimating the value on an instrument’s readout scale; failing to cali- brate glassware and instrumentation; and misinterpreting procedural directions. Personal errors can be minimized with proper care. Identifying Determinate Errors Determinate errors can be difficult to detect. Without knowing the true value for an analysis, the usual situation in any analysis with meaning, there is no accepted value with which the experimental result can be compared. Nevertheless, a few strategies can be used to discover the presence of a determinate error. Some determinate errors can be detected experimentally by analyzing several samples of different size. The magnitude of a constant determinate error is the same for all samples and, therefore, is more significant when analyzing smaller sam- ples. The presence of a constant determinate error can be detected by running sev- eral analyses using different amounts of sample, and looking for a systematic change in the property being measured. For example, consider a quantitative analysis in which we separate the analyte from its matrix and determine the analyte’s mass. Let’s assume that the sample is 50.0% w/w analyte; thus, if we analyze a 0.100-g sample, the analyte’s true mass is 0.050 g. The first two columns of Table 4.5 give the true mass of analyte for several additional samples. If the analysis has a positive constant determinate error of 0.010 g, then the experimentally determined mass for 60 Modern Analytical Chemistry Table 4 . 5 Effect of Constant Positive Determinate Error on Analysis of Sample Containing 50% Analyte (%w/w) Mass Sample True Mass of Analyte Constant Error Mass of Analyte Determined Percent Analyte Reported (g) (g) (g) (g) (%w/w) 0.100 0.050 0.010 0.060 60.0 0.200 0.100 0.010 0.110 55.0 0.400 0.200 0.010 0.210 52.5 0.800 0.400 0.010 0.410 51.2 1.000 0.500 0.010 0.510 51.0 constant determinate error A determinate error whose value is the same for all samples. personal error An error due to biases introduced by the analyst. 1400-CH04 9/8/99 3:53 PM Page 60 any sample will always be 0.010 g, larger than its true mass (column four of Table 4.5). The analyte’s reported weight percent, which is shown in the last column of Table 4.5, becomes larger when we analyze smaller samples. A graph of % w/w ana- lyte versus amount of sample shows a distinct upward trend for small amounts of sample (Figure 4.1). A smaller concentration of analyte is obtained when analyzing smaller samples in the presence of a constant negative determinate error. A proportional determinate error, in which the error’s magnitude depends on the amount of sample, is more difficult to detect since the result of an analysis is in- dependent of the amount of sample. Table 4.6 outlines an example showing the ef- fect of a positive proportional error of 1.0% on the analysis of a sample that is 50.0% w/w in analyte. In terms of equations 4.4 and 4.5, the reagent blank, S reag , is an example of a constant determinate error, and the sensitivity, k, may be affected by proportional errors. Potential determinate errors also can be identified by analyzing a standard sam- ple containing a known amount of analyte in a matrix similar to that of the samples being analyzed. Standard samples are available from a variety of sources, such as the National Institute of Standards and Technology (where they are called standard reference materials) or the American Society for Testing and Materials. For exam- ple, Figure 4.2 shows an analysis sheet for a typical reference material. Alternatively, the sample can be analyzed by an independent method known to give accurate results, and the re- sults of the two methods can be compared. Once identified, the source of a determinate error can be corrected. The best prevention against errors affect- ing accuracy, however, is a well-designed procedure that identifies likely sources of determinate errors, coupled with careful laboratory work. The data in Table 4.1 were obtained using a calibrated balance, certified by the manufacturer to have a tolerance of less than ±0.002 g. Suppose the Treasury Department reports that the mass of a 1998 U.S. penny is approximately 2.5 g. Since the mass of every penny in Table 4.1 exceeds the re- ported mass by an amount significantly greater than the balance’s tolerance, we can safely conclude that the error in this analysis is not due to equip- ment error. The actual source of the error is re- vealed later in this chapter. Chapter 4 Evaluating Analytical Data 61 Amount of sample % w/w anal yte Negative constant error Positive constant error True % w/w analyte Figure 4.1 Effect of a constant determinate error on the reported concentration of analyte. Table 4 .6 Effect of Proportional Positive Determinate Error on Analysis of Sample Containing 50% Analyte (%w/w) Mass Sample True Mass of Analyte Proportional Error Mass of Analyte Determined Percent Analyte Reported (g) (g) (%) (g) (%w/w) 0.200 0.100 1.00 0.101 50.5 0.400 0.200 1.00 0.202 50.5 0.600 0.300 1.00 0.303 50.5 0.800 0.400 1.00 0.404 50.5 1.000 0.500 1.00 0.505 50.5 proportional determinate error A determinate error whose value depends on the amount of sample analyzed. standard reference material A material available from the National Institute of Standards and Technology certified to contain known concentrations of analytes. 1400-CH04 9/8/99 3:53 PM Page 61 4 B.2 Precision Precision is a measure of the spread of data about a central value and may be ex- pressed as the range, the standard deviation, or the variance. Precision is commonly divided into two categories: repeatability and reproducibility. Repeatability is the precision obtained when all measurements are made by the same analyst during a single period of laboratory work, using the same solutions and equipment. Repro- ducibility, on the other hand, is the precision obtained under any other set of con- ditions, including that between analysts, or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the repro- ducibility of an analysis can be no better than its repeatability. Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magni- tude and direction. Indeterminate errors need not affect the accuracy of an analy- sis. Since indeterminate errors are randomly scattered around a central value, posi- tive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. Sources of Indeterminate Error Indeterminate errors can be traced to several sources, including the collection of samples, the manipulation of samples during the analysis, and the making of measurements. When collecting a sample, for instance, only a small portion of the available material is taken, increasing the likelihood that small-scale inhomogeneities in the sample will affect the repeatability of the analysis. Individual pennies, for example, are expected to show variation from several sources, including the manufacturing process, and the loss of small amounts of metal or the addition of dirt during circu- lation. These variations are sources of indeterminate error associated with the sam- pling process. 62 Modern Analytical Chemistry SRM Type Unit of issue 2694a Simulated rainwater Set of 4: 2 of 50 mL at each of 2 levels Constituent element parameter 2694a-I 2694a-II pH, 25°C 4.30 3.60 Electrolytic Conductivity (S/cm, 25°C) 25.4 129.3 Acidity, meq/L 0.0544 0.283 Fluoride, mg/L 0.057 0.108 Chloride, mg/L (0.23)* (0.94)* Nitrate, mg/L (0.53)* 7.19 Sulfate, mg/L 2.69 10.6 Sodium, mg/L 0.208 0.423 Potassium, mg/L 0.056 0.108 Ammonium, mg/L (0.12)* (1.06)* Calcium, mg/L 0.0126 0.0364 Magnesium, mg/L 0.0242 0.0484 * Values in parentheses are not certified and are given for information only. Figure 4.2 Analysis sheet for Simulated Rainwater (SRM 2694a). Adapted from NIST Special Publication 260: Standard Reference Materials Catalog 1995–96, p. 64; U.S. Department of Commerce, Technology Administration, National Institute of Standards and Technology. Simulated Rainwater (liquid form) This SRM was developed to aid in the analysis of acidic rainwater by providing a stable, homogeneous material at two levels of acidity. repeatability The precision for an analysis in which the only source of variability is the analysis of replicate samples. reproducibility The precision when comparing results for several samples, for several analysts or several methods. indeterminate error Any random error that causes some measurements or results to be too high while others are too low. 1400-CH04 9/8/99 3:53 PM Page 62 During the analysis numerous opportunities arise for random variations in the way individual samples are treated. In determining the mass of a penny, for exam- ple, each penny should be handled in the same manner. Cleaning some pennies but not cleaning others introduces an indeterminate error. Finally, any measuring device is subject to an indeterminate error in reading its scale, with the last digit always being an estimate subject to random fluctuations, or background noise. For example, a buret with scale divisions every 0.1 mL has an in- herent indeterminate error of ±0.01 – 0.03 mL when estimating the volume to the hundredth of a milliliter (Figure 4.3). Background noise in an electrical meter (Fig- ure 4.4) can be evaluated by recording the signal without analyte and observing the fluctuations in the signal over time. Evaluating Indeterminate Error Although it is impossible to eliminate indetermi- nate error, its effect can be minimized if the sources and relative magnitudes of the indeterminate error are known. Indeterminate errors may be estimated by an ap- propriate measure of spread. Typically, a standard deviation is used, although in some cases estimated values are used. The contribution from analytical instruments and equipment are easily measured or estimated. Inde- terminate errors introduced by the analyst, such as in- consistencies in the treatment of individual samples, are more difficult to estimate. To evaluate the effect of indeterminate error on the data in Table 4.1, ten replicate determinations of the mass of a single penny were made, with results shown in Table 4.7. The standard deviation for the data in Table 4.1 is 0.051, and it is 0.0024 for the data in Table 4.7. The significantly better precision when determining the mass of a single penny sug- gests that the precision of this analysis is not limited by the balance used to measure mass, but is due to a significant variability in the masses of individual pennies. Chapter 4 Evaluating Analytical Data 63 Signal Time Figure 4.4 Background noise in a meter obtained by measuring signal over time in the absence of analyte. Table 4 . 7 Replicate Determinations of the Mass of a Single United States Penny in Circulation Mass Replicate Number (g) 1 3.025 2 3.024 3 3.028 4 3.027 5 3.028 6 3.023 7 3.022 8 3.021 9 3.026 10 3.024 Figure 4.3 Close-up of buret, showing difficulty in estimating volume. With scale divisions every 0.1 mL it is difficult to read the actual volume to better than ±0.01 – 0.03 mL. 30 31 1400-CH04 9/8/99 3:53 PM Page 63 64 Modern Analytical Chemistry uncertainty The range of possible values for a measurement. 4 B. 3 Error and Uncertainty Analytical chemists make a distinction between error and uncertainty. 3 Error is the difference between a single measurement or result and its true value. In other words, error is a measure of bias. As discussed earlier, error can be divided into de- terminate and indeterminate sources. Although we can correct for determinate error, the indeterminate portion of the error remains. Statistical significance testing, which is discussed later in this chapter, provides a way to determine whether a bias resulting from determinate error might be present. Uncertainty expresses the range of possible values that a measurement or result might reasonably be expected to have. Note that this definition of uncertainty is not the same as that for precision. The precision of an analysis, whether reported as a range or a standard deviation, is calculated from experimental data and provides an estimation of indeterminate error affecting measurements. Uncertainty accounts for all errors, both determinate and indeterminate, that might affect our result. Al- though we always try to correct determinate errors, the correction itself is subject to random effects or indeterminate errors. To illustrate the difference between precision and un- certainty, consider the use of a class A 10-mL pipet for de- livering solutions. A pipet’s uncertainty is the range of volumes in which its true volume is expected to lie. Sup- pose you purchase a 10-mL class A pipet from a labora- tory supply company and use it without calibration. The pipet’s tolerance value of ±0.02 mL (see Table 4.2) repre- sents your uncertainty since your best estimate of its vol- ume is 10.00 mL ±0.02 mL. Precision is determined ex- perimentally by using the pipet several times, measuring the volume of solution delivered each time. Table 4.8 shows results for ten such trials that have a mean of 9.992 mL and a standard deviation of 0.006. This standard devi- ation represents the precision with which we expect to be able to deliver a given solution using any class A 10-mL pipet. In this case the uncertainty in using a pipet is worse than its precision. Interestingly, the data in Table 4.8 allow us to calibrate this specific pipet’s delivery volume as 9.992 mL. If we use this vol- ume as a better estimate of this pipet’s true volume, then the uncertainty is ±0.006. As expected, calibrating the pipet allows us to lower its uncertainty. 4 C Propagation of Uncertainty Suppose that you need to add a reagent to a flask by several successive transfers using a class A 10-mL pipet. By calibrating the pipet (see Table 4.8), you know that it delivers a volume of 9.992 mL with a standard deviation of 0.006 mL. Since the pipet is calibrated, we can use the standard deviation as a measure of uncertainty. This uncertainty tells us that when we use the pipet to repetitively deliver 10 mL of solution, the volumes actually delivered are randomly scattered around the mean of 9.992 mL. If the uncertainty in using the pipet once is 9.992 ± 0.006 mL, what is the un- certainty when the pipet is used twice? As a first guess, we might simply add the un- certainties for each delivery; thus (9.992 mL + 9.992 mL) ± (0.006 mL + 0.006 mL) = 19.984 ± 0.012 mL Table 4 .8 Experimentally Determined Volumes Delivered by a 10-mL Class A Pipet Volume Volume Delivered Delivered Trial (mL) Trial (mL) 1 10.002 6 9.983 2 9.993 7 9.991 3 9.984 8 9.990 4 9.996 9 9.988 5 9.989 10 9.999 error A measure of bias in a result or measurement. 1400-CH04 9/8/99 3:54 PM Page 64 Chapter 4 Evaluating Analytical Data 65 It is easy to see that combining uncertainties in this way overestimates the total un- certainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that both volumes are either greater than 9.992 mL or less than 9.992 mL. At the other extreme, we might assume that the two deliveries will always be on op- posite sides of the pipet’s mean volume. In this case we subtract the uncertainties for the two deliveries, (9.992 mL + 9.992 mL) ± (0.006 mL – 0.006 mL) = 19.984 ± 0.000 mL underestimating the total uncertainty. So what is the total uncertainty when using this pipet to deliver two successive volumes of solution? From the previous discussion we know that the total uncer- tainty is greater than ±0.000 mL and less than ±0.012 mL. To estimate the cumula- tive effect of multiple uncertainties, we use a mathematical technique known as the propagation of uncertainty. Our treatment of the propagation of uncertainty is based on a few simple rules that we will not derive. A more thorough treatment can be found elsewhere. 4 4 C.1 A Few Symbols Propagation of uncertainty allows us to estimate the uncertainty in a calculated re- sult from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are s R , s A , s B , and s C . The uncertainties for A, B, and C can be reported in several ways, in- cluding calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 4 C.2 Uncertainty When Adding or Subtracting When measurements are added or subtracted, the absolute uncertainty in the result is the square root of the sum of the squares of the absolute uncertainties for the in- dividual measurements. Thus, for the equations R = A + B + C or R = A + B – C, or any other combination of adding and subtracting A, B, and C, the absolute uncer- tainty in R is 4.6 EXAMPLE 4 . 5 The class A 10-mL pipet characterized in Table 4.8 is used to deliver two successive volumes. Calculate the absolute and relative uncertainties for the total delivered volume. SOLUTION The total delivered volume is obtained by adding the volumes of each delivery; thus V tot = 9.992 mL + 9.992 mL = 19.984 mL Using the standard deviation as an estimate of uncertainty, the uncertainty in the total delivered volume is s R =+=(. ) (. ) .0 006 0 006 0 0085 22 ssss R ABC =++ 222 1400-CH04 9/8/99 3:54 PM Page 65 66 Modern Analytical Chemistry Thus, we report the volume and its absolute uncertainty as 19.984 ± 0.008 mL. The relative uncertainty in the total delivered volume is 4 C. 3 Uncertainty When Multiplying or Dividing When measurements are multiplied or divided, the relative uncertainty in the result is the square root of the sum of the squares of the relative uncertainties for the indi- vidual measurements. Thus, for the equations R = A · B · C or R = A · B/C, or any other combination of multiplying and dividing A, B, and C, the relative uncertainty in R is 4.7 EXAMPLE 4 .6 The quantity of charge, Q, in coulombs passing through an electrical circuit is Q = I · t where I is the current in amperes and t is the time in seconds. When a current of 0.15 ± 0.01 A passes through the circuit for 120 ± 1 s, the total charge is Q = (0.15 A) · (120 s) = 18 C Calculate the absolute and relative uncertainties for the total charge. SOLUTION Since charge is the product of current and time, its relative uncertainty is or ±6.7%. The absolute uncertainty in the charge is s R = R · 0.0672 = (18) · (±0.0672) = ±1.2 Thus, we report the total charge as 18 C ± 1 C. 4 C. 4 Uncertainty for Mixed Operations Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing. As shown in the following example, the propagation of un- certainty is easily calculated by treating each operation separately using equations 4.6 and 4.7 as needed. s R . . R = ? ? ? ? ? ? + ? ? ? ? ? ? =± 001 015 1 120 0 0672 22 . s R s A s B s C RABC = ? ? ? ? ? ? + ? ? ? ? ? ? + ? ? ? ? ? ? 222 0 0085 19 984 100 0 043 . . .%×= 1400-CH04 9/8/99 3:54 PM Page 66 Chapter 4 Evaluating Analytical Data 67 EXAMPLE 4 . 7 For a concentration technique the relationship between the measured signal and an analyte’s concentration is given by equation 4.5 S meas = kC A + S reag Calculate the absolute and relative uncertainties for the analyte’s concentration if S meas is 24.37 ± 0.02, S reag is 0.96 ± 0.02, and k is 0.186 ± 0.003 ppm –1 . SOLUTION Rearranging equation 4.5 and solving for C A gives the analyte’s concentration as 126 ppm. To estimate the uncertainty in C A , we first determine the uncertainty for the numerator, S meas – S reag , using equation 4.6 The numerator, therefore, is 23.41 ± 0.028 (note that we retain an extra significant figure since we will use this uncertainty in further calculations). To complete the calculation, we estimate the relative uncertainty in C A using equation 4.7, giving or a percent relative uncertainty of 1.6%. The absolute uncertainty in the analyte’s concentration is s R = (125.9 ppm) · (0.0162) = ±2.0 ppm giving the analyte’s concentration as 126 ± 2 ppm. 4 C. 5 Uncertainty for Other Mathematical Functions Many other mathematical operations are commonly used in analytical chemistry, including powers, roots, and logarithms. Equations for the propagation of uncer- tainty for some of these functions are shown in Table 4.9. EXAMPLE 4 .8 The pH of a solution is defined as pH = –log[H + ] where [H + ] is the molar concentration of H + . If the pH of a solution is 3.72 with an absolute uncertainty of ±0.03, what is the [H + ] and its absolute uncertainty? s R . . . R = ? ? ? ? ? ? + ? ? ? ? ? ? = 0 028 23 41 0 003 0 186 0 0162 22 . . s. . R =+=()() .0 02 0 02 0 028 22 C SS k A meas reag ppm ppm= ? = ? = 24 37 0 96 0 186 125 9 .. . . -1 1400-CH04 9/8/99 3:54 PM Page 67 SOLUTION The molar concentration of H + for this pH is [H + ]=10 –pH =10 –3.72 = 1.91 · 10 –4 M or 1.9 · 10 –4 M to two significant figures. From Table 4.9 the relative uncertainty in [H + ] is and the absolute uncertainty is (1.91 · 10 –4 M) · (0.069) = 1.3 · 10 –5 M We report the [H + ] and its absolute uncertainty as 1.9 (±0.1) · 10 –4 M. s R s R =×=×=2 303 2 303 0 03 0 069.... A 68 Modern Analytical Chemistry Table 4 . 9 Propagation of Uncertainty for Selected Functions a Function s R RkA s ks RAB s s s RAB s s s RAB s R s A s B R A B s R s A s B RA s s A RA s RA R AB R AB RA RA R A R = =+ + =× = ? ? ? ? ? ? + ? ? ? ? ? ? == ? ? ? ? ? ? + ? ? ? ? ? ? = = = = = = = 2 2 2 2 22 22 – ln( ) log( ) B B 0.4343 × == == ? ? ? ? ? ? s A Re s R s R s R s RA s R k s A A A R A A R A k RA 10 2 303. a These equations assume that the measurements A and B are uncorrelated; that is, s A is independent of s B. 4 C.6 Is Calculating Uncertainty Actually Useful? Given the complexity of determining a result’s uncertainty when several mea- surements are involved, it is worth examining some of the reasons why such cal- culations are useful. A propagation of uncertainty allows us to estimate an ex- 1400-CH04 9/8/99 3:54 PM Page 68 Chapter 4 Evaluating Analytical Data 69 pected uncertainty for an analysis. Comparing the expected uncertainty to that which is actually obtained can provide useful information. For example, in de- termining the mass of a penny, we estimated the uncertainty in measuring mass as ±0.002 g based on the balance’s tolerance. If we measure a single penny’s mass several times and obtain a standard deviation of ±0.020 g, we would have reason to believe that our measurement process is out of control. We would then try to identify and correct the problem. A propagation of uncertainty also helps in deciding how to improve the un- certainty in an analysis. In Example 4.7, for instance, we calculated the concen- tration of an analyte, obtaining a value of 126 ppm with an absolute uncertainty of ±2 ppm and a relative uncertainty of 1.6%. How might we improve the analy- sis so that the absolute uncertainty is only ±1 ppm (a relative uncertainty of 0.8%)? Looking back on the calculation, we find that the relative uncertainty is determined by the relative uncertainty in the measured signal (corrected for the reagent blank) and the relative uncertainty in the method’s sensitivity, k, Of these two terms, the sensitivity’s uncertainty dominates the total uncertainty. Measuring the signal more carefully will not improve the overall uncertainty of the analysis. On the other hand, the desired improvement in uncertainty can be achieved if the sensitivity’s absolute uncertainty can be decreased to ±0.0015 ppm –1 . As a final example, a propagation of uncertainty can be used to decide which of several procedures provides the smallest overall uncertainty. Preparing a solu- tion by diluting a stock solution can be done using several different combina- tions of volumetric glassware. For instance, we can dilute a solution by a factor of 10 using a 10-mL pipet and a 100-mL volumetric flask, or by using a 25-mL pipet and a 250-mL volumetric flask. The same dilution also can be accom- plished in two steps using a 50-mL pipet and a 100-mL volumetric flask for the first dilution, and a 10-mL pipet and a 50-mL volumetric flask for the second di- lution. The overall uncertainty, of course, depends on the uncertainty of the glassware used in the dilutions. As shown in the following example, we can use the tolerance values for volumetric glassware to determine the optimum dilution strategy. 5 EXAMPLE 4 . 9 Which of the following methods for preparing a 0.0010 M solution from a 1.0 M stock solution provides the smallest overall uncertainty? (a) A one-step dilution using a 1-mL pipet and a 1000-mL volumetric flask. (b) A two-step dilution using a 20-mL pipet and a 1000-mL volumetric flask for the first dilution and a 25-mL pipet and a 500-mL volumetric flask for the second dilution. 0 003 0 186 0 016 1 6 . . ., .%=± ± or 0 028 23 41 0 0012 0 12 . . ., .%=± ± or 1400-CH04 9/8/99 3:54 PM Page 69 70 Modern Analytical Chemistry SOLUTION Letting M a and M b represent the molarity of the final solutions from method (a) and method (b), we can write the following equations Using the tolerance values for pipets and volumetric flasks given in Table 4.2, the overall uncertainties in M a and M b are Since the relative uncertainty for M b is less than that for M a , we find that the two-step dilution provides the smaller overall uncertainty. 4 D The Distribution of Measurements and Results An analysis, particularly a quantitative analysis, is usually performed on several replicate samples. How do we report the result for such an experiment when results for the replicates are scattered around a central value? To complicate matters fur- ther, the analysis of each replicate usually requires multiple measurements that, themselves, are scattered around a central value. Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny’s mass. Including the standard deviation, or other measure of spread, pro- vides the necessary information about the uncertainty in measuring mass. Never- theless, the central tendency and spread together do not provide a definitive state- ment about a penny’s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. How we report the result of an experiment is further complicated by the need to compare the results of different experiments. For example, Table 4.10 shows re- sults for a second, independent experiment to determine the mass of a U.S. penny in circulation. Although the results shown in Tables 4.1 and 4.10 are similar, they are not identical; thus, we are justified in asking whether the results are in agree- ment. Unfortunately, a definitive comparison between these two sets of data is not possible based solely on their respective means and standard deviations. Developing a meaningful method for reporting an experiment’s result requires the ability to predict the true central value and true spread of the population under investigation from a limited sampling of that population. In this section we will take a quantitative look at how individual measurements and results are distributed around a central value. s R .. s R .. . . R M R M ? ? ? ? ? ? = ? ? ? ? ? ? + ? ? ? ? ? ? = ? ? ? ? ? ? = ? ? ? ? ? ? + ? ? ? ? ? ? + ? ? ? ? ? ? + ? ? ? ? ? ? = a b 0 006 1 000 03 1000 0 0 006 003 20 00 003 25 00 02 500 0 03 1000 0 0 002 22 222 2 . . M M a b M MmL mL M MmLmL mL mL == == 0 0010 10 1000 1000 0 0 0010 1 0 20 00 25 00 1000 0 500 0 . (. )(. ) . . ( . )( . )( . ) (. )(. ) 1400-CH04 9/8/99 3:54 PM Page 70 4 D.1 Populations and Samples In the previous section we introduced the terms “population” and “sample” in the context of reporting the result of an experiment. Before continuing, we need to un- derstand the difference between a population and a sample. A population is the set of all objects in the system being investigated. These objects, which also are mem- bers of the population, possess qualitative or quantitative characteristics, or values, that can be measured. If we analyze every member of a population, we can deter- mine the population’s true central value, m , and spread, s . The probability of occurrence for a particular value, P(V), is given as where V is the value of interest, M is the value’s frequency of occurrence in the pop- ulation, and N is the size of the population. In determining the mass of a circulating United States penny, for instance, the members of the population are all United States pennies currently in circulation, while the values are the possible masses that a penny may have. In most circumstances, populations are so large that it is not feasible to analyze every member of the population. This is certainly true for the population of circulating U.S. pennies. Instead, we select and analyze a limited subset, or sample, of the popula- tion. The data in Tables 4.1 and 4.10, for example, give results for two samples drawn at random from the larger population of all U.S. pennies currently in circulation. 4 D.2 Probability Distributions for Populations To predict the properties of a population on the basis of a sample, it is necessary to know something about the population’s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called probability distributions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability dis- tribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. PV M N ()= Chapter 4 Evaluating Analytical Data 71 Table 4 .10 Results for a Second Determination of the Mass of a United States Penny in Circulation Penny Mass (g) 1 3.052 2 3.141 3 3.083 4 3.083 5 3.048 – X 3.081 s 0.037 population All members of a system. sample Those members of a population that we actually collect and analyze. probability distribution Plot showing frequency of occurrence for members of a population. 1400-CH04 9/8/99 3:54 PM Page 71 72 Modern Analytical Chemistry binomial distribution Probability distribution showing chance of obtaining one of two specific outcomes in a fixed number of trials. *N! is read as N-factorial and is the product N · (N –1)· (N –2)· ... · 1. For example, 4! is 4 · 3 · 2 · 1, or 24. Your calculator probably has a key for calculating factorials. Binomial Distribution The binomial distribution describes a population in which the values are the number of times a particular outcome occurs during a fixed num- ber of trials. Mathematically, the binomial distribution is given as where P(X,N) is the probability that a given outcome will occur X times during N trials, and p is the probability that the outcome will occur in a single trial.* If you flip a coin five times, P(2,5) gives the probability that two of the five trials will turn up “heads.” A binomial distribution has well-defined measures of central tendency and spread. The true mean value, for example, is given as m = Np and the true spread is given by the variance s 2 = Np(1 – p) or the standard deviation The binomial distribution describes a population whose members have only certain, discrete values. A good example of a population obeying the binomial dis- tribution is the sampling of homogeneous materials. As shown in Example 4.10, the binomial distribution can be used to calculate the probability of finding a particular isotope in a molecule. EXAMPLE 4 .10 Carbon has two common isotopes, 12 C and 13 C, with relative isotopic abundances of, respectively, 98.89% and 1.11%. (a) What are the mean and standard deviation for the number of 13 C atoms in a molecule of cholesterol? (b) What is the probability of finding a molecule of cholesterol (C 27 H 44 O) containing no atoms of 13 C? SOLUTION The probability of finding an atom of 13 C in cholesterol follows a binomial distribution, where X is the sought for frequency of occurrence of 13 C atoms, N is the number of C atoms in a molecule of cholesterol, and p is the probability of finding an atom of 13 C. (a) The mean number of 13 C atoms in a molecule of cholesterol is m = Np =27· 0.0111 = 0.300 with a standard deviation of (b) Since the mean is less than one atom of 13 C per molecule, most molecules of cholesterol will not have any 13 C. To calculate σ= ? =( )( . )( . ) .27 0 0111 1 0 0111 0 172 σ= ?Np p()1 PX,N N XN X pp XNX () ! !( )! ()= ? ××? ? 1 homogeneous Uniform in composition. 1400-CH04 9/8/99 3:54 PM Page 72 the probability, we substitute appropriate values into the binomial equation There is therefore a 74.0% probability that a molecule of cholesterol will not have an atom of 13 C. A portion of the binomial distribution for atoms of 13 C in cholesterol is shown in Figure 4.5. Note in particular that there is little probability of finding more than two atoms of 13 C in any molecule of cholesterol. P(, ) ! !( )! (. ) ( . ) .027 27 027 0 0 0111 1 0 0111 0 740 027 = ? ××? = ? Chapter 4 Evaluating Analytical Data 73 Pr obability Number of atoms of carbon-13 in a molecule of cholesterol 0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 12345 Figure 4.5 Portion of the binomial distribution for the number of naturally occurring 13 C atoms in a molecule of cholesterol. Normal Distribution The binomial distribution describes a population whose members have only certain, discrete values. This is the case with the number of 13 C atoms in a molecule, which must be an integer number no greater then the number of carbon atoms in the molecule. A molecule, for example, cannot have 2.5 atoms of 13 C. Other populations are considered continuous, in that members of the popula- tion may take on any value. The most commonly encountered continuous distribution is the Gaussian, or normal distribution, where the frequency of occurrence for a value, X, is given by The shape of a normal distribution is determined by two parameters, the first of which is the population’s central, or true mean value, m , given as where n is the number of members in the population. The second parameter is the population’s variance, s 2 , which is calculated using the following equation* 4.8 σ μ 2 1 2 = ? = ∑ ()X n i i N μ= = ∑ X n i i N 1 fX X ( ) exp () = ?? ? ? ? ? ? ? 1 2 2 2 2 2 piσ μ σ normal distribution “Bell-shaped” probability distribution curve for measurements and results showing the effect of random error. *Note the difference between the equation for a population’s variance, which includes the term n in the denominator, and the similar equation for the variance of a sample (the square of equation 4.3), which includes the term n – 1 in the denominator. The reason for this difference is discussed later in the chapter. 1400-CH04 9/8/99 3:54 PM Page 73 74 Modern Analytical Chemistry f ( x ) Value of x –50 –40 –30 –20 –10 0 (a) (b) (c) 10 20 30 40 50 Figure 4.6 Normal distributions for (a) m = 0 and s 2 = 25; (b) m = 0 and s 2 = 100; and (c) m = 0 and s 2 = 400. Examples of normal distributions with m = 0 and s 2 = 25, 100 or 400, are shown in Figure 4.6. Several features of these normal distributions deserve atten- tion. First, note that each normal distribution contains a single maximum corre- sponding to m and that the distribution is symmetrical about this value. Second, increasing the population’s variance increases the distribution’s spread while de- creasing its height. Finally, because the normal distribution depends solely on m and s 2 , the area, or probability of occurrence between any two limits defined in terms of these parameters is the same for all normal distribution curves. For ex- ample, 68.26% of the members in a normally distributed population have values within the range m ±1s , regardless of the actual values of m and s . As shown in Example 4.11, probability tables (Appendix 1A) can be used to determine the probability of occurrence between any defined limits. EXAMPLE 4 .11 The amount of aspirin in the analgesic tablets from a particular manufacturer is known to follow a normal distribution, with m = 250 mg and s 2 = 25. In a random sampling of tablets from the production line, what percentage are expected to contain between 243 and 262 mg of aspirin? SOLUTION The normal distribution for this example is shown in Figure 4.7, with the shaded area representing the percentage of tablets containing between 243 and 262 mg of aspirin. To determine the percentage of tablets between these limits, we first determine the percentage of tablets with less than 243 mg of aspirin, and the percentage of tablets having more than 262 mg of aspirin. This is accomplished by calculating the deviation, z, of each limit from m , using the following equation where X is the limit in question, and s , the population standard deviation, is 5. Thus, the deviation for the lower limit is z X = ?μ σ 1400-CH04 9/8/99 3:54 PM Page 74 Figure 4.7 Normal distribution for population of aspirin tablets with m = 250 mg aspirin and s 2 = 25. The shaded area shows the percentage of tablets containing between 243 and 262 mg of aspirin. Chapter 4 Evaluating Analytical Data 75 f (mg aspirin) Aspirin (mg) 290280220 240 250 260 270210 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 230 and the deviation for the upper limit is Using the table in Appendix 1A, we find that the percentage of tablets with less than 243 mg of aspirin is 8.08%, and the percentage of tablets with more than 262 mg of aspirin is 0.82%. The percentage of tablets containing between 243 and 262 mg of aspirin is therefore 100.00% – 8.08% – 0.82 % = 91.10% 4 D. 3 Confidence Intervals for Populations If we randomly select a single member from a pop- ulation, what will be its most likely value? This is an important question, and, in one form or another, it is the fundamental problem for any analysis. One of the most important features of a population’s probability distribution is that it provides a way to answer this question. Earlier we noted that 68.26% of a normally distrib- uted population is found within the range of m ±1s . Stat- ing this another way, there is a 68.26% probability that a member selected at random from a normally distributed population will have a value in the interval of m ±1s . In general, we can write X i = m ±zs 4.9 where the factor z accounts for the desired level of confidence. Values reported in this fashion are called confidence intervals. Equation 4.9, for example, is the confidence interval for a single member of a population. Confidence intervals can be quoted for any desired probability level, several examples of which are shown in Table 4.11. For reasons that will be discussed later in the chapter, a 95% confidence interval frequently is reported. z up = ? =+ 262 250 5 24. z low = ? =? 243 250 5 14. Table 4 .11 Confidence Intervals for Normal Distribution Curves Between the Limits m ± zs z Confidence Interval (%) 0.50 38.30 1.00 68.26 1.50 86.64 1.96 95.00 2.00 95.44 2.50 98.76 3.00 99.73 3.50 99.95 confidence interval Range of results around a mean value that could be explained by random error. 1400-CH04 9/8/99 3:54 PM Page 75 76 Modern Analytical Chemistry EXAMPLE 4 .12 What is the 95% confidence interval for the amount of aspirin in a single analgesic tablet drawn from a population where m is 250 mg and s 2 is 25? SOLUTION According to Table 4.11, the 95% confidence interval for a single member of a normally distributed population is X i = m– 1.96s = 250 mg ± (1.96)(5) = 250 mg ± 10 mg Thus, we expect that 95% of the tablets in the population contain between 240 and 260 mg of aspirin. Alternatively, a confidence interval can be expressed in terms of the popula- tion’s standard deviation and the value of a single member drawn from the popu- lation. Thus, equation 4.9 can be rewritten as a confidence interval for the popula- tion mean m = X i ± zs 4.10 EXAMPLE 4 .1 3 The population standard deviation for the amount of aspirin in a batch of analgesic tablets is known to be 7 mg of aspirin. A single tablet is randomly selected, analyzed, and found to contain 245 mg of aspirin. What is the 95% confidence interval for the population mean? SOLUTION The 95% confidence interval for the population mean is given as m = X i ± zs = 245 ± (1.96)(7) = 245 mg ± 14 mg There is, therefore, a 95% probability that the population’s mean, m , lies within the range of 231–259 mg of aspirin. Confidence intervals also can be reported using the mean for a sample of size n, drawn from a population of known s . The standard deviation for the mean value, s – X , which also is known as the standard error of the mean, is The confidence interval for the population’s mean, therefore, is 4.11 μ σ =±X z n σ σ X n = 1400-CH04 9/8/99 3:54 PM Page 76 Chapter 4 Evaluating Analytical Data 77 EXAMPLE 4 .1 4 What is the 95% confidence interval for the analgesic tablets described in Example 4.13, if an analysis of five tablets yields a mean of 245 mg of aspirin? SOLUTION In this case the confidence interval is given as Thus, there is a 95% probability that the population’s mean is between 239 and 251 mg of aspirin. As expected, the confidence interval based on the mean of five members of the population is smaller than that based on a single member. 4 D. 4 Probability Distributions for Samples In Section 4D.2 we introduced two probability distributions commonly encoun- tered when studying populations. The construction of confidence intervals for a normally distributed population was the subject of Section 4D.3. We have yet to ad- dress, however, how we can identify the probability distribution for a given popula- tion. In Examples 4.11–4.14 we assumed that the amount of aspirin in analgesic tablets is normally distributed. We are justified in asking how this can be deter- mined without analyzing every member of the population. When we cannot study the whole population, or when we cannot predict the mathematical form of a popu- lation’s probability distribution, we must deduce the distribution from a limited sampling of its members. Sample Distributions and the Central Limit Theorem Let’s return to the problem of determining a penny’s mass to explore the relationship between a population’s distribution and the distribution of samples drawn from that population. The data shown in Tables 4.1 and 4.10 are insufficient for our purpose because they are not large enough to give a useful picture of their respective probability distributions. A better picture of the probability distribution requires a larger sample, such as that shown in Table 4.12, for which – X is 3.095 and s 2 is 0.0012. The data in Table 4.12 are best displayed as a histogram, in which the fre- quency of occurrence for equal intervals of data is plotted versus the midpoint of each interval. Table 4.13 and Figure 4.8 show a frequency table and histogram for the data in Table 4.12. Note that the histogram was constructed such that the mean value for the data set is centered within its interval. In addition, a normal distribu- tion curve using – X and s 2 to estimate m and s 2 is superimposed on the histogram. It is noteworthy that the histogram in Figure 4.8 approximates the normal dis- tribution curve. Although the histogram for the mass of pennies is not perfectly symmetrical, it is roughly symmetrical about the interval containing the greatest number of pennies. In addition, we know from Table 4.11 that 68.26%, 95.44%, and 99.73% of the members of a normally distributed population are within, re- spectively, ±1s , ±2s, and ±3s . If we assume that the mean value, 3.095 g, and the sample variance, 0.0012, are good approximations for m and s 2 , we find that 73%, μ= ± = ±245 196 7 5 245 6 ( . )( ) mg mg histogram A plot showing the number of times an observation occurs as a function of the range of observed values. 1400-CH04 9/8/99 3:54 PM Page 77 78 Modern Analytical Chemistry Table 4 .1 3 Frequency Distribution for the Data in Table 4.12 Interval Frequency 2.991–3.009 2 3.010–3.028 0 3.029–3.047 4 3.048–3.066 19 3.067–3.085 15 3.086–3.104 23 3.105–3.123 19 3.124–3.142 12 3.143–3.161 13 3.162–3.180 1 3.181–3.199 2 Table 4 .12 Individual Masses for a Large Sample of U.S. Pennies in Circulation a Penny Weight Penny Weight Penny Weight Penny Weight (g) (g) (g) (g) 1 3.126 26 3.073 51 3.101 76 3.086 2 3.140 27 3.084 52 3.049 77 3.123 3 3.092 28 3.148 53 3.082 78 3.115 4 3.095 29 3.047 54 3.142 79 3.055 5 3.080 30 3.121 55 3.082 80 3.057 6 3.065 31 3.116 56 3.066 81 3.097 7 3.117 32 3.005 57 3.128 82 3.066 8 3.034 33 3.115 58 3.112 83 3.113 9 3.126 34 3.103 59 3.085 84 3.102 10 3.057 35 3.086 60 3.086 85 3.033 11 3.053 36 3.103 61 3.084 86 3.112 12 3.099 37 3.049 62 3.104 87 3.103 13 3.065 38 2.998 63 3.107 88 3.198 14 3.059 39 3.063 64 3.093 89 3.103 15 3.068 40 3.055 65 3.126 90 3.126 16 3.060 41 3.181 66 3.138 91 3.111 17 3.078 42 3.108 67 3.131 92 3.126 18 3.125 43 3.114 68 3.120 93 3.052 19 3.090 44 3.121 69 3.100 94 3.113 20 3.100 45 3.105 70 3.099 95 3.085 21 3.055 46 3.078 71 3.097 96 3.117 22 3.105 47 3.147 72 3.091 97 3.142 23 3.063 48 3.104 73 3.077 98 3.031 24 3.083 49 3.146 74 3.178 99 3.083 25 3.065 50 3.095 75 3.054 100 3.104 a Pennies are identified in the order in which they were sampled and weighed. 1400-CH04 9/8/99 3:54 PM Page 78 Figure 4.8 Histogram for data in Table 4.12. A normal distribution curve for the data, based on – X and s 2 , is superimposed on the histogram. 95%, and 100% of the pennies fall within these limits. It is easy to imagine that in- creasing the number of pennies in the sample will result in a histogram that even more closely approximates a normal distribution. We will not offer a formal proof that the sample of pennies in Table 4.12 and the population from which they were drawn are normally distributed; how- ever, the evidence we have seen strongly suggests that this is true. Although we cannot claim that the results for all analytical experiments are normally distrib- uted, in most cases the data we collect in the laboratory are, in fact, drawn from a normally distributed population. That this is generally true is a consequence of the central limit theorem. 6 According to this theorem, in systems subject to a variety of indeterminate errors, the distribution of results will be approximately normal. Furthermore, as the number of contributing sources of indeterminate error increases, the results come even closer to approximating a normal distribu- tion. The central limit theorem holds true even if the individual sources of in- determinate error are not normally distributed. The chief limitation to the central limit theorem is that the sources of indeterminate error must be indepen- dent and of similar magnitude so that no one source of error dominates the final distribution. Estimating m and s 2 Our comparison of the histogram for the data in Table 4.12 to a normal distribution assumes that the sample’s mean, – X, and variance, s 2 , are appropriate estimators of the population’s mean, m , and variance, s 2 . Why did we select – X and s 2 , as opposed to other possible measures of central tendency and spread? The explanation is simple; – X and s 2 are considered unbiased estimators of m and s 2 . 7,8 If we could analyze every possible sample of equal size for a given popula- tion (e.g., every possible sample of five pennies), calculating their respective means and variances, the average mean and the average variance would equal m and s 2 . Al- though – X and s 2 for any single sample probably will not be the same as m or s 2 , they provide a reasonable estimate for these values. Chapter 4 Evaluating Analytical Data 79 Frequenc y Weight of pennies (g) 3.181 to 3.199 3.162 to 3.180 3.048 to 3.066 3.010 to 3.028 3.029 to 3.047 3.086 to 3.104 3.105 to 3.123 3.124 to 3.142 3.143 to 3.161 2.991 to 3.009 0 5 10 15 20 25 3.067 to 3.085 central limit theorem The distribution of measurements subject to indeterminate errors is often a normal distribution. 1400-CH04 9/8/99 3:54 PM Page 79 80 Modern Analytical Chemistry degrees of freedom The number of independent values on which a result is based (n ). Degrees of Freedom Unlike the population’s variance, the variance of a sample in- cludes the term n – 1 in the denominator, where n is the size of the sample 4.12 Defining the sample’s variance with a denominator of n, as in the case of the popu- lation’s variance leads to a biased estimation of s 2 . The denominators of the vari- ance equations 4.8 and 4.12 are commonly called the degrees of freedom for the population and the sample, respectively. In the case of a population, the degrees of freedom is always equal to the total number of members, n, in the population. For the sample’s variance, however, substituting – X for m removes a degree of freedom from the calculation. That is, if there are n members in the sample, the value of the n th member can always be deduced from the remaining n – 1 members and – X. For example, if we have a sample with five members, and we know that four of the members are 1, 2, 3, and 4, and that the mean is 3, then the fifth member of the sample must be ( – X · n)–X 1 – X 2 – X 3 – X 4 =(3· 5)–1–2–3–4=5 4 D. 5 Confidence Intervals for Samples Earlier we introduced the confidence interval as a way to report the most probable value for a population’s mean, m , when the population’s standard deviation, s , is known. Since s 2 is an unbiased estimator of s 2 , it should be possible to construct confidence intervals for samples by replacing s in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define s 2 for a single member of a population. Consequently, equation 4.10 cannot be extended to situa- tions in which s 2 is used as an estimator of s 2 . In other words, when s is unknown, we cannot construct a confidence interval for m by sampling only a single member of the population. The second complication is that the values of z shown in Table 4.11 are derived for a normal distribution curve that is a function of s 2 , not s 2 . Although s 2 is an un- biased estimator of s 2 , the value of s 2 for any randomly selected sample may differ significantly from s 2 . To account for the uncertainty in estimating s 2 , the term z in equation 4.11 is replaced with the variable t, where t is defined such that t ? z at all confidence levels. Thus, equation 4.11 becomes 4.13 Values for t at the 95% confidence level are shown in Table 4.14. Note that t be- comes smaller as the number of the samples (or degrees of freedom) increase, ap- proaching z as n approaches infinity. Additional values of t for other confidence lev- els can be found in Appendix 1B. EXAMPLE 4 .1 5 What is the 95% confidence interval for the data in Table 4.1? SOLUTION The mean and standard deviation for this sample are, respectively, 3.117 g and 0.051 g. Since the sample consists of seven measurements, there are six degrees μ= ±X ts n s XX n i i n 2 2 1 1 = ? ? = ∑ () 1400-CH04 9/8/99 3:54 PM Page 80 of freedom. The value of t from Table 4.14, is 2.45. Substituting into equation 4.13 gives Thus, there is a 95% probability that the population’s mean is between 3.070 and 3.164 g. μ= ± = ± = ±X ts n 3 117 2 45 0 051 7 3 117 0 047. (. )(. ) .. g Chapter 4 Evaluating Analytical Data 81 Table 4 .1 4 Values of t for the 95% Confidence Interval Degrees of Freedom t 1 12.71 2 4.30 3 3.18 4 2.78 5 2.57 6 2.45 7 2.36 8 2.31 9 2.26 10 2.23 12 2.18 14 2.14 16 2.12 18 2.10 20 2.09 30 2.04 50 2.01 ∞ 1.96 4 D.6 A Cautionary Statement There is a temptation when analyzing data to plug numbers into an equation, carry out the calculation, and report the result. This is never a good idea, and you should develop the habit of constantly reviewing and evaluating your data. For example, if analyzing five samples gives an analyte’s mean concentration as 0.67 ppm with a standard deviation of 0.64 ppm, then the 95% confidence interval is This confidence interval states that the analyte’s true concentration lies within the range of –0.16 ppm to 1.44 ppm. Including a negative concentration within the con- fidence interval should lead you to reevaluate your data or conclusions. On further investigation your data may show that the standard deviation is larger than expected, μ= ± = ±067 278 064 5 064 080. (. )(. ) .. pmp 1400-CH04 9/8/99 3:54 PM Page 81 Figure 4.9 Three examples of possible relationships between the probability distributions for two populations. (a) Completely separate distributions; (b) Distributions with a great deal of overlap; (c) Distributions with some overlap. 82 Modern Analytical Chemistry *The topic of detection limits is discussed at the end of this chapter. making the confidence interval too broad, or you may conclude that the analyte’s concentration is too small to detect accurately.* A second example is also informative. When samples are obtained from a nor- mally distributed population, their values must be random. If results for several samples show a regular pattern or trend, then the samples cannot be normally dis- tributed. This may reflect the fact that the underlying population is not normally distributed, or it may indicate the presence of a time-dependent determinate error. For example, if we randomly select 20 pennies and find that the mass of each penny exceeds that of the preceding penny, we might suspect that the balance on which the pennies are being weighed is drifting out of calibration. 4 E Statistical Analysis of Data In the previous section we noted that the result of an analysis is best expressed as a confidence interval. For example, a 95% confidence interval for the mean of five re- sults gives the range in which we expect to find the mean for 95% of all samples of equal size, drawn from the same population. Alternatively, and in the absence of de- terminate errors, the 95% confidence interval indicates the range of values in which we expect to find the population’s true mean. The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample’s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as “Does a newly devel- oped method for the analysis of cholesterol in blood give results that are signifi- cantly different from those obtained when using a standard method?” or “Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant?” In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 4 E.1 Significance Testing Let’s consider the following problem. Two sets of blood samples have been collected from a patient receiving medication to lower her concentration of blood glucose. One set of samples was drawn immediately before the medication was adminis- tered; the second set was taken several hours later. The samples are analyzed and their respective means and variances reported. How do we decide if the medication was successful in lowering the patient’s concentration of blood glucose? One way to answer this question is to construct probability distribution curves for each sample and to compare the curves with each other. Three possible out- comes are shown in Figure 4.9. In Figure 4.9a, the probability distribution curves are completely separated, strongly suggesting that the samples are significantly dif- ferent. In Figure 4.9b, the probability distributions for the two samples are highly overlapped, suggesting that any difference between the samples is insignificant. Fig- ure 4.9c, however, presents a dilemma. Although the means for the two samples ap- pear to be different, the probability distributions overlap to an extent that a signifi- cant number of possible outcomes could belong to either distribution. In this case we can, at best, only make a statement about the probability that the samples are significantly different. Values (a) Values (b) Values (c) 1400-CH04 9/8/99 3:54 PM Page 82 The process by which we determine the probability that there is a significant difference between two samples is called significance testing or hypothesis testing. Before turning to a discussion of specific examples, however, we will first establish a general approach to conducting and interpreting significance tests. 4 E.2 Constructing a Significance Test A significance test is designed to determine whether the difference between two or more values is too large to be explained by indeterminate error. The first step in constructing a significance test is to state the experimental problem as a yes- or-no question, two examples of which were given at the beginning of this sec- tion. A null hypothesis and an alternative hypothesis provide answers to the ques- tion. The null hypothesis, H 0 , is that indeterminate error is sufficient to explain any difference in the values being compared. The alternative hypothesis, H A , is that the difference between the values is too great to be explained by random error and, therefore, must be real. A significance test is conducted on the null hy- pothesis, which is either retained or rejected. If the null hypothesis is rejected, then the alternative hypothesis must be accepted. When a null hypothesis is not rejected, it is said to be retained rather than accepted. A null hypothesis is re- tained whenever the evidence is insufficient to prove it is incorrect. Because of the way in which significance tests are conducted, it is impossible to prove that a null hypothesis is true. The difference between retaining a null hypothesis and proving the null hy- pothesis is important. To appreciate this point, let us return to our example on de- termining the mass of a penny. After looking at the data in Table 4.12, you might pose the following null and alternative hypotheses H 0 : Any U.S. penny in circulation has a mass that falls in the range of 2.900–3.200 g H A : Some U.S. pennies in circulation have masses that are less than 2.900 g or more than 3.200 g. To test the null hypothesis, you reach into your pocket, retrieve a penny, and deter- mine its mass. If the mass of this penny is 2.512 g, then you have proved that the null hypothesis is incorrect. Finding that the mass of your penny is 3.162 g, how- ever, does not prove that the null hypothesis is correct because the mass of the next penny you sample might fall outside the limits set by the null hypothesis. After stating the null and alternative hypotheses, a significance level for the analysis is chosen. The significance level is the confidence level for retaining the null hypothesis or, in other words, the probability that the null hypothesis will be incor- rectly rejected. In the former case the significance level is given as a percentage (e.g., 95%), whereas in the latter case, it is given as a , where a is defined as Thus, for a 95% confidence level, a is 0.05. Next, an equation for a test statistic is written, and the test statistic’s critical value is found from an appropriate table. This critical value defines the breakpoint between values of the test statistic for which the null hypothesis will be retained or rejected. The test statistic is calculated from the data, compared with the critical value, and the null hypothesis is either rejected or retained. Finally, the result of the significance test is used to answer the original question. α= ?1 100 confidence level Chapter 4 Evaluating Analytical Data 83 significance test A statistical test to determine if the difference between two values is significant. null hypothesis A statement that the difference between two values can be explained by indeterminate error; retained if the significance test does not fail (H 0 ). alternative hypothesis A statement that the difference between two values is too great to be explained by indeterminate error; accepted if the significance test shows that null hypothesis should be rejected (H A ). 1400-CH04 9/8/99 3:54 PM Page 83 Figure 4.10 Examples of (a) two-tailed, (b) and (c) one- tailed, significance tests. The shaded areas in each curve represent the values for which the null hypothesis is rejected. 4 E. 3 One-Tailed and Two-Tailed Significance Tests Consider the situation when the accuracy of a new analytical method is evaluated by analyzing a standard reference material with a known m . A sample of the standard is analyzed, and the sample’s mean is determined. The null hypothesis is that the sam- ple’s mean is equal to m H 0 : – X = m If the significance test is conducted at the 95% confidence level (a = 0.05), then the null hypothesis will be retained if a 95% confidence interval around – X contains m . If the alternative hypothesis is H A : – X ≠ m then the null hypothesis will be rejected, and the alternative hypothesis accepted if m lies in either of the shaded areas at the tails of the sample’s probability distribution (Figure 4.10a). Each of the shaded areas accounts for 2.5% of the area under the probability distribution curve. This is called a two-tailed significance test because the null hypothesis is rejected for values of m at either extreme of the sample’s prob- ability distribution. The alternative hypothesis also can be stated in one of two additional ways H A : – X > m H A : – X < m for which the null hypothesis is rejected if m falls within the shaded areas shown in Figure 4.10(b) and Figure 4.10(c), respectively. In each case the shaded area repre- sents 5% of the area under the probability distribution curve. These are examples of one-tailed significance tests. For a fixed confidence level, a two-tailed test is always the more conservative test because it requires a larger difference between – X and m to reject the null hypothesis. Most significance tests are applied when there is no a priori expectation about the relative magnitudes of the parameters being compared. A two-tailed significance test, therefore, is usually the appropriate choice. One-tailed significance tests are reserved for situations when we have reason to expect one parameter to be larger or smaller than the other. For example, a one-tailed significance test would be appropriate for our earlier example regarding a medication’s effect on blood glucose levels since we believe that the medication will lower the concentration of glucose. 4 E. 4 Errors in Significance Testing Since significance tests are based on probabilities, their interpretation is naturally subject to error. As we have already seen, significance tests are carried out at a sig- nificance level, a , that defines the probability of rejecting a null hypothesis that is true. For example, when a significance test is conducted at a = 0.05, there is a 5% probability that the null hypothesis will be incorrectly rejected. This is known as a type 1 error, and its risk is always equivalent to a . Type 1 errors in two-tailed and one-tailed significance tests are represented by the shaded areas under the probabil- ity distribution curves in Figure 4.10. The second type of error occurs when the null hypothesis is retained even though it is false and should be rejected. This is known as a type 2 error, and its probability of occurrence is b . Unfortunately, in most cases b cannot be easily cal- culated or estimated. 84 Modern Analytical Chemistry Values (a) Values (b) Values (c) two-tailed significance test Significance test in which the null hypothesis is rejected for values at either end of the normal distribution. type 1 error The risk of falsely rejecting the null hypothesis (a ). type 2 error The risk of falsely retaining the null hypothesis (b ). one-tailed significance test Significance test in which the null hypothesis is rejected for values at only one end of the normal distribution. 1400-CH04 9/8/99 3:54 PM Page 84 Figure 4.11 Relationship between confidence intervals and results of a significance test. (a) The shaded area under the normal distribution curves shows the apparent confidence intervals for the sample based on t exp . The solid bars in (b) and (c) show the actual confidence intervals that can be explained by indeterminate error using the critical value of (a ,n ). In part (b) the null hypothesis is rejected and the alternative hypothesis is accepted. In part (c) the null hypothesis is retained. Chapter 4 Evaluating Analytical Data 85 The probability of a type 1 error is inversely related to the probability of a type 2 error. Minimizing a type 1 error by decreasing a , for example, increases the likeli- hood of a type 2 error. The value of a chosen for a particular significance test, therefore, represents a compromise between these two types of error. Most of the examples in this text use a 95% confidence level, or a = 0.05, since this is the most frequently used confidence level for the majority of analytical work. It is not unusual, however, for more stringent (e.g. a = 0.01) or for more lenient (e.g. a = 0.10) confidence levels to be used. 4 F Statistical Methods for Normal Distributions The most commonly encountered probability distribution is the normal, or Gauss- ian, distribution. A normal distribution is characterized by a true mean, m , and vari- ance, s 2 , which are estimated using – X and s 2 . Since the area between any two limits of a normal distribution is well defined, the construction and evaluation of signifi- cance tests are straightforward. 4 F.1 Comparing — X to m One approach for validating a new analytical method is to analyze a standard sample containing a known amount of analyte, m . The method’s accuracy is judged by determining the average amount of analyte in several samples, – X, and using a significance test to compare it with m . The null hypothesis is that – X and m are the same and that any difference between the two values can be explained by in- determinate errors affecting the determination of – X.The alternative hypothesis is that the difference between – X and m is too large to be explained by indeterminate error. The equation for the test (experimental) statistic, t exp , is derived from the confi- dence interval for m 4.14 Rearranging equation 4.14 4.15 gives the value of t exp when m is at either the right or left edge of the sample’s ap- parent confidence interval (Figure 4.11a). The value of t exp is compared with a critical value, t(a ,n ), which is determined by the chosen significance level, a , the degrees of freedom for the sample, n , and whether the significance test is one- tailed or two-tailed. Values for t(a ,n ) are found in Appendix 1B. The critical value t(a ,n ) defines the confidence interval that can be explained by indetermi- nate errors. If t exp is greater than t(a ,n ), then the confidence interval for the data is wider than that expected from indeterminate errors (Figure 4.11b). In this case, the null hypothesis is rejected and the alternative hypothesis is accepted. If t exp is less than or equal to t(a ,n ), then the confidence interval for the data could be at- tributed to indeterminate error, and the null hypothesis is retained at the stated significance level (Figure 4.11c). A typical application of this significance test, which is known as a t-test of – X to m , is outlined in the following example. t Xn s exp = ?×μ μ= ±X ts n exp (a) X – t exp s n X + t exp s n (b) X – t exp s n X – t(a ,n )s n X + t(a ,n )s n X + t exp s n (c) X – t exp s n X – t(a,n )s n X + t(a,n )s n X + t exp s n t-test Statistical test for comparing two mean values to see if their difference is too large to be explained by indeterminate error. 1400-CH04 9/8/99 3:54 PM Page 85 86 Modern Analytical Chemistry EXAMPLE 4 .16 Before determining the amount of Na 2 CO 3 in an unknown sample, a student decides to check her procedure by analyzing a sample known to contain 98.76% w/w Na 2 CO 3 . Five replicate determinations of the %w/w Na 2 CO 3 in the standard were made with the following results 98.71% 98.59% 98.62% 98.44% 98.58% Is the mean for these five trials significantly different from the accepted value at the 95% confidence level (a = 0.05)? SOLUTION The mean and standard deviation for the five trials are – X = 98.59 s = 0.0973 Since there is no reason to believe that – X must be either larger or smaller than m , the use of a two-tailed significance test is appropriate. The null and alternative hypotheses are H 0 : – X = m H A : – X ≠ m The test statistic is The critical value for t(0.05,4), as found in Appendix 1B, is 2.78. Since t exp is greater than t(0.05, 4), we must reject the null hypothesis and accept the alternative hypothesis. At the 95% confidence level the difference between – X and m is significant and cannot be explained by indeterminate sources of error. There is evidence, therefore, that the results are affected by a determinate source of error. If evidence for a determinate error is found, as in Example 4.16, its source should be identified and corrected before analyzing additional samples. Failing to reject the null hypothesis, however, does not imply that the method is accurate, but only indicates that there is insufficient evidence to prove the method inaccurate at the stated confidence level. The utility of the t-test for – X and m is improved by optimizing the conditions used in determining – X. Examining equation 4.15 shows that increasing the num- ber of replicate determinations, n, or improving the precision of the analysis en- hances the utility of this significance test. A t-test can only give useful results, however, if the standard deviation for the analysis is reasonable. If the standard deviation is substantially larger than the expected standard deviation, s , the con- fidence interval around – X will be so large that a significant difference between – X and m may be difficult to prove. On the other hand, if the standard deviation is significantly smaller than expected, the confidence interval around – X will be too small, and a significant difference between – X and m may be found when none ex- ists. A significance test that can be used to evaluate the standard deviation is the subject of the next section. t Xn s exp .. . .= ?× = ?× = μ 98 76 98 59 5 0 0973 391 1400-CH04 9/8/99 3:54 PM Page 86 Chapter 4 Evaluating Analytical Data 87 4 F.2 Comparing s 2 to s 2 When a particular type of sample is analyzed on a regular basis, it may be possible to determine the expected, or true variance, s 2 , for the analysis. This often is the case in clinical labs where hundreds of blood samples are analyzed each day. Repli- cate analyses of any single sample, however, results in a sample variance, s 2 . A statis- tical comparison of s 2 to s 2 provides useful information about whether the analysis is in a state of “statistical control.” The null hypothesis is that s 2 and s 2 are identical, and the alternative hypothesis is that they are not identical. The test statistic for evaluating the null hypothesis is called an F-test, and is given as either 4.16 (s 2 > s 2 )(s 2 > s 2 ) depending on whether s 2 is larger or smaller than s 2 . Note that F exp is defined such that its value is always greater than or equal to 1. If the null hypothesis is true, then F exp should equal 1. Due to indeterminate er- rors, however, the value for F exp usually is greater than 1. A critical value, F(a , n num , n den ), gives the largest value of F that can be explained by indeterminate error. It is chosen for a specified significance level, a , and the degrees of freedom for the vari- ances in the numerator, n num , and denominator, n den . The degrees of freedom for s 2 is n – 1, where n is the number of replicates used in determining the sample’s vari- ance. Critical values of F for a = 0.05 are listed in Appendix 1C for both one-tailed and two-tailed significance tests. EXAMPLE 4 .1 7 A manufacturer’s process for analyzing aspirin tablets has a known variance of 25. A sample of ten aspirin tablets is selected and analyzed for the amount of aspirin, yielding the following results 254 249 252 252 249 249 250 247 251 252 Determine whether there is any evidence that the measurement process is not under statistical control at a = 0.05. SOLUTION The variance for the sample of ten tablets is 4.3. A two-tailed significance test is used since the measurement process is considered out of statistical control if the sample’s variance is either too good or too poor. The null hypothesis and alternative hypotheses are H 0 : s 2 = s 2 H A : s 2 ≠ s 2 The test statistic is The critical value for F(0.05, ￥ , 9) from Appendix 1C is 3.33. Since F is greater than F(0.05,￥ , 9), we reject the null hypothesis and accept the alternative hypothesis that the analysis is not under statistical control. One explanation for the unreasonably small variance could be that the aspirin tablets were not selected randomly. F s exp . .=== σ 2 2 25 43 58 F s F s exp exp == 2 2 2 2 σ σ or F-test Statistical test for comparing two variances to see if their difference is too large to be explained by indeterminate error. 1400-CH04 9/8/99 3:54 PM Page 87 88 Modern Analytical Chemistry 4 F. 3 Comparing Two Sample Variances The F-test can be extended to the comparison of variances for two samples, A and B, by rewriting equation 4.16 as where A and B are defined such that s 2 A is greater than or equal to s 2 B . An example of this application of the F-test is shown in the following example. EXAMPLE 4 .18 Tables 4.1 and 4.8 show results for two separate experiments to determine the mass of a circulating U.S. penny. Determine whether there is a difference in the precisions of these analyses at a = 0.05. SOLUTION Letting A represent the results in Table 4.1 and B represent the results in Table 4.8, we find that the variances are s 2 A = 0.00259 and s 2 B = 0.00138. A two-tailed significance test is used since there is no reason to suspect that the results for one analysis will be more precise than that of the other. The null and alternative hypotheses are H 0 : s 2 A = s 2 B H A : s 2 A ? s 2 B and the test statistic is The critical value for F(0.05, 6, 4) is 9.197. Since F exp is less than F(0.05, 6, 4), the null hypothesis is retained. There is no evidence at the chosen significance level to suggest that the difference in precisions is significant. 4 F. 4 Comparing Two Sample Means The result of an analysis is influenced by three factors: the method, the sample, and the analyst. The influence of these factors can be studied by conducting a pair of ex- periments in which only one factor is changed. For example, two methods can be compared by having the same analyst apply both methods to the same sample and examining the resulting means. In a similar fashion, it is possible to compare two analysts or two samples. Significance testing for comparing two mean values is divided into two cate- gories depending on the source of the data. Data are said to be unpaired when each mean is derived from the analysis of several samples drawn from the same source. Paired data are encountered when analyzing a series of samples drawn from differ- ent sources. Unpaired Data Consider two samples, A and B, for which mean values, – X A and – X B , and standard deviations, s A and s B , have been measured. Confidence intervals for m A and m B can be written for both samples F s s exp . . .== = A B 2 2 0 00259 0 00138 188 F s s exp = A B 2 2 unpaired data Two sets of data consisting of results obtained using several samples drawn from a single source. paired data Two sets of data consisting of results obtained using several samples drawn from different sources. 1400-CH04 9/8/99 3:55 PM Page 88 4.17 4.18 where n A and n B are the number of replicate trials conducted on samples A and B. A comparison of the mean values is based on the null hypothesis that – X A and – X B are identical, and an alternative hypothesis that the means are significantly different. A test statistic is derived by letting m A equal m B , and combining equations 4.17 and 4.18 to give Solving for – X A – – X B and using a propagation of uncertainty, gives Finally, solving for t, which we replace with t exp , leaves us with 4.19 The value of t exp is compared with a critical value, t(a , n ), as determined by the cho- sen significance level, a , the degrees of freedom for the sample, n , and whether the significance test is one-tailed or two-tailed. It is unclear, however, how many degrees of freedom are associated with t(a , n ) since there are two sets of independent measurements. If the variances s 2 A and s 2 B esti- mate the same s 2 , then the two standard deviations can be factored out of equation 4.19 and replaced by a pooled standard deviation, s pool , which provides a better esti- mate for the precision of the analysis. Thus, equation 4.19 becomes 4.20 with the pooled standard deviation given as 4.21 As indicated by the denominator of equation 4.21, the degrees of freedom for the pooled standard deviation is n A + n B –2. If s A and s B are significantly different, however, then t exp must be calculated using equation 4.19. In this case, the degrees of freedom is calculated using the fol- lowing imposing equation. 4.22 Since the degrees of freedom must be an integer, the value of n obtained using equation 4.22 is rounded to the nearest integer. Regardless of whether equation 4.19 or 4.20 is used to calculate t exp , the null hy- pothesis is rejected if t exp is greater than t(a , n ), and retained if t exp is less than or equal to t(a , n ). ν= + ++ + [( / ) ( / )] [( / ) /( )] [( / ) /( )] – sn sn sn n sn n A A B B A AA B BB 222 22 22 11 2 s nsns nn pool A A B B AB = ?+? +? ()()11 2 22 t XX snn exp (/ ) (/ ) = ? + AB pool 11 AB t XX sn sn exp (/ ) (/) = ? + AB A 2 A B 2 B XXt s n s n AB ?=× + A 2 A B 2 B X ts n X ts n A A A B B B ±=± Chapter 4 Evaluating Analytical Data 89 μ μ AA A A BB B B =± =± X ts n X ts n 1400-CH04 9/8/99 3:55 PM Page 89 90 Modern Analytical Chemistry EXAMPLE 4 .1 9 Tables 4.1 and 4.8 show results for two separate experiments to determine the mass of a circulating U.S. penny. Determine whether there is a difference in the means of these analyses at a = 0.05. SOLUTION To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an F-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated where the subscript A indicates the data in Table 4.1, and the subscript B indicates the data in Table 4.8. The comparison of the means for the two analyses is based on the null hypothesis H 0 : – X A = – X B and a two-tailed alternative hypothesis H A : – X A ? – X B Since the standard deviations can be pooled, the test statistic is calculated using equation 4.20 The critical value for t(0.05, 10), from Appendix 1B, is 2.23. Since t exp is less than t(0.05, 10) the null hypothesis is retained, and there is no evidence that the two sets of pennies are significantly different at the chosen significance level. EXAMPLE 4 .20 The %w/w Na 2 CO 3 in soda ash can be determined by an acid–base titration. The results obtained by two analysts are shown here. Determine whether the difference in their mean values is significant at a = 0.05. Analyst A Analyst B 86.82 81.01 87.04 86.15 86.93 81.73 87.01 83.19 86.20 80.27 87.00 83.94 t XX snn exp (/ / ) .. .(//) .= ? + = ? + = AB pool A B 11 3 117 3 081 0 0459 1 7 1 5 134 s nsns nn pool A A B B AB = ?+? +? = ?+? +? = ()() ()(. )()(. ) . 11 2 7 1 0 00259 5 1 0 00138 752 0 0459 22 1400-CH04 9/8/99 3:55 PM Page 90 SOLUTION We begin by summarizing the mean and standard deviation for the data reported by each analyst. These values are – X A = 86.83% s A = 0.32 – X B = 82.71% s B = 2.16 A two-tailed F-test of the following null and alternative hypotheses H 0 : s 2 A = s 2 B H A : s 2 A ? s 2 B is used to determine whether a pooled standard deviation can be calculated. The test statistic is Since F exp is larger than the critical value of 7.15 for F(0.05, 5, 5), the null hypothesis is rejected and the alternative hypothesis that the variances are significantly different is accepted. As a result, a pooled standard deviation cannot be calculated. The mean values obtained by the two analysts are compared using a two- tailed t-test. The null and alternative hypotheses are H 0 : – X A = – X B H A: – X A ≠ – X B Since a pooled standard deviation could not be calculated, the test statistic, t exp , is calculated using equation 4.19 and the degrees of freedom are calculated using equation 4.22 The critical value for t(0.05, 5) is 2.57. Since the calculated value of t exp is greater than t(0.05, 5) we reject the null hypothesis and accept the alternative hypothesis that the mean values for %w/w Na 2 CO 3 reported by the two analysts are significantly different at the chosen significance level. Paired Data In some situations the variation within the data sets being compared is more significant than the difference between the means of the two data sets. This is commonly encountered in clinical and environmental studies, where the data being compared usually consist of a set of samples drawn from several populations. For example, a study designed to investigate two procedures for monitoring the concentration of glucose in blood might involve blood samples drawn from ten pa- tients. If the variation in the blood glucose levels among the patients is significantly larger than the anticipated variation between the methods, then an analysis in which the data are treated as unpaired will fail to find a significant difference between the ν= + ++ + ?= ≈ [(./)(./)] {( . / ) /( )} {( . / ) /( )} . 032 6 216 6 032 6 6 1 216 6 6 1 2535 222 22 22 t XX sn sn exp (/ ) (/) .. [( . ) / ] [( . ) / ] .= ? + = ? + = AB A A B B 22 2 2 86 83 82 71 032 6 216 6 462 F s s exp (. ) (. ) .== = B A 2 2 2 2 216 032 45 6 Chapter 4 Evaluating Analytical Data 91 1400-CH04 9/8/99 3:55 PM Page 91 92 Modern Analytical Chemistry methods. In general, paired data sets are used whenever the variation being investi- gated is smaller than other potential sources of variation. In a study involving paired data the difference, d i , between the paired values for each sample is calculated. The average difference, – d, and standard deviation of the differences, s d , are then calculated. The null hypothesis is that – d is 0, and that there is no difference in the results for the two data sets. The alternative hypothesis is that the results for the two sets of data are significantly different, and, therefore, – d is not equal to 0. The test statistic, t exp , is derived from a confidence interval around – d where n is the number of paired samples. Replacing t with t exp and rearranging gives The value of t exp is then compared with a critical value, t(a , n ), which is determined by the chosen significance level, a , the degrees of freedom for the sample, n , and whether the significance test is one-tailed or two-tailed. For paired data, the degrees of freedom is n – 1. If t exp is greater than t(a , n ), then the null hypothesis is rejected and the alternative hypothesis is accepted. If t exp is less than or equal to t(a , n ), then the null hypothesis is retained, and a significant difference has not been demon- strated at the stated significance level. This is known as the paired t-test. EXAMPLE 4 .21 Marecek and colleagues developed a new electrochemical method for the rapid quantitative analysis of the antibiotic monensin in the fermentation vats used during its production. 9 The standard method for the analysis, which is based on a test for microbiological activity, is both difficult and time-consuming. As part of the study, samples taken at different times from a fermentation production vat were analyzed for the concentration of monensin using both the electrochemical and microbiological procedures. The results, in parts per thousand (ppt),* are reported in the following table. Sample Microbiological Electrochemical 1 129.5 132.3 2 89.6 91.0 3 76.6 73.6 4 52.2 58.2 5 110.8 104.2 6 50.4 49.9 7 72.4 82.1 8 141.4 154.1 9 75.0 73.4 10 34.1 38.1 11 60.3 60.1 Determine whether there is a significant difference between the methods at a = 0.05. t dn s d exp = 0 =±d ts n d *1 ppt is equivalent to 0.1%. paired t-test Statistical test for comparing paired data to determine if their difference is too large to be explained by indeterminate error. 1400-CH04 9/8/99 3:55 PM Page 92 SOLUTION This is an example of a paired data set since the acquisition of samples over an extended period introduces a substantial time-dependent change in the concentration of monensin. The comparison of the two methods must be done with the paired t-test, using the following null and two-tailed alternative hypotheses H 0 : – d =0 H A: – d ≠ 0 Defining the difference between the methods as d = X elect – X micro we can calculate the difference for each sample Sample 123456789101 d 2.8 1.4 –3.0 6.0 –6.6 –0.5 9.7 12.7 –1.6 4.0 –0.2 The mean and standard deviation for the differences are 2.25 and 5.63, respectively. The test statistic is which is smaller than the critical value of 2.23 for t(0.05, 10). Thus, the null hypothesis is retained, and there is no evidence that the two methods yield different results at the stated significance level. A paired t-test can only be applied when the individual differences, d i , belong to the same population. This will only be true if the determinate and indeterminate errors affecting the results are independent of the concentration of analyte in the samples. If this is not the case, a single sample with a larger error could result in a value of d i that is substantially larger than that for the remaining samples. Including this sample in the calculation of – d and s d leads to a biased estimate of the true mean and standard deviation. For samples that span a limited range of analyte concentra- tions, such as that in Example 4.21, this is rarely a problem. When paired data span a wide range of concentrations, however, the magnitude of the determinate and in- determinate sources of error may not be independent of the analyte’s concentra- tion. In such cases the paired t-test may give misleading results since the paired data with the largest absolute determinate and indeterminate errors will dominate – d. In this situation a comparison is best made using a linear regression, details of which are discussed in the next chapter. 4 F. 5 Outliers On occasion, a data set appears to be skewed by the presence of one or more data points that are not consistent with the remaining data points. Such values are called outliers. The most commonly used significance test for identifying outliers is Dixon’s Q-test. The null hypothesis is that the apparent outlier is taken from the same popula- tion as the remaining data. The alternative hypothesis is that the outlier comes from a different population, and, therefore, should be excluded from consideration. The Q-test compares the difference between the suspected outlier and its near- est numerical neighbor to the range of the entire data set. Data are ranked from smallest to largest so that the suspected outlier is either the first or the last data t dn s d exp . . .== = 225 11 563 133 Chapter 4 Evaluating Analytical Data 93 outlier Data point whose value is much larger or smaller than the remaining data. Dixon’s Q-test Statistical test for deciding if an outlier can be removed from a set of data. 1400-CH04 9/8/99 3:55 PM Page 93 94 Modern Analytical Chemistry point. The test statistic, Q exp , is calculated using equation 4.23 if the suspected out- lier is the smallest value (X 1 ) 4.23 or using equation 4.24 if the suspected outlier is the largest value (X n ) 4.24 where n is the number of members in the data set, including the suspected outlier. It is important to note that equations 4.23 and 4.24 are valid only for the detection of a single outlier. Other forms of Dixon’s Q-test allow its extension to the detection of multiple outliers. 10 The value of Q exp is compared with a critical value, Q(a , n), at a significance level of a . The Q-test is usually applied as the more conservative two- tailed test, even though the outlier is the smallest or largest value in the data set. Values for Q(a , n) can be found in Appendix 1D. If Q exp is greater than Q(a , n), then the null hypothesis is rejected and the outlier may be rejected. When Q exp is less than or equal to Q(a , n) the suspected outlier must be retained. EXAMPLE 4 .22 The following masses, in grams, were recorded in an experiment to determine the average mass of a U.S. penny. 3.067 3.049 3.039 2.514 3.048 3.079 3.094 3.109 3.102 Determine if the value of 2.514 g is an outlier at a = 0.05. SOLUTION To begin with, place the masses in order from smallest to largest 2.514 3.039 3.048 3.049 3.067 3.079 3.094 3.102 3.109 and calculate Q exp The critical value for Q(0.05, 9) is 0.493. Since Q exp > Q(0.05, 9) the value is assumed to be an outlier, and can be rejected. The Q-test should be applied with caution since there is a probability, equal to a , that an outlier identified by the Q-test actually is not an outlier. In addition, the Q-test should be avoided when rejecting an outlier leads to a precision that is un- reasonably better than the expected precision determined by a propagation of un- certainty. Given these two concerns it is not surprising that some statisticians cau- tion against the removal of outliers. 11 On the other hand, testing for outliers can provide useful information if we try to understand the source of the suspected out- lier. For example, the outlier identified in Example 4.22 represents a significant change in the mass of a penny (an approximately 17% decrease in mass), due to a change in the composition of the U.S. penny. In 1982, the composition of a U.S. penny was changed from a brass alloy consisting of 95% w/w Cu and 5% w/w Zn, to a zinc core covered with copper. 12 The pennies in Example 4.22 were therefore drawn from different populations. Q XX XX exp .. .. .= ? ? = ? ? = 21 91 3 039 2 514 3 109 2 514 0 882 Q XX XX nn n exp = ? ? ?1 1 Q XX XX n exp = ? ? 21 1 1400-CH04 9/8/99 3:55 PM Page 94 Figure 4.12 Normal distribution curves showing the definition of detection limit and limit of identification (LOI). The probability of a type 1 error is indicated by the dark shading, and the probability of a type 2 error is indicated by light shading. Chapter 4 Evaluating Analytical Data 95 (a) Probability distribution for blank S reag (S A ) DL Probability distribution for blank (b) Probability distribution for sample S reag (S A ) DL Probability distribution for blank (c) Probability distribution for sample S reag (S A ) LOI limit of identification The smallest concentration or absolute amount of analyte such that the probability of type 1 and type 2 errors are equal (LOI). detection limit The smallest concentration or absolute amount of analyte that can be reliably detected. 4 G Detection Limits The focus of this chapter has been the evaluation of analytical data, including the use of statistics. In this final section we consider how statistics may be used to char- acterize a method’s ability to detect trace amounts of an analyte. A method’s detection limit is the smallest amount or concentration of analyte that can be detected with statistical confidence. The International Union of Pure and Applied Chemistry (IUPAC) defines the detection limit as the smallest concen- tration or absolute amount of analyte that has a signal significantly larger than the signal arising from a reagent blank. Mathematically, the analyte’s signal at the detec- tion limit, (S A ) DL , is (S A ) DL = S reag + zs reag 4.25 where S reag is the signal for a reagent blank, s reag is the known standard devia- tion for the reagent blank’s signal, and z is a factor accounting for the desired confidence level. The concentration, (C A ) DL , or absolute amount of analyte, (n A ) DL , at the detection limit can be determined from the signal at the detection limit. The value for z depends on the desired significance level for reporting the detection limit. Typically, z is set to 3, which, from Appendix 1A, corresponds to a signifi- cance level of a = 0.00135. Consequently, only 0.135% of measurements made on the blank will yield signals that fall outside this range (Figure 4.12a). When s reag is unknown, the term zs reag may be replaced with ts reag , where t is the appropriate value from a t-table for a one-tailed analysis. 13 In analyzing a sample to determine whether an analyte is present, the signal for the sample is compared with the signal for the blank. The null hypothesis is that the sample does not contain any analyte, in which case (S A ) DL and S reag are identical. The alternative hypothesis is that the analyte is present, and (S A ) DL is greater than S reag . If (S A ) DL exceeds S reag by zs (or ts), then the null hypothesis is rejected and there is evidence for the analyte’s presence in the sample. The proba- bility that the null hypothesis will be falsely rejected, a type 1 error, is the same as the significance level. Selecting z to be 3 minimizes the probability of a type 1 error to 0.135%. Significance tests, however, also are subject to type 2 errors in which the null hypothesis is falsely retained. Consider, for example, the situation shown in Figure 4.12b, where S A is exactly equal to (S A ) DL. In this case the probability of a type 2 error is 50% since half of the signals arising from the sample’s population fall below the detection limit. Thus, there is only a 50:50 probability that an analyte at the IUPAC detection limit will be detected. As defined, the IUPAC definition for the detection limit only indicates the smallest signal for which we can say, at a signifi- cance level of a , that an analyte is present in the sample. Failing to detect the ana- lyte, however, does not imply that it is not present. An alternative expression for the detection limit, which minimizes both type 1 and type 2 errors, is the limit of identification, (S A ) LOI , which is defined as 14 (S A ) LOI = S reag + zs reag + zs samp () () () () C S k n S k ADL ADL ADL ADL = = 1400-CH04 9/8/99 3:55 PM Page 95 As shown in Figure 4.12c, the limit of identification is selected such that there is an equal probability of type 1 and type 2 errors. The American Chemical Society’s Committee on Environmental Analytical Chemistry recommends the limit of quantitation, (S A ) LOQ , which is defined as 15 (S A ) LOQ = S reag +10s reag Other approaches for defining the detection limit have also been developed. 16 The detection limit is often represented, particularly when used in debates over public policy issues, as a distinct line separating analytes that can be detected from those that cannot be detected. 17 This use of a detection limit is incorrect. Defining the detection limit in terms of statistical confidence levels implies that there may be a gray area where the analyte is sometimes detected and sometimes not detected. This is shown in Figure 4.13 where the upper and lower confidence limits are de- fined by the acceptable probabilities for type 1 and type 2 errors. Analytes produc- ing signals greater than that defined by the upper confidence limit are always de- tected, and analytes giving signals smaller than the lower confidence limit are never detected. Signals falling between the upper and lower confidence limits, however, are ambiguous because they could belong to populations representing either the reagent blank or the analyte. Figure 4.12c represents the smallest value of S A for which no such ambiguity exists. 96 Modern Analytical Chemistry S reag S A ??? Always detected Never detected Lower confidence interval Upper confidence interval Figure 4.13 Establishment of areas where the signal is never detected, always detected, and where results are ambiguous. The upper and lower confidence limits are defined by the probability of a type 1 error (dark shading), and the probability of a type 2 error (light shading). 4 H KEY TERMS alternative hypothesis (p. 83) binomial distribution (p. 72) central limit theorem (p. 79) confidence interval (p. 75) constant determinate error (p. 60) degrees of freedom (p. 80) detection limit (p. 95) determinate error (p. 58) Dixon’s Q-test (p. 93) error (p. 64) F-test (p. 87) heterogeneous (p. 58) histogram (p. 77) homogeneous (p. 72) indeterminate error (p. 62) limit of identification (p. 95) limit of quantitation (p. 96) mean (p. 54) measurement error (p. 58) median (p. 55) method error (p. 58) normal distribution (p. 73) null hypothesis (p. 83) one-tailed significance test (p. 84) outlier (p. 93) paired data (p. 88) paired t-test (p. 92) personal error (p. 60) population (p. 71) probability distribution (p. 71) proportional determinate error (p. 61) range (p. 56) repeatability (p. 62) reproducibility (p. 62) sample (p. 71) sampling error (p. 58) significance test (p. 83) standard deviation (p. 56) standard reference material (p. 61) tolerance (p. 58) t-test (p. 85) two-tailed significance test (p. 84) type 1 error (p. 84) type 2 error (p. 84) uncertainty (p. 64) unpaired data (p. 88) variance (p. 57) The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is re- ported by stating the mean or median. The range, standard devia- tion, or variance may be used to report the data’s spread. Data also are characterized by their errors, which include determinate errors affecting the data’s accuracy, and indeterminate errors affecting the data’s precision. A propagation of uncertainty allows us to es- timate the affect of these determinate and indeterminate errors on results determined from our data. The distribution of the results of an analysis around a central value is often described by a probability distribution, two examples 4 I SUMMARY limit of quantitation The smallest concentration or absolute amount of analyte that can be reliably determined (LOQ). 1400-CH04 9/8/99 3:55 PM Page 96 Chapter 4 Evaluating Analytical Data 97 of which are the binomial distribution and the normal distribution. Knowing the type of distribution allows us to determine the proba- bility of obtaining results within a specified range. For a normal distribution this range is best expressed as a confidence interval. A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analy- sis, or for other samples. A t-test is used to compare mean val- ues, and an F-test to compare precisions. Comparisons between two sets of data require an initial evaluation of whether the data is paired or unpaired. For unpaired data it is also necessary to decide if the standard deviations can be pooled. A decision about whether to retain an outlying value can be made using Dixon’s Q-test. Finally, we have seen that the detection limit is a statistical statement about the smallest amount of analyte that can be de- tected with confidence. A detection limit is not exact because its value depends on how willing we are to falsely report the analyte’s presence or absence in a sample. When reporting a detection limit, you should clearly indicate how you arrived at its value. Experiments Cunningham, C. C.; Brown, G. R.; St Pierre, L. E. “Evaluation of Experimental Data,” J. Chem. Educ. 1981, 58, 509–511. In this experiment students determine the density of glass marbles and the radius of the bore of a glass capillary tube. Density is determined by measuring a marble’s mass and volume, the latter by measuring a marble’s diameter and assuming a spherical shape. Results are compared with those expected for a normal distribution. The radius of a glass capillary tube is determined using Poiseuille’s equation by measuring the volume flow rate of water as a function of the hydrostatic head. In both experiments the experimentally obtained standard deviation is compared with that estimated by a propagation of uncertainty. Gordus, A. A. “Statistical Evaluation of Class Data for Two Buret Readings,” J. Chem. Educ. 1987, 64, 376–377. The volumes of water in two burets are read, and the difference between the volumes are calculated. Students analyze the data by drawing histograms for each of the three volumes, comparing results with those predicted for a normal distribution. Harvey, D. T. “Statistical Evaluation of Acid/Base Indicators,” J. Chem. Educ. 1991, 68, 329–331. In this experiment students standardize a solution of HCl by titration using several different indicators to signal the titration’s end point. A statistical analysis of the data using t- tests and F-tests allows students to compare results obtained using the same indicator, with results obtained using different indicators. The results of this experiment can be used later when discussing the selection of appropriate indicators. O’Reilley, J. E. “The Length of a Pestle,” J. Chem. Educ. 1986, 63, 894–896. In this experiment students measure the length of a pestle using a wooden meter stick, a stainless-steel ruler, and a vernier caliper. The data collected in this experiment provide an opportunity to discuss significant figures and sources of error. Statistical analysis includes the Q-test, t-test, and F-test. Paselk, R. A. “An Experiment for Introducing Statistics to Students of Analytical and Clinical Chemistry,” J. Chem. Educ. 1985, 62, 536. Students use a commercial diluter to prepare five sets of dilutions of a stock dye solution (each set contains ten replicates) using two different diluters. Results are compared using t-tests and F-tests. Richardson, T. H. “Reproducible Bad Data for Instruction in Statistical Methods,” J. Chem. Educ. 1991, 68, 310–311. This experiment uses the change in the mass of a U.S. penny to create data sets with outliers. Students are given a sample of ten pennies, nine of which are from one population. The Q-test is used to verify that the outlier can be rejected. Class data from each of the two populations of pennies are pooled and compared with results predicted for a normal distribution. Sheeran, D. “Copper Content in Synthetic Copper Carbonate: A Statistical Comparison of Experimental and Expected Results,” J. Chem. Educ. 1998, 75, 453–456. In this experiment students synthesize basic copper(II) carbonate and determine the %w/w Cu by reducing the copper to Cu. A statistical analysis of the results shows that the synthesis does not produce CuCO 3 , the compound that many predict to be the product (although it does not exist). Results are shown to be consistent with a hemihydrate of malachite, Cu 2 (OH) 2 (CO 3 ) ? 1/2H 2 O, or azurite, Cu 3 (OH) 2 (CO 3 ) 2 . The following experiments may be used to introduce the statistical analysis of data in the analytical chemistry laboratory. Each experiment is annotated with a brief description of the data collected and the type of statistical analysis used in evaluating the data. Experiments —Continued 4 J Suggested EXPERIMENTS 1400-CH04 9/8/99 3:55 PM Page 97 98 Modern Analytical Chemistry Spencer, R. D. “The Dependence of Strength in Plastics upon Polymer Chain Length and Chain Orientation,” J. Chem. Educ. 1984, 61, 555–563. The stretching properties of polymers are investigated by examining the effect of polymer orientation, polymer chain length, stretching rate, and temperature. Homogeneity of polymer films and consistency between lots of polymer films also are investigated. Statistical analysis of data includes Q- tests and t-tests. Thomasson, K.; Lofthus-Merschman, S.; Humbert, M.; et al. “Applying Statistics in the Undergraduate Chemistry Laboratory: Experiments with Food Dyes,” J. Chem. Educ. 1998, 75, 231–233. The absorbance of solutions of food dyes is used to explore the treatment of outliers and the application of the t-test for comparing means. Vitha, M. F.; Carr, P. W. “A Laboratory Exercise in Statistical Analysis of Data,” J. Chem. Educ. 1997, 74, 998–1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. Experiments Continued from page 97 1. The following masses were recorded for 12 different U.S. quarters (all given in grams): 5.683 5.549 5.548 5.552 5.620 5.536 5.539 5.684 5.551 5.552 5.554 5.632 Report the mean, median, range, standard deviation, and variance for these data. 2. Shown in the following rows are results for the determination of acetaminophen (in milligrams) in ten separate tablets of Excedrin Extra Strength Pain Reliever. 18 224.3 240.4 246.3 239.4 253.1 261.7 229.4 255.5 235.5 249.7 (a) Report the mean, median, range, standard deviation, and variance for these data. (b) Assuming that – X and s 2 are good approximations for m and s 2 , and that the population is normally distributed, what percentage of tablets are expected to contain more than the standard amount of 250 mg acetaminophen per tablet? 3. Salem and Galan have developed a new method for determining the amount of morphine hydrochloride in tablets. 19 Results, in milligrams, for several tablets containing different nominal dosages follow 100-mg 60-mg 30-mg 10-mg tablets tablets tablets tablets 99.17 54.21 28.51 19.06 94.31 55.62 26.25 8.83 95.92 57.40 25.92 9.08 94.55 57.51 28.62 93.83 52.59 24.93 (a) For each dosage, calculate the mean and standard deviation for the milligrams of morphine hydrochloride per tablet. (b) Assuming that – X and s 2 are good approximations for m and s 2 , and that the population is normally distributed, what percentage of tablets at each dosage level are expected to contain more than the nominal amount of morphine hydrochloride per tablet? 4. Daskalakis and co-workers recently evaluated several procedures for digesting the tissues of oysters and mussels prior to analyzing the samples for silver. 20 One of the methods used to evaluate the procedure is a spike recovery in which a known amount of silver is added to the tissue sample and the percent of the added silver found on analysis is reported. Ideally, spike recoveries should fall within the range 100 ± 15%. The results for one method are 106% 108% 92% 99% 104% 101% 93% 93% Assuming that the spike recoveries are normally distributed, what is the probability that any single spike recovery will be within the accepted range? 5. The formula weight (FW) of a gas can be determined using the following form of the ideal gas law where g is the mass in grams, R is the gas constant, T is the temperature in kelvins, P is the pressure in atmospheres, and V is the volume in liters. In a typical analysis the following data are obtained (with estimated uncertainties in parentheses) FW gRT PV = 4 K PROBLEMS 1400-CH04 9/8/99 3:55 PM Page 98 g = 0.118 (±0.002) R = 0.082056 (±0.000001) T = 298.2 (±0.1) P = 0.724 (±0.005) V = 0.250 (±0.005) (a) What is the compound’s formula weight and its estimated uncertainty? (b) To which variable(s) should you direct your attention if you wish to improve the uncertainty in the compound’s molecular weight? 6. A standard solution of Mn 2+ was prepared by dissolving 0.250 g of Mn in 10 mL of concentrated HNO 3 (measured with a graduated cylinder). The resulting solution was quantitatively transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10-mL aliquot of the solution was pipeted into a 500-mL volumetric flask and diluted to volume. (a) Express the concentration of Mn in parts per million, and estimate uncertainty by a propagation of uncertainty calculation. (b) Would the uncertainty in the solution’s concentration be improved by using a pipet to measure the HNO 3 , instead of a graduated cylinder? 7. Hydroscopic materials often are measured by the technique of weighing by difference. In this technique the material is placed in a sealed container and weighed. A portion of the material is removed, and the container and the remaining material are reweighed. The difference between the two masses gives the amount of material that was sampled. A solution of a hydroscopic material with a gram formula weight of 121.34 (±0.01) was prepared in the following manner. A sample of the compound and its container has a mass of 23.5811 g. A portion of the compound was transferred to a 100-mL volumetric flask and diluted to volume. The mass of the compound and container after the transfer is 22.1559 g. Calculate the molarity of the solution, and estimate its uncertainty by a propagation of uncertainty calculation. 8. Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/ — n. 9. What is the smallest mass that can be measured on an analytical balance with a tolerance of ±0.1 mg, such that the relative error is less than 0.1%? 10. Which of the following is the best way to dispense 100.0 mL of a reagent: (a) use a 50-mL pipet twice; (b) use a 25-mL pipet four times; or (c) use a 10-mL pipet ten times? 11. A solution can be diluted by a factor of 200 using readily available pipets (1-mL to 100-mL) and volumetric flasks (10-mL to 1000-mL) in either one, two, or three steps. Limiting yourself to glassware listed in Table 4.2, determine the proper combination of glassware to accomplish each dilution, and rank them in order of their most probable uncertainties. 12. Explain why changing all values in a data set by a constant amount will change – X but will have no effect on s. 13. Obtain a sample of a metal from your instructor, and determine its density by one or both of the following methods: Method A: Obtain the sample’s mass with a balance. Calculate the sample’s volume using appropriate linear dimensions. Method B: Obtain the sample’s mass with a balance. Calculate the sample’s volume by measuring the amount of water that it displaces. This can be done by adding water to a graduated cylinder, reading the volume, adding the object, and reading the new volume. The difference in volumes is equal to the object’s volume. Determine the density at least five times. (a) Report the mean, the standard deviation, and the 95% confidence interval for your results. (b) Find the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density. (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results? If not, suggest some possible reasons for this disagreement. 14. How many carbon atoms must a molecule have if the mean number of 13 C atoms per molecule is 1.00? What percent of such molecules will have no atoms of 13 C? 15. In Example 4.10 we determined the probability that a molecule of cholesterol, C 27 H 44 O, had no atoms of 13 C. (a) Calculate the probability that a molecule of cholesterol, has one atom of 13 C. (b) What is the probability that a molecule of cholesterol will have two or more atoms of 13 C? 16. Berglund and Wichart investigated the quantitative determination of Cr in high-alloy steels by a potentiometric titration of Cr 6+ . 21 Before titrating the steel was dissolved in acid and the chromium oxidized to Cr 6+ by peroxydisulfate. Following are their results (%w/w Cr) for the analysis of a single reference steel. 16.968 16.922 16.840 16.883 16.887 16.977 16.857 16.728 Calculate the mean, the standard deviation, and the 95% confidence interval about the mean. What does this confidence interval mean? 17. Ketkar and co-workers developed a new analytical method for measuring trace levels of atmospheric gases. 22 The analysis of a sample containing 40.0 parts per thousand (ppt) 2-chloroethylsulfide yielded the following results 43.3 34.8 31.9 37.8 34.4 31.9 42.1 33.6 35.3 (a)Determine whether there is a significant difference between the experimental mean and the expected value at a = 0.05. (b) As part of this study a reagent blank was analyzed 12 times, giving a mean of 0.16 ppt and a standard deviation of 1.20 ppt. What are the IUPAC detection limit, the limit of identification, and limit of quantitation for this method assuming a = 0.05? Chapter 4 Evaluating Analytical Data 99 1400-CH04 9/8/99 3:55 PM Page 99 18. To test a spectrophotometer for its accuracy, a solution of 60.06 ppm K 2 Cr 2 O 7 in 5.0 mM H 2 SO 4 is prepared and analyzed. This solution has a known absorbance of 0.640 at 350.0 nm in a 1.0-cm cell when using 5.0 mM H 2 SO 4 as a reagent blank. Several aliquots of the solution are analyzed with the following results 0.639 0.638 0.640 0.639 0.640 0.639 0.638 Determine whether there is a significant difference between the experimental mean and the expected value at a = 0.01. 19. Monna and co-workers studied the use of radioactive isotopes as a means of dating sediments collected from the bottom of lakes and estuaries. 23 To verify this method they analyzed a 208 Po standard known to have an activity of 77.5 decays/min, obtaining the following results 77.09 75.37 72.42 76.84 77.84 76.69 78.03 74.96 77.54 76.09 81.12 75.75 Determine whether there is a significant difference between the mean and the expected value at a = 0.05. 20. A 2.6540-g sample of an iron ore known to contain 53.51% w/w Fe is dissolved in a small portion of concentrated HCl and diluted to volume in a 250-mL volumetric flask. A spectrophotometric method is used to determine the concentration of Fe in this solution, yielding results of 5840, 5770, 5650, and 5660 ppm. Determine whether there is a significant difference between the experimental mean and the expected value at a = 0.05. 21. Horvat and colleagues investigated the application of atomic absorption spectroscopy to the analysis of Hg in coal fly ash. 24 Of particular interest was the development of an appropriate procedure for digesting the samples in order to release the Hg for analysis. As part of their study they tested several reagents for digesting samples. Results obtained with HNO 3 and with a 1 + 3 mixture of HNO 3 and HCl are shown here. All concentrations are given as nanograms of Hg per gram of sample. HNO 3 : 161 165 160 167 166 1 + 3 HNO 3 –HCl: 159 145 140 147 143 156 Determine whether there is a significant difference between these methods at a = 0.05. 22. Lord Rayleigh, John William Strutt (1842–1919) was one of the most well-known scientists of the late nineteenth and early twentieth centuries, publishing over 440 papers and receiving the Nobel Prize in chemistry in 1904 for the discovery of argon. An important turning point in the discovery of Ar was Rayleigh’s experimental measurements of the density of N 2 . Rayleigh approached this experiment in two ways: first by taking atmospheric air and removing any O 2 and H 2 that was present; and second, by chemically producing N 2 by decomposing nitrogen-containing compounds (NO, N 2 O, and NH 4 NO 3 ) and again removing any O 2 and H 2 . His results for the density of N 2 , published in Proc. Roy. Soc. 1894, LV, 340 (publication 210), follow (all values are for grams of gas at equivalent volume, pressure, and temperature). Atmospheric 2.31017 2.30986 2.31010 2.31001 Origin: 2.31024 2.31010 2.31028 Chemical 2.30143 2.29890 2.29816 2.30182 Origin: 2.29869 2.29940 2.29849 2.29889 Explain why these data led Rayleigh to look for and discover Ar. 23. Gács and Ferraroli reported a new method for monitoring the concentration of SO 2 in air. 25 They compared their method with the standard method by sampling and analyzing urban air from a single location. Air samples were collected by drawing air through a collection solution for 6 min. Following is a summary of their results with SO 2 concentrations reported in microliters per cubic meter. Standard method: 21.62 22.20 24.27 23.54 24.25 23.09 21.02 New method: 21.54 20.51 22.31 21.30 24.62 25.72 21.54 Using an appropriate statistical test, determine whether there is any significant difference between the standard and new methods at a = 0.05. 24. The accuracy of a spectrophotometer can be checked by measuring absorbances for a series of standard dichromate solutions that can be obtained in sealed cuvettes from the National Institute of Standards and Technology. Absorbances are measured at 257 nm and compared with the accepted values. The results obtained when testing a newly purchased spectrophotometer are shown here. Determine if the tested spectrophotometer is accurate at a = 0.05. Standard: 1 2 3 4 5 Measured absorbance: 0.2872 0.5773 0.8674 1.1623 1.4559 Accepted absorbance: 0.2871 0.5760 0.8677 1.1608 1.4565 25. Maskarinec and associates investigated the stability of volatile organics in environmental water samples. 26 Of particular interest was establishing proper conditions for maintaining the sample’s integrity between its collection and analysis. Two preservatives were investigated (ascorbic acid and sodium bisulfate), and maximum holding times were determined for a number of volatile organics and water matrices. Results (in days) for surface waters follow. Ascorbic Sodium acid bisulfate methylene chloride 77 62 carbon disulfide 23 54 trichloroethane 52 51 benzene 62 42 1,1,2-trichloroethane 57 53 1,1,2,2-tetrachloroethane 33 85 tetrachloroethene 41 63 toluene 32 94 chlorobenzene 36 86 100 Modern Analytical Chemistry 1400-CH04 9/8/99 3:55 PM Page 100 Determine whether there is a significant difference in the effectiveness of the two preservatives at a = 0.10. 26. Using X-ray diffraction, Karstang and Kvalhein reported a new method for determining the weight percent of kalonite in complex clay minerals. 27 To test the method, nine samples containing known amounts of kalonite were prepared and analyzed. The results (as %w/w kalonite) are shown. Actual: 5.0 10.0 20.0 40.0 50.0 60.0 80.0 90.0 95.0 Found: 6.8 11.7 19.8 40.5 53.6 61.7 78.9 91.7 94.7 Evaluate the accuracy of the method at a = 0.05. 27. Mizutani and colleagues reported the development of a new method for the analysis of l-malate. 28 As part of their study they analyzed a series of beverages using both their method and a standard spectrophotometric procedure based on a clinical kit purchased from Boerhinger Scientific. A summary follows of their results (in parts per million). Sample Electrode Spectrophotometric Apple juice 1 34.0 33.4 Apple juice 2 22.6 28.4 Apple juice 3 29.7 29.5 Apple juice 4 24.9 24.8 Grape juice 1 17.8 18.3 Grape juice 2 14.8 15.4 Mixed fruit juice 1 8.6 8.5 Mixed fruit juice 2 31.4 31.9 White wine 1 10.8 11.5 White wine 2 17.3 17.6 White wine 3 15.7 15.4 White wine 4 18.4 18.3 Determine whether there is a significant difference between the methods at a = 0.05. 28. Alexiev and associates describe an improved photometric method for the determination of Fe 3+ based on its catalytic effect on the oxidation of sulphanilic acid by KIO 4 . 29 As part of their study the concentration of Fe 3+ in human serum samples was determined by the proposed method and the standard method. Following are the results, with concentrations in micromoles/L. Sample Proposed Method Standard Method 1 8.25 8.06 2 9.75 8.84 3 9.75 8.36 4 9.75 8.73 5 10.75 13.13 6 11.25 13.65 7 13.88 13.85 8 14.25 13.43 Determine whether there is a significant difference between the two methods at a = 0.05. 29. The following data were collected during a study of the concentration of Zn in samples drawn from several locations in Lake Erie (all concentrations in parts per million). [Zn 2+ ] at the [Zn 2+ ] at the air–water sediment–water Location interface interface 1 0.430 0.415 2 0.266 0.238 3 0.567 0.390 4 0.531 0.410 5 0.707 0.605 6 0.716 0.609 Determine whether there is a significant difference between the concentration of Zn 2+ at the air–water interface and the sediment–water interface at a = 0.05. 30. Ten laboratories were asked to determine the concentration of an analyte A in three standard test samples. Following are the results, in parts per million. 30 Laboratory Sample 1 Sample 2 Sample 3 1 22.6 13.6 16.0 2 23.0 14.2 15.9 3 21.5 13.9 16.3 4 21.9 13.9 16.9 5 21.3 13.5 16.7 6 22.1 13.5 17.4 7 23.1 13.9 17.5 8 21.7 13.5 16.8 9 22.2 12.9 17.2 10 21.7 13.8 16.7 Determine if there are any potential outliers in Sample 1, Sample 2, or Sample 3 at a significance level of a = 0.05. 31. When copper metal and powdered sulfur are placed in a crucible and ignited, the product is a sulfide with an empirical formula of Cu x S. The value of x can be determined by weighing the Cu and S before ignition, and finding the mass of Cu x S when the reaction is complete. Following are the Cu/S ratios from 62 such experiments. 1.764 1.838 1.865 1.866 1.872 1.877 1.890 1.891 1.891 1.897 1.899 1.900 1.906 1.908 1.910 1.911 1.916 1.919 1.920 1.922 1.927 1.931 1.935 1.936 1.936 1.937 1.939 1.939 1.940 1.941 1.941 1.942 1.943 1.948 1.953 1.955 1.957 1.957 1.957 1.959 1.962 1.963 1.963 1.963 1.966 1.968 1.969 1.973 1.975 1.976 1.977 1.981 1.981 1.988 1.993 1.993 1.995 1.995 1.995 2.017 2.029 2.042 (a) Calculate the mean and standard deviation for these data. (b) Construct a histogram for this data set. From a visual inspection of your histogram, do the data appear to be normally distributed? (c) In a normally distributed population, 68.26% of all members lie within the range m ±1s . What percentage of the data lies within the range – X ±1s? Does this support your answer to the previous Chapter 4 Evaluating Analytical Data 101 1400-CH04 9/8/99 3:55 PM Page 101 question? (d) Assuming that – X and s 2 are good approximations for m and s 2 , what percentage of all experimentally determined Cu/S ratios will be greater than 2? How does this compare with the experimental data? Does this support your conclusion about whether the data are normally distributed? (e) It has been reported that this method for preparing copper sulfide results in a nonstoichiometric compound with a Cu/S ratio of less than 2. Determine if the mean value for these data is significantly less than 2 at a significance level of a = 0.01. 102 Modern Analytical Chemistry A more comprehensive discussion of the analysis of data, covering all topics considered in this chapter as well as additional material, can be found in any textbook on statistics or data analysis; following are several such texts. Anderson, R. L. Practical Statistics for Analytical Chemists. Van Nostrand Reinhold: New York, 1987. Graham, R. C. Data Analysis for the Chemical Sciences. VCH Publishers: New York, 1993. Mark, H.; Workman, J. Statistics in Spectroscopy. Academic Press: Boston, 1991. Mason, R. L.; Gunst, R. F.; Hess, J. L. Statistical Design and Analysis of Experiments. Wiley: New York, 1989. Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed. Ellis Horwood PTR Prentice-Hall: New York, 1993. Sharaf, M. H.; Illman, D. L.; Kowalski, B. R. Chemometrics. Wiley- Interscience: New York, 1986. The difference between precision and accuracy and a discussion of indeterminate and determinate sources of error is covered in the following paper. Treptow, R. S. “Precision and Accuracy in Measurements,” J. Chem. Educ. 1998, 75, 992–995. The detection of outliers, particularly when working with a small number of samples, is discussed in the following papers. Efstathiou, C. “Stochastic Calculation of Critical Q-Test Values for the Detection of Outliers in Measurements,” J. Chem. Educ. 1992, 69, 773–736. Kelly, P. C. “Outlier Detection in Collaborative Studies,” Anal. Chem. 1990, 73, 58–64. Mitschele, J. “Small Sample Statistics,” J. Chem. Educ. 1991, 68, 470–473. The following papers provide additional information on error and uncertainty, including the propagation of uncertainty. Andraos, J. “On the Propagation of Statistical Errors for a Function of Several Variables,” J. Chem. Educ. 1996, 73, 150–154. Donato, H.; Metz, C. “A Direct Method for the Propagation of Error Using a Personal Computer Spreadsheet Program,” J. Chem. Educ. 1988, 65, 867–868. Gordon, R.; Pickering, M.; Bisson, D. “Uncertainty Analysis by the ‘Worst Case’ Method,” J. Chem. Educ. 1984, 61, 780–781. Guare, C. J. “Error, Precision and Uncertainty,” J. Chem. Educ. 1991, 68, 649–652. Guedens, W. J.; Yperman, J.; Mullens, J.; et al. “Statistical Analysis of Errors: A Practical Approach for an Undergraduate Chemistry Lab,” Part 1. The Concept, J. Chem. Educ. 1993, 70, 776–779; Part 2. Some Worked Examples, J. Chem. Educ. 1993, 70, 838–841. Heydorn, K. “Detecting Errors in Micro and Trace Analysis by Using Statistics,” Anal. Chim. Acta 1993, 283, 494–499. Taylor, B. N.; Kuyatt, C. E. “Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,” NIST Technical Note 1297, 1994. A further discussion of detection limits is found in Currie, L. A., ed. Detection in Analytical Chemistry: Importance, Theory and Practice. American Chemical Society: Washington, DC, 1988. 4 L SUGGESTED READINGS 1. Goedhart, M. J.; Verdonk, A. H. J. Chem. Educ. 1991, 68, 1005–1009. 2. Rousseeuw, P. J. J. Chemom. 1991, 5, 1–20. 3. Ellison, S.; Wegscheider, W.; Williams, A. Anal. Chem. 1997, 69, 607A–613A. 4. Shoemaker, D. P.; Garland, C. W.; Nibler, J. W. Experiments in Physical Chemistry, 5th ed. McGraw-Hill: New York, 1989, pp. 55–63. 5. Lam, R. B.; Isenhour, T. L. Anal. Chem. 1980, 52, 1158–1161. 6. Mark, H.; Workman, J. Spectroscopy, 1988, 3(1), 44–48. 7. Winn, R. L. Statistics for Scientists and Engineers, Prentice-Hall: Englewood Cliffs, NJ, 1964; pp. 165–174. 8. Mark, H.; Workman, J. Spectroscopy, 1989, 4(3), 56–58. 9. Marecek, V.; Janchenova, H.; Brezina, M.; et al. Anal. Chim. Acta 1991, 244, 15–19. 10. Rorabacher, D. B. Anal. Chem. 1991, 63, 139–146. 11. Deming, W. E. Statistical Adjustment of Data. Wiley: New York, 1943 (republished by Dover: New York, 1961); p. 171. 12. Richardson, T. H. J. Chem. Educ. 1991, 68, 310–311. 13. Kirchner, C. J. “Estimation of Detection Limits for Environmental Analytical Procedures,” In Currie, L. A., ed. Detection in Analytical Chemistry: Importance, Theory and Practice. American Chemical Society: Washington, DC, 1988. 14. Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A–724A. 4 M REFERENCES 1400-CH04 9/8/99 3:55 PM Page 102 15. “Guidelines for Data Acquisition and Data Quality Control Evaluation in Environmental Chemistry,” Anal. Chem. 1980, 52, 2242–2249. 16. (a) Ferrus, R.; Egea, M. R. Anal. Chim. Acta 1994, 287, 119–145; (b) Glaser, J. A.; Foerst, D. L.; McKee, G. D.; et al. Environ. Sci. Technol. 1981, 15, 1426–1435; (c) Boumans, P. W. J. M. Anal. Chem. 1994, 66, 459A–467A; (d) Kimbrough, D. E.; Wakakuwa, J. Environ. Sci. Technol. 1994, 28, 338–345; (e) Currie, L. A. Anal. Chem. 1968, 40, 586–593. 17. (a) Rogers, L. B. J. Chem. Educ. 1986, 63, 3–6; (b) Mark, H.; Workman, J. Spectroscopy, 1989, 4(1), 52–55. 18. Simonian, M. H.; Dinh, S.; Fray, L. A. Spectroscopy 1993, 8(6), 37–47. 19. Salem, I. I.; Galan, A. C. Anal. Chim. Acta 1993, 283, 334–337. 20. Daskalakis, K. D.; O’Connor, T. P.; Crecelius, E. A. Environ. Sci. Technol. 1997, 31, 2303–2306. 21. Berglund, B.; Wichardt, C. Anal. Chim. Acta 1990, 236, 399–410. 22. Ketkar, S. N.; Dulak, J. G.; Dheandhanou, S.; et al. Anal. Chim. Acta. 1991, 245, 267–270. 23. Monna, F.; Mathieu, D.; Marques Jr., et al. Anal. Chim. Acta 1996, 330, 107–116. 24. Horvat, M.; Lupsina, V.; Pihlar, B. Anal. Chim. Acta. 1991, 243, 71–79. 25. Gács, I.; Ferraroli, R. Anal. Chim. Acta 1992, 269, 177–185. 26. Maskarinec, M. P.; Johnson, L. H.; Holladay, S. K.; et al. Environ. Sci. Technol. 1990, 24, 1665–1670. 27. Karstang, T. V.; Kvalhein, O. M. Anal. Chem. 1991, 63, 767–772. 28. Mizutani, F.; Yabuki, S.; Asai, M. Anal. Chim. Acta. 1991, 245, 145–150. 29. Alexiev, A.; Rubino, S.; Deyanova, M.; et al. Anal. Chem. Acta 1994, 295, 211–219. 30. These data are adapted from Steiner, E. H. “Planning and Analysis of Results of Collaborative Tests” published in Statistical Manual of the Association of Official Analytical Chemists, Association of Official Analytical Chemists: Washington, DC, 1975. Chapter 4 Evaluating Analytical Data 103 1400-CH04 9/8/99 3:55 PM Page 103 104 Chapter 5 Calibrations, Standardizations, and Blank Corrections In Chapter 3 we introduced a relationship between the measured signal, S meas , and the absolute amount of analyte S meas = kn A +S reag 5.1 or the relative amount of analyte in a sample S meas = kC A + S reag 5.2 where n A is the moles of analyte, C A is the analyte s concentration, k is the method s sensitivity, and S reag is the contribution to S meas from constant errors introduced by the reagents used in the analysis. To obtain an accurate value for n A or C A it is necessary to avoid determinate errors affecting S meas , k, and S reag . This is accomplished by a combination of calibrations, standardizations, and reagent blanks. 1400-CH05 9/8/99 3:58 PM Page 104 Chapter 5 Calibrations, Standardizations, and Blank Corrections 105 5 A Calibrating Signals Signals are measured using equipment or instruments that must be properly cali- brated if S meas is to be free of determinate errors. Calibration is accomplished against a standard, adjusting S meas until it agrees with the standard’s known signal. Several common examples of calibration are discussed here. When the signal is a measurement of mass, S meas is determined with an analyti- cal balance. Before a balance can be used, it must be calibrated against a reference weight meeting standards established by either the National Institute for Standards and Technology or the American Society for Testing and Materials. With an elec- tronic balance the sample’s mass is determined by the current required to generate an upward electromagnetic force counteracting the sample’s downward gravita- tional force. The balance’s calibration procedure invokes an internally programmed calibration routine specifying the reference weight to be used. The reference weight is placed on the balance’s weighing pan, and the relationship between the displace- ment of the weighing pan and the counteracting current is automatically adjusted. Calibrating a balance, however, does not eliminate all sources of determinate error. Due to the buoyancy of air, an object’s weight in air is always lighter than its weight in vacuum. If there is a difference between the density of the object being weighed and the density of the weights used to calibrate the balance, then a correc- tion to the object’s weight must be made. 1 An object’s true weight in vacuo, W v , is related to its weight in air, W a , by the equation where D o is the object’s density, D w is the density of the calibration weight, and 0.0012 is the density of air under normal laboratory conditions (all densities are in units of g/cm 3 ). Clearly the greater the difference between D o and D w the more seri- ous the error in the object’s measured weight. The buoyancy correction for a solid is small, and frequently ignored. It may be significant, however, for liquids and gases of low density. This is particularly impor- tant when calibrating glassware. For example, a volumetric pipet is calibrated by carefully filling the pipet with water to its calibration mark, dispensing the water into a tared beaker and determining the mass of water transferred. After correcting for the buoyancy of air, the density of water is used to calculate the volume of water dispensed by the pipet. EXAMPLE 5 .1 A 10-mL volumetric pipet was calibrated following the procedure just outlined, using a balance calibrated with brass weights having a density of 8.40 g/cm 3 . At 25 °C the pipet was found to dispense 9.9736 g of water. What is the actual volume dispensed by the pipet? SOLUTION At 25 °C the density of water is 0.99705 g/cm 3 . The water’s true weight, therefore, is W v g=×+ ? ? ? ? ? ? × ? ? ? ? ? ? ? ? =9 9736 1 1 0 99705 1 840 0 0012 9 9842. . – . .. g WW DD va ow =×+ ? ? ? ? ? ? × ? ? ? ? ? ? ? ? 1 11 0 0012–. 1400-CH05 9/8/99 3:58 PM Page 105 106 Modern Analytical Chemistry and the actual volume of water dispensed by the pipet is If the buoyancy correction is ignored, the pipet’s volume is reported as introducing a negative determinate error of –0.11%. Balances and volumetric glassware are examples of laboratory equipment. Lab- oratory instrumentation also must be calibrated using a standard providing a known response. For example, a spectrophotometer’s accuracy can be evaluated by measuring the absorbance of a carefully prepared solution of 60.06 ppm K 2 Cr 2 O 7 in 0.0050 M H 2 SO 4 , using 0.0050 M H 2 SO 4 as a reagent blank. 2 The spectrophotome- ter is considered calibrated if the resulting absorbance at a wavelength of 350.0 nm is 0.640 ± 0.010 absorbance units. Be sure to read and carefully follow the calibra- tion instructions provided with any instrument you use. 5 B Standardizing Methods The American Chemical Society’s Committee on Environmental Improvement de- fines standardization as the process of determining the relationship between the measured signal and the amount of analyte. 3 A method is considered standardized when the value of k in equation 5.1 or 5.2 is known. In principle, it should be possible to derive the value of k for any method by considering the chemical and physical processes responsible for the signal. Unfortu- nately, such calculations are often of limited utility due either to an insufficiently developed theoretical model of the physical processes or to nonideal chemical be- havior. In such situations the value of k must be determined experimentally by ana- lyzing one or more standard solutions containing known amounts of analyte. In this section we consider several approaches for determining the value of k. For sim- plicity we will assume that S reag has been accounted for by a proper reagent blank, allowing us to replace S meas in equations 5.1 and 5.2 with the signal for the species being measured. 5 B.1 Reagents Used as Standards The accuracy of a standardization depends on the quality of the reagents and glass- ware used to prepare standards. For example, in an acid–base titration, the amount of analyte is related to the absolute amount of titrant used in the analysis by the stoichiometry of the chemical reaction between the analyte and the titrant. The amount of titrant used is the product of the signal (which is the volume of titrant) and the titrant’s concentration. Thus, the accuracy of a titrimetric analysis can be no better than the accuracy to which the titrant’s concentration is known. Primary Reagents Reagents used as standards are divided into primary reagents and secondary reagents. A primary reagent can be used to prepare a standard con- taining an accurately known amount of analyte. For example, an accurately weighed sample of 0.1250 g K 2 Cr 2 O 7 contains exactly 4.249 · 10 –4 mol of K 2 Cr 2 O 7 . If this 9 9736 10 003 10 003 . .. 0.99705 g/cm cm mL 3 3 g == 9 9842 10 014 10 014 . .. 0.99705 g/cm cm mL 3 3 g == primary reagent A reagent of known purity that can be used to make a solution of known concentration. 1400-CH05 9/8/99 3:58 PM Page 106 Figure 5.1 Examples of typical packaging labels from reagent grade chemicals. Label (a) provides the actual lot assay for the reagent as determined by the manufacturer. Note that potassium has been flagged with an asterisk (*) because its assay exceeds the maximum limit established by the American Chemical Society (ACS). Label (b) does not provide assayed values, but indicates that the reagent meets the specifications of the ACS for the listed impurities. An assay for the reagent also is provided. ? David Harvey/Marilyn Culler, photographer. same sample is placed in a 250-mL volumetric flask and diluted to volume, the con- centration of the resulting solution is exactly 1.700 · 10 –3 M. A primary reagent must have a known stoichiometry, a known purity (or assay), and be stable during long-term storage both in solid and solution form. Because of the difficulty in es- tablishing the degree of hydration, even after drying, hydrated materials usually are not considered primary reagents. Reagents not meeting these criteria are called sec- ondary reagents. The purity of a secondary reagent in solid form or the concentra- tion of a standard prepared from a secondary reagent must be determined relative to a primary reagent. Lists of acceptable primary reagents are available. 4 Appendix 2 contains a selected listing of primary standards. Other Reagents Preparing a standard often requires additional substances that are not primary or secondary reagents. When a standard is prepared in solution, for ex- ample, a suitable solvent and solution matrix must be used. Each of these solvents and reagents is a potential source of additional analyte that, if unaccounted for, leads to a determinate error. If available, reagent grade chemicals conforming to standards set by the American Chemical Society should be used. 5 The packaging label included with a reagent grade chemical (Figure 5.1) lists either the maximum allowed limit for specific impurities or provides the actual assayed values for the im- purities as reported by the manufacturer. The purity of a reagent grade chemical can be improved by purification or by conducting a more accurate assay. As dis- cussed later in the chapter, contributions to S meas from impurities in the sample ma- trix can be compensated for by including an appropriate blank determination in the analytical procedure. Chapter 5 Calibrations, Standardizations, and Blank Corrections 107 reagent grade Reagents conforming to standards set by the American Chemical Society. secondary reagent A reagent whose purity must be established relative to a primary reagent. (a) (b) 1400-CH05 9/8/99 3:59 PM Page 107 Preparing Standard Solutions Solutions of primary standards generally are pre- pared in class A volumetric glassware to minimize determinate errors. Even so, the relative error in preparing a primary standard is typically ±0.1%. The relative error can be improved if the glassware is first calibrated as described in Example 5.1. It also is possible to prepare standards gravimetrically by taking a known mass of stan- dard, dissolving it in a solvent, and weighing the resulting solution. Relative errors of ±0.01% can typically be achieved in this fashion. It is often necessary to prepare a series of standard solutions, each with a differ- ent concentration of analyte. Such solutions may be prepared in two ways. If the range of concentrations is limited to only one or two orders of magnitude, the solu- tions are best prepared by transferring a known mass or volume of the pure stan- dard to a volumetric flask and diluting to volume. When working with larger con- centration ranges, particularly those extending over more than three orders of magnitude, standards are best prepared by a serial dilution from a single stock solu- tion. In a serial dilution a volume of a concentrated stock solution, which is the first standard, is diluted to prepare a second standard. A portion of the second standard is then diluted to prepare a third standard, and the process is repeated until all nec- essary standards have been prepared. Serial dilutions must be prepared with extra care because a determinate error in the preparation of any single standard is passed on to all succeeding standards. 5 B.2 Single-Point versus Multiple-Point Standardizations* The simplest way to determine the value of k in equation 5.2 is by a single- point standardization. A single standard containing a known concentration of analyte, C S , is prepared and its signal, S stand , is measured. The value of k is calcu- lated as 5.3 A single-point standardization is the least desirable way to standardize a method. When using a single standard, all experimental errors, both de- terminate and indeterminate, are carried over into the calculated value for k. Any uncertainty in the value of k increases the uncertainty in the ana- lyte’s concentration. In addition, equation 5.3 establishes the standardiza- tion relationship for only a single concentration of analyte. Extending equation 5.3 to samples containing concentrations of analyte different from that in the standard assumes that the value of k is constant, an as- sumption that is often not true. 6 Figure 5.2 shows how assuming a con- stant value of k may lead to a determinate error. Despite these limitations, single-point standardizations are routinely used in many laboratories when the analyte’s range of expected concentrations is limited. Under these con- ditions it is often safe to assume that k is constant (although this assump- tion should be verified experimentally). This is the case, for example, in clinical laboratories where many automated analyzers use only a single standard. The preferred approach to standardizing a method is to prepare a se- ries of standards, each containing the analyte at a different concentration. Standards are chosen such that they bracket the expected range for the k S C = stand S 108 Modern Analytical Chemistry single-point standardization Any standardization using a single standard containing a known amount of analyte. *The following discussion of standardizations assumes that the amount of analyte is expressed as a concentration. It also applies, however, when the absolute amount of analyte is given in grams or moles. Assumed relationship Actual relationship Actual concentration Concentration reported S stand C s C A Signal Figure 5.2 Example showing how an improper use of a single-point standardization can lead to a determinate error in the reported concentration of analyte. 1400-CH05 9/8/99 3:59 PM Page 108 Figure 5.3 Examples of (a) straight-line and (b) curved normal calibration curves. Chapter 5 Calibrations, Standardizations, and Blank Corrections 109 multiple-point standardization Any standardization using two or more standards containing known amounts of analyte. *Linear regression, also known as the method of least squares, is covered in Section 5C. analyte’s concentration. Thus, a multiple-point standardization should use at least three standards, although more are preferable. A plot of S stand versus C S is known as a calibration curve. The exact standardization, or calibration relationship, is deter- mined by an appropriate curve-fitting algorithm.* Several approaches to standard- ization are discussed in the following sections. 5 B. 3 External Standards The most commonly employed standardization method uses one or more external standards containing known concentrations of analyte. These standards are identi- fied as external standards because they are prepared and analyzed separately from the samples. A quantitative determination using a single external standard was described at the beginning of this section, with k given by equation 5.3. Once standardized, the concentration of analyte, C A , is given as 5.4 EXAMPLE 5 .2 A spectrophotometric method for the quantitative determination of Pb 2+ levels in blood yields an S stand of 0.474 for a standard whose concentration of lead is 1.75 ppb. How many parts per billion of Pb 2+ occur in a sample of blood if S samp is 0.361? SOLUTION Equation 5.3 allows us to calculate the value of k for this method using the data for the standard Once k is known, the concentration of Pb 2+ in the sample of blood can be calculated using equation 5.4 A multiple-point external standardization is accomplished by constructing a calibration curve, two examples of which are shown in Figure 5.3. Since this is the most frequently employed method of standardization, the resulting relation- ship often is called a normal calibration curve. When the calibration curve is a linear (Figure 5.3a), the slope of the line gives the value of k. This is the most de- sirable situation since the method’s sensitivity remains constant throughout the standard’s concentration range. When the calibration curve is nonlinear, the method’s sensitivity is a function of the analyte’s concentration. In Figure 5.3b, for example, the value of k is greatest when the analyte’s concentration is small and decreases continuously as the amount of analyte is increased. The value of k at any point along the calibration curve is given by the slope at that point. In C S k A samp == = 0 361 0 2709 133 . . . ppb ppb –1 k S C == = stand –1 ppb S ppb 0 474 175 0 2709 . . . C S k A samp = normal calibration curve A calibration curve prepared using several external standards. external standard A standard solution containing a known amount of analyte, prepared separately from samples containing the analyte. (a) C A S stand (b) C A S stand 1400-CH05 9/8/99 3:59 PM Page 109 110 Modern Analytical Chemistry either case, the calibration curve provides a means for relating S samp to the ana- lyte’s concentration. EXAMPLE 5 . 3 A second spectrophotometric method for the quantitative determination of Pb 2+ levels in blood gives a linear normal calibration curve for which S stand = (0.296 ppb –1 ) · C S + 0.003 What is the Pb 2+ level (in ppb) in a sample of blood if S samp is 0.397? SOLUTION To determine the concentration of Pb 2+ in the sample of blood, we replace S stand in the calibration equation with S samp and solve for C A It is worth noting that the calibration equation in this problem includes an extra term that is not in equation 5.3. Ideally, we expect the calibration curve to give a signal of zero when C S is zero. This is the purpose of using a reagent blank to correct the measured signal. The extra term of +0.003 in our calibration equation results from uncertainty in measuring the signal for the reagent blank and the standards. An external standardization allows a related series of samples to be ana- lyzed using a single calibration curve. This is an important advantage in labo- ratories where many samples are to be analyzed or when the need for a rapid throughput of samples is critical. Not surprisingly, many of the most com- monly encountered quantitative analytical methods are based on an external standardization. There is a serious limitation, however, to an external standardization. The relationship between S stand and C S in equation 5.3 is determined when the analyte is present in the external standard’s matrix. In using an exter- nal standardization, we assume that any difference between the matrix of the standards and the sample’s matrix has no effect on the value of k. A proportional determinate error is introduced when differences between the two matrices cannot be ignored. This is shown in Figure 5.4, where the re- lationship between the signal and the amount of analyte is shown for both the sample’s matrix and the standard’s matrix. In this example, using a normal calibration curve results in a negative determinate error. When matrix problems are expected, an effort is made to match the matrix of the standards to that of the sample. This is known as matrix matching. When the sample’s matrix is unknown, the matrix effect must be shown to be negligi- ble, or an alternative method of standardization must be used. Both approaches are discussed in the following sections. 5 B. 4 Standard Additions The complication of matching the matrix of the standards to that of the sample can be avoided by conducting the standardization in the sample. This is known as the method of standard additions. The simplest version of a standard addi- C S A samp ppb === –. . .–. . . – 0 003 0 296 0 397 0 003 0 296 133 1 ppb ppb –1 Reported Calibration curve obtained in standard’s matrix Calibration curve obtained in sample’s matrix Actual Amount of analyte Signal Figure 5.4 Effect of the sample’s matrix on a normal calibration curve. matrix matching Adjusting the matrix of an external standard so that it is the same as the matrix of the samples to be analyzed. method of standard additions A standardization in which aliquots of a standard solution are added to the sample. Colorplate 1 shows an example of a set of external standards and their corresponding normal calibration curve. 1400-CH05 9/8/99 3:59 PM Page 110 Figure 5.5 Illustration showing the method of standard additions in which separate aliquots of sample are diluted to the same final volume. One aliquot of sample is spiked with a known volume of a standard solution of analyte before diluting to the final volume. tion is shown in Figure 5.5. A volume, V o , of sample is diluted to a final volume, V f , and the signal, S samp is measured. A second identical aliquot of sample is spiked with a volume, V s , of a standard solution for which the analyte’s concen- tration, C S , is known. The spiked sample is diluted to the same final volume and its signal, S spike , is recorded. The following two equations relate S samp and S spike to the concentration of analyte, C A , in the original sample 5.5 5.6 where the ratios V o /V f and V s /V f account for the dilution. As long as V s is small rela- tive to V o , the effect of adding the standard to the sample’s matrix is insignificant, and the matrices of the sample and the spiked sample may be considered identical. Under these conditions the value of k is the same in equations 5.5 and 5.6. Solving both equations for k and equating gives 5.7 Equation 5.7 can be solved for the concentration of analyte in the original sample. S CVV S CVV CVV samp Ao f spike Ao f Ss f (/) (/) (/) = + SkC V V C V V spike A o f S s f =+ ? ? ? ? ? ? SkC V V samp A o f = Chapter 5 Calibrations, Standardizations, and Blank Corrections 111 Add V o of C A Dilute to V f Total concentration of analyte C A V o V f Add V o of C A Dilute to V f Total concentration of analyte Add V S of C S C A V o V f + C S V S V f aliquot A portion of a solution. 1400-CH05 9/8/99 3:59 PM Page 111 Figure 5.6 Illustration showing an alternative form of the method of standard additions. In this case a sample containing the analyte is spiked with a known volume of a standard solution of analyte without further diluting either the sample or the spiked sample. 112 Modern Analytical Chemistry EXAMPLE 5 . 4 A third spectrophotometric method for the quantitative determination of the concentration of Pb 2+ in blood yields an S samp of 0.193 for a 1.00-mL sample of blood that has been diluted to 5.00 mL. A second 1.00-mL sample is spiked with 1.00 m L of a 1560-ppb Pb 2+ standard and diluted to 5.00 mL, yielding an S spike of 0.419. Determine the concentration of Pb 2+ in the original sample of blood. SOLUTION The concentration of Pb 2+ in the original sample of blood can be determined by making appropriate substitutions into equation 5.7 and solving for C A . Note that all volumes must be in the same units, thus V s is converted from 1.00 m L to 1.00 · 10 –3 mL. Thus, the concentration of Pb 2+ in the original sample of blood is 1.33 ppb. It also is possible to make a standard addition directly to the sample after mea- suring S samp (Figure 5.6). In this case, the final volume after the standard addition is V o + V s and equations 5.5–5.7 become S samp = kC A 5.8 SkC V VV C V VV spike A o os S s os = + + + ? ? ? ? ? ? 0 193 100 500 0 419 100 500 1560 100 10 0 193 0 200 0 419 0 200 0 312 0 0386 0 0602 0 0838 0 0452 0 0602 3 . . . . . . . . . . .. .. . .. – C C CC CC C C A A AA AA A mL mL mL mL ppb ppb ppb ppb mL 5.00 mL ? ? ? ? ? ? = ? ? ? ? ? ? + × ? ? ? ? ? ? = + += = A ppb= 133. Total concentration of analyte C A Total concentration of analyte Add V S of C S V o V o C A V o V o + V S + C S V S V o + V S 1400-CH05 9/8/99 3:59 PM Page 112 Colorplate 2 shows an example of a set of standard additions and their corresponding standard additions calibration curve. Chapter 5 Calibrations, Standardizations, and Blank Corrections 113 5.9 EXAMPLE 5 . 5 A fourth spectrophotometric method for the quantitative determination of the concentration of Pb 2+ in blood yields an S samp of 0.712 for a 5.00-mL sample of blood. After spiking the blood sample with 5.00 m L of a 1560-ppb Pb 2+ standard, an S spike of 1.546 is measured. Determine the concentration of Pb 2+ in the original sample of blood. SOLUTION The concentration of Pb 2+ in the original sample of blood can be determined by making appropriate substitutions into equation 5.9 and solving for C A . Thus, the concentration of Pb 2+ in the original sample of blood is 1.33 ppb. The single-point standard additions outlined in Examples 5.4 and 5.5 are easily adapted to a multiple-point standard addition by preparing a series of spiked sam- ples containing increasing amounts of the standard. A calibration curve is prepared by plotting S spike versus an appropriate measure of the amount of added standard. Figure 5.7 shows two examples of a standard addition calibration curve based on equation 5.6. In Figure 5.7(a) S spike is plotted versus the volume of the standard so- lution spikes, V s . When k is constant, the calibration curve is linear, and it is easy to show that the x-intercept’s absolute value is C A V o /C S . EXAMPLE 5 .6 Starting with equation 5.6, show that the equations for the slope, y-intercept, and x-intercept in Figure 5.7(a) are correct. SOLUTION We begin by rewriting equation 5.6 as which is in the form of the linear equation Y = y-intercept + slope · X S kC V V kC V V spike Ao f S f s =+× S C S CVVV CVVV samp A spike Aoos Ssos = ++ +[/( )] [/( )] 0 712 1 546 500 500 500 1560 500 10 10 0 712 1 546 0 9990 1 558 0 7113 1 109 1 546 133 3 3 .. . (. . . ( .. .. .. . . – – C C CC C A A AA A mL mL ppb ppb ppb ppb = +× + × × = + += = ? ? ? ? ? ? ? ? ? ? ? ? 10 mL) mL 5.00 mL + 5.00 mL) –3 1400-CH05 9/8/99 3:59 PM Page 113 Figure 5.7 Examples of calibration curves for the method of standard additions. In (a) the signal is plotted versus the volume of the added standard, and in (b) the signal is plotted versus the concentration of the added standard after dilution. 114 Modern Analytical Chemistry V S S spiked x-intercept = (a) –C A V o C S y-intercept = kC A V o V f Slope = kC S V f S spiked x-intercept = (b) –C A V o V f y-intercept = kC A V o V f Slope = k C S V S V f ( ) where Y is S spike and X is V s . The slope of the line, therefore, is kC S /V f , and the y-intercept is kC A V o /V f . The x-intercept is the value of X when Y is 0, or Thus, the absolute value of the x-intercept is C A V o /C S . Since both V o and C S are known, the x-intercept can be used to calculate the ana- lyte’s concentration. EXAMPLE 5 . 7 A fifth spectrophotometric method for the quantitative determination of the concentration of Pb 2+ in blood uses a multiple-point standard addition based on equation 5.6. The original blood sample has a volume of 1.00 mL, and the standard used for spiking the sample has a concentration of 1560 ppb Pb 2+ . All samples were diluted to 5.00 mL before measuring the signal. A calibration curve of S spike versus V s is described by 0 =+× == kC V V kC V x x kC V V kC V CV C Ao f S f Ao f Sf Ao S ( -intercept) -intercept – (/) (/) – 1400-CH05 9/8/99 3:59 PM Page 114 S spike = 0.266 + 312 mL –1 · V s Determine the concentration of Pb 2+ in the original sample of blood. SOLUTION To find the x-intercept we let S spike equal 0 0 = 0.266 + 312 mL –1 · (x-intercept) and solve for the x-intercept’s absolute value, giving a value of 8.526 · 10 –4 mL. Thus and the concentration of Pb 2+ in the blood sample, C A , is 1.33 ppb. Figure 5.7(b) shows the relevant relationships when S spike is plotted versus the con- centrations of the spiked standards after dilution. Standard addition calibration curves based on equation 5.8 are also possible. Since a standard additions calibration curve is constructed in the sample, it cannot be extended to the analysis of another sample. Each sample, therefore, re- quires its own standard additions calibration curve. This is a serious drawback to the routine application of the method of standard additions, particularly in labora- tories that must handle many samples or that require a quick turnaround time. For example, suppose you need to analyze ten samples using a three-point calibration curve. For a normal calibration curve using external standards, only 13 solutions need to be analyzed (3 standards and 10 samples). Using the method of standard additions, however, requires the analysis of 30 solutions, since each of the 10 sam- ples must be analyzed three times (once before spiking and two times after adding successive spikes). The method of standard additions can be used to check the validity of an exter- nal standardization when matrix matching is not feasible. To do this, a normal cali- bration curve of S stand versus C S is constructed, and the value of k is determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k. If the two val- ues of k are identical, then any difference between the sample’s matrix and that of the external standards can be ignored. When the values of k are different, a propor- tional determinate error is introduced if the normal calibration curve is used. 5 B. 5 Internal Standards The successful application of an external standardization or the method of standard additions, depends on the analyst’s ability to handle samples and standards repro- ducibly. When a procedure cannot be controlled to the extent that all samples and standards are treated equally, the accuracy and precision of the standardization may suffer. For example, if an analyte is present in a volatile solvent, its concentration will increase if some solvent is lost to evaporation. Suppose that you have a sample and a standard with identical concentrations of analyte and identical signals. If both experience the same loss of solvent their concentrations of analyte and signals will continue to be identical. In effect, we can ignore changes in concentration due to evaporation provided that the samples and standards experience an equivalent loss of solvent. If an identical standard and sample experience different losses of solvent, x CV C C -intercept mL = =× = × 8 526 10 100 1560 4 . (. ) – Ao S A mL ppb Chapter 5 Calibrations, Standardizations, and Blank Corrections 115 1400-CH05 9/8/99 3:59 PM Page 115 116 Modern Analytical Chemistry internal standard A standard, whose identity is different from the analyte’s, that is added to all samples and standards containing the analyte. however, their concentrations and signals will no longer be equal. In this case, an external standardization or standard addition results in a determinate error. A standardization is still possible if the analyte’s signal is referenced to a signal generated by another species that has been added at a fixed concentration to all samples and standards. The added species, which must be different from the ana- lyte, is called an internal standard. Since the analyte and internal standard in any sample or standard receive the same treatment, the ratio of their signals will be unaffected by any lack of repro- ducibility in the procedure. If a solution contains an analyte of concentration C A , and an internal standard of concentration, C IS , then the signals due to the analyte, S A , and the internal standard, S IS , are S A = k A C A S IS = k IS C IS where k A and k IS are the sensitivities for the analyte and internal standard, respec- tively. Taking the ratio of the two signals gives 5.10 Because equation 5.10 is defined in terms of a ratio, K, of the analyte’s sensitivity and the internal standard’s sensitivity, it is not necessary to independently deter- mine values for either k A or k IS . In a single-point internal standardization, a single standard is prepared, and K is determined by solving equation 5.10 5.11 Once the method is standardized, the analyte’s concentration is given by EXAMPLE 5 .8 A sixth spectrophotometric method for the quantitative determination of Pb 2+ levels in blood uses Cu 2+ as an internal standard. A standard containing 1.75 ppb Pb 2+ and 2.25 ppb Cu 2+ yields a ratio of S A /S IS of 2.37. A sample of blood is spiked with the same concentration of Cu 2+ , giving a signal ratio of 1.80. Determine the concentration of Pb 2+ in the sample of blood. SOLUTION Equation 5.11 allows us to calculate the value of K using the data for the standard The concentration of Pb 2+ , therefore, is C C K S S A IS A IS samp ppb= ? ? ? ? ? ? ? ? ? ? ? ? =×= 225 305 180 133 . . . . Pb 2+ K C C S S = ? ? ? ? ? ? ? ? ? ? ? ? =×= IS A A IS stand 225 175 237 305 . . .. C C K S S A IS A IS samp = ? ? ? ? ? ? ? ? ? ? ? ? K C C S S = ? ? ? ? ? ? ? ? ? ? ? ? IS A A IS stand S S k k C C K C C A IS A IS A IS A IS =× =× 1400-CH05 9/8/99 3:59 PM Page 116 Chapter 5 Calibrations, Standardizations, and Blank Corrections 117 A single-point internal standardization has the same limitations as a single- point normal calibration. To construct an internal standard calibration curve, it is necessary to prepare several standards containing different concentrations of ana- lyte. These standards are usually prepared such that the internal standard’s concen- tration is constant. Under these conditions a calibration curve of (S A /S IS ) stand versus C A is linear with a slope of K/C IS . EXAMPLE 5 . 9 A seventh spectrophotometric method for the quantitative determination of Pb 2+ levels in blood gives a linear internal standards calibration curve for which What is the concentration (in ppb) of Pb 2+ in a sample of blood if (S A /S IS ) samp is 2.80? SOLUTION To determine the concentration of Pb 2+ in the sample of blood, we replace (S A /S IS ) stand in the calibration equation with (S A /S IS ) samp and solve for C A The concentration of Pb 2+ in the sample of blood is 1.33 ppb. When the internal standard’s concentration cannot be held constant the data must be plotted as (S A /S IS ) stand versus C A /C IS , giving a linear calibration curve with a slope of K. 5 C Linear Regression and Calibration Curves In a single-point external standardization, we first determine the value of k by measuring the signal for a single standard containing a known concentration of analyte. This value of k and the signal for the sample are then used to calculate the concentration of analyte in the sample (see Example 5.2). With only a single determination of k, a quantitative analysis using a single-point external stan- dardization is straightforward. This is also true for a single-point standard addi- tion (see Examples 5.4 and 5.5) and a single-point internal standardization (see Example 5.8). A multiple-point standardization presents a more difficult problem. Consider the data in Table 5.1 for a multiple-point external standardiza- tion. What is the best estimate of the relationship between S meas and C S ? It is tempting to treat this data as five separate single-point standardizations, determining k for each stan- dard and reporting the mean value. Despite its simplicity, this is not an appropriate way to treat a multiple-point standardization. In a single-point standardization, we assume that the reagent blank (the first row in Table 5.1) corrects for all constant sources of determinate error. If this is not the case, then the value of k determined by a single- point standardization will have a determinate error. C SS A A IS samp ppb ppb ppb= + = + = (/) . . .. . . –– 0 006 211 2 80 0 006 211 133 11 S S C A IS A ppb ? ? ? ? ? ? =× stand (. ) – . – 2 11 0 006 1 Table 5 .1 Data for Hypothetical Multiple- Point External Standardization C S S meas 0.000 0.00 0.100 12.36 0.200 24.83 0.300 35.91 0.400 48.79 0.500 60.42 1400-CH05 9/8/99 3:59 PM Page 117 Table 5.2 demonstrates how an uncorrected constant error affects our determination of k. The first three columns show the concentration of analyte, the true measured signal (no constant error) and the true value of k for five standards. As expected, the value of k is the same for each standard. In the fourth column a constant determinate error of +0.50 has been added to the measured signals. The corresponding val- ues of k are shown in the last column. Note that a different value of k is obtained for each standard and that all values are greater than the true value. As we noted in Section 5B.2, this is a significant limitation to any single-point standardization. How do we find the best estimate for the relationship be- tween the measured signal and the concentration of analyte in a multiple-point standardization? Figure 5.8 shows the data in Table 5.1 plotted as a normal calibration curve. Although the data appear to fall along a straight line, the actual calibration curve is not intuitively obvious. The process of mathemati- cally determining the best equation for the calibration curve is called regression. 5 C.1 Linear Regression of Straight-Line Calibration Curves A calibration curve shows us the relationship between the measured signal and the analyte’s concentration in a series of standards. The most useful calibration curve is a straight line since the method’s sensitivity is the same for all concentrations of an- alyte. The equation for a linear calibration curve is y = b 0 + b 1 x 5.12 where y is the signal and x is the amount of analyte. The constants b 0 and b 1 are the true y-intercept and the true slope, respectively. The goal of linear regres- sion is to determine the best estimates for the slope, b 1 , and y-intercept, b 0 . This is accomplished by minimizing the residual error between the experimental val- ues, y i , and those values, ? y i , predicted by equation 5.12 (Figure 5.9). For obvious reasons, a regression analysis is also called a least-squares treatment. Several ap- proaches to the linear regression of equation 5.12 are discussed in the following sections. 118 Modern Analytical Chemistry Table 5 .2 Effect of a Constant Determinate Error on the Value of k Calculated Using a Single-Point Standardization C A S meas kS meas k (true) (true) (with constant error) (apparent) 1.00 1.00 1.00 1.50 1.50 2.00 2.00 1.00 2.50 1.25 3.00 3.00 1.00 3.50 1.17 4.00 4.00 1.00 4.50 1.13 5.00 5.00 1.00 5.50 1.10 mean k(true) = 1.00 mean k (apparent) = 1.23 80 60 40 20 0 S meas C A 0.0 0.20.1 0.3 0.4 0.5 0.6 Figure 5.8 Normal calibration plot of hypothetical data from Table 5.1. linear regression A mathematical technique for fitting an equation, such as that for a straight line, to experimental data. residual error The difference between an experimental value and the value predicted by a regression equation. 1400-CH05 9/8/99 3:59 PM Page 118 Figure 5.9 Residual error in linear regression, where the filled circle shows the experimental value y i , and the open circle shows the predicted value ? y i . Chapter 5 Calibrations, Standardizations, and Blank Corrections 119 Regression line Residual error = y i – ?y i Total residual error = ∑ (y i – ?y i ) 2 ?y i y i 5 C.2 Unweighted Linear Regression with Errors in y The most commonly used form of linear regression is based on three assump- tions: (1) that any difference between the experimental data and the calculated regression line is due to indeterminate errors affecting the values of y, (2) that these indeterminate errors are normally distributed, and (3) that the indetermi- nate errors in y do not depend on the value of x. Because we assume that indeter- minate errors are the same for all standards, each standard contributes equally in estimating the slope and y-intercept. For this reason the result is considered an unweighted linear regression. The second assumption is generally true because of the central limit theorem outlined in Chapter 4. The validity of the two remaining assumptions is less cer- tain and should be evaluated before accepting the results of a linear regression. In particular, the first assumption is always suspect since there will certainly be some indeterminate errors affecting the values of x. In preparing a calibration curve, however, it is not unusual for the relative standard deviation of the mea- sured signal (y) to be significantly larger than that for the concentration of ana- lyte in the standards (x). In such circumstances, the first assumption is usually reasonable. Finding the Estimated Slope and y-Intercept The derivation of equations for cal- culating the estimated slope and y-intercept can be found in standard statistical texts 7 and is not developed here. The resulting equation for the slope is given as 5.13 and the equation for the y-intercept is 5.14 Although equations 5.13 and 5.14 appear formidable, it is only necessary to evaluate four summation terms. In addition, many calculators, spreadsheets, and other com- puter software packages are capable of performing a linear regression analysis based on this model. To save time and to avoid tedious calculations, learn how to use one of these tools. For illustrative purposes, the necessary calculations are shown in de- tail in the following example. EXAMPLE 5 .10 Using the data from Table 5.1, determine the relationship between S meas and C S by an unweighted linear regression. SOLUTION Equations 5.13 and 5.14 are written in terms of the general variables x and y. As you work through this example, remember that x represents the concentration of analyte in the standards (C S ), and that y corresponds to the signal (S meas ). We begin by setting up a table to help in the calculation of the summation terms S x i , S y i , S x 2 i , and S x i y i which are needed for the calculation of b 0 and b 1 b ybx n ii 0 1 = ∑∑ – b nxy x y nx x ii i i i i 1 2 = ∑∑∑ ∑∑ – –( ) 2 1400-CH05 9/8/99 3:59 PM Page 119 Figure 5.10 Normal calibration curve for the hypothetical data in Table 5.1, showing the regression line. x i y i x i 2 x i y i 0.000 0.00 0.000 0.000 0.100 12.36 0.010 1.236 0.200 24.83 0.040 4.966 0.300 35.91 0.090 10.773 0.400 48.79 0.160 19.516 0.500 60.42 0.250 30.210 Adding the values in each column gives S x i = 1.500 S y i = 182.31 S x 2 i = 0.550 S x i y i = 66.701 Substituting these values into equations 5.12 and 5.13 gives the estimated slope and the estimated y-intercept The relationship between the signal and the analyte, therefore, is S meas = 120.70 · C S + 0.21 Note that for now we keep enough significant figures to match the number of decimal places to which the signal was measured. The resulting calibration curve is shown in Figure 5.10. b 0 182 31 120 706 1 500 6 0 209== .–( . )(. ) . b 1 2 6 66 701 1 500 182 31 6 0 550 1 500 120 706== ()( . )–(. )( . ) ()(. )–(. ) . 120 Modern Analytical Chemistry 80 60 40 20 0 S meas C A 0.0 0.20.1 0.3 0.4 0.5 0.6 Uncertainty in the Regression Analysis As shown in Figure 5.10, the regression line need not pass through the data points (this is the consequence of indeter- minate errors affecting the signal). The cumulative deviation of the data from the regression line is used to calculate the uncertainty in the regression due to 1400-CH05 9/8/99 3:59 PM Page 120 Chapter 5 Calibrations, Standardizations, and Blank Corrections 121 standard deviation about the regression The uncertainty in a regression analysis due to indeterminate error (s r ). indeterminate error. This is called the standard deviation about the regression, s r , and is given as 5.15 where y i is the i th experimental value, and ? y i is the corresponding value predicted by the regression line ? y i = b 0 + b 1 x i There is an obvious similarity between equation 5.15 and the standard deviation in- troduced in Chapter 4, except that the sum of squares term for s r is determined rela- tive to ? y i instead of – y , and the denominator is n – 2 instead of n – 1; n – 2 indicates that the linear regression analysis has only n – 2 degrees of freedom since two pa- rameters, the slope and the intercept, are used to calculate the values of ? y i . A more useful representation of uncertainty is to consider the effect of indeter- minate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as 5.16 5.17 These standard deviations can be used to establish confidence intervals for the true slope and the true y-intercept b 1 = b 1 ± ts b 1 5.18 b o = b o ± ts b 0 5.19 where t is selected for a significance level of a and for n – 2 degrees of freedom. Note that the terms ts b 1 and ts b 0 do not contain a factor of because the con- fidence interval is based on a single regression line. Again, many calculators, spread- sheets, and computer software packages can handle the calculation of s b 0 and s b 1 and the corresponding confidence intervals for b 0 and b 1. Example 5.11 illustrates the calculations. EXAMPLE 5 .11 Calculate the 95% confidence intervals for the slope and y-intercept determined in Example 5.10. SOLUTION Again, as you work through this example, remember that x represents the concentration of analyte in the standards (C S ), and y corresponds to the signal (S meas ). To begin with, it is necessary to calculate the standard deviation about the regression. This requires that we first calculate the predicted signals, ? y i , using the slope and y-intercept determined in Example 5.10. Taking the first standard as an example, the predicted signal is ? y i = b 0 + b 1 x = 0.209 + (120.706)(0.100) = 12.280 () – n 1 s sx nx x sx nxx b i i i i i 0 22 = ∑ ∑∑ = ∑ ∑ r 22 2 r 22 –( ) ( – ) s ns nx x s xx b i ii 1 r 2 2 r 2 = ∑∑ = ∑–( ) ( – ) 22 s y–y n ii i n r = ∑ = ( ? ) – 2 1 2 1400-CH05 9/8/99 3:59 PM Page 121 The results for all six solutions are shown in the following table. x i y i ?y i (y i –?y i ) 2 0.000 0.00 0.209 0.0437 0.100 12.36 12.280 0.0064 0.200 24.83 24.350 0.2304 0.300 35.91 36.421 0.2611 0.400 48.79 48.491 0.0894 0.500 60.42 60.562 0.0202 Adding together the data in the last column gives the numerator of equation 5.15, S (y i – ? y i ) 2 , as 0.6512. The standard deviation about the regression, therefore, is Next we calculate s b1 and s b0 using equations 5.16 and 5.17. Values for the summation terms S x 2 i and S x i are found in Example 5.10. Finally, the 95% confidence intervals (a = 0.05, 4 degrees of freedom) for the slope and y-intercept are b 1 = b 1 ± ts b1 = 120.706 ± (2.78)(0.965) = 120.7 ± 2.7 b 0 = b 0 ± ts b0 = 0.209 ± (2.78)(0.292) = 0.2 ± 0.8 The standard deviation about the regression, s r , suggests that the measured signals are precise to only the first decimal place. For this reason, we report the slope and intercept to only a single decimal place. To minimize the uncertainty in the predicted slope and y-intercept, calibration curves are best prepared by selecting standards that are evenly spaced over a wide range of concentrations or amounts of analyte. The reason for this can be rational- ized by examining equations 5.16 and 5.17. For example, both s b0 and s b1 can be minimized by increasing the value of the term S (x i – – x) 2 , which is present in the de- nominators of both equations. Thus, increasing the range of concentrations used in preparing standards decreases the uncertainty in the slope and the y-intercept. Fur- thermore, to minimize the uncertainty in the y-intercept, it also is necessary to de- crease the value of the term S x 2 i in equation 5.17. This is accomplished by spreading the calibration standards evenly over their range. Using the Regression Equation Once the regression equation is known, we can use it to determine the concentration of analyte in a sample. When using a normal cali- bration curve with external standards or an internal standards calibration curve, we measure an average signal for our sample, – Y X , and use it to calculate the value of X s sx nx x b i i i 0 2 2 2 0 4035 0 550 6 0 550 1 500 0 292= ∑ ∑∑ == r 22 2 –( ) (. )(. ) ()(. )–(. ) . s ns nx x b i i 1 r 2 2 = ∑∑ == –( ) ()(. ) ()(. )–(. ) . 2 2 2 6 0 4035 6 0 550 1 500 0 965 s r == 0 6512 62 0 4035 . – . 122 Modern Analytical Chemistry 1400-CH05 9/8/99 3:59 PM Page 122 Chapter 5 Calibrations, Standardizations, and Blank Corrections 123 5.20 The standard deviation for the calculated value of X is given by the following equation 5.21 where m is the number of replicate samples used to establish – Y X , n is the number of calibration standards, – y is the average signal for the standards, and x i and – x are the individual and mean concentrations of the standards. 8 Once s X is known the confi- dence interval for the analyte’s concentration can be calculated as m X = X ± ts X where m X is the expected value of X in the absence of determinate errors, and the value of t is determined by the desired level of confidence and for n – 2 degrees of freedom. The following example illustrates the use of these equations for an analysis using a normal calibration curve with external standards. EXAMPLE 5 .12 Three replicate determinations are made of the signal for a sample containing an unknown concentration of analyte, yielding values of 29.32, 29.16, and 29.51. Using the regression line from Examples 5.10 and 5.11, determine the analyte’s concentration, C A , and its 95% confidence interval. SOLUTION The equation for a normal calibration curve using external standards is S meas = b 0 + b 1 · C A thus, – Y X is the average signal of 29.33, and X is the analyte’s concentra- tion. Substituting the value of – Y X into equation 5.20 along with the estimated slope and the y-intercept for the regression line gives the analyte’s concentration as To calculate the standard deviation for the analyte’s concentration, we must determine the values for – y and S (x i – – x) 2 . The former is just the average signal for the standards used to construct the calibration curve. From the data in Table 5.1, we easily calculate that – y is 30.385. Calculating S (x i – – x) 2 looks formidable, but we can simplify the calculation by recognizing that this sum of squares term is simply the numerator in a standard deviation equation; thus, S (x i – – x) 2 = s 2 (n –1) where s is the standard deviation for the concentration of analyte in the standards used to construct the calibration curve. Using the data in Table 5.1, we find that s is 0.1871 and S (x i – – x) 2 = (0.1871) 2 (6 – 1) = 0.175 CX Y–b b A X == = = 0 1 29 33 0 209 120 706 0 241 .–. . . s s bm n Yy bx–x X X i =++ ∑ ? ? ? ? ? ? ? ? ? ? r 1 2 1 22 12 11 (–) () / X Yb b X = – 0 1 1400-CH05 9/8/99 3:59 PM Page 123 Figure 5.11 Plot of the residual error in y as a function of x. The distribution of the residuals in (a) indicates that the regression model was appropriate for the data, and the distributions in (b) and (c) indicate that the model does not provide a good fit for the data. Substituting known values into equation 5.21 gives Finally, the 95% confidence interval for 4 degrees of freedom is m A = C A ± ts A = 0.241 – (2.78)(0.0024) = 0.241 ± 0.007 In a standard addition the analyte’s concentration is determined by extrapolat- ing the calibration curve to find the x-intercept. In this case the value of X is and the standard deviation in X is where n is the number of standards used in preparing the standard additions cali- bration curve (including the sample with no added standard), and – y is the average signal for the n standards. Because the analyte’s concentration is determined by ex- trapolation, rather than by interpolation, s X for the method of standard additions generally is larger than for a normal calibration curve. A linear regression analysis should not be accepted without evaluating the validity of the model on which the calculations were based. Perhaps the simplest way to evaluate a regression analysis is to calculate and plot the residual error for each value of x. The residual error for a single calibration standard, r i , is given as r i = y i – ? y i If the regression model is valid, then the residual errors should be randomly distrib- uted about an average residual error of 0, with no apparent trend toward either smaller or larger residual errors (Figure 5.11a). Trends such as those shown in Fig- ures 5.11b and 5.11c provide evidence that at least one of the assumptions on which the regression model is based are incorrect. For example, the trend toward larger residual errors in Figure 5.11b suggests that the indeterminate errors affecting y are not independent of the value of x. In Figure 5.11c the residual errors are not ran- domly distributed, suggesting that the data cannot be modeled with a straight-line relationship. Regression methods for these two cases are discussed in the following sections. 5 C. 3 Weighted Linear Regression with Errors in y Equations 5.13 for the slope, b 1 , and 5.14 for the y-intercept, b 0 , assume that indeterminate errors equally affect each value of y. When this assumption is false, as shown in Figure 5.11b, the variance associated with each value of y must be included when estimating b 0 and b 1 . In this case the predicted slope and inter- cept are 5.22 b n wxy wx wy nwx wx ii ii ii i i ii 1 22 = ∑∑∑ ∑∑ i – –( ) s s bn y bx–x X i =+ ∑ ? ? ? ? ? ? ? ? ? ? r 1 2 1 22 12 1() () / Xx b b ==-intercept – 0 1 ss XA == ++ ? ? ? ? ? ? ? ? = 0 4035 120 706 1 3 1 6 29 33 30 385 120 706 0 175 0 0024 2 2 12 . . (. – . ) (.)(.) . / 124 Modern Analytical Chemistry 0 (a) x i 0 (b) x i 0 (c) x i Residual err or Residual err or Residual err or 1400-CH05 9/8/99 3:59 PM Page 124 Chapter 5 Calibrations, Standardizations, and Blank Corrections 125 and 5.23 where w i is a weighting factor accounting for the variance in measuring y i . Values of w i are calculated using equation 5.24. 5.24 where s i is the standard deviation associated with y i . The use of a weighting factor ensures that the contribution of each pair of xy values to the regression line is pro- portional to the precision with which y i is measured. EXAMPLE 5 .1 3 The following data were recorded during the preparation of a calibration curve, where – S meas and s are the mean and standard deviation, respectively, for three replicate measurements of the signal. C A – S meas s 0.000 0.00 0.02 0.100 12.36 0.02 0.200 24.83 0.07 0.300 35.91 0.13 0.400 48.79 0.22 0.500 60.42 0.33 Determine the relationship between – S meas and C A using a weighted linear regression model. SOLUTION Once again, as you work through this example, remember that x represents the concentration of analyte in the standards (C S ), and y corresponds to the average signal ( – S meas ). We begin by setting up a table to aid in the calculation of the weighting factor. x i y i s i s i –2 w i 0.000 0.00 0.02 2500.00 2.8339 0.100 12.36 0.02 2500.00 2.8339 0.200 24.83 0.07 204.08 0.2313 0.300 35.91 0.13 59.17 0.0671 0.400 48.79 0.22 20.66 0.0234 0.500 60.42 0.33 9.18 0.0104 Adding together the values in the forth column gives which is used to calculate the weights in the last column. As a check on the calculation, the sum of the weights in the last column should equal the number of calibration standards, n. In this case S w i = 6.0000 s i – . 2 5293 09 ∑ = w ns s i i i = ∑ – – 2 2 b wy b wx n ii ii 0 1 = ∑∑ – 1400-CH05 9/8/99 3:59 PM Page 125 After the individual weights have been calculated, a second table is used to aid in calculating the four summation terms in equations 5.22 and 5.23. x i y i w i w i x i w i y i w i x i 2 w i x i y i 0.000 0.00 2.8339 0.0000 0.0000 0.0000 0.0000 0.100 12.36 2.8339 0.2834 35.0270 0.0283 3.5027 0.200 24.83 0.2313 0.0463 5.7432 0.0093 1.1486 0.300 35.91 0.0671 0.0201 2.4096 0.0060 0.7229 0.400 48.79 0.0234 0.0094 1.1417 0.0037 0.4567 0.500 60.42 0.0104 0.0052 0.6284 0.0026 0.3142 Adding the values in the last four columns gives S w i x i = 0.3644 S w i y i = 44.9499 S w i x i 2 = 0.0499 S w i x i y i = 6.1451 Substituting these values into the equations 5.22 and 5.23 gives the estimated slope and the estimated y-intercept The relationship between the signal and the concentration of the analyte, therefore, is – S meas = 122.98 · C A + 0.02 with the calibration curve shown in Figure 5.12. b 0 44 9499 122 985 0 3644 6 0 0224== .–(.)(.) . b 1 2 6 6 1451 0 3644 44 9499 6 0 0499 0 3644 122 985== ()(. )–(. )( . ) ()(. )–(. ) . 126 Modern Analytical Chemistry 80 60 40 20 0 S meas C A 0.0 0.20.1 0.3 0.4 0.5 0.6 Figure 5.12 Weighted normal calibration curve for the data in Example 5.13. The lines through the data points show the standard deviation of the signal for the standards. These lines have been scaled by a factor of 50 so that they can be seen on the same scale as the calibration curve. 1400-CH05 9/8/99 3:59 PM Page 126 Equations for calculating confidence intervals for the slope, the y-intercept, and the concentration of analyte when using a weighted linear regression are not as easily defined as for an unweighted linear regression. 9 The confidence in- terval for the concentration of an analyte, however, will be at its optimum value when the analyte’s signal is near the weighted centroid, – y, of the calibration curve 5 C. 4 Weighted Linear Regression with Errors in Both x and y If we remove the assumption that the indeterminate errors affecting a calibration curve are found only in the signal (y), then indeterminate errors affecting the preparation of standards containing known amounts of analyte (x) must be fac- tored into the regression model. The solution for the resulting regression line is computationally more involved than that for either the unweighted or weighted regression lines, and is not presented in this text. The suggested readings at the end of the chapter list several papers discussing algorithms for this regression method. 5 C. 5 Curvilinear and Multivariate Regression Regression models based on a straight line, despite their apparent complexity, use the simplest functional relationship between two variables. In many cases, calibra- tion curves show a pronounced curvature at high concentrations of analyte (see Fig- ure 5.3b). One approach to constructing a calibration curve when curvature exists is to seek a transformation function that will make the data linear. Logarithms, expo- nentials, reciprocals, square roots, and trigonometric functions have all been used in this capacity. A plot of y versus log x is a typical example. Such transformations are not without complications. Perhaps the most obvious is that data that originally has a uniform variance for the y values will not maintain that uniform variance when the variable is transformed. A more rigorous approach to developing a regression model for a nonlinear calibration curve is to fit a polynomial equation such as y = a + bx + cx 2 to the data. Equations for calculating the parameters a, b, and c are derived in the same manner as that described earlier for the straight-line model. 10 When a single polynomial equation cannot be fitted to the calibration data, it may be possible to fit separate polynomial equations to short segments of the calibration curve. The result is a sin- gle continuous calibration curve known as a spline function. The regression models considered earlier apply only to functions containing a single independent variable. Analytical methods, however, are frequently subject to determinate sources of error due to interferents that contribute to the measured sig- nal. In the presence of a single interferent, equations 5.1 and 5.2 become S meas = k A n A + k I n I + S reag S meas = k A C A + k I C I + S reag where k I is the interferent’s sensitivity, n I is the moles of interferent, and C I is the in- terferent’s concentration. Multivariate calibration curves can be prepared using standards that contain known amounts of analyte and interferent. 11 y n w ii = ∑ 1 y Chapter 5 Calibrations, Standardizations, and Blank Corrections 127 1400-CH05 9/8/99 3:59 PM Page 127 128 Modern Analytical Chemistry Table 5 . 3 Hypothetical Data Used to Study Procedures for Method Blanks Sample W s a S stand Number W x b S samp 1.6667 0.2500 1 62.4746 0.8000 5.0000 0.5000 2 82.7915 1.0000 8.3333 0.7500 3 103.1085 1.2000 9.5507 0.8413 11.6667 1.0000 analyte-free c 0.1000 18.1600 1.4870 19.9333 1.6200 Calibration equation: S stand = 0.0750 · W s + 0.1250 Source: Modified from Cardone, M. J. Anal. Chem. 1986, 58, 433–438. a W s = weight of analyte used to prepare standard solution by diluting to a fixed volume, V. b W x = weight of sample used to prepare sample solution by diluting to a fixed volume, V. c Analyte-free sample prepared in the same fashion as samples, but without the analyte being present. 5 D Blank Corrections In discussing ways to standardize a method, we assumed that an appropriate reagent blank had been used to correct S meas for signals originating from sources other than the analyte. At that time we did not ask an important question— “What constitutes an appropriate reagent blank?” Surprisingly, the answer is not intuitively obvious. In one study, 12 analytical chemists were asked to evaluate a data set consist- ing of a normal calibration curve, three samples of different size but drawn from the same source, and an analyte-free sample (Table 5.3). At least four different approaches for correcting the signals were used by the participants: (1) ignore the correction entirely, which clearly is incorrect; (2) use the y-intercept of the calibration curve as a calibration blank, CB; (3) use the analyte-free sample as a reagent blank, RB; and (4) use both the calibration and reagent blanks. Equa- tions for calculating the concentration of analyte using each approach are shown in Table 5.4, along with the resulting concentration for the analyte in each of the three samples. That all four methods give a different result for the concentration of analyte underscores the importance of choosing a proper blank but does not tell us which of the methods is correct. In fact, the variation within each method for the reported concentration of analyte indicates that none of these four methods has adequately corrected for the blank. Since the three samples were drawn from the same source, they must have the same true concentration of analyte. Since all four methods predict concentrations of analyte that are dependent on the size of the sample, we can conclude that none of these blank corrections has accounted for an underlying constant source of determinate error. To correct for all constant method errors, a blank must account for signals due to the reagents and solvent used in the analysis and any bias due to interac- 1400-CH05 9/8/99 3:59 PM Page 128 tions between the analyte and the sample matrix. Both the calibration blank and the reagent blank correct for signals due to the reagents and solvents. Any differ- ence in their values is due to the number and composition of samples contribut- ing to the determination of the blank. Unfortunately, neither the calibration blank nor the reagent blank can cor- rect for bias due to analyte–matrix interactions because the analyte is missing in the reagent blank, and the sample’s matrix is missing from the calibration blank. The true method blank must include both the matrix and the analyte and, conse- quently, can only be determined using the sample itself. One approach is to mea- sure the signal for samples of different size and determine the regression line from a plot of signal versus the amount of sample. The resulting y-intercept gives the signal for the condition of no sample and is known as the total Youden blank. 13 This is the true blank correction. The regression line for the sample data in Table 5.3 is S samp = 0.009844 · W x + 0.185 giving a true blank correction of 0.185. Using this value to correct the signals gives identical values for the concentration of analyte in all three samples (see Table 5.4, bottom row). The total Youden blank is not encountered frequently in analytical work, because most chemists rely on a calibration blank when using calibration curves and rely on reagent blanks when using a single-point standardization. As long as any constant bias due to analyte–matrix interactions can be ignored, which is often the case, the accuracy of the method will not suffer. It is always a good idea, however, to check for constant sources of error, by analyzing samples of different sizes, before relying on either a calibration or reagent blank. Chapter 5 Calibrations, Standardizations, and Blank Corrections 129 Table 5 . 4 Equations and Resulting Concentrations for Different Approaches to Correcting for the Method Blank Concentration of Analyte in Approach for Correcting Method Blank Equation a Sample 1 Sample 2 Sample 3 Ignore blank corrections 0.1707 0.1610 0.1552 Use calibration blank 0.1441 0.1409 0.1390 Use reagent blank 0.1494 0.1449 0.1422 Use both calibration and reagent blank 0.1227 0.1248 0.1261 Use total Youden blank 0.1313 0.1313 0.1313 a C A = concentration of analyte; W a = weight of analyte; W x = weight of sample; k = slope of calibration curve = 0.075 (see Table 5.3). Abbreviations: CB = calibration blank = 0.125 (see Table 5.3); RB = reagent blank = 0.100 (see Table 5.3); TYB = total Youden blank = 0.185 (see text). C W W S kW C W W S kW C W W S kW C W W S kW C W W S kW a xx a xx a xx a xx a xx A samp A samp A samp A samp A samp == == == == == – – –– – CB RB CB RB TYB total Youden blank A blank that corrects the signal for analyte–matrix interactions. 1400-CH05 9/8/99 3:59 PM Page 129 130 Modern Analytical Chemistry 5 E KEY TERMS aliquot (p. 111) external standard (p. 109) internal standard (p. 116) linear regression (p. 118) matrix matching (p. 110) method of standard additions (p. 110) multiple-point standardization (p. 109) normal calibration curve (p. 109) primary reagent (p. 106) reagent grade (p. 107) residual error (p. 118) secondary reagent (p. 107) single-point standardization (p. 108) standard deviation about the regression (p. 121) total Youden blank (p. 129) In a quantitative analysis, we measure a signal and calculate the amount of analyte using one of the following equations. S meas = kn A + S reag S meas = kC A + S reag To obtain accurate results we must eliminate determinate errors affecting the measured signal, S meas , the method’s sensitivity, k, and any signal due to the reagents, S reag . To ensure that S meas is determined accurately, we calibrate the equipment or instrument used to obtain the signal. Balances are calibrated using standard weights. When necessary, we can also correct for the buoyancy of air. Volumetric glassware can be calibrated by measuring the mass of water contained or de- livered and using the density of water to calculate the true vol- ume. Most instruments have calibration standards suggested by the manufacturer. An analytical method is standardized by determining its sensi- tivity. There are several approaches to standardization, including the use of external standards, the method of standard addition, and the use of an internal standard. The most desirable standard- ization strategy is an external standardization. The method of standard additions, in which known amounts of analyte are added to the sample, is used when the sample’s matrix complicates the analysis. An internal standard, which is a species (not analyte) added to all samples and standards, is used when the procedure does not allow for the reproducible handling of samples and standards. Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple- point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regres- sion analysis can provide an equation for the standardization. A reagent blank corrects the measured signal for signals due to reagents other than the sample that are used in an analysis. The most common reagent blank is prepared by omitting the sample. When a simple reagent blank does not compensate for all constant sources of determinate error, other types of blanks, such as the total Youden blank, can be used. 5 F SUMMARY Calibration—Volumetric glassware (burets, pipets, and volumetric flasks) can be calibrated in the manner described in Example 5.1. Most instruments have a calibration sample that can be prepared to verify the instrument’s accuracy and precision. For example, as described in this chapter, a solution of 60.06 ppm K 2 Cr 2 O 7 in 0.0050 M H 2 SO 4 should give an absorbance of 0.640 ± 0.010 at a wavelength of 350.0 nm when using 0.0050 M H 2 SO 4 as a reagent blank. These exercises also provide practice with using volumetric glassware, weighing samples, and preparing solutions. Standardization—External standards, standard additions, and internal standards are a common feature of many quantitative analyses. Suggested experiments using these standardization methods are found in later chapters. A good project experiment for introducing external standardization, standard additions, and the importance of the sample’s matrix is to explore the effect of pH on the quantitative analysis of an acid–base indicator. Using bromothymol blue as an example, external standards can be prepared in a pH 9 buffer and used to analyze samples buffered to different pHs in the range of 6–10. Results can be compared with those obtained using a standard addition. 5 G Suggested EXPERIMENTS The following exercises and experiments help connect the material in this chapter to the analytical laboratory. Experiments 1400-CH05 9/8/99 3:59 PM Page 130 Chapter 5 Calibrations, Standardizations, and Blank Corrections 131 1. In calibrating a 10-mL pipet, a measured volume of water was transferred to a tared flask and weighed, yielding a mass of 9.9814 g. (a) Calculate, with and without correcting for buoyancy, the volume of water delivered by the pipet. Assume that the density of water is 0.99707 g/cm 3 and that the density of the weights is 8.40 g/cm 3 . (b) What are the absolute and relative errors introduced by failing to account for the effect of buoyancy? Is this a significant source of determinate error for the calibration of a pipet? Explain. 2. Repeat the questions in problem 1 for the case when a mass of 0.2500 g is measured for a solid that has a density of 2.50 g/cm 3 . 3. Is the failure to correct for buoyancy a constant or proportional source of determinate error? 4. What is the minimum density of a substance necessary to keep the buoyancy correction to less than 0.01% when using brass calibration weights with a density of 8.40 g/cm 3 ? 5. Describe how you would use a serial dilution to prepare 100 mL each of a series of standards with concentrations of 1.000 · 10 –5 , 1.000 · 10 –4 , 1.000 · 10 –3 , and 1.000 · 10 –2 M from a 0.1000 M stock solution. Calculate the uncertainty for each solution using a propagation of uncertainty, and compare to the uncertainty if each solution was prepared by a single dilution of the stock solution. Tolerances for different types of volumetric glassware and digital pipets are found in Tables 4.2 and 4.4. Assume that the uncertainty in the molarity of the stock solution is ±0.0002. 6. Three replicate determinations of the signal for a standard solution of an analyte at a concentration of 10.0 ppm give values of 0.163, 0.157, and 0.161 (arbitrary units), respectively. The signal for a method blank was found to be 0.002. Calculate the concentration of analyte in a sample that gives a signal of 0.118. 7. A 10.00-g sample containing an analyte was transferred to a 250-mL volumetric flask and diluted to volume. When a 10.00-mL aliquot of the resulting solution was diluted to 25.00 mL it was found to give a signal of 0.235 (arbitrary units). A second 10.00-mL aliquot was spiked with 10.00 mL of a 1.00- ppm standard solution of the analyte and diluted to 25.00 mL. The signal for the spiked sample was found to be 0.502. Calculate the weight percent of analyte in the original sample. 8. A 50.00-mL sample containing an analyte gives a signal of 11.5 (arbitrary units). A second 50-mL aliquot of the sample, which is spiked with 1.00-mL of a 10.0-ppm standard solution of the analyte, gives a signal of 23.1. What is the concentration of analyte in the original sample? 9. An appropriate standard additions calibration curve based on equation 5.8 plots S spike (V o + V s ) on the y-axis and C s V s on the x-axis. Clearly explain why you cannot plot S spike on the y- axis and C s [V s /(V o + V s )] on the x-axis. Derive equations for the slope and y-intercept, and explain how the amount of analyte in a sample can be determined from the calibration curve. 10. A standard sample was prepared containing 10.0 ppm of an analyte and 15.0 ppm of an internal standard. Analysis of the sample gave signals for the analyte and internal standard of 0.155 and 0.233 (arbitrary units), respectively. Sufficient internal standard was added to a sample to make it 15.0 ppm in the internal standard. Analysis of the sample yielded signals for the analyte and internal standard of 0.274 and 0.198, respectively. Report the concentration of analyte in the sample. 11. For each of the pairs of calibration curves in Figure 5.13 on page 132, select the calibration curve with the better set of standards. Briefly explain the reasons for your selections. The scales for the x-axes and y-axes are the same for each pair. 12. The following standardization data were provided for a series of external standards of Cd 2+ that had been buffered to a pH of 4.6. 14 [Cd 2+ ] (nM) 15.4 30.4 44.9 59.0 72.7 86.0 S meas (nA) 4.8 11.4 18.2 26.6 32.3 37.7 (a) Determine the standardization relationship by a linear regression analysis, and report the confidence intervals for the slope and y-intercept. (b) Construct a plot of the residuals, and comment on their significance. At a pH of 3.7 the following data were recorded [Cd 2+ ] (nM) 15.4 30.4 44.9 59.0 72.7 86.0 S meas (nA) 15.0 42.7 58.5 77.0 101 118 (c) How much more or less sensitive is this method at the lower pH? (d) A single sample is buffered to a pH of 3.7 and analyzed for cadmium, yielding a signal of 66.3. Report the concentration of Cd 2+ in the sample and its 95% confidence interval. 13. To determine the concentration of analyte in a sample, a standard additions was performed. A 5.00-mL portion of the sample was analyzed and then successive 0.10-mL spikes of a 600.0-ppb standard of the analyte 5 H PROBLEMS 1400-CH05 9/8/99 3:59 PM Page 131 132 Modern Analytical Chemistry Signal Signal C A (a) C A Signal Signal C A (b) C A Signal Signal C A (c) C A Figure 5.13 1400-CH05 9/8/99 3:59 PM Page 132 were added, analyzing after each spike. The following results were obtained Volume of Spike Signal (mL) (arbitrary units) 0.00 0.119 0.10 0.231 0.20 0.339 0.30 0.442 Construct an appropriate standard additions calibration curve, and use a linear regression analysis to determine the concentration of analyte in the original sample and its 95% confidence interval. 14. Troost and Olavesen investigated the application of an internal standardization to the quantitative analysis of polynuclear aromatic hydrocarbons. 15 The following results were obtained for the analysis of the analyte phenanthrene using isotopically labeled phenanthrene as an internal standard S A /S IS C A /C IS Replicate 1 Replicate 2 0.50 0.514 0.522 1.25 0.993 1.024 2.00 1.486 1.471 3.00 2.044 2.080 4.00 2.342 2.550 (a) Determine the standardization relationship by a linear regression, and report the confidence intervals for the slope and y-intercept. (b) Based on your results, explain why the authors of this paper concluded that the internal standardization was inappropriate. 15. In Chapter 4 we used a paired t-test to compare two methods that had been used to independently analyze a series of samples of variable composition. An alternative approach is to plot the results for one method versus those for the other. If the two methods yield identical results, then the plot should have a true slope (b 1 ) of 1.00 and a true y-intercept (b 0 ) of 0.0. A t-test can be used to compare the actual slope and y- intercept with these ideal values. The appropriate test statistic for the y-intercept is found by rearranging equation 5.18 Rearranging equation 5.17 gives the test statistic for the slope Reevaluate the data in problem 24 in Chapter 4 using the same significance level as in the original problem.* 16. Franke and co-workers evaluated a standard additions method for a voltammetric determination of Tl. 16 A summary of their results is tabulated here. ppm Tl Instrument Response for Replicates added (m A) 0.000 2.53 2.50 2.70 2.63 2.70 2.80 2.52 0.387 8.42 7.96 8.54 8.18 7.70 8.34 7.98 1.851 29.65 28.70 29.05 28.30 29.20 29.95 28.95 5.734 84.8 85.6 86.0 85.2 84.2 86.4 87.8 Determine the standardization relationship using a weighted linear regression. t b s b s bb exp 11 == β 11 1 100–.– t b s b s bb exp 00 == β 00 0 – Chapter 5 Calibrations, Standardizations, and Blank Corrections 133 In addition to the texts listed as suggested readings in Chapter 4, the following text provides additional details on regression Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd. ed. Wiley: New York, 1981. Several articles providing more details about linear regression follow. Boqué, R.; Rius, F. X.; Massart, D. L. “Straight Line Calibration: Something More Than Slopes, Intercepts, and Correlation Coefficients,” J. Chem. Educ. 1993, 70, 230–232. Henderson, G. “Lecture Graphic Aids for Least-Squares Analysis,” J. Chem. Educ. 1988, 65, 1001–1003. Renman, L., Jagner, D. “Asymmetric Distribution of Results in Calibration Curve and Standard Addition Evaluations,” Anal. Chim. Acta 1997, 357, 157–166. Two useful papers providing additional details on the method of standard additions are Bader, M. “A Systematic Approach to Standard Addition Methods in Instrumental Analysis,” J. Chem. Educ. 1980, 57, 703–706. 5 I SUGGESTED READINGS *Although this is a commonly used procedure for comparing two methods, it does violate one of the assumptions of an ordinary linear regression. Since both methods are expected to have indeterminate errors, an unweighted regression with errors in y may produce a biased result, with the slope being underestimated and the y-intercept being overestimated. This limitation can be minimized by placing the more precise method on the x-axis, using ten or more samples to increase the degrees of freedom in the analysis, and by using samples that uniformly cover the range of concentrations. For more information see Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed. Ellis Horwood PTR Prentice-Hall: New York, 1993. Alternative approaches are discussed in Hartman, C.; Smeyers-Verbeke, J.; Penninckx, W.; Massart, D. L. Anal. Chim. Acta 1997, 338, 19–40 and Zwanziger, H. W.; Sarbu, C. Anal. Chem. 1998, 70, 1277–1280. 1400-CH05 9/8/99 3:59 PM Page 133 Nimura, Y.; Carr, M. R. “Reduction of the Relative Error in the Standard Additions Method,” Analyst 1990, 115, 1589–1595. The following paper discusses the importance of weighting experimental data when using linear regression Karolczak, M. “To Weight or Not to Weight? An Analyst’s Dilemma,” Curr. Separations 1995, 13, 98–104. Algorithms for performing a linear regression with errors in both x and y are discussed in Irvin, J. A.; Quickenden, T. L. “Linear Least Squares Treatment When There Are Errors in Both x and y,” J. Chem. Educ. 1983, 60, 711–712. Kalantar, A. H. “Kerrich’s Method for y = a x Data When Both y and x Are Uncertain,” J. Chem. Educ. 1991, 68, 368–370. Macdonald, J. R.; Thompson, W. J. “Least-Squares Fitting When Both Variables Contain Errors: Pitfalls and Possibilities,” Am. J. Phys. 1992, 60, 66–73. Ogren, P. J.; Norton, J. R. “Applying a Simple Linear Least- Squares Algorithm to Data with Uncertainties in Both Variables,” J. Chem. Educ. 1992, 69, A130–A131. The following paper discusses some of the problems that may be encountered when using linear regression to model data that have been mathematically transformed into a linear form. Chong, D. P. “On the Use of Least Squares to Fit Data in Linear Form,” J. Chem. Educ. 1994, 71, 489–490. The analysis of nonlinear data is covered in the following papers. Harris, D. C. “Nonlinear Least-Squares Curve Fitting with Microsoft Excel Solver,” J. Chem. Educ. 1998, 75, 119–121. Lieb, S. G. “Simplex Method of Nonlinear Least-Squares—A Logical Complementary Method to Linear Least-Squares Analysis of Data,” J. Chem. Educ. 1997, 74, 1008–1011. Machuca-Herrera, J. G. “Nonlinear Curve Fitting with Spreadsheets,” J. Chem. Educ. 1997, 74, 448–449. Zielinski, T. J.; Allendoerfer, R. D. “Least Squares Fitting of Nonlinear Data in the Undergraduate Laboratory,” J. Chem. Educ. 1997, 74, 1001–1007. More information on multivariate regression can be found in Lang, P. M.; Kalivas, J. H. “A Global Perspective on Multivariate Calibration Methods,” J. Chemometrics 1993, 7, 153–164. Kowalski, B. R.; Seasholtz, M. B. “Recent Developments in Multivariate Calibration,” J. Chemometrics 1991, 5 129–145. An additional discussion on method blanks is found in the following two papers. Cardone, M. J. “Detection and Determination of Error in Analytical Methodology. Part II. Correction for Corrigible Systematic Error in the Course of Real Sample Analysis,” J. Assoc. Off. Anal. Chem. 1983, 66, 1283–1294. Cardone, M. J. “Detection and Determination of Error in Analytical Methodology. Part IIB. Direct Calculational Technique for Making Corrigible Systematic Error Corrections,” J. Assoc. Off. Anal. Chem. 1985, 68, 199–202. 134 Modern Analytical Chemistry 1. Battino, R.; Williamson, A. G. J. Chem. Educ. 1984, 61, 51–52. 2. Ebel, S. Fresenius J. Anal. Chem. 1992, 342, 769. 3. ACS Committee on Environmental Improvement “Guidelines for Data Acquisition and Data Quality Evaluation in Environmental Chemistry,” Anal. Chem. 1980, 52, 2242–2249. 4. Moody, J. R.; Greenburg, P. R.; Pratt, K. W.; et al. Anal. Chem. 1988, 60, 1203A–1218A. 5. Committee on Analytical Reagents, Reagent Chemicals, 8th ed., American Chemical Society: Washington, DC, 1993. 6. Cardone, M. J.; Palmero, P. J.; Sybrandt, L. B. Anal. Chem. 1980, 52, 1187–1191. 7. Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd ed. Wiley: New York, 1981. 8. (a) Miller, J. N. Analyst 1991, 116, 3–14; and (b) Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Chemometrics, Wiley-Interscience: New York, 1986; pp. 126–127. 9. Bonate, P. J. Anal. Chem. 1993, 65, 1367–1372. 10. (a) Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Chemometrics, Wiley-Interscience: New York, 1986; (b) Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach, Elsevier: Amsterdam, 1987. 11. Beebe, K. R.; Kowalski, B. R. Anal. Chem. 1987, 59, 1007A–1017A. 12. Cardone, M. J. Anal. Chem. 1986, 58, 433–438. 13. Cardone, M. J. Anal. Chem. 1986, 58, 438–445. 14. Wojciechowski, M; Balcerzak, J. Anal. Chim. Acta 1991, 249, 433–445. 15. Troost, J. R.; Olavesen, E. Y. Anal. Chem. 1996, 68, 708–711. 16. Franke, J. P.; de Zeeuw, R. A.; Hakkert, R. Anal. Chem. 1978, 50, 1374–1380. 5 J REFERENCES 1400-CH05 9/8/99 4:00 PM Page 134 Chapter 6 135 Equilibrium Chemistry Regardless of the problem on which an analytical chemist is working, its solution ultimately requires a knowledge of chemistry and the ability to reason with that knowledge. For example, an analytical chemist developing a method for studying the effect of pollution on spruce trees needs to know, or know where to find, the structural and chemical differences between p-hydroxybenzoic acid and p-hydroxyacetophenone, two common phenols found in the needles of spruce trees (Figure 6.1). Chemical reasoning is a product of experience and is constructed from knowledge acquired in the classroom, the laboratory, and the chemical literature. The material in this text assumes familiarity with topics covered in the courses and laboratory work you have already completed. This chapter provides a review of equilibrium chemistry. Much of the material in this chapter should be familiar to you, but other ideas are natural extensions of familiar topics. 1400-CH06 9/9/99 7:40 AM Page 135 136 Modern Analytical Chemistry equilibrium A system is at equilibrium when the concentrations of reactants and products remain constant. CaCO 3 Ca 2+ Time Grams Figure 6.2 Change in mass of undissolved Ca 2+ and solid CaCO 3 over time during the precipitation of CaCO 3 . 6A Reversible Reactions and Chemical Equilibria In 1798, the chemist Claude Berthollet (1748–1822) accompanied a French military expedition to Egypt. While visiting the Natron Lakes, a series of salt water lakes carved from limestone, Berthollet made an observation that contributed to an im- portant discovery. Upon analyzing water from the Natron Lakes, Berthollet found large quantities of common salt, NaCl, and soda ash, Na 2 CO 3 , a result he found sur- prising. Why would Berthollet find this result surprising and how did it contribute to an important discovery? Answering these questions provides an example of chemical reasoning and introduces the topic of this chapter. Berthollet “knew” that a reaction between Na 2 CO 3 and CaCl 2 goes to comple- tion, forming NaCl and a precipitate of CaCO 3 as products. Na 2 CO 3 + CaCl 2 fi 2NaCl + CaCO 3 Understanding this, Berthollet expected that large quantities of NaCl and Na 2 CO 3 could not coexist in the presence of CaCO 3 . Since the reaction goes to completion, adding a large quantity of CaCl 2 to a solution of Na 2 CO 3 should produce NaCl and CaCO 3 , leaving behind no unreacted Na 2 CO 3 . In fact, this result is what he ob- served in the laboratory. The evidence from Natron Lakes, where the coexistence of NaCl and Na 2 CO 3 suggests that the reaction has not gone to completion, ran counter to Berthollet’s expectations. Berthollet’s important insight was recognizing that the chemistry occurring in the Natron Lakes is the reverse of what occurs in the laboratory. CaCO 3 + 2NaCl fi Na 2 CO 3 + CaCl 2 Using this insight Berthollet reasoned that the reaction is reversible, and that the relative amounts of “reactants” and “products” determine the direction in which the reaction occurs, and the final composition of the reaction mixture. We recog- nize a reaction’s ability to move in both directions by using a double arrow when writing the reaction. Na 2 CO 3 + CaCl 2 t 2NaCl + CaCO 3 Berthollet’s reasoning that reactions are reversible was an important step in understanding chemical reactivity. When we mix together solutions of Na 2 CO 3 and CaCl 2 , they react to produce NaCl and CaCO 3 . If we monitor the mass of dissolved Ca 2+ remaining and the mass of CaCO 3 produced as a function of time, the result will look something like the graph in Figure 6.2. At the start of the reaction the mass of dissolved Ca 2+ decreases and the mass of CaCO 3 in- creases. Eventually, however, the reaction reaches a point after which no further changes occur in the amounts of these species. Such a condition is called a state of equilibrium. Although a system at equilibrium appears static on a macroscopic level, it is important to remember that the forward and reverse reactions still occur. A reac- tion at equilibrium exists in a “steady state,” in which the rate at which any species forms equals the rate at which it is consumed. 6B Thermodynamics and Equilibrium Chemistry Thermodynamics is the study of thermal, electrical, chemical, and mechanical forms of energy. The study of thermodynamics crosses many disciplines, including physics, engineering, and chemistry. Of the various branches of thermodynamics, Figure 6.1 Structures of (a) p-hydroxybenzoic acid and (b) p-hydroxyacetophenone. O OH OH (a) (b) O OH CH 3 1400-CH06 9/9/99 7:40 AM Page 136 the most important to chemistry is the study of the changes in energy occurring during a chemical reaction. Consider, for example, the general equilibrium reaction shown in equation 6.1, involving the solutes A, B, C, and D, with stoichiometric coefficients a, b, c, and d. aA+bB t cC+dD 6.1 By convention, species to the left of the arrows are called reactants, and those on the right side of the arrows are called products. As Berthollet discovered, writing a reac- tion in this fashion does not guarantee that the reaction of A and B to produce C and D is favorable. Depending on initial conditions, the reaction may move to the left, to the right, or be in a state of equilibrium. Understanding the factors that determine the final position of a reaction is one of the goals of chemical thermodynamics. Chemical systems spontaneously react in a fashion that lowers their overall free energy. At a constant temperature and pressure, typical of many bench-top chemi- cal reactions, the free energy of a chemical reaction is given by the Gibb’s free en- ergy function ?G = ?H – T ?S 6.2 where T is the temperature in kelvins, and ?G, ?H, and ?S are the differences in the Gibb’s free energy, the enthalpy, and the entropy between the products and reactants. Enthalpy is a measure of the net flow of energy, as heat, during a chemical re- action. Reactions in which heat is produced have a negative ?H and are called exothermic. Endothermic reactions absorb heat from their surroundings and have a positive ?H. Entropy is a measure of randomness, or disorder. The entropy of an individual species is always positive and tends to be larger for gases than for solids and for more complex rather than simpler molecules. Reactions that result in a large number of simple, gaseous products usually have a positive ?S. The sign of ?G can be used to predict the direction in which a reaction moves to reach its equilibrium position. A reaction is always thermodynamically favored when enthalpy decreases and entropy increases. Substituting the inequalities ?H <0 and ?S > 0 into equation 6.2 shows that ?G is negative when a reaction is thermo- dynamically favored. When ?G is positive, the reaction is unfavorable as written (although the reverse reaction is favorable). Systems at equilibrium have a ?G of zero. As a system moves from a nonequilibrium to an equilibrium position, ?G must change from its initial value to zero. At the same time, the species involved in the reaction undergo a change in their concentrations. The Gibb’s free energy, there- fore, must be a function of the concentrations of reactants and products. As shown in equation 6.3, the Gibb’s free energy can be divided into two terms. ?G = ?G°+RT ln Q 6.3 The first term, ?G°, is the change in Gibb’s free energy under standard-state condi- tions; defined as a temperature of 298 K, all gases with partial pressures of 1 atm, all solids and liquids pure, and all solutes present with 1 M concentrations. The second term, which includes the reaction quotient, Q, accounts for nonstandard-state pres- sures or concentrations. For reaction 6.1 the reaction quotient is 6.4 where the terms in brackets are the molar concentrations of the solutes. Note that the reaction quotient is defined such that the concentrations of products are placed Q cd ab = [][] [][] CD AB Chapter 6 Equilibrium Chemistry 137 Gibb’s free energy A thermodynamic function for systems at constant temperature and pressure that indicates whether or not a reaction is favorable (?G < 0), unfavorable (?G > 0), or at equilibrium (?G = 0). enthalpy A change in enthalpy indicates the heat absorbed or released during a chemical reaction at constant pressure. entropy A measure of disorder. standard state Condition in which solids and liquids are in pure form, gases have partial pressures of 1 atm, solutes have concentrations of 1 M, and the temperature is 298 K. 1400-CH06 9/9/99 7:40 AM Page 137 138 Modern Analytical Chemistry in the numerator, and the concentrations of reactants are placed in the denominator. In addition, each concentration term is raised to a power equal to its stoichiometric coefficient in the balanced chemical reaction. Partial pressures are substituted for concentrations when the reactant or product is a gas. The concentrations of pure solids and pure liquids do not change during a chemical reaction and are excluded from the reaction quotient. At equilibrium the Gibb’s free energy is zero, and equation 6.3 simplifies to ?G°=–RT ln K where K is an equilibrium constant that defines the reaction’s equilibrium posi- tion. The equilibrium constant is just the numerical value obtained when substitut- ing the concentrations of reactants and products at equilibrium into equation 6.4; thus, 6.5 where the subscript “eq” indicates a concentration at equilibrium. Although the subscript “eq” is usually omitted, it is important to remember that the value of K is determined by the concentrations of solutes at equilibrium. As written, equation 6.5 is a limiting law that applies only to infinitely dilute solutions, in which the chemical behavior of any species in the system is unaffected by all other species. Corrections to equation 6.5 are possible and are discussed in more detail at the end of the chapter. 6C Manipulating Equilibrium Constants We will use two useful relationships when working with equilibrium constants. First, if we reverse a reaction’s direction, the equilibrium constant for the new reac- tion is simply the inverse of that for the original reaction. For example, the equilib- rium constant for the reaction is the inverse of that for the reaction Second, if we add together two reactions to obtain a new reaction, the equilibrium constant for the new reaction is the product of the equilibrium constants for the original reactions. AC AC AC AC AC C AC AC AC C ACAC AC AC AC AC C AC AC += += +==×= t t t [] [][] [] [][] [] [][] [] [][] [] [][] K K KKK 1 22 2 2312 22 2 2 AB A B AB AB 22 1 2 2 2 1 t +== [][] [] K K ABAB AB AB +=2 21 2 2 t [] [][] K K= cd ab [][] [][] CD AB eq eq eq eq equilibrium constant For a reaction at equilibrium, the equilibrium constant determines the relative concentrations of products and reactants. 1400-CH06 9/9/99 7:40 AM Page 138 Chapter 6 Equilibrium Chemistry 139 EXAMPLE 6.1 Calculate the equilibrium constant for the reaction 2A+B t C+3D given the following information Rxn 1: A + B t D K 1 = 0.40 Rxn 2: A + E t C+D+F K 2 = 0.10 Rxn 3: C + E t B K 3 = 2.0 Rxn 4: F + C t D+B K 4 = 5.0 SOLUTION The overall reaction is given as Rxn 1 + Rxn 2 – Rxn 3 + Rxn 4 If Rxn 3 is reversed, giving then the overall reaction is Rxn 1 + Rxn 2 + Rxn 5 + Rxn 4 and the overall equilibrium constant is K overall = K 1 · K 2 · K 5 · K 4 = 0.40 · 0.10 · 0.50 · 5.0 = 0.10 6D Equilibrium Constants for Chemical Reactions Several types of reactions are commonly used in analytical procedures, either in preparing samples for analysis or during the analysis itself. The most important of these are precipitation reactions, acid–base reactions, complexation reactions, and oxidation–reduction reactions. In this section we review these reactions and their equilibrium constant expressions. 6D.1 Precipitation Reactions A precipitation reaction occurs when two or more soluble species combine to form an insoluble product that we call a precipitate. The most common precipitation re- action is a metathesis reaction, in which two soluble ionic compounds exchange parts. When a solution of lead nitrate is added to a solution of potassium chloride, for example, a precipitate of lead chloride forms. We usually write the balanced re- action as a net ionic equation, in which only the precipitate and those ions involved in the reaction are included. Thus, the precipitation of PbCl 2 is written as Pb 2+ (aq) + 2Cl – (aq) t PbCl 2 (s) In the equilibrium treatment of precipitation, however, the reverse reaction de- scribing the dissolution of the precipitate is more frequently encountered. PbCl 2 (s) t Pb 2+ (aq) + 2Cl – (aq) Rxn B C E 5 11 20 050 5 3 : . .t +===K K precipitate An insoluble solid that forms when two or more soluble reagents are combined. 1400-CH06 9/9/99 7:40 AM Page 139 The equilibrium constant for this reaction is called the solubility product, K sp , and is given as K sp = [Pb 2+ ][Cl – ] 2 = 1.7 · 10 –5 6.6 Note that the precipitate, which is a solid, does not appear in the K sp expression. It is important to remember, however, that equation 6.6 is valid only if PbCl 2 (s) is present and in equilibrium with the dissolved Pb 2+ and Cl – . Values for selected solu- bility products can be found in Appendix 3A. 6D.2 Acid—Base Reactions A useful definition of acids and bases is that independently introduced by Jo- hannes Br?nsted (1879–1947) and Thomas Lowry (1874–1936) in 1923. In the Br?nsted-Lowry definition, acids are proton donors, and bases are proton accep- tors. Note that these definitions are interrelated. Defining a base as a proton accep- tor means an acid must be available to provide the proton. For example, in reac- tion 6.7 acetic acid, CH 3 COOH, donates a proton to ammonia, NH 3 , which serves as the base. CH 3 COOH(aq)+NH 3 (aq) t CH 3 COO – (aq)+NH 4 + (aq) 6.7 When an acid and a base react, the products are a new acid and base. For exam- ple, the acetate ion, CH 3 COO – , in reaction 6.7 is a base that reacts with the acidic ammonium ion, NH 4 + , to produce acetic acid and ammonia. We call the acetate ion the conjugate base of acetic acid, and the ammonium ion is the conjugate acid of ammonia. Strong and Weak Acids The reaction of an acid with its solvent (typically water) is called an acid dissociation reaction. Acids are divided into two categories based on the ease with which they can donate protons to the solvent. Strong acids, such as HCl, almost completely transfer their protons to the solvent molecules. HCl(aq)+H 2 O(l) fi H 3 O + (aq)+Cl – (aq) In this reaction H 2 O serves as the base. The hydronium ion, H 3 O + , is the conju- gate acid of H 2 O, and the chloride ion is the conjugate base of HCl. It is the hy- dronium ion that is the acidic species in solution, and its concentration deter- mines the acidity of the resulting solution. We have chosen to use a single arrow (fi ) in place of the double arrows (t) to indicate that we treat HCl as if it were completely dissociated in aqueous solutions. A solution of 0.10 M HCl is effec- tively 0.10 M in H 3 O + and 0.10 M in Cl – . In aqueous solutions, the common strong acids are hydrochloric acid (HCl), hydroiodic acid (HI), hydrobromic acid (HBr), nitric acid (HNO 3 ), perchloric acid (HClO 4 ), and the first proton of sulfu- ric acid (H 2 SO 4 ). Weak acids, of which aqueous acetic acid is one example, cannot completely donate their acidic protons to the solvent. Instead, most of the acid remains undis- sociated, with only a small fraction present as the conjugate base. CH 3 COOH(aq)+H 2 O(l) t H 3 O + (aq)+CH 3 COO – (aq) The equilibrium constant for this reaction is called an acid dissociation constant, K a , and is written as K a ==× + [][ ] . – – H O CH COO CH COOH 33 3 5 175 10 140 Modern Analytical Chemistry solubility product The equilibrium constant for a reaction in which a solid dissociates into its ions (K sp ). acid A proton donor. base A proton acceptor. acid dissociation constant The equilibrium constant for a reaction in which an acid donates a proton to the solvent (K a ). 1400-CH06 9/9/99 7:40 AM Page 140 Note that the concentration of H 2 O is omitted from the K a expression because its value is so large that it is unaffected by the dissociation reaction.* The magnitude of K a provides information about the relative strength of a weak acid, with a smaller K a corresponding to a weaker acid. The ammonium ion, for example, with a K a of 5.70 · 10 –10 , is a weaker acid than acetic acid. Monoprotic weak acids, such as acetic acid, have only a single acidic proton and a single acid dissociation constant. Some acids, such as phosphoric acid, can donate more than one proton and are called polyprotic weak acids. Polyprotic acids are described by a series of acid dissociation steps, each characterized by it own acid dissociation constant. Phosphoric acid, for example, has three acid dissociation re- actions and acid dissociation constants. The decrease in the acid dissociation constant from K a1 to K a3 tells us that each suc- cessive proton is harder to remove. Consequently, H 3 PO 4 is a stronger acid than H 2 PO 4 – , and H 2 PO 4 – is a stronger acid than HPO 4 2– . Strong and Weak Bases Just as the acidity of an aqueous solution is a measure of the concentration of the hydronium ion, H 3 O + , the basicity of an aqueous solution is a measure of the concentration of the hydroxide ion, OH – . The most common example of a strong base is an alkali metal hydroxide, such as sodium hydroxide, which completely dissociates to produce the hydroxide ion. NaOH(aq) fi Na + (aq)+OH – (aq) Weak bases only partially accept protons from the solvent and are characterized by a base dissociation constant, K b . For example, the base dissociation reaction and base dissociation constant for the acetate ion are Polyprotic bases, like polyprotic acids, also have more than one base dissociation re- action and base dissociation constant. Amphiprotic Species Some species can behave as either an acid or a base. For ex- ample, the following two reactions show the chemical reactivity of the bicarbonate ion, HCO 3 – , in water. CH COO H O OH CH COOH CH COOH OH CH COO b 32 3 3 3 10 571 10 –– – – () () () () [][] . aq aq aq K ++ ==× ? l t HPO HO HO HPO HPO HO HPO H PO H O H O HPO HPO H O HPO 34 2 3 2 4 1 2 4 3 34 3 2 4 23 4 2 4 3 2 4 711 10 () () () () [][] [] . () () () () [][] [ – – – – aq aq aq K aq aq aq K ++ ==× ++ = + + +? ?+ l l t t a 2 a 2 – – – ] . () () () () [][] [] . =× ++ ==× ?+? ?+ ? 632 10 45 10 8 4 23 4 3 4 3 4 13 HPO H O H O PO PO H O HPO 2 3 a 3 2 aq aq aq K l t Chapter 6 Equilibrium Chemistry 141 *The concentration of pure water is approximately 55.5 M base dissociation constant The equilibrium constant for a reaction in which a base accepts a proton from the solvent (K b ). 1400-CH06 9/9/99 7:40 AM Page 141 142 Modern Analytical Chemistry amphiprotic A species capable of acting as both an acid and a base. HCO 3 – (aq)+H 2 O(l) t H 3 O + (aq)+CO 3 2– (aq) 6.8 HCO 3 – (aq)+H 2 O(l) t OH – (aq)+H 2 CO 3 (aq) 6.9 A species that can serve as both a proton donor and a proton acceptor is called am- phiprotic. Whether an amphiprotic species behaves as an acid or as a base depends on the equilibrium constants for the two competing reactions. For bicarbonate, the acid dissociation constant for reaction 6.8 K a2 = 4.69 · 10 –11 is smaller than the base dissociation constant for reaction 6.9. K b2 = 2.25 · 10 –8 Since bicarbonate is a stronger base than it is an acid (k b2 > k a2 ), we expect that aqueous solutions of HCO 3 – will be basic. Dissociation of Water Water is an amphiprotic solvent in that it can serve as an acid or a base. An interesting feature of an amphiprotic solvent is that it is capable of reacting with itself as an acid and a base. H 2 O(l)+H 2 O(l) t H 3 O + (aq)+OH – (aq) The equilibrium constant for this reaction is called water’s dissociation con- stant, K w , K w =[H 3 O + ][OH – ] 6.10 which has a value of 1.0000 · 10 –14 at a temperature of 24 °C. The value of K w varies substantially with temperature. For example, at 20 °C, K w is 6.809 · 10 –15 , but at 30 °C K w is 1.469 · 10 –14 . At the standard state temperature of 25 °C, K w is 1.008 · 10 –14 , which is sufficiently close to 1.00 · 10 –14 that the latter value can be used with negligible error. The pH Scale An important consequence of equation 6.10 is that the concentra- tions of H 3 O + and OH – are related. If we know [H 3 O + ] for a solution, then [OH – ] can be calculated using equation 6.10. EXAMPLE 6.2 What is the [OH – ] if the [H 3 O + ] is 6.12 · 10 –5 M? SOLUTION Equation 6.10 also allows us to develop a pH scale that indicates the acidity of a so- lution. When the concentrations of H 3 O + and OH – are equal, a solution is neither acidic nor basic; that is, the solution is neutral. Letting [H 3 O + ] = [OH – ] and substituting into equation 6.10 leaves us with K w =[H 3 O + ] 2 = 1.00 · 10 –14 [] [] . . . – – – – OH HO w == × × =× + K 3 14 5 10 100 10 612 10 163 10 pH Defined as pH = –log[H 3 O + ]. 1400-CH06 9/9/99 7:40 AM Page 142 Chapter 6 Equilibrium Chemistry 143 Solving for [H 3 O + ] gives A neutral solution has a hydronium ion concentration of 1.00 · 10 –7 M and a pH of 7.00.* For a solution to be acidic, the concentration of H 3 O + must be greater than that for OH – , or [H 3 O + ] > 1.00 · 10 –7 M The pH of an acidic solution, therefore, must be less than 7.00. A basic solution, on the other hand, will have a pH greater than 7.00. Figure 6.3 shows the pH scale along with pH values for some representative solutions. Tabulating Values for K a and K b A useful observation about acids and bases is that the strength of a base is inversely proportional to the strength of its conjugate acid. Consider, for example, the dissociation reactions of acetic acid and acetate. CH 3 COOH(aq)+H 2 O(l) t H 3 O + (aq)+CH 3 COO – (aq) 6.11 CH 3 COO – (aq)+H 2 O(l) t CH 3 COOH(aq)+OH – (aq) 6.12 Adding together these two reactions gives 2H 2 O(l) t H 3 O + (aq)+OH – (aq) 6.13 The equilibrium constant for equation 6.13 is K w . Since equation 6.13 is obtained by adding together reactions 6.11 and 6.12, K w may also be expressed as the product of K a for CH 3 COOH and K b for CH 3 COO – . Thus, for a weak acid, HA, and its con- jugate weak base, A – , K w = K a · K b 6.14 This relationship between K a and K b simplifies the tabulation of acid and base dis- sociation constants. Acid dissociation constants for a variety of weak acids are listed in Appendix 3B. The corresponding values of K b for their conjugate weak bases are determined using equation 6.14. EXAMPLE 6. 3 Using Appendix 3B, calculate the following equilibrium constants (a) K b for pyridine, C 5 H 5 N (b) K b for dihydrogen phosphate, H 2 PO 4 – SOLUTION () . . . () . . . , , – – – , , – – – – a b bC H N w aC H NH bH PO w aH PO K K K K K K 55 55 24 34 100 10 590 10 169 10 100 10 711 10 141 10 14 6 9 14 3 12 == × × =× == × × =× + []. . –– HO 3 14 7 1 00 10 1 00 10 + =×=× *The use of a p-function to express a concentration is covered in Chapter 2. Gastric juice 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Vinegar Milk Neutral Blood Seawater Milk of magnesia Household bleach “Pure” rain pH Figure 6.3 pH scale showing values for representative solutions. 1400-CH06 9/9/99 7:40 AM Page 143 144 Modern Analytical Chemistry 6D. 3 Complexation Reactions A more general definition of acids and bases was proposed by G. N. Lewis (1875–1946) in 1923. The Br?nsted–Lowry definition of acids and bases focuses on an acid’s proton-donating ability and a base’s proton-accepting ability. Lewis the- ory, on the other hand, uses the breaking and forming of covalent bonds to describe acid–base characteristics. In this treatment, an acid is an electron pair acceptor, and a base is an electron pair donor. Although Lewis theory can be applied to the treat- ment of acid–base reactions, it is more useful for treating complexation reactions between metal ions and ligands. The following reaction between the metal ion Cd 2+ and the ligand NH 3 is typi- cal of a complexation reaction. Cd 2+ (aq) + 4(:NH 3 )(aq) t Cd(:NH 3 ) 4 2+ (aq) 6.15 The product of this reaction is called a metal–ligand complex. In writing the equa- tion for this reaction, we have shown ammonia as :NH 3 to emphasize the pair of electrons it donates to Cd 2+ . In subsequent reactions we will omit this notation. The formation of a metal–ligand complex is described by a formation con- stant, K f . The complexation reaction between Cd 2+ and NH 3 , for example, has the following equilibrium constant 6.16 The reverse of reaction 6.15 is called a dissociation reaction and is characterized by a dissociation constant, K d , which is the reciprocal of K f . Many complexation reactions occur in a stepwise fashion. For example, the re- action between Cd 2+ and NH 3 involves four successive reactions Cd 2+ (aq)+NH 3 (aq) t Cd(NH 3 ) 2+ (aq) 6.17 Cd(NH 3 ) 2+ (aq)+NH 3 (aq) t Cd(NH 3 ) 2 2+ (aq) 6.18 Cd(NH 3 ) 2 2+ (aq)+NH 3 (aq) t Cd(NH 3 ) 3 2+ (aq) 6.19 Cd(NH 3 ) 3 2+ (aq)+NH 3 (aq) t Cd(NH 3 ) 4 2+ (aq) 6.20 This creates a problem since it no longer is clear what reaction is described by a for- mation constant. To avoid ambiguity, formation constants are divided into two cat- egories. Stepwise formation constants, which are designated as K i for the ith step, describe the successive addition of a ligand to the metal–ligand complex formed in the previous step. Thus, the equilibrium constants for reactions 6.17–6.20 are, re- spectively, K 1 , K 2 , K 3 , and K 4 . Overall, or cumulative formation constants, which are designated as b i , describe the addition of i ligands to the free metal ion. The equilibrium constant expression given in equation 6.16, therefore, is correctly iden- tified as b 4 , where b 4 = K 1 · K 2 · K 3 · K 4 In general b i = K 1 · K 2 · ... · K i Stepwise and cumulative formation constants for selected metal–ligand complexes are given in Appendix 3C. K f Cd NH Cd NH ==× + + [( )] [][] . 3 4 2 2 3 4 7 55 10 stepwise formation constant The formation constant for a metal–ligand complex in which only one ligand is added to the metal ion or to a metal–ligand complex (K i ). ligand A Lewis base that binds with a metal ion. formation constant The equilibrium constant for a reaction in which a metal and a ligand bind to form a metal–ligand complex (K f ). dissociation constant The equilibrium constant for a reaction in which a metal–ligand complex dissociates to form uncomplexed metal ion and ligand (K d ). cumulative formation constant The formation constant for a metal–ligand complex in which two or more ligands are simultaneously added to a metal ion or to a metal–ligand complex (b i ). 1400-CH06 9/9/99 7:40 AM Page 144 Chapter 6 Equilibrium Chemistry 145 Equilibrium constants for complexation reactions involving solids are defined by combining appropriate K sp and K f expressions. For example, the solubility of AgCl increases in the presence of excess chloride as the result of the following com- plexation reaction AgCl(s)+Cl – (aq) t AgCl 2 – (aq) 6.21 This reaction can be separated into three reactions for which equilibrium constants are known—the solubility of AgCl, described by its K sp AgCl(s) t Ag + (aq)+Cl – (aq) and the stepwise formation of AgCl 2 – , described by K 1 and K 2 Ag + (aq)+Cl – (aq) t AgCl(aq) AgCl(aq)+Cl – (aq) t AgCl 2 – (aq) The equilibrium constant for reaction 6.21, therefore, is equal to K sp · K 1 · K 2 . EXAMPLE 6. 4 Determine the value of the equilibrium constant for the reaction PbCl 2 (s) t PbCl 2 (aq) SOLUTION This reaction can be broken down into three reactions. The first of these reactions is the solubility of PbCl 2 , described by its K sp PbCl 2 (s) t Pb 2+ (aq) + 2Cl – (aq) and the second and third are the stepwise formation of PbCl 2 (aq), described by K 1 and K 2 Pb 2+ (aq)+Cl – (aq) t PbCl + (aq) PbCl + (aq)+Cl – (aq) t PbCl 2 (aq) Using values for K sp , K 1 , and K 2 from Appendices 3A and 3C, we find the equilibrium constant to be K = K sp · K 1 · K 2 = (1.7 · 10 –5 )(38.9)(1.62) = 1.1 · 10 –3 6D. 4 Oxidation—Reduction Reactions In a complexation reaction, a Lewis base donates a pair of electrons to a Lewis acid. In an oxidation–reduction reaction, also known as a redox reaction, electrons are not shared, but are transferred from one reactant to another. As a result of this elec- tron transfer, some of the elements involved in the reaction undergo a change in ox- idation state. Those species experiencing an increase in their oxidation state are oxi- dized, while those experiencing a decrease in their oxidation state are reduced. For example, in the following redox reaction between Fe 3+ and oxalic acid, H 2 C 2 O 4 , iron is reduced since its oxidation state changes from +3 to +2. 2Fe 3+ (aq)+H 2 C 2 O 4 (aq)+2H 2 O(l) t 2Fe 2+ (aq) + 2CO 2 (g)+2H 3 O + (aq) 6.22 redox reaction An electron-transfer reaction. 1400-CH06 9/9/99 7:40 AM Page 145 Oxalic acid, on the other hand, is oxidized since the oxidation state for carbon in- creases from +3 in H 2 C 2 O 4 to +4 in CO 2 . Redox reactions, such as that shown in equation 6.22, can be divided into sepa- rate half-reactions that individually describe the oxidation and the reduction processes. H 2 C 2 O 4 (aq)+2H 2 O(l) fi 2CO 2 (g)+2H 3 O + (aq)+2e – Fe 3+ (aq)+e – fi Fe 2+ (aq) It is important to remember, however, that oxidation and reduction reactions al- ways occur in pairs.* This relationship is formalized by the convention of calling the species being oxidized a reducing agent, because it provides the electrons for the re- duction half-reaction. Conversely, the species being reduced is called an oxidizing agent. Thus, in reaction 6.22, Fe 3+ is the oxidizing agent and H 2 C 2 O 4 is the reducing agent. The products of a redox reaction also have redox properties. For example, the Fe 2+ in reaction 6.22 can be oxidized to Fe 3+ , while CO 2 can be reduced to H 2 C 2 O 4 . Borrowing some terminology from acid–base chemistry, we call Fe 2+ the conjugate reducing agent of the oxidizing agent Fe 3+ and CO 2 the conjugate oxidizing agent of the reducing agent H 2 C 2 O 4. Unlike the reactions that we have already considered, the equilibrium position of a redox reaction is rarely expressed by an equilibrium constant. Since redox reac- tions involve the transfer of electrons from a reducing agent to an oxidizing agent, it is convenient to consider the thermodynamics of the reaction in terms of the electron. The free energy, ?G , associated with moving a charge, Q, under a potential, E, is given by ?G = EQ Charge is proportional to the number of electrons that must be moved. For a reac- tion in which one mole of reactant is oxidized or reduced, the charge, in coulombs, is Q = nF where n is the number of moles of electrons per mole of reactant, and F is Faraday’s constant (96,485 C mol –1 ). The change in free energy (in joules per mole; J/mol) for a redox reaction, therefore, is ?G =–nFE 6.23 where ?G has units of joules per mole. The appearance of a minus sign in equation 6.23 is due to a difference in the conventions for assigning the favored direction for reactions. In thermodynamics, reactions are favored when ?G is negative, and redox reactions are favored when E is positive. The relationship between electrochemical potential and the concentrations of reactants and products can be determined by substituting equation 6.23 into equation 6.3 –nFE =–nFE°+RT ln Q where E° is the electrochemical potential under standard-state conditions. Dividing through by –nF leads to the well-known Nernst equation. 146 Modern Analytical Chemistry *Separating a redox reaction into its half-reactions is useful if you need to balance the reaction. One method for balancing redox reactions is reviewed in Appendix 4. oxidation A loss of electrons. reduction A gain of electrons. reducing agent A species that donates electrons to another species. oxidizing agent A species that accepts electrons from another species. Nernst equation An equation relating electrochemical potential to the concentrations of products and reactants. 1400-CH06 9/9/99 7:40 AM Page 146 Chapter 6 Equilibrium Chemistry 147 *ln(x) = 2.303 log(x) Substituting appropriate values for R and F, assuming a temperature of 25 °C (298 K), and switching from ln to log* gives the potential in volts as 6.24 The standard-state electrochemical potential, E°, provides an alternative way of expressing the equilibrium constant for a redox reaction. Since a reaction at equilib- rium has a ?G of zero, the electrochemical potential, E, also must be zero. Substi- tuting into equation 6.24 and rearranging shows that 6.25 Standard-state potentials are generally not tabulated for chemical reactions, but are calculated using the standard-state potentials for the oxidation, E° ox , and reduction half-reactions, E° red . By convention, standard-state potentials are only listed for re- duction half-reactions, and E° for a reaction is calculated as E° reac = E° red – E° ox where both E° red and E° ox are standard-state reduction potentials. Since the potential for a single half-reaction cannot be measured, a reference half- reaction is arbitrarily assigned a standard-state potential of zero. All other reduction potentials are reported relative to this reference. The standard half-reaction is 2H 3 O + (aq)+2e – t 2H 2 O(l)+H 2 (g) Appendix 3D contains a listing of the standard-state reduction potentials for se- lected species. The more positive the standard-state reduction potential, the more favorable the reduction reaction will be under standard-state conditions. Thus, under standard-state conditions, the reduction of Cu 2+ to Cu (E° = +0.3419) is more favorable than the reduction of Zn 2+ to Zn (E° = –0.7618). EXAMPLE 6. 5 Calculate (a) the standard-state potential, (b) the equilibrium constant, and (c) the potential when [Ag + ] = 0.020 M and [Cd 2+ ] = 0.050 M, for the following reaction taking place at 25 °C. Cd(s) + 2Ag + (aq) t Cd 2+ (aq) + 2Ag(s) SOLUTION (a) In this reaction Cd is undergoing oxidation, and Ag + is undergoing reduction. The standard-state cell potential, therefore, is (b) To calculate the equilibrium constant, we substitute the values for the standard-state potential and number of electrons into equation 6.25. EE E oo Ag Ag o Cd Cd VVV== =++ // –.–(.).2 0 7996 0 4030 1 2026 E RT nF K o = log E=E n Q o – . log 0 05916 E=E RT nF Q o –ln 1 2026 0 05916 2 . . log= K 1400-CH06 9/9/99 7:40 AM Page 147 148 Modern Analytical Chemistry Solving for K gives the equilibrium constant as log K = 40.6558 K = 4.527 · 10 40 (c) The potential when the [Ag + ] is 0.020 M and the [Cd 2+ ] is 0.050 M is calculated using equation 6.24 employing the appropriate relationship for the reaction quotient Q. 6E Le Ch telier s Principle The equilibrium position for any reaction is defined by a fixed equilibrium con- stant, not by a fixed combination of concentrations for the reactants and products. This is easily appreciated by examining the equilibrium constant expression for the dissociation of acetic acid. 6.26 As a single equation with three variables, equation 6.26 does not have a unique so- lution for the concentrations of CH 3 COOH, CH 3 COO – , and H 3 O + . At constant temperature, different solutions of acetic acid may have different values for [H 3 O + ], [CH 3 COO – ] and [CH 3 COOH], but will always have the same value of K a . If a solution of acetic acid at equilibrium is disturbed by adding sodium acetate, the [CH 3 COO – ] increases, suggesting an apparent increase in the value of K a . Since K a must remain constant, however, the concentration of all three species in equa- tion 6.26 must change in a fashion that restores K a to its original value. In this case, equilibrium is reestablished by the partial reaction of CH 3 COO – and H 3 O + to pro- duce additional CH 3 COOH. The observation that a system at equilibrium responds to a stress by reequili- brating in a manner that diminishes the stress, is formalized as Le Chatelier’s prin- ciple. One of the most common stresses that we can apply to a reaction at equilib- rium is to change the concentration of a reactant or product. We already have seen, in the case of sodium acetate and acetic acid, that adding a product to a reaction mixture at equilibrium converts a portion of the products to reactants. In this in- stance, we disturb the equilibrium by adding a product, and the stress is diminished by partially reacting the excess product. Adding acetic acid has the opposite effect, partially converting the excess acetic acid to acetate. In our first example, the stress to the equilibrium was applied directly. It is also possible to apply a concentration stress indirectly. Consider, for example, the fol- lowing solubility equilibrium involving AgCl AgCl(s) t Ag + (aq)+Cl – (aq) 6.27 K a ==× + [][ ] . – – H O CH COO CH COOH 33 3 5 175 10 EE n = = = ° + + – . log [] [] .– . log (. ) (. ) . 0 05916 1 2026 0 05916 2 0 050 0 020 114 2 2 2 Cd Ag V Le Chatelier’s principle When stressed, a system that was at equilibrium returns to its equilibrium state by reacting in a manner that relieves the stress. 1400-CH06 9/9/99 7:40 AM Page 148 Chapter 6 Equilibrium Chemistry 149 The effect on the solubility of AgCl of adding AgNO 3 is obvious,* but what is the ef- fect of adding a ligand that forms a stable, soluble complex with Ag + ? Ammonia, for example, reacts with Ag + as follows Ag + (aq) + 2NH 3 (aq) t Ag(NH 3 ) 2 + (aq) 6.28 Adding ammonia decreases the concentration of Ag + as the Ag(NH 3 ) 2 + complex forms. In turn, decreasing the concentration of Ag + increases the solubility of AgCl as reaction 6.27 reestablishes its equilibrium position. Adding together reactions 6.27 and 6.28 clarifies the effect of ammonia on the solubility of AgCl, by showing that ammonia is a reactant. AgCl(s) + 2NH 3 (aq) t Ag(NH 3 ) 2 + (aq)+Cl – (aq) 6.29 EXAMPLE 6.6 What is the effect on the solubility of AgCl if HNO 3 is added to the equilibrium solution defined by reaction 6.29? SOLUTION Nitric acid is a strong acid that reacts with ammonia as shown here HNO 3 (aq)+NH 3 (aq) t NH 4 + (aq)+NO 3 – (aq) Adding nitric acid lowers the concentration of ammonia. Decreasing ammonia’s concentration causes reaction 6.29 to move from products to reactants, decreasing the solubility of AgCl. Increasing or decreasing the partial pressure of a gas is the same as increasing or decreasing its concentration. ? The effect on a reaction’s equilibrium position can be analyzed as described in the preceding example for aqueous solutes. Since the concentration of a gas depends on its partial pressure, and not on the total pressure of the system, adding or removing an inert gas has no effect on the equilibrium po- sition of a gas-phase reaction. Most reactions involve reactants and products that are dispersed in a solvent. If the amount of solvent is changed, either by diluting or concentrating the solu- tion, the concentrations of all reactants and products either decrease or increase. The effect of these changes in concentration is not as intuitively obvious as when the concentration of a single reactant or product is changed. As an example, let’s consider how dilution affects the equilibrium position for the formation of the aqueous silver-amine complex (reaction 6.28). The equilibrium constant for this reaction is 6.30 β 2 3 2 3 2 = + + [( )] [][ ] Ag NH Ag NH eq eq eq *Adding AgNO 3 decreases the solubility of AgCl. ?The relationship between pressure and concentration can be deduced from the ideal gas law. Starting with PV = nRT, we solve for the molar concentration Of course, this assumes an ideal gas (which is usually a reasonable assumption under normal laboratory conditions). Molar concentration == n V P RT 1400-CH06 9/9/99 7:40 AM Page 149 150 Modern Analytical Chemistry ladder diagram A visual tool for evaluating systems at equilibrium. where the subscript “eq” is included for clarification. If a portion of this solution is diluted with an equal volume of water, each of the concentration terms in equa- tion 6.30 is cut in half. Thus, the reaction quotient becomes which we can rewrite as Since Q is greater than b 2 , equilibrium must be reestablished by shifting the reac- tion to the left, decreasing the concentration of Ag(NH 3 ) 2 + . Furthermore, this new equilibrium position lies toward the side of the equilibrium reaction with the greatest number of solutes (one Ag + ion and two molecules of NH 3 versus the sin- gle metal–ligand complex). If the solution of Ag(NH 3 ) 2 + is concentrated, by evapo- rating some of the solvent, equilibrium is reestablished in the opposite direction. This is a general conclusion that can be applied to any reaction, whether gas-phase, liquid-phase, or solid-phase. Increasing volume always favors the direction pro- ducing the greatest number of particles, and decreasing volume always favors the direction producing the fewest particles. If the number of particles is the same on both sides of the equilibrium, then the equilibrium position is unaffected by a change in volume. 6F Ladder Diagrams When developing or evaluating an analytical method, we often need to under- stand how the chemistry taking place affects our results. We have already seen, for example, that adding NH 3 to a solution of Ag + is a poor idea if we intend to isolate the Ag + as a precipitate of AgCl (reaction 6.29). One of the primary sources of determinate method errors is a failure to account for potential chemi- cal interferences. In this section we introduce the ladder diagram as a simple graphical tool for evaluating the chemistry taking place during an analysis. 1 Using ladder dia- grams, we will be able to determine what reactions occur when several reagents are combined, estimate the approximate composition of a system at equilibrium, and evaluate how a change in solution conditions might affect our results. 6F.1 Ladder Diagrams for Acid—Base Equilibria To see how a ladder diagram is constructed, we will use the acid–base equilibrium between HF and F – HF(aq)+H 2 O(l) t H 3 O + (aq)+F – (aq) for which the acid dissociation constant is Taking the log of both sides and multiplying through by –1 gives – log – log [ ] – log [] [] , – K aHF HO F HF = + 3 K aHF HO F HF , – [][] [] = + 3 Q = ? ? ? ? ? ? ? ? ? ? ? ? =× + + 05 05 4 3 3 2 3 2 2 . (.) [( )] [][ ] Ag NH Ag NH eq eq β Q = + + ( . )[ ( ) ] (.)[ ] (.)[ ] 05 05 05 3 2 2 3 2 Ag NH Ag NH eq eq eq 1400-CH06 9/9/99 7:40 AM Page 150 Figure 6.4 Ladder diagram for HF, showing areas of predominance for HF and F – . Finally, replacing the negative log terms with p-functions and rearranging leaves us with 6.31 Examining equation 6.31 tells us a great deal about the relationship between pH and the relative amounts of F – and HF at equilibrium. If the concentrations of F – and HF are equal, then equation 6.31 reduces to pH=pK a,HF = –log(K a,HF ) = –log(6.8 · 10 –4 ) = 3.17 For concentrations of F – greater than that of HF, the log term in equation 6.31 is positive and pH>pK a,HF or pH > 3.17 This is a reasonable result since we expect the concentration of hydrofluoric acid’s conjugate base, F – , to increase as the pH increases. Similar reasoning shows that the concentration of HF exceeds that of F – when pH < pK a,HF or pH < 3.17 Now we are ready to construct the ladder diagram for HF (Figure 6.4). The ladder diagram consists of a vertical scale of pH values oriented so that smaller (more acidic) pH levels are at the bottom and larger (more basic) pH levels are at the top. A horizontal line is drawn at a pH equal to pK a,HF . This line, or step, separates the solution into regions where each of the two conjugate forms of HF predominate. By referring to the ladder diagram, we see that at a pH of 2.5 hydrofluoric acid will exist predominately as HF. If we add sufficient base to the solution such that the pH increases to 4.5, the predominate form be- comes F – . Figure 6.5 shows a second ladder diagram containing information about HF/F – and NH 4 + /NH 3 . From this ladder diagram we see that if the pH is less than 3.17, the predominate species are HF and NH 4 + . For pH’s between 3.17 and 9.24 the predominate species are F – and NH 4 + , whereas above a pH of 9.24 the predominate species are F – and NH 3 . Ladder diagrams are particularly useful for evaluating the reactivity of acids and bases. An acid and a base cannot coexist if their respective areas of predominance do not overlap. If we mix together solutions of NH 3 and HF, the reaction HF(aq)+NH 3 (aq) t NH 4 + (aq)+F – (aq) 6.32 occurs because the predominance areas for HF and NH 3 do not overlap. Be- fore continuing, let us show that this conclusion is reasonable by calculating the equilibrium constant for reaction 6.32. To do so we need the following three reactions and their equilibrium constants. HF H O H O F NH H O OH NH HO OH HO a b w () () () () . () () () () . () () () . –– – – aq aq aq K aq aq aq K aq aq K K ++=× +== × + + + 23 4 32 4 5 14 68 10 175 10 2 11 100 10 l l l t t t pH p F HF a =+K log [] [] – Chapter 6 Equilibrium Chemistry 151 pH = pK a,HF = 3.17 HF F – pH pH = pK a,HF = 3.17 HF NH 3 NH 4 + F – pH pH = pK a,NH 3 = 9.24 Figure 6.5 Ladder diagram for HF and NH 3 . 1400-CH06 9/9/99 7:40 AM Page 151 152 Modern Analytical Chemistry Adding together these reactions gives us reaction 6.32, for which the equilibrium constant is Since the equilibrium constant is significantly greater than 1, the reaction’s equilib- rium position lies far to the right. This conclusion is general and applies to all lad- der diagrams. The following example shows how we can use the ladder diagram in Figure 6.5 to evaluate the composition of any solution prepared by mixing together solutions of HF and NH 3 . EXAMPLE 6. 7 Predict the pH and composition of a solution prepared by adding 0.090 mol of HF to 0.040 mol of NH 3 . SOLUTION Since HF is present in excess and the reaction between HF and NH 3 is favorable, the NH 3 will react to form NH 4 + . At equilibrium, essentially no NH 3 remains and Moles NH 4 + = 0.040 mol Converting NH 3 to NH 4 + consumes 0.040 mol of HF; thus Moles HF = 0.090 – 0.040 = 0.050 mol Moles F – = 0.040 mol According to the ladder diagram for this system (see Figure 6.5), a pH of 3.17 results when there is an equal amount of HF and F – . Since we have more HF than F – , the pH will be slightly less than 3.17. Similar reasoning will show you that mixing together 0.090 mol of NH 3 and 0.040 mol of HF will result in a solution whose pH is slightly larger than 9.24. If the areas of predominance for an acid and a base overlap each other, then practically no reaction occurs. For example, if we mix together solutions of NaF and NH 4 Cl, we expect that there will be no significant change in the moles of F – and NH 4 + . Furthermore, the pH of the mixture must be between 3.17 and 9.24. Because F – and NH 4 + can coexist over a range of pHs we cannot be more specific in estimat- ing the solution’s pH. The ladder diagram for HF/F – also can be used to evaluate the effect of pH on other equilibria that include either HF or F – . For example, the solubility of CaF 2 CaF 2 (s) t Ca 2+ (aq)+2F – (aq) is affected by pH because F – is a weak base. Using Le Chatelier’s principle, if F – is converted to HF, the solubility of CaF 2 will increase. To minimize the solubility of CaF 2 we want to control the solution’s pH so that F – is the predominate species. From the ladder diagram we see that maintaining a pH of more than 3.17 ensures that solubility losses are minimal. K KK K == ×× × =× ab w (. )(. ) (. ) . –– – 68 10 175 10 100 10 119 10 45 14 6 1400-CH06 9/9/99 7:40 AM Page 152 Figure 6.6 Ladder diagram for metal–ligand complexes of Cd 2+ and NH 3 . Chapter 6 Equilibrium Chemistry 153 Cd(NH 3 ) 3 2+ Cd(NH 3 ) 2 2+ Cd(NH 3 ) 2+ Cd 2+ log K 1 = 2.55 log K 4 = 0.84 log K 3 = 1.34 log K 2 = 2.01 Cd(NH 3 ) 4 2+ p NH 3 6F.2 Ladder Diagrams for Complexation Equilibria The same principles used in constructing and interpreting ladder diagrams for acid–base equilibria can be applied to equilibria involving metal–ligand com- plexes. For complexation reactions the ladder diagram’s scale is defined by the concentration of uncomplexed, or free ligand, pL. Using the formation of Cd(NH 3 ) 2+ as an example Cd 2+ (aq)+NH 3 (aq) t Cd(NH 3 ) 2+ (aq) we can easily show that the dividing line between the predominance regions for Cd 2+ and Cd(NH 3 ) 2+ is log(K 1 ). Since K 1 for Cd(NH 3 ) 2+ is 3.55 · 10 2 , log(K 1 ) is 2.55. Thus, for a pNH 3 greater than 2.55 (concentrations of NH 3 less than 2.8 · 10 –3 M), Cd 2+ is the predominate species. A complete ladder diagram for the metal–ligand complexes of Cd 2+ and NH 3 is shown in Figure 6.6. EXAMPLE 6.8 Using the ladder diagram in Figure 6.7, predict the result of adding 0.080 mol of Ca 2+ to 0.060 mol of Mg(EDTA) 2– . EDTA is an abbreviation for the ligand ethylenediaminetetraacetic acid. SOLUTION The predominance regions for Ca 2+ and Mg(EDTA) 2– do not overlap, therefore, the reaction Ca 2+ + Mg(EDTA) 2– t Mg 2+ + Ca(EDTA) 2– will take place. Since there is an excess of Ca 2+ , the composition of the final solution is approximately Moles Ca 2+ = 0.080 – 0.060 = 0.020 mol Moles Ca(EDTA) 2– = 0.060 mol K K K K 1 3 2 2 3 1 3 2 2 3 1 3 2 2 3 31 2 3 2 = = =+ =+ + + + + + + + + [( )] [][] log( ) log [( )] [] – log[ ] log( ) log [( )] [] log( ) log [] [( )] Cd NH Cd NH Cd NH Cd NH Cd NH Cd pNH pNH Cd Cd NH 1400-CH06 9/9/99 7:40 AM Page 153 Figure 6.7 Ladder diagram for metal–ligand complexes of ethylenediaminetetraacetic acid (EDTA) with Ca 2+ and Mg 2+ . Moles Mg 2+ = 0.060 mol Moles Mg(EDTA) 2– = 0 mol 154 Modern Analytical Chemistry Mg 2+ Ca 2+ Ca(EDTA) 2– Mg(EDTA) 2– log K f,Mg(EDTA) 2– = 8.79 log K f,Ca(EDTA) 2– = 10.69 p EDTA log log [( )] [] – log [ ]β 3 3 3 2 2 3 3= + + Zn NH Zn NH We can also construct ladder diagrams using cumulative formation constants in place of stepwise formation constants. The first three stepwise formation con- stants for the reaction of Zn 2+ with NH 3 Zn 2+ (aq)+NH 3 (aq) t Zn(NH 3 ) 2+ (aq) K 1 = 1.6 · 10 2 Zn(NH 3 ) 2+ (aq)+NH 3 (aq) t Zn(NH 3 ) 2 2+ (aq) K 2 = 1.95 · 10 2 Zn(NH 3 ) 2 2+ (aq)+NH 3 (aq) t Zn(NH 3 ) 3 2+ (aq) K 3 = 2.3 · 10 2 show that the formation of Zn(NH 3 ) 3 2+ is more favorable than the formation of Zn(NH 3 ) 2+ or Zn(NH 3 ) 2 2+ . The equilibrium, therefore, is best represented by the cumulative formation reaction Zn 2+ (aq) + 3NH 3 (aq) t Zn(NH 3 ) 3 2+ (aq) for which Taking the log of each side gives β 3 3 3 2 2 3 3 6 72 10==× + + [( )] [][] . Zn NH Zn NH 1400-CH06 9/9/99 7:40 AM Page 154 Figure 6.8 Ladder diagram for Zn 2+ , Zn(NH 3 ) 3 2+ , and Zn(NH 3 ) 4 2+ , showing how cumulative formation constants are included. or The concentrations of Zn 2+ and Zn(NH 3 ) 3 2+ , therefore, are equal when A complete ladder diagram for the Zn 2+ –NH 3 system is shown in Figure 6.8. 6F. 3 Ladder Diagram for Oxidation—Reduction Equilibria Ladder diagrams can also be used to evaluate equilibrium reactions in redox sys- tems. Figure 6.9 shows a typical ladder diagram for two half-reactions in which the scale is the electrochemical potential, E. Areas of predominance are defined by the Nernst equation. Using the Fe 3+ /Fe 2+ half-reaction as an example, we write For potentials more positive than the standard-state potential, the predominate species is Fe 3+ , whereas Fe 2+ predominates for potentials more negative than E°. When coupled with the step for the Sn 4+ /Sn 2+ half-reaction, we see that Sn 2+ can be used to reduce Fe 3+ . If an excess of Sn 2+ is added, the potential of the resulting solu- tion will be near +0.154 V. Using standard-state potentials to construct a ladder diagram can present problems if solutes are not at their standard-state concentra- tions. Because the concentrations of the reduced and oxidized species are in a logarithmic term, deviations from standard-state concentra- tions can usually be ignored if the steps being compared are separated by at least 0.3 V. 1b A trickier problem occurs when a half-reaction’s po- tential is affected by the concentration of another species. For example, the potential for the following half-reaction UO 2 2+ (aq)+4H 3 O + (aq)+2e – t U 4+ (aq)+6H 2 O(l) depends on the pH of the solution. To define areas of predominance in this case, we begin with the Nernst equation and factor out the concentration of H 3 O + . From this equation we see that the areas of predominance for UO 2 2+ and U 4+ are defined by a step whose potential is Figure 6.10 shows how a change in pH affects the step for the UO 2 2+ /U 4+ half-reaction. E =+ = + 0 327 0 05916 2 0 327 0 1183 3 4 . . log [ ] . – .HO pH E =+ + + + 0 327 0 05916 2 0 05916 2 3 4 4 2 2 . . log [ ] – . log [] [] HO U UO E = + ++ 0 327 0 05916 2 4 2 2 3 4 .– . log [] [][] U UO H O EE==+ ° + + + + – . log [] [] . – . log [] [] 0 05916 0 771 0 05916 2 3 2 3 Fe Fe V Fe Fe pNH 33 6 1 3 1 3 72 10 229== ×=log log( . ) .β pNH Zn Zn NH 33 2 3 3 2 1 3 1 3 =+ + + log log [] [( )] β Chapter 6 Equilibrium Chemistry 155 Zn 2+ Zn(NH 3 ) 3 2+ Zn(NH 3 ) 4 2+ log K 4 = 2.03 pNH 3 3 log b 3 = 2.29 Fe 3+ Fe 2+ E E ° Fe 3+ /Fe 2+ = +0.771 Sn 4+ Sn 2+ E ° Sn 4+ /Sn 2+ = +0.154 Figure 6.9 Ladder diagram for the Fe 3+ /Fe 2+ and Sn 4+ /SN 2+ half- reactions. 1400-CH06 9/9/99 7:40 AM Page 155 Figure 6.10 Ladder diagram showing the effect of a change in pH on the areas of predominance for the UO 2 2+ /U 4+ half-reaction. 156 Modern Analytical Chemistry UO 2 2+ U 4+ +0.090 V (pH = 2) +0.209 V (pH = 1) +0.327 V (pH = 0) E 6G Solving Equilibrium Problems Ladder diagrams are a useful tool for evaluating chemical reactivity, usually provid- ing a reasonable approximation of a chemical system’s composition at equilibrium. When we need a more exact quantitative description of the equilibrium condition, a ladder diagram may not be sufficient. In this case we can find an algebraic solution. Perhaps you recall solving equilibrium problems in your earlier coursework in chemistry. In this section we will learn how to set up and solve equilibrium prob- lems. We will start with a simple problem and work toward more complex ones. 6G.1 A Simple Problem: Solubility of Pb(IO 3 ) 2 in Water When an insoluble compound such as Pb(IO 3 ) 2 is added to a solution a small por- tion of the solid dissolves. Equilibrium is achieved when the concentrations of Pb 2+ and IO 3 – are sufficient to satisfy the solubility product for Pb(IO 3 ) 2 . At equilibrium the solution is saturated with Pb(IO 3 ) 2 . How can we determine the concentrations of Pb 2+ and IO 3 – , and the solubility of Pb(IO 3 ) 2 in a saturated solution prepared by adding Pb(IO 3 ) 2 to distilled water? We begin by writing the equilibrium reaction Pb(IO 3 ) 2 (s) t Pb 2+ (aq) + 2IO 3 – (aq) and its equilibrium constant K sp = [Pb 2+ ][IO 3 – ] 2 = 2.5 · 10 –13 6.33 As equilibrium is established, two IO 3 – ions are produced for each ion of Pb 2+ . If we assume that the molar concentration of Pb 2+ at equilibrium is x then the molar con- centration of IO 3 – is 2x. To help keep track of these relationships, we can use the following table. PbI 2 (s) t Pb 2+ (aq) + 2IO 3 – (aq) Initial concentration solid 0 0 Change in concentration solid +x +2x Equilibrium concentration solid 0 + x = x 0+2x =2x Substituting the equilibrium concentrations into equation 6.33 (x)(2x) 2 = 2.5 · 10 –13 and solving gives 4x 3 = 2.5 · 10 –13 x = 3.97 · 10 –5 The equilibrium concentrations of Pb 2+ and IO 3 – , therefore, are [Pb 2+ ]=x = 4.0 · 10 –5 M [I – ]=2x = 7.9 · 10 –5 M Since one mole of Pb(IO 3 ) 2 contains one mole of Pb 2+ , the solubility of Pb(IO 3 ) 2 is the same as the concentration of Pb 2+ ; thus, the solubility of Pb(IO 3 ) 2 is 4.0 · 10 –5 M. 1400-CH06 9/9/99 7:40 AM Page 156 6G.2 A More Complex Problem: The Common Ion Effect Calculating the solubility of Pb(IO 3 ) 2 in distilled water is a straightforward prob- lem since the dissolution of the solid is the only source of Pb 2+ or IO 3 – . How is the solubility of Pb(IO 3 ) 2 affected if we add Pb(IO 3 ) 2 to a solution of 0.10 M Pb(NO 3 ) 2 ? Before we set up and solve the problem algebraically, think about the chemistry occurring in this system, and decide whether the solubility of Pb(IO 3 ) 2 will increase, decrease, or remain the same. This is a good habit to develop. Knowing what answers are reasonable will help you spot errors in your calcula- tions and give you more confidence that your solution to a problem is correct. We begin by setting up a table to help us keep track of the concentrations of Pb 2+ and IO 3 – in this system. PbI 2 (s) t Pb 2+ (aq) + 2IO 3 – (aq) Initial concentration solid 0.10 0 Change in concentration solid +x +2x Equilibrium concentration solid 0.10 + x = x 0+2x =2x Substituting the equilibrium concentrations into the solubility product expression (equation 6.33) (0.10 + x)(2x) 2 = 2.5 · 10 –13 and multiplying out the terms on the left leaves us with 4x 3 + 0.40x 2 = 2.5 · 10 –13 6.34 This is a more difficult equation to solve than that for the solubility of Pb(IO 3 ) 2 in dis- tilled water, and its solution is not immediately obvious. A rigorous solution to equa- tion 6.34 can be found using available computer software packages and spreadsheets. How might we solve equation 6.34 if we do not have access to a computer? One possibility is that we can apply our understanding of chemistry to simplify the algebra. From Le Chatelier’s principle, we expect that the large initial concentration of Pb 2+ will significantly decrease the solubility of Pb(IO 3 ) 2 . In this case we can reasonably expect the equilibrium concentration of Pb 2+ to be very close to its initial concentration; thus, the following approximation for the equilibrium concentration of Pb 2+ seems reasonable [Pb 2+ ] = 0.10 + x ? 0.10 M Substituting into equation 6.34 (0.10)(2x) 2 = 2.5 · 10 –13 and solving for x gives 0.40x 2 = 2.5 · 10 –13 x = 7.91 · 10 –7 Before accepting this answer, we check to see if our approximation was reasonable. In this case the approximation 0.10 + x ? 0.10 seems reasonable since the difference between the two values is negligible. The equilibrium concentrations of Pb 2+ and IO 3 – , therefore, are [Pb 2+ ] = 0.10 + x ? 0.10 M [I – ]=2x = 1.6 · 10 –6 M Chapter 6 Equilibrium Chemistry 157 1400-CH06 9/9/99 7:40 AM Page 157 158 Modern Analytical Chemistry common ion effect The solubility of an insoluble salt decreases when it is placed in a solution already containing one of the salt’s ions. The solubility of Pb(IO 3 ) 2 is equal to the additional concentration of Pb 2+ in solu- tion, or 7.9 · 10 –7 mol/L. As expected, the solubility of Pb(IO 3 ) 2 decreases in the presence of a solution that already contains one of its ions. This is known as the common ion effect. As outlined in the following example, the process of making and evaluating ap- proximations can be extended if the first approximation leads to an unacceptably large error. EXAMPLE 6. 9 Calculate the solubility of Pb(IO 3 ) 2 in 1.0 · 10 –4 M Pb(NO 3 ) 2 . SOLUTION Letting x equal the change in the concentration of Pb 2+ , the equilibrium concentrations are [Pb 2+ ] = 1.0 · 10 –4 + x [IO 3 – ]=2x and (1.0 · 10 –4 + x)(2x) 2 = 2.5 · 10 –13 We start by assuming that [Pb 2+ ] = 1.0 · 10 –4 + x ≈ 1.0 · 10 –4 M and solve for x, obtaining a value of 2.50 · 10 –5 . Substituting back gives the calculated concentration of Pb 2+ at equilibrium as [Pb 2+ ] = 1.0 · 10 –4 + 2.50 · 10 –5 = 1.25 · 10 –4 M a value that differs by 25% from our approximation that the equilibrium concentration is 1.0 · 10 –4 M. This error seems unreasonably large. Rather than shouting in frustration, we make a new assumption. Our first assumption that the concentration of Pb 2+ is 1.0 · 10 –4 M was too small. The calculated concentration of 1.25 · 10 –4 M, therefore, is probably a little too large. Let us assume that [Pb 2+ ] = 1.0 · 10 –4 + x ≈1.2 · 10 –4 M Substituting into the solubility product equation and solving for x gives us x = 2.28 · 10 –5 or a concentration of Pb 2+ at equilibrium of [Pb 2+ ] = 1.0 · 10 –4 + (2.28 · 10 –5 ) = 1.23 · 10 –4 M which differs from our assumed concentration of 1.2 · 10 –4 M by 2.5%. This seems to be a reasonable error since the original concentration of Pb 2+ is given to only two significant figures. Our final solution, to two significant figures, is [Pb 2+ ] = 1.2 · 10 –4 M [IO 3 – ] = 4.6 · 10 –5 M and the solubility of Pb(IO 3 ) 2 is 2.3 · 10 –5 mol/L. This iterative approach to solving an equation is known as the method of successive approximations. 1400-CH06 9/9/99 7:40 AM Page 158 6G. 3 Systematic Approach to Solving Equilibrium Problems Calculating the solubility of Pb(IO 3 ) 2 in a solution of Pb(NO 3 ) 2 was more com- plicated than calculating its solubility in distilled water. The necessary calcula- tions, however, were still relatively easy to organize, and the assumption used to simplify the problem was fairly obvious. This problem was reasonably straight- forward because it involved only a single equilibrium reaction, the solubility of Pb(IO 3 ) 2 . Calculating the equilibrium composition of a system with multiple equilibrium reactions can become quite complicated. In this section we will learn how to use a systematic approach to setting up and solving equilibrium problems. As its name implies, a systematic approach involves a series of steps: 1. Write all relevant equilibrium reactions and their equilibrium constant expressions. 2. Count the number of species whose concentrations appear in the equilibrium constant expressions; these are your unknowns. If the number of unknowns equals the number of equilibrium constant expressions, then you have enough information to solve the problem. If not, additional equations based on the conservation of mass and charge must be written. Continue to add equations until you have the same number of equations as you have unknowns. 3. Decide how accurate your final answer needs to be. This decision will influence your evaluation of any assumptions you use to simplify the problem. 4. Combine your equations to solve for one unknown (usually the one you are most interested in knowing). Whenever possible, simplify the algebra by making appropriate assumptions. 5. When you obtain your final answer, be sure to check your assumptions. If any of your assumptions prove invalid, then return to the previous step and continue solving. The problem is complete when you have an answer that does not violate any of your assumptions. Besides equilibrium constant equations, two other types of equations are used in the systematic approach to solving equilibrium problems. The first of these is a mass balance equation, which is simply a statement of the conservation of matter. In a solution of a monoprotic weak acid, for example, the combined concentrations of the conjugate weak acid, HA, and the conjugate weak base, A – , must equal the weak acid’s initial concentration, C HA .* The second type of equation is a charge balance equation. A charge balance equation is a statement of solution electroneutrality. Total positive charge from cations = total negative charge from anions Mathematically, the charge balance expression is expressed as where [M z+ ] i and [A z– ] j are, respectively, the concentrations of the ith cation and the jth anion, and (z + ) i and (z – ) j are the charges of the ith cation and the jth anion. Note that the concentration terms are multiplied by the absolute values of each ion’s charge, since electroneutrality is a conservation of charge, not con- centration. Every ion in solution, even those not involved in any equilibrium () [ ] () [ ] –– zz ii i n z j z j j m + == ∑∑ ×= × 11 MA + Chapter 6 Equilibrium Chemistry 159 charge balance equation An equation stating that the total concentration of positive charge in a solution must equal the total concentration of negative charge. mass balance equation An equation stating that matter is conserved, and that the total amount of a species added to a solution must equal the sum of the amount of each of its possible forms present in solution. *You may recall that this is the difference between a formal concentration and a molar concentration. 1400-CH06 9/9/99 7:40 AM Page 159 160 Modern Analytical Chemistry reactions, must be included in the charge balance equation. The charge balance equation for an aqueous solution of Ca(NO 3 ) 2 is 2 · [Ca 2+ ]+[H 3 O + ] = [OH – ] + [NO 3 – ] Note that the concentration of Ca 2+ is multiplied by 2, and that the concentrations of H 3 O + and OH – are also included. Charge balance equations must be written carefully since every ion in solution must be included. This presents a problem when the concentration of one ion in solution is held constant by a reagent of un- specified composition. For example, in many situations pH is held constant using a buffer. If the composition of the buffer is not specified, then a charge balance equa- tion cannot be written. EXAMPLE 6.10 Write a mass balance and charge balance equations for a 0.10 M solution of NaHCO 3 . SOLUTION It is easier to keep track of what species are in solution if we write down the reactions that control the solution’s composition. These reactions are the dissolution of a soluble salt NaHCO 3 (s) fi Na + (aq) + HCO 3 – (aq) and the acid–base dissociation reactions of HCO 3 – and H 2 O HCO 3 – (aq)+H 2 O(l) t H 3 O + (aq)+CO 3 2– (aq) HCO 3 – (aq)+H 2 O(l) t OH – (aq)+H 2 CO 3 (aq) 2H 2 O(l) t H 3 O + (aq)+OH – (aq) The mass balance equations are 0.10 M = [H 2 CO 3 ] + [HCO 3 – ] + [CO 3 2– ] 0.10 M = [Na + ] The charge balance equation is [Na + ]+[H 3 O + ] = [OH – ] + [HCO 3 – ]+2· [CO 3 2– ] 6G. 4 pH of a Monoprotic Weak Acid To illustrate the systematic approach, let us calculate the pH of 1.0 M HF. Two equilbria affect the pH of this system. The first, and most obvious, is the acid disso- ciation reaction for HF HF(aq)+H 2 O(l) t H 3 O + (aq)+F – (aq) for which the equilibrium constant expression is 6.35 K a HO F HF ==× + [][] [] . – – 3 4 68 10 1400-CH06 9/9/99 7:40 AM Page 160 The second equilibrium reaction is the dissociation of water, which is an obvious yet easily disregarded reaction 2H 2 O(l) t H 3 O + (aq)+OH – (aq) K w =[H 3 O + ][OH – ] = 1.00 · 10 –14 6.36 Counting unknowns, we find four ([HF], [F – ], [H 3 O + ], and [OH – ]). To solve this problem, therefore, we need to write two additional equations involving these un- knowns. These equations are a mass balance equation C HF =[HF]+[F – ] 6.37 and a charge balance equation [H 3 O + ]=[F – ] + [OH – ] 6.38 We now have four equations (6.35, 6.36, 6.37, and 6.38) and four unknowns ([HF], [F – ], [H 3 O + ], and [OH – ]) and are ready to solve the problem. Before doing so, however, we will simplify the algebra by making two reasonable assumptions. First, since HF is a weak acid, we expect the solution to be acidic; thus it is reason- able to assume that [H 3 O + ] >> [OH – ] simplifying the charge balance equation (6.38) to [H 3 O + ]=[F – ] 6.39 Second, since HF is a weak acid we expect that very little dissociation occurs, and [HF] >> [F – ] Thus, the mass balance equation (6.36) simplifies to C HF = [HF] 6.40 For this exercise we will accept our assumptions if the error introduced by each as- sumption is less than ±5%. Substituting equations 6.39 and 6.40 into the equilibrium constant expression for the dissociation of HF (equation 6.35) and solving for the concentration of H 3 O + gives us Before accepting this answer, we must verify that our assumptions are acceptable. The first assumption was that the [OH – ] is significantly smaller than the [H 3 O + ]. To calculate the concentration of OH – we use the K w expression (6.36) Clearly this assumption is reasonable. The second assumption was that the [F – ] is significantly smaller than the [HF]. From equation 6.39 we have [F – ] = 2.6 · 10 –2 M [] [] . . . – – – – OH HO M w == × × =× + K 3 14 2 13 100 10 26 10 38 10 [ ] ( . )( . ) . –– HO M aHF3 42 68 10 10 26 10 + ==× =×KC K C a HF HO HO = ++ [][] 33 Chapter 6 Equilibrium Chemistry 161 1400-CH06 9/9/99 7:40 AM Page 161 162 Modern Analytical Chemistry Since the [F – ] is 2.6% of C HF , this assumption is also within our limit that the error be no more than ±5%. Accepting our solution for the concentration of H 3 O + , we find that the pH of 1.0 M HF is 1.59. How does the result of this calculation change if we require our assumptions to have an error of less than ±1%. In this case we can no longer assume that [HF] >> [F – ]. Solving the mass balance equation (6.37) for [HF] [HF] = C HF –[F – ] and substituting into the K a expression along with equation 6.39 gives Rearranging leaves us with a quadratic equation [H 3 O + ] 2 = K a C HF – K a [H 3 O + ] [H 3 O + ] 2 + K a [H 3 O + ]–K a C HF =0 which we solve using the quadratic formula where a, b, and c are the coefficients in the quadratic equation ax 2 + bx + c =0. Solving the quadratic formula gives two roots, only one of which has any chemical significance. For our problem the quadratic formula gives roots of Only the positive root has any chemical significance since the negative root implies that the concentration of H 3 O + is negative. Thus, the [H 3 O + ] is 2.6 · 10 –2 M, and the pH to two significant figures is still 1.59. This same approach can be extended to find the pH of a monoprotic weak base, replacing K a with K b , C HF with the weak base’s concentration, and solving for the [OH – ] in place of [H 3 O + ]. EXAMPLE 6.11 Calculate the pH of 0.050 M NH 3 . State any assumptions made in simplifying the calculation, and verify that the error is less than 5%. SOLUTION Since NH 3 is a weak base (K b = 1.75 · 10 –5 ), we assume that [OH – ] >> [H 3 O + ] and C NH 3 = 0.050 M With these assumptions, we find (be sure to check the derivation) [ ] ( . )( . ) . ––– OH M bNH ==× =×KC 3 175 10 0050 935 10 54 x = ×± × × = ×±× =× × – . ( . ) – ( )( )(– . )( . ) () –. . .–. –– – –– –– 68 10 68 10 4 1 68 10 10 21 68 10 522 10 2 2 57 10 2 63 10 442 4 42 22 or x bb ac a = ±–– 2 4 2 K C a HF HO HO = + + [] –[ ] 3 2 3 1400-CH06 9/9/99 7:41 AM Page 162 Figure 6.12 Ladder diagram for the amino acid alanine. Both assumptions are acceptable (again, verify that this is true). The concentration of H 3 O + is calculated using K w giving a pH of 10.97. [] [] . . . – – – – HO OH w 3 14 4 11 100 10 935 10 107 10 + == × × =× K Chapter 6 Equilibrium Chemistry 163 CH 3 H 2 L + COOH + H 3 N C H CH 3 HL pK a1 = 2.348 COO –+ H 3 N C H pK a2 = 9.867 CH 3 L – COO – H 2 N C H 2.348 9.867 pH CH 3 H 2 L + COOH + H 3 N C H CH 3 HL COO –+ H 3 N C H CH 3 L – COO – H 2 N C H Figure 6.11 Acid–base equilibria for the amino acid alanine. 6G. 5 pH of a Polyprotic Acid or Base A more challenging problem is to find the pH of a solution prepared from a polyprotic acid or one of its conjugate species. As an example, we will use the amino acid alanine whose structure and acid dissociation constants are shown in Figure 6.11. pH of 0.10 M H 2 L + Alanine hydrochloride is a salt consisting of the diprotic weak acid H 2 L + and Cl – . Because H 2 L + has two acid dissociation reactions, a complete systematic solution to this problem will be more complicated than that for a mono- protic weak acid. Using a ladder diagram (Figure 6.12) can help us simplify the problem. Since the areas of predominance for H 2 L + and L – are widely separated, we can assume that any solution containing an appreciable quantity of H 2 L + will con- tain essentially no L – . In this case, HL is such a weak acid that H 2 L + behaves as if it were a monoprotic weak acid. To find the pH of 0.10 M H 2 L + , we assume that [H 3 O + ] >> [OH – ] Because H 2 L + is a relatively strong weak acid, we cannot simplify the problem fur- ther, leaving us with Solving the resulting quadratic equation gives the [H 3 O + ] as 1.91 · 10 –2 M or a pH of 1.72. Our assumption that [H 3 O + ] is significantly greater than [OH – ] is acceptable. pH of 0.10 M L — The alaninate ion is a diprotic weak base, but using the ladder dia- gram as a guide shows us that we can treat it as if it were a monoprotic weak base. Following the steps in Example 6.11 (which is left as an exercise), we find that the pH of 0.10 M alaninate is 11.42. K C a HL HO HO = + + + [] –[ ] 3 2 3 2 1400-CH06 9/9/99 7:41 AM Page 163 pH of 0.1 M HL Finding the pH of a solution of alanine is more complicated than that for H 2 L + or L – because we must consider two equilibrium reactions involving HL. Alanine is an amphiprotic species, behaving as an acid HL(aq)+H 2 O(l) t H 3 O + (aq)+L – (aq) and a base HL(aq)+H 2 O(l) t OH – (aq)+H 2 L + (aq) As always, we must also consider the dissociation of water 2H 2 O(l) t H 3 O + (aq)+OH – (aq) This leaves us with five unknowns ([H 2 L + ], [HL], [L – ], [H 3 O + ], and [OH – ]), for which we need five equations. These equations are K a2 and K b2 for HL, the K w equation, K w =[H 3 O + ][OH – ] a mass balance equation on HL, C HL =[H 2 L + ]+[HL]+[L – ] and a charge balance equation [H 2 L + ]+[H 3 O + ] = [OH – ]+[L – ] From the ladder diagram it appears that we may safely assume that the concentra- tions of H 2 L + and L – are significantly smaller than that for HL, allowing us to sim- plify the mass balance equation to C HL = [HL] Next we solve K b2 for [H 2 L + ] and K a2 for [L – ] Substituting these equations, along with the equation for K w , into the charge bal- ance equation gives us which simplifies to 6.41 [] [] () [] (/) [] HO HO HO HO HL a waHL aHL w HL a aaHL aw HL a 3 1 3 2 3 2 2 1 3 12 1 1 1 1 1 + + + + + ? ? ? ? ? ? =+ = + + = + + C K KKC KC K CK KKC KK CK C K KKC HL a waHL HO HO HO HO [] [] [][] 3 1 3 3 2 3 + + ++ += + [] [] [][] – L HL HO HO aaHL == ++ KKC 2 3 2 3 [] [] [] [][ ] [ ] – HL HL OH HL HO HO w a a HL a 2 1 3 1 3 1 + ++ == = K K K C K K K K K a b w a HO L HL OH H L HL 2 3 2 1 2 = == + + [][] [] [][ ] [] – – 164 Modern Analytical Chemistry 1400-CH06 9/9/99 7:41 AM Page 164 Chapter 6 Equilibrium Chemistry 165 We can simplify this equation further if K a1 K w << K a1 K a2 C HL , and if K a1 << C HL , giving 6.42 For a solution of 0.10 M alanine, the [H 3 O + ] is or a pH of 6.11. Verifying that the assumptions are acceptable is left as an exercise. Triprotic Acids and Bases, and Beyond The treatment of a diprotic acid or base is easily extended to acids and bases having three or more acid–base sites. For a tripro- tic weak acid such as H 3 PO 4 , for example, we can treat H 3 PO 4 as if it was a mono- protic weak acid, H 2 PO 4 – and HPO 4 2– as if they were intermediate forms of diprotic weak acids, and PO 4 3– as if it was a monoprotic weak base. EXAMPLE 6.12 Calculate the pH of 0.10 M Na 2 HPO 4 . SOLUTION We treat HPO 4 2– as the intermediate form of a diprotic weak acid H 2 PO 4 – (aq) t HPO 4 2– (aq) t PO 4 3– (aq) where the equilibrium constants are K a2 = 6.32 · 10 –8 and K a3 = 4.5 · 10 –13 . Since the value of K a3 is so small, we use equation 6.41 instead of equation 6.42. or a pH of 9.73. 6G.6 Effect of Complexation on Solubility The solubility of a precipitate can be improved by adding a ligand capable of forming a soluble complex with one of the precipitate’s ions. For example, the solubility of AgI increases in the presence of NH 3 due to the formation of the soluble Ag(NH 3 ) 2 + complex. As a final illustration of the systematic approach to solving equilibrium problems, let us find the solubility of AgI in 0.10 M NH 3 . We begin by writing the equilibria that we need to consider AgI(s) t Ag + (aq)+I – (aq) Ag + (aq) + 2NH 3 (aq) t Ag(NH 3 ) 2 + (aq) NH 3 (aq)+H 2 O(l) t OH – (aq)+NH 4 + (aq) 2H 2 O(l) t H 3 O + (aq)+OH – (aq) [] ( . )( . )( . ) ( . )( . ) .. . –– – – – – HO 3 813 8 14 8 10 6 32 10 4 5 10 0 10 6 32 10 1 00 10 0 10 6 32 10 186 10 + = ×× +× × +× =× [ ] ( . )( . ) . –– – HO M 3 310 7 4 487 10 1 358 10 7 807 10 + =× ×=× []HO aa312 + = KK 1400-CH06 9/9/99 7:41 AM Page 165 Counting unknowns, we find that there are seven—[Ag + ], [I – ], [Ag(NH 3 ) 2 + ], [NH 3 ], [NH 4 + ], [OH – ], and [H 3 O + ]. Four of the equations needed to solve this problem are given by the equilibrium constant expressions Three additional equations are needed. The first of these equations is a mass balance for NH 3 . C NH 3 = [NH 3 ] + [NH 4 + ]+2· [Ag(NH 3 ) 2 + ] Note that in writing this mass balance equation, the concentration of Ag(NH 3 ) 2 + must be multiplied by 2 since two moles of NH 3 occurs per mole of Ag(NH 3 ) 2 + . The second additional equation is a mass balance on iodide and silver. Since AgI is the only source of I – and Ag + , every iodide in solution must have an associated silver ion; thus [I – ] = [Ag + ] + [Ag(NH 3 ) 2 + ] Finally, the last equation is a charge balance equation [Ag + ] + [Ag(NH 3 ) 2 + ] + [NH 4 + ]+[H 3 O + ]=[I – ] + [OH – ] Our problem looks challenging, but several assumptions greatly simplify the al- gebra. First, since the formation of the Ag(NH 3 ) 2 + complex is favorable, we will as- sume that [Ag + ] << [Ag(NH 3 ) 2 + ] Second, since NH 3 is a base, we will assume that [H 3 O + ] << [OH – ] [NH 4 + ] << [NH 3 ] + [Ag(NH 3 ) 2 + ] Finally, since K sp is significantly smaller than b 2 , it seems likely that the solubility of AgI is small and [Ag(NH 3 ) 2 + ] << [NH 3 ] Using these assumptions allows us to simplify several equations. The mass bal- ance for NH 3 is now C NH 3 = [NH 3 ] and the mass balance for I – is [I – ] = [Ag(NH 3 ) 2 + ] Simplifying the charge balance expression by dropping [H 3 O + ] and [Ag + ], and re- placing [Ag(NH 3 ) 2 + ] with [I – ] gives K K K sp b w Ag I Ag NH Ag NH NH OH NH HO OH ==× ==× ==× ==× + + + + + [][]. [( )] [][ ] . [][] [] . [][]. –– – – –– 83 10 17 10 175 10 100 10 17 2 3 2 3 2 7 4 3 5 3 14 β 166 Modern Analytical Chemistry 1400-CH06 9/9/99 7:41 AM Page 166 Chapter 6 Equilibrium Chemistry 167 [NH 4 + ] = [OH – ] We continue by multiplying together the equations for K sp and b 2 , giving Substituting in the new mass balance equations for NH 3 and I – and solving for the [I – ] gives Before accepting this answer, we first check our assumptions. Using the K sp equation we calculate the [Ag + ] to be From the simplified mass balance equation for I – , we have [Ag(NH 3 ) 2 + ]=[I – ] = 3.7 · 10 –6 M Our first assumption that the [Ag + ] is significantly smaller than the [Ag(NH 3 ) 2 + ], therefore, is reasonable. Furthermore, our third assumption that the [Ag(NH 3 ) 2 + ] is significantly less than the [NH 3 ] also is reasonable. Our second assumption was [NH 4 + ] << [NH 3 ] + [Ag(NH 3 ) 2 + ] To verify this assumption, we solve the K b equation for [NH 4 + ] giving [NH 4 + ] = 1.3 · 10 –3 M Although the [NH 4 + ] is not significantly smaller than the combined concentrations of NH 3 and Ag(NH 3 ) 2 + , the error is only about 1%. Since this is not an excessively large error, we will accept this approximation as reasonable. Since one mole of AgI produces one mole of I – , the solubility of AgI is the same as the concentration of iodide, or 3.7 · 10 –6 mol/L. 6H Buffer Solutions Adding as little as 0.1 mL of concentrated HCl to a liter of H 2 O shifts the pH from 7.0 to 3.0. The same addition of HCl to a liter solution that is 0.1 M in both a weak acid and its conjugate weak base, however, results in only a negligible change in pH. Such solutions are called buffers, and their buffering action is a consequence of the relationship between pH and the relative concentrations of the conjugate weak acid/weak base pair. [][] [] [] . . – – NH OH NH NH M 4 3 4 2 5 010 175 10 ++ ==× [] [] . . . – – – – Ag I M sp + == × × =× K 83 10 37 10 22 10 17 6 11 [] (. ) . [] . – – –– I M IM 2 2 9 6 010 14 10 37 10 =× =× [] () . – – I NH 2 2 9 3 14 10 C =× K sp Ag NH I NH β 2 3 2 3 2 9 14 10==× + [( )][] [] . – – buffer A solution containing a conjugate weak acid/weak base pair that is resistant to a change in pH when a strong acid or strong base is added. 1400-CH06 9/9/99 7:41 AM Page 167 A mixture of acetic acid and sodium acetate is one example of an acid/base buffer. The equilibrium position of the buffer is governed by the reaction CH 3 COOH(aq)+H 2 O(l) t H 3 O + (aq)+CH 3 COO – (aq) and its acid dissociation constant 6.43 The relationship between the pH of an acid–base buffer and the relative amounts of CH 3 COOH and CH 3 COO – is derived by taking the negative log of both sides of equation 6.43 and solving for the pH 6.44 Buffering occurs because of the logarithmic relationship between pH and the ratio of the weak base and weak acid concentrations. For example, if the equilibrium concentrations of CH 3 COOH and CH 3 COO – are equal, the pH of the buffer is 4.76. If sufficient strong acid is added such that 10% of the acetate ion is converted to acetic acid, the concentration ratio [CH 3 COO – ]/[CH 3 COOH] changes to 0.818, and the pH decreases to 4.67. 6H.1 Systematic Solution to Buffer Problems Equation 6.44 is written in terms of the concentrations of CH 3 COOH and CH 3 COO – at equilibrium. A more useful relationship relates the buffer’s pH to the initial concentrations of weak acid and weak base. A general buffer equation can be derived by considering the following reactions for a weak acid, HA, and the salt of its conjugate weak base, NaA. NaA(aq) fi Na + (aq)+A – (aq) HA(aq)+H 2 O(l) t H 3 O + (aq)+A – (aq) 2H 2 O(l) t H 3 O + (aq)+OH – (aq) Since the concentrations of Na + , A – , HA, H 3 O + , and OH – are unknown, five equa- tions are needed to uniquely define the solution’s composition. Two of these equa- tions are given by the equilibrium constant expressions The remaining three equations are given by mass balance equations on HA and Na + C HA + C NaA =[HA]+[A – ] 6.45 C NaA = [Na + ] 6.46 and a charge balance equation [H 3 O + ] + [Na + ] = [OH – ]+[A – ] K K a w HO A HA HO OH = = + + [][] [] [][] – – 3 3 pH p CH COO CH COOH CH COO CH COOH a =+ =+K log [] . log [] –– 3 3 3 3 476 K a H O CH COO CH COOH ==× + [][ ] . – – 33 3 5 175 10 168 Modern Analytical Chemistry 1400-CH06 9/9/99 7:41 AM Page 168 Chapter 6 Equilibrium Chemistry 169 Substituting equation 6.46 into the charge balance equation and solving for [A – ] gives [A – ]=C NaA – [OH – ]+[H 3 O + ] 6.47 which is substituted into equation 6.45 to give the concentration of HA [HA] = C HA + [OH – ]–[H 3 O + ] 6.48 Finally, substituting equations 6.47 and 6.48 into the K a equation for HA and solv- ing for pH gives the general buffer equation If the initial concentrations of weak acid and weak base are greater than [H 3 O + ] and [OH – ], the general equation simplifies to the Henderson–Hasselbalch equation. 6.49 As in Example 6.13, the Henderson–Hasselbalch equation provides a simple way to calculate the pH of a buffer and to determine the change in pH upon adding a strong acid or strong base. EXAMPLE 6.1 3 Calculate the pH of a buffer that is 0.020 M in NH 3 and 0.030 M in NH 4 Cl. What is the pH after adding 1.00 mL of 0.10 M NaOH to 0.10 L of this buffer? SOLUTION The acid dissociation constant for NH 4 + is 5.70 · 10 –10 ; thus the initial pH of the buffer is Adding NaOH converts a portion of the NH 4 + to NH 3 due to the following reaction NH 4 + (aq)+OH – (aq) t NH 3 (aq)+H 2 O(l) Since the equilibrium constant for this reaction is large, we may treat the reaction as if it went to completion. The new concentrations of NH 4 + and NH 3 are therefore C V C V NH tot NH tot moles NH moles OH ML M L L M moles NH moles OH ML M L L M 4 3 4 3 3 3 0 030 0 10 0 10 1 00 10 0 101 0 029 0 020 0 10 0 10 1 00 10 0 101 0 021 + = = × = = + = +× = + – ( . )( . ) – ( . )( . ) . . (. )(. ) (. )(. ) . . – – – – pH NH NH =+ =+ = + 924 924 0 020 0 030 906 3 4 . log . log . . . C C pH p a NaA HA =+K C C log pH p OH H O OH H O a NaA HA =+ + + + + K C C log –[ ] [ ] []–[ ] – – 3 3 Henderson–Hasselbalch equation Equation showing the relationship between a buffer’s pH and the relative amounts of the buffer’s conjugate weak acid and weak base. 1400-CH06 9/9/99 7:41 AM Page 169 Substituting the new concentrations into the Henderson–Hasselbalch equation gives a pH of Multiprotic weak acids can be used to prepare buffers at as many different pH’s as there are acidic protons. For example, a diprotic weak acid can be used to prepare buffers at two pH’s and a triprotic weak acid can be used to prepare three different buffers. The Henderson–Hasselbalch equation applies in each case. Thus, buffers of malonic acid (pK a1 = 2.85 and pK a2 = 5.70) can be prepared for which where H 2 M, HM – , and M 2– are the different forms of malonic acid. The capacity of a buffer to resist a change in pH is a function of the absolute concentration of the weak acid and the weak base, as well as their relative propor- tions. The importance of the weak acid’s concentration and the weak base’s con- centration is obvious. The more moles of weak acid and weak base that a buffer has, the more strong base or strong acid it can neutralize without significantly changing the buffer’s pH. The relative proportions of weak acid and weak base af- fect the magnitude of the change in pH when adding a strong acid or strong base. Buffers that are equimolar in weak acid and weak base require a greater amount of strong acid or strong base to effect a change in pH of one unit. 2 Consequently, buffers are most effective to the addition of either acid or base at pH values near the pK a of the weak acid. Buffer solutions are often prepared using standard “recipes” found in the chemical literature. 3 In addition, computer programs have been developed to aid in the preparation of other buffers. 4 Perhaps the simplest means of preparing a buffer, however, is to prepare a solution containing an appropriate conjugate weak acid and weak base and measure its pH. The pH is easily adjusted to the desired pH by adding small portions of either a strong acid or a strong base. Although this treatment of buffers was based on acid–base chemistry, the idea of a buffer is general and can be extended to equilibria involving complexation or redox reactions. For example, the Nernst equation for a solution containing Fe 2+ and Fe 3+ is similar in form to the Henderson–Hasselbalch equation. Consequently, solutions of Fe 2+ and Fe 3+ are buffered to a potential near the standard-state reduction potential for Fe 3+ . 6H.2 Representing Buffer Solutions with Ladder Diagrams Ladder diagrams provide a simple graphical description of a solution’s predominate species as a function of solution conditions. They also provide a convenient way to show the range of solution conditions over which a buffer is most effective. For ex- EE=?++ ° + + Fe Fe Fe Fe 32 0 05916 2 3 / . log [] [] pH M HM =+570 2 . log – – C C pH HM HM =+285 2 . log –C C pH =+ =924 0 021 0 029 910. log . . . 170 Modern Analytical Chemistry 1400-CH06 9/9/99 7:41 AM Page 170 Figure 6.13 Ladder diagrams showing buffer regions for (a) HF/F – acid–base buffer; (b) Ca 2+ /Ca(EDTA) 2– metal–ligand complexation buffer; and (c) SN 4+ /Sn 2+ oxidation–reduction buffer. Chapter 6 Equilibrium Chemistry 171 F – 4.17 2.17 (a) HF pK a,HF = 3.17 Ca 2+ 11.69 9.69 (b) Ca(EDTA) 2– logK f,Ca(EDTA) 2– = 10.69 Sn 4+ 0.184 0.124 (c) Sn 2+ E ° Sn 4+ /Sn 2+ = 0.154 p EDTA EpH ample, an acid–base buffer can only exist when the relative abundance of the weak acid and its conjugate weak base are similar. For convenience, we will assume that an acid–base buffer exists when the concentration ratio of weak base to weak acid is between 0.1 and 10. Applying the Henderson–Hasselbalch equation shows that acid–base buffer exists within the range of pH = pK a ± 1. In the same manner, it is easy to show that a complexation buffer for the metal–ligand complex ML n exists when pL = log K f ± 1, and that a redox buffer exists for E = E° ± (0.05916/n). Ladder diagrams showing buffer regions for several equilibria are shown in Figure 6.13. 6I Activity Effects Suppose you need to prepare a buffer with a pH of 9.36. Using the Henderson– Hasselbalch equation, you calculate the amounts of acetic acid and sodium acetate needed and prepare the buffer. When you measure the pH, however, you find that it is 9.25. If you have been careful in your calculations and measurements, what can account for the difference between the obtained and expected pHs? In this section, we will examine an important limitation to our use of equilibrium constants and learn how this limitation can be corrected. Careful measurements of the solubility of AgIO 3 show that it increases in the presence of KNO 3 , even though neither K + or NO 3 – participates in the solubility re- action. 5 Clearly the equilibrium position for the reaction AgIO 3 (s) t Ag + (aq)+IO 3 – (aq) pH p p pH p p aa aa =+ = =+ =+ KK KK log – log 1 10 1 10 1 1 1400-CH06 9/9/99 7:41 AM Page 171 172 Modern Analytical Chemistry ionic strength A quantitative method for reporting the ionic composition of a solution that takes into account the greater effect of more highly charged ions (m ). depends on the composition of the solution. When the solubility product for AgIO 3 is calculated using the equilibrium concentrations of Ag + and IO 3 – K sp = [Ag + ][IO 3 – ] its apparent value increases when an inert electrolyte such as KNO 3 is added. Why should adding an inert electrolyte affect the equilibrium position of a chemical reaction? We can explain the effect of KNO 3 on the solubility of AgIO 3 by considering the reaction on a microscopic scale. The solution in which equilibrium is established contains a variety of cations and anions—K + , Ag + , H 3 O + , NO 3 – , IO 3 – and OH – . Although the solution is homogeneous, on the average, there are more anions in regions near Ag + ions, and more cations in regions near IO 3 – ions. Thus, Ag + and IO 3 – are surrounded by charged ionic atmospheres that partially screen the ions from each other. The formation of AgIO 3 requires the disruption of the ionic atmospheres surrounding the Ag + and IO 3 – ions. Increasing the concentrations of ions in solution, by adding KNO 3 , increases the size of these ionic atmospheres. Since more energy is now required to disrupt the ionic atmospheres, there is a decrease in the formation of AgIO 3 , and an apparent increase in the equilibrium constant. The ionic composition of a solution frequently is expressed by its ionic strength, m where c i and z i are the concentration and charge of the ith ion. EXAMPLE 6.1 4 Calculate the ionic strength of 0.10 M NaCl. Repeat the calculation for a solution of 0.10 M Na 2 SO 4 . SOLUTION The ionic strength for 0.10 M NaCl is For 0.10 M Na 2 SO 4 , the ionic strength is Note that the unit for ionic strength is molarity, but that the molar ionic strength need not match the molar concentration of the electrolyte. For a 1:1 electrolyte, such as NaCl, ionic strength and molar concentration are identical. The ionic strength of a 2:1 electrolyte, such as Na 2 SO 4 , is three times larger than the elec- trolyte’s molar concentration. The true thermodynamic equilibrium constant is a function of activity rather than concentration. The activity of a species, a A , is defined as the prod- uct of its molar concentration, [A], and a solution-dependent activity coeffi- cient, g A . a A = [A]g A μ= + + = + + = + 1 2 12 1 2 020 1 010 2 030 2 4 2 22 ([ ]( ) [ ](– ) ) [( . )( ) ( . )(– ) ] . – Na SO M μ= + + = + + = + 1 2 11 1 2 010 1 010 1 010 22 2 2 ([ ]( ) [ ](– ) ) [( . )( ) ( . )(– ) ] . – Na Cl M μ= ∑ 1 2 2 cz i i i activity True thermodynamic constants use a species activity in place of its molar concentration (a). activity coefficient The number that when multiplied by a species’ concentration gives that species’ activity (g ). 1400-CH06 9/9/99 7:41 AM Page 172 The true thermodynamic equilibrium constant, K sp , for the solubility of AgIO 3 , therefore, is K sp =(a Ag +)(a IO 3 – ) = [Ag + ][IO 3 – ](g Ag +)(g IO 3 – ) To accurately calculate the solubility of AgIO 3 , we must know the activity coeffi- cients for Ag + and IO 3 – . For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients are approximately unity under most reasonable experimental conditions. For reac- tions involving only these species, differences between activity and concentration are negligible. Activity coefficients for ionic solutes, however, depend on the ionic composition of the solution. It is possible, using the extended Debye–Hückel the- ory,* to calculate activity coefficients using equation 6.50 6.50 where Z A is the charge of the ion, a A is the effective diameter of the hydrated ion in nanometers (Table 6.1), m is the solution’s ionic strength, and 0.51 and 3.3 are con- stants appropriate for aqueous solutions at 25 °C. Several features of equation 6.50 deserve mention. First, as the ionic strength approaches zero, the activity coefficient approaches a value of one. Thus, in a solu- tion where the ionic strength is zero, an ion’s activity and concentration are identi- cal. We can take advantage of this fact to determine a reaction’s thermodynamic equilibrium constant. The equilibrium constant based on concentrations is mea- sured for several increasingly smaller ionic strengths and the results extrapolated – log . . γ μ αμ A A A = ×× +×× 051 133 2 z Chapter 6 Equilibrium Chemistry 173 *See any standard textbook on physical chemistry for more information on the Debye–Hückel theory and its application to solution equilibrium Table 6.1 Effective Diameters (a ) for Selected Inorganic Cations and Anions Effective Diameter Ion (nm) H 3 O + 0.9 Li + 0.6 Na + , IO 3 – , HSO 3 – , HCO 3 – , H 2 PO 4 – 0.45 OH – , F – , SCN – , HS – , ClO 3 – , ClO 4 – , MnO 4 – 0.35 K + , Cl – , Br – , I – , CN – , NO 2 – , NO 3 – 0.3 Cs + , Tl + , Ag + , NH 4 + 0.25 Mg 2+ , Be 2+ 0.8 Ca 2+ , Cu 2+ , Zn 2+ , Sn 2+ , Mn 2+ , Fe 2+ , Ni 2+ , Co 2+ 0.6 Sr 2+ , Ba 2+ , Cd 2+ , Hg 2+ , S 2– 0.5 Pb 2+ , CO 3 2– , SO 3 2– 0.45 Hg 2 2+ , SO 4 2– , S 2 O 3 2– , CrO 4 2– , HPO 4 2– 0.40 Al 3+ , Fe 3+ , Cr 3+ 0.9 PO 4 3– , Fe(CN) 6 3– 0.4 Zr 4+ , Ce 4+ , Sn 4+ 1.1 Fe(CN) 6 4– 0.5 Source: Values from Kielland, J. J. Am. Chem. Soc. 1937, 59, 1675. 1400-CH06 9/9/99 7:41 AM Page 173 Colorplate 3 provides a visual demonstration of the effect of ionic strength on the equilibrium reaction Fe 3+ (aq) + SCN — (aq) t Fe(SCN) 2+ (aq) 174 Modern Analytical Chemistry back to zero ionic strength to give the thermodynamic equilibrium constant. Sec- ond, activity coefficients are smaller, and thus activity effects are more important, for ions with higher charges and smaller effective diameters. Finally, the extended Debye–Hückel equation provides reasonable activity coefficients for ionic strengths of less than 0.1. Modifications to the extended Debye–Hückel equation, which extend the calculation of activity coefficients to higher ionic strength, have been proposed. 6 EXAMPLE 6.1 5 Calculate the solubility of Pb(IO 3 ) 2 in a matrix of 0.020 M Mg(NO 3 ) 2 . SOLUTION We begin by calculating the ionic strength of the solution. Since Pb(IO 3 ) 2 is only sparingly soluble, we will assume that its contribution to the ionic strength can be ignored; thus Activity coefficients for Pb 2+ and I – are calculated using equation 6.50 giving an activity coefficient for Pb 2+ of 0.43. A similar calculation for IO 3 – gives its activity coefficient as 0.81. The equilibrium constant expression for the solubility of PbI 2 is Letting [Pb 2+ ]=x and [IO 3 – ]=2x we have (x)(2x) 2 (0.45)(0.81) 2 = 2.5 · 10 –13 Solving for x gives a value of 6.0 · 10 –5 or a solubility of 6.0 · 10 –5 mol/L. This compares to a value of 4.0 · 10 –5 mol/L when activity is ignored. Failing to correct for activity effects underestimates the solubility of PbI 2 in this case by 33%. As this example shows, failing to correct for the effect of ionic strength can lead to significant differences between calculated and actual concentrations. Neverthe- less, it is not unusual to ignore activities and assume that the equilibrium constant is expressed in terms of concentrations. There is a practical reason for this—in an analysis one rarely knows the composition, much less the ionic strength of a sample solution. Equilibrium calculations are often used as a guide when developing an an- alytical method. Only by conducting the analysis and evaluating the results can we judge whether our theory matches reality. K sp Pb IO Pb IO==× + + ?[][] . ––2 3 213 2 3 25 10 γγ – log .() . .. . . γ Pb 2 051 2 0060 1 3 3 0 45 0 060 0 366 2 + = ×+ × +× × = μ= + + = 1 2 0 20 2 0 040 1 0 060 22 [( . )( ) ( . )(– ) ] .MMM 1400-CH06 9/9/99 7:41 AM Page 174 Chapter 6 Equilibrium Chemistry 175 6J Two Final Thoughts About Equilibrium Chemistry In this chapter we have reviewed and extended our understanding of equilibrium chemistry. We also have developed several tools for evaluating the composition of a system at equilibrium. These tools differ in how accurately they allow us to answer questions involving equilibrium chemistry. They also differ in their ease of use. An important part of having several tools available to you is knowing when to use them. If you need to know whether a reaction is favorable, or the approximate pH of a solution, a ladder diagram may be sufficient to meet your needs. On the other hand, if you require an accurate estimate of a compound’s solubility, a rigorous cal- culation using the systematic approach and activity coefficients is necessary. Finally, a consideration of equilibrium chemistry can only help us decide what reactions are favorable. Knowing that a reaction is favorable does not guarantee that the reaction will occur. How fast a reaction approaches its equilibrium position does not depend on the magnitude of the equilibrium constant. The rate of a chem- ical reaction is a kinetic, not a thermodynamic, phenomenon. Kinetic effects and their application in analytical chemistry are discussed in Chapter 13. 6K KEY TERMS acid (p. 140) acid dissociation constant (p. 140) activity (p. 172) activity coefficient (p. 172) amphiprotic (p. 142) base (p. 140) base dissociation constant (p. 141) buffer (p. 167) charge balance equation (p. 159) common ion effect (p. 158) cumulative formation constant (p. 144) dissociation constant (p. 144) enthalpy (p. 137) entropy (p. 137) equilibrium (p. 136) equilibrium constant (p. 138) formation constant (p. 144) Gibb’s free energy (p. 137) Henderson–Hasselbalch equation (p. 169) ionic strength (p. 172) ladder diagram (p. 150) Le Chatelier’s principle (p. 148) ligand (p. 144) mass balance equation (p. 159) Nernst equation (p. 146) oxidation (p. 146) oxidizing agent (p. 146) pH (p. 142) precipitate (p. 139) redox reaction (p. 145) reducing agent (p. 146) reduction (p. 146) solubility product (p. 140) standard state (p. 137) stepwise formation constant (p. 144) Analytical chemistry is more than a collection of techniques; it is the application of chemistry to the analysis of samples. As you will see in later chapters, almost all analytical methods use chemical re- activity to accomplish one or more of the following—dissolve the sample, separate analytes and interferents, transform the analyte to a more useful form, or provide a signal. Equilibrium chemistry and thermodynamics provide us with a means for predicting which reactions are likely to be favorable. The most important types of reactions are precipitation reac- tions, acid–base reactions, metal–ligand complexation reactions, and redox reactions. In a precipitation reaction two or more solu- ble species combine to produce an insoluble product called a pre- cipitate. The equilibrium properties of a precipitation reaction are described by a solubility product. Acid–base reactions occur when an acid donates a proton to a base. The equilibrium position of an acid–base reaction is de- scribed using either the dissociation constant for the acid, K a , or the dissociation constant for the base, K b . The product of K a and K b for an acid and its conjugate base is K w (water’s dissociation constant). Ligands have electron pairs that they can donate to a metal ion, forming a metal–ligand complex. The formation of the metal– ligand complex ML 2 , for example, may be described by a stepwise formation constant in which each ligand is added one at a time; thus, K 1 represents the addition of the first ligand to M, and K 2 represents the addition of the second ligand to ML. Alternatively, the formation of ML 2 can be described by a cumulative, or overall formation constant, b 2 , in which both ligands are added to M. 6L SUMMARY 1400-CH06 9/9/99 7:41 AM Page 175 176 Modern Analytical Chemistry In a redox reaction, one of the reactants is oxidized while an- other reactant is reduced. Equilibrium constants are rarely used when characterizing redox reactions. Instead, we use the electro- chemical potential, positive values of which indicate a favorable reaction. The Nernst equation relates this potential to the concen- trations of reactants and products. Le Chatelier’s principle provides a means for predicting how systems at equilibrium respond to a change in conditions. When a stress is applied to an equilibrium by adding a reactant or product, by adding a reagent that reacts with one of the reactants or prod- ucts, or by changing the volume, the system responds by moving in the direction that relieves the stress. You should be able to describe a system at equilibrium both qualitatively and quantitatively. Rigorous solutions to equilibrium problems can be developed by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. When a less rigorous an- swer is needed, a ladder diagram may help you decide the equilib- rium system’s composition. Solutions containing a weak acid and its conjugate base show only a small change in pH upon the addition of small amounts of strong acid or strong base. Such solutions are called buffers. Buffers can also be formed using a metal and its metal–ligand complex, or an oxidizing agent and its conjugate reducing agent. Both the systematic approach to solving equilibrium problems and ladder diagrams can be used to characterize a buffer. A quantitative solution to an equilibrium problem may give an answer that does not agree with the value measured experimen- tally. This result occurs when the equilibrium constant based on concentrations is matrix-dependent. The true, thermodynamic equilibrium constant is based on the activities, a, of the reactants and products. A species’ activity is related to its molar concentra- tion by an activity coefficient, g , where a i = g i [] i . Activity coeffi- cients often can be calculated, making possible a more rigorous treatment of equilibria. “The Effect of Ionic Strength on an Equilibrium Constant (A Class Study).” In J. A. Bell, ed. Chemical Principles in Practice. Addison-Wesley: Reading, MA, 1967. In this experiment the equilibrium constant for the dissociation of bromocresol green is measured at several ionic strengths. Results are extrapolated to zero ionic strength to find the thermodynamic equilibrium constant. “Equilibrium Constants for Calcium Iodate Solubility and Iodic Acid Dissociation.” In J. A. Bell, ed. Chemical Principles in Practice. Addison-Wesley: Reading, MA, 1967. The effect of pH on the solubility of Ca(IO 3 ) 2 is studied in this experiment. “The Solubility of Silver Acetate.” In J. A. Bell, ed. Chemical Principles in Practice. Addison-Wesley: Reading, MA, 1967. In this experiment the importance of the soluble silver acetate complexes AgCH 3 COO(aq) and Ag(CH 3 COO) 2 – (aq) in describing the solubility of AgCH 3 COO(s) is investigated. Green, D. B.; Rechtsteiner, G.; Honodel, A. “Determination of the Thermodynamic Solubility Product, K sp , of PbI 2 Assuming Nonideal Behavior,” J. Chem. Educ. 1996, 73, 789–792. The thermodynamic solubility product for PbI 2 is determined in this experiment by measuring its solubility at several ionic strengths. 6M Suggested EXPERIMENTS The following experiments involve the experimental determination of equilibrium constants and, in some cases, demonstrate the importance of activity effects. Experiments 1. Write equilibrium constant expressions for the following reactions. Determine the value for the equilibrium constant for each reaction using appropriate equilibrium constants from Appendix 3. a. NH 3 (aq) + HCl(aq) t NH 4 + (aq)+Cl – (aq) b. PbI 2 (s)+S 2– (aq) t PbS(s)+2I – (aq) c. CdY 2– (aq) + 4CN – (aq) t Cd(CN) 4 2– (aq)+Y 4– (aq) [Y 4– is EDTA] d. AgCl(s) + 2NH 3 (aq) t Ag(NH 3 ) 2 + (aq)+Cl – (aq) e. BaCO 3 (s)+2H 3 O + (aq) t Ba 2+ (aq)+H 2 CO 3 (aq)+2H 2 O(l) 2. Using a ladder diagram, explain why the following reaction H 3 PO 4 (aq)+F – (aq) t HF(aq)+H 2 PO 4 – (aq) is favorable, whereas H 3 PO 4 (aq)+2F – (aq) t 2HF(aq)+H 2 PO 4 2– (aq) is unfavorable. Determine the equilibrium constant for these reactions, and verify that they are consistent with your ladder diagram. 6N PROBLEMS 1400-CH06 9/9/99 7:41 AM Page 176 3. Calculate the potential for the following redox reaction when the [Fe 3+ ] = 0.050 M, [Fe 2+ ] = 0.030 M, [Sn 2+ ] = 0.015 M and [Sn 4+ ] = 0.020 M 2Fe 3+ (aq)+Sn 2+ (aq) t Sn 4+ (aq) + 2Fe 2+ (aq) 4. Balance the following redox reactions, and calculate the standard-state potential and the equilibrium constant for each. Assume that the [H 3 O + ] is 1 M for acidic solutions, and that the [OH – ] is 1 M for basic solutions. a. MnO 4 – (aq)+H 2 SO 3 (aq) t Mn 2+ (aq)+ SO 4 2– (aq) (acidic solution) b. IO 3 – (aq)+I – (aq) t I 2 (s) (acidic solution) c. ClO – (aq)+I – t IO 3 – (aq)+Cl – (aq) (basic solution) 5. Sulfur can be determined quantitatively by oxidizing to SO 4 2– and precipitating as BaSO 4 . The solubility reaction for BaSO 4 is BaSO 4 (s) t Ba 2+ (aq)+SO 4 2– (aq) How will the solubility of BaSO 4 be affected by (a) decreasing the pH of the solution; (b) adding BaCl 2 ; (c) decreasing the volume of the solution? 6. Write charge balance and mass balance equations for the following solutions a. 0.1 M NaCl b. 0.1 M HCl c. 0.1 M HF d. 0.1 M NaH 2 PO 4 e. MgCO 3 (saturated solution) f. 0.1 M Ag(CN) 2 – (from AgNO 3 and KCN) g. 0.1 M HCl and 0.050 M NaNO 2 7. Using the systematic approach, calculate the pH of the following solutions a. 0.050 M HClO 4 b. 1.00 · 10 –7 M HCl c. 0.025 M HClO d. 0.010 M HCOOH e. 0.050 M Ba(OH) 2 f. 0.010 M C 5 H 5 N 8. Construct ladder diagrams for the following diprotic weak acids (H 2 L), and estimate the pH of 0.10 M solutions of H 2 L, HL – , and L 2– . Using the systematic approach, calculate the pH of each of these solutions. a. maleic acid b. malonic acid c. succinic acid 9. Ignoring activity effects, calculate the solubility of Hg 2 Cl 2 in the following a. A saturated solution of Hg 2 Cl 2 b. 0.025 M Hg 2 (NO 3 ) 2 saturated with Hg 2 Cl 2 c. 0.050 M NaCl saturated with Hg 2 Cl 2 10. The solubility of CaF 2 is controlled by the following two reactions CaF 2 (s) t Ca 2+ (aq)+F – (aq) HF(aq)+H 2 O(l) t H 3 O + (aq) +F – (aq) Calculate the solubility of CaF 2 in a solution buffered to a pH of 7.00. Use a ladder diagram to help simplify the calculations. How would your approach to this problem change if the pH is buffered to 2.00? What is the solubility of CaF 2 at this pH? 11. Calculate the solubility of Mg(OH) 2 in a solution buffered to a pH of 7.00. How does this compare with its solubility in unbuffered water? 12. Calculate the solubility of Ag 3 PO 4 in a solution buffered to a pH of 9.00. 13. Determine the equilibrium composition of saturated solution of AgCl. Assume that the solubility of AgCl is influenced by the following reactions. AgCl(s) t Ag + (aq)+Cl – (aq) Ag + (aq)+Cl – (aq) t AgCl(aq) AgCl(aq)+Cl – (aq) t AgCl 2 – (aq) 14. Calculate the ionic strength of the following solutions a. 0.050 M NaCl b. 0.025 M CuCl 2 c. 0.10 M Na 2 SO 4 15. Repeat the calculations in problem 9, this time correcting for activity effects. 16. With the permission of your instructor, carry out the following experiment. In a beaker, mix equal volumes of 0.001 M NH 4 SCN and 0.001 M FeCl 3 (the latter solution must be acidified with concentrated HNO 3 at a ratio of 4 drops/L to prevent the precipitation of Fe(OH) 3 ). Divide solution in half, and add solid KNO 3 to one portion at a ratio of 4 g per 100 mL. Compare the colors of the two solutions (see Color Plate 3), and explain why they are different. The relevant reaction is Fe 3+ (aq) + SCN – (aq) t Fe(SCN) 2+ (aq) 17. Over what pH range do you expect Ca 3 (PO 4 ) 2 to have its minimum solubility? 18. Construct ladder diagrams for the following systems, and describe the information that can be obtained from each a. HF and H 3 PO 4 b. Ag(CN) 2 – , Ni(CN) 4 2– and Fe(CN) 6 4– c. Cr 2 O 7 2– /Cr 3+ and Fe 3+ /Fe 2+ 19. Calculate the pH of the following acid–base buffers a. 100 mL of 0.025 M formic acid and 0.015 M sodium formate b. 50.00 mL of 0.12 M NH 3 and 3.50 mL of 1.0 M HCl c. 5.00 g of Na 2 CO 3 and 5.00 g of NaHCO 3 in 0.100 L 20. Calculate the pH of the buffers in problem 19 after adding 5.0 · 10 –4 mol of HCl. 21. Calculate the pH of the buffers in problem 19 after adding 5.0 · 10 –4 mol of NaOH. 22. Consider the following hypothetical complexation reaction between a metal, M, and a ligand, L M(aq)+L(aq) t ML(aq) Chapter 6 Equilibrium Chemistry 177 1400-CH06 9/9/99 7:41 AM Page 177 178 Modern Analytical Chemistry with a formation constant of 1.5 · 10 8 . Derive an equation, similar to the Henderson–Hasselbalch equation, which relates pM to the concentrations of L and ML. What will be the pM for a solution containing 0.010 mol of M and 0.020 mol of L? What will the pM be if 0.002 mol of M are added? 23. A redox buffer contains an oxidizing agent and its conjugate reducing agent. Calculate the potential of a solution containing 0.010 mol of Fe 3+ and 0.015 mol of Fe 2+ . What is the potential if sufficient oxidizing agent is added such that 0.002 mol of Fe 2+ is converted to Fe 3+ ? A lucid discussion of Berthollet’s discovery of the reversibility of reactions is found in Roots-Bernstein, R. S. Discovering. Harvard University Press: Cambridge, MA, 1989. The following texts and articles provide additional coverage of equilibrium chemistry and the systematic approach to solving equilibrium problems. Butler, J. N. Ionic Equilibria: A Mathematical Approach. Addison- Wesley: Reading, MA, 1964. Butler, J. N. Solubility and pH Calculations. Addison-Wesley: Reading, MA, 1973. Chaston, S. “Calculating Complex Equilibrium Concentrations by a Next Guess Factor Method,” J. Chem. Educ. 1993, 70, 622–624. Fernando, Q.; Ryan, M. D. Calculations in Analytical Chemistry, Harcourt Brace Jovanovich: New York, 1982. Freiser, H. Concepts and Calculations in Analytical Chemistry, CRC Press: Boca Raton, 1992. Freiser, H.; Fernando, Q. Ionic Equilibria in Analytical Chemistry, Wiley: New York, 1963. Gordus, A. A. “Chemical Equilibrium I. The Thermodynamic Equilibrium Concept,” J. Chem. Educ. 1991, 68, 138–140. Gordus, A. A. “Chemical Equilibrum II. Deriving an Exact Equilibrium Equation,” J. Chem. Educ. 1991, 68, 215–217. Gordus, A. A. “Chemical Equilibrium III. A Few Math Tricks,” J. Chem. Educ. 1991, 68, 291–293. Gordus, A. A. “Chemical Equilibrium IV. Weak Acids and Bases,” J. Chem. Educ. 1991, 68, 397–399. Gordus, A. A. “Chemical Equilibrium VI. Buffer Solutions,” J. Chem. Educ. 1991, 68, 656–658. Gordus, A. A. “Chemical Equilibrium VII. Precipitates,” J. Chem. Educ. 1991, 68, 927–930. Gordus, A. A. Schaum’s Outline of Analytical Chemistry. McGraw- Hill: New York, 1985. Olivieri, A. C. “Solution of Acid–Base Equilibria by Successive Approximations,” J. Chem. Educ. 1990, 67, 229–231. Ramette, R. W. Chemical Equilibrium and Analysis. Addison- Wesley: Reading, MA, 1981. Thomson, B. M.; Kessick, M. A. “On the Preparation of Buffer Solutions,” J. Chem. Educ. 1981, 58, 743–746. Weltin, E. “Are the Equilibrium Concentrations for a Chemical Reaction Always Uniquely Determined by the Initial Concentrations?” J. Chem. Educ. 1990, 67, 548. Weltin, E. “A Numerical Method to Calculate Equilibrium Concentrations for Single-Equation Systems,” J. Chem. Educ. 1991, 68, 486–487. Weltin, E. “Calculating Equilibrium Concentrations,” J. Chem. Educ. 1992, 69, 393–396. Weltin, E. “Calculating Equilibrium Concentrations for Stepwise Binding of Ligands and Polyprotic Acid-Base Systems,” J. Chem. Educ. 1993, 70, 568–571. Weltin, E. “Equilibrium Calculations are Easier Than You Think— But You Do Have to Think!” J. Chem. Educ. 1993, 70, 571–573. 6O SUGGESTED READINGS 1. (a) Runo, J. R.; Peters, D. G. J. Chem. Educ. 1993, 70, 708–713; (b) Vale, J.; Fernandez-Pereira, C.; Alcalde, M. J. Chem. Educ. 1993, 70, 790–795. 2. Van Slyke, D. D. J. Biol. Chem. 1922, 52, 525–570. 3. (a) Bower, V. E.; Bates, R. G. J. Res. Natl. Bur. Stand. (U. S.) 1955, 55, 197–200; (b) Bates, R. G. Ann. N. Y. Acad. Sci. 1961, 92, 341–356; (c) Bates, R. G. Determination of pH, 2nd ed. Wiley-Interscience: New York, 1973, p. 73. 4. Lambert, W. J. J. Chem. Educ. 1990, 67, 150–153. 5. Kolthoff, I. M.; Lingane, J. J. Phys. Chem. 1938, 42, 133–140. 6. Davies, C. W. Ion Association. Butterworth: London, 1962. 6P REFERENCES 1400-CH06 9/9/99 7:41 AM Page 178 Chapter 7 179 Obtaining and Preparing Samples for Analysis When we first use an analytical method to solve a problem, it is not unusual to find that our results are of questionable accuracy or so imprecise as to be meaningless. Looking back we may find that nothing in the method seems amiss. In designing the method we considered sources of determinate and indeterminate error and took appropriate steps, such as including a reagent blank and calibrating our instruments, to minimize their effect. Why, then, might a carefully designed method give such poor results? One explanation is that we may not have accounted for errors associated with the sample. When we collect the wrong sample or lose analyte while preparing the sample for analysis, we introduce a determinate source of error. If we do not collect enough samples or collect samples of the wrong size, the precision of the analysis may suffer. In this chapter we consider how collecting samples and preparing them for analysis can affect the accuracy and precision of our results. 1400-CH07 9/8/99 4:02 PM Page 179 Figure 7.1 Percent of overall variance (s 2 o ) due to the method as a function of the relative magnitudes of the standard deviation of the method and the standard deviation of sampling (s m /s s ). The dotted lines show that the variance due to the method accounts for 10% of the overall variance when s s =3· s m . 180 Modern Analytical Chemistry *Values for t can be found in Appendix 1B. 7 A The Importance of Sampling When a manufacturer produces a chemical they wish to list as ACS Reagent Grade, they must demonstrate that it conforms to specifications established by the Ameri- can Chemical Society (ACS). For example, ACS specifications for NaHCO 3 require that the concentration of iron be less than or equal to 0.001% w/w. To verify that a production lot meets this standard, the manufacturer performs a quantitative analy- sis, reporting the result on the product’s label. Because it is impractical to analyze the entire production lot, its properties are estimated from a limited sampling. Sev- eral samples are collected and analyzed, and the resulting mean, – X, and standard de- viation, s, are used to establish a confidence interval for the production lot’s true mean, m 7.1 where n is the number of samples, and t is a statistical factor whose value is deter- mined by the number of samples and the desired confidence level.* Selecting a sample introduces a source of determinate error that cannot be cor- rected during the analysis. If a sample does not accurately represent the population from which it is drawn, then an analysis that is otherwise carefully conducted will yield inaccurate results. Sampling errors are introduced whenever we extrapolate from a sample to its target population. To minimize sampling errors we must col- lect the right sample. Even when collecting the right sample, indeterminate or random errors in sam- pling may limit the usefulness of our results. Equation 7.1 shows that the width of a confidence interval is directly proportional to the standard deviation. The overall standard deviation for an analysis, s o , is determined by ran- dom errors affecting each step of the analysis. For convenience, we di- vide the analysis into two steps. Random errors introduced when collect- ing samples are characterized by a standard deviation for sampling, s s . The standard deviation for the analytical method, s m , accounts for ran- dom errors introduced when executing the method’s procedure. The re- lationship among s o , s s , and s m is given by a propagation of random error s 2 o = s 2 m + s 2 s 7.2 Equation 7.2 shows that an analysis’ overall variance may be lim- ited by either the analytical method or sample collection. Unfortu- nately, analysts often attempt to minimize overall variance by im- proving only the method’s precision. This is futile, however, if the standard deviation for sampling is more than three times greater than that for the method. 1 Figure 7.1 shows how the ratio s m /s s affects the percentage of overall variance attributable to the method. When the method’s standard deviation is one third of that for sampling, indeterminate method errors explain only 10% of the overall variance. Attempting to improve the analysis by decreasing s m provides only a nominal change in the overall variance. μ= ±X ts n P e r centa g e of s 2 o due to s 2 m s m /s s 0.5 1.5 20 0 10 20 30 40 50 60 70 80 1 1400-CH07 9/8/99 4:02 PM Page 180 Chapter 7 Obtaining and Preparing Samples for Analysis 181 EXAMPLE 7 .1 A quantitative analysis for an analyte gives a mean concentration of 12.6 ppm. The standard deviation for the method is found to be 1.1 ppm, and that due to sampling is 2.1 ppm. (a) What is the overall variance for the analysis? (b) By how much does the overall variance change if s m is improved by 10% to 0.99 ppm? (c) By how much does the overall variance change if s s is improved by 10% to 1.9 ppm? SOLUTION (a) The overall variance is s o 2 = s m 2 + s s 2 = (1.1) 2 + (2.1) 2 = 1.21 + 4.41 = 5.62 ? 5.6 (b) Improving the method’s standard deviation changes the overall variance to s o 2 = (0.99) 2 + (2.1) 2 = 0.98 + 4.41 = 5.39 ? 5.4 Thus, a 10% improvement in the method’s standard deviation changes the overall variance by approximately 4%. (c) Changing the standard deviation for sampling s o 2 = (1.1) 2 + (1.9) 2 = 1.21 + 3.61 = 4.82 ? 4.8 improves the overall variance by almost 15%. As expected, since s s is larger than s m , a more significant improvement in the overall variance is realized when we focus our attention on sampling problems. To determine which step has the greatest effect on the overall variance, both s m 2 and s s 2 must be known. The analysis of replicate samples can be used to estimate the overall variance. The variance due to the method is determined by analyzing a stan- dard sample, for which we may assume a negligible sampling variance. The variance due to sampling is then determined by difference. EXAMPLE 7 .2 The following data were collected as part of a study to determine the effect of sampling variance on the analysis of drug animal-feed formulations. 2 % Drug (w/w) % Drug (w/w) 0.0114 0.0099 0.0105 0.0105 0.0109 0.0107 0.0102 0.0106 0.0087 0.0103 0.0103 0.0104 0.0100 0.0095 0.0098 0.0101 0.0101 0.0103 0.0105 0.0095 0.0097 The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 1400-CH07 9/8/99 4:03 PM Page 181 182 Modern Analytical Chemistry sampling plan A plan that ensures that a representative sample is collected. SOLUTION The overall variance, s o 2 , is determined using the data on the left and is equal to 4.71 · 10 –7 . The method’s contribution to the overall variance, s m 2 , is determined using the data on the right and is equal to 7.00 · 10 –8 . The variance due to sampling, s s 2 , is therefore s s 2 = s o 2 – s m 2 = 4.71 · 10 –7 – 7.00 · 10 –8 = 4.01 · 10 –7 7 B Designing A Sampling Plan A sampling plan must support the goals of an analysis. In characterization studies a sample’s purity is often the most important parameter. For example, a material sci- entist interested in the surface chemistry of a metal is more likely to select a freshly exposed surface, created by fracturing the sample under vacuum, than a surface that has been exposed to the atmosphere for an extended time. In a qualitative analysis the sample’s composition does not need to be identical to that of the substance being analyzed, provided that enough sample is taken to ensure that all components can be detected. In fact, when the goal of an analysis is to identify components present at trace levels, it may be desirable to discriminate against major components when sampling. In a quantitative analysis, however, the sample’s composition must accurately represent the target population. The focus of this section, therefore, is on designing a sampling plan for a quantitative analysis. Five questions should be considered when designing a sampling plan: 1. From where within the target population should samples be collected? 2. What type of samples should be collected? 3. What is the minimum amount of sample needed for each analysis? 4. How many samples should be analyzed? 5. How can the overall variance be minimized? Each of these questions is considered below in more detail. 7 B.1 Where to Sample the Target Population Sampling errors occur when a sample’s composition is not identical to that of the population from which it is drawn. When the material being sampled is homoge- neous, individual samples can be taken without regard to possible sampling errors. Unfortunately, in most situations the target population is heterogeneous in either time or space. As a result of settling, for example, medications available as oral sus- pensions may have a higher concentration of their active ingredients at the bottom of the container. Before removing a dose (sample), the suspension is shaken to min- imize the effect of this spatial heterogeneity. Clinical samples, such as blood or urine, frequently show a temporal heterogeneity. A patient’s blood glucose level, for instance, will change in response to eating, medication, or exercise. Other systems show both spatial and temporal heterogeneities. The concentration of dissolved O 2 in a lake shows a temporal heterogeneity due to the change in seasons, whereas point sources of pollution may produce a spatial heterogeneity. When the target population’s heterogeneity is of concern, samples must be ac- quired in a manner that ensures that determinate sampling errors are insignificant. If the target population can be thoroughly homogenized, then samples can be taken without introducing sampling errors. In most cases, however, homogenizing the 1400-CH07 9/8/99 4:03 PM Page 182 Chapter 7 Obtaining and Preparing Samples for Analysis 183 random sample A sample collected at random from the target population. target population is impracticable. Even more important, homogenization destroys information about the analyte’s spatial or temporal distribution within the target population. Random Sampling The ideal sampling plan provides an unbiased estimate of the target population’s properties. This requirement is satisfied if the sample is collected at random from the target population. 3 Despite its apparent simplicity, a true ran- dom sample is difficult to obtain. Haphazard sampling, in which samples are col- lected without a sampling plan, is not random and may reflect an analyst’s uninten- tional biases. The best method for ensuring the collection of a random sample is to divide the target population into equal units, assign a unique number to each unit, and use a random number table (Appendix 1E) to select the units from which to sample. Example 7.3 shows how this is accomplished. EXAMPLE 7 . 3 To analyze the properties of a 100 cm · 100 cm polymer sheet, ten 1 cm · 1 cm samples are to be selected at random and removed for analysis. Explain how a random number table can be used to ensure that samples are drawn at random. SOLUTION As shown in the following grid, we divide the polymer sheet into 10,000 1 cm · 1 cm squares, each of which can be identified by its row number and its column number. For example, the highlighted square is in row 1 and column 2. To pick ten squares at random, we enter the random number table at an arbitrary point, and let that number represent the row for the first sample. We then move through the table in a predetermined fashion, selecting random numbers for the column of the first sample, the row of the second sample, and so on until all ten samples have been selected. Since our random number table (Appendix 1E) uses five-digit numbers we will use only the last two digits. Let’s begin with the fifth entry and use every other entry after that. The fifth entry is 65423 making the first row number 23. The next entry we use is 41812, giving the first column number as 12. Continuing in this manner, the ten samples are as follows: Sample Row Column Sample Row Column 1 23 12 6 93 83 2 45 80 7 91 17 3 81 12 8 45 13 4 66 17 9 12 92 54 0 109752 0 0 1 2 98 99 1 2 98 99 1400-CH07 9/8/99 4:03 PM Page 183 A randomly collected sample makes no assumptions about the target popula- tion, making it the least biased approach to sampling. On the other hand, random sampling requires more time and expense than other sampling methods since a greater number of samples are needed to characterize the target population. Judgmental Sampling The opposite of random sampling is selective, or judg- mental sampling, in which we use available information about the target popula- tion to help select samples. Because assumptions about the target population are included in the sampling plan, judgmental sampling is more biased than random sampling; however, fewer samples are required. Judgmental sampling is common when we wish to limit the number of independent variables influencing the re- sults of an analysis. For example, a researcher studying the bioaccumulation of polychlorinated biphenyls (PCBs) in fish may choose to exclude fish that are too small or that appear diseased. Judgmental sampling is also encountered in many protocols in which the sample to be collected is specifically defined by the regula- tory agency. Systematic Sampling Random sampling and judgmental sampling represent ex- tremes in bias and the number of samples needed to accurately characterize the tar- get population. Systematic sampling falls in between these extremes. In systematic sampling the target population is sampled at regular intervals in space or time. For a system exhibiting a spatial heterogeneity, such as the distribution of dissolved O 2 in a lake, samples can be systematically collected by dividing the system into discrete units using a two- or three-dimensional grid pattern (Figure 7.2). Samples are collected from the center of each unit, or at the intersection of grid lines. When a heterogeneity is time-dependent, as is common in clinical studies, samples are drawn at regular intervals. When a target population’s spatial or temporal heterogeneity shows a periodic trend, a systematic sampling leads to a significant bias if samples are not collected frequently enough. This is a common problem when sampling electronic signals, in which case the problem is known as alias- ing. Consider, for example, a signal consisting of a simple sine wave. Fig- ure 7.3a shows how an insufficient sampling frequency underestimates the signal’s true frequency. According to the Nyquist theorem, to determine a periodic signal’s true fre- quency, we must sample the signal at a rate that is at least twice its frequency (Fig- ure 7.3b); that is, the signal must be sampled at least twice during a single cycle or period. When samples are collected at an interval of D t, the highest frequency that can be accurately monitored has a frequency of (2 D t) –1 . For example, if samples are collected every hour, the highest frequency that we can monitor is 0.5 h –1 , or a peri- odic cycle lasting 2 h. A signal with a cycling period of less than 2 h (a frequency of more than 0.5 h –1 ) cannot be monitored. Ideally, the sampling frequency should be at least three to four times that of the highest frequency signal of interest. Thus, if an hourly periodic cycle is of interest, samples should be collected at least every 15–20 min. Systematic—Judgmental Sampling Combinations of the three primary approaches to sampling are also possible. 4 One such combination is systematic–judgmental sampling, which is encountered in environmental studies when a spatial or tempo- 184 Modern Analytical Chemistry judgmental sampling Samples collected from the target population using available information about the analyte’s distribution within the population. Nyquist theorem Statement that a periodic signal must be sampled at least twice each period to avoid a determinate error in measuring its frequency. systematic–judgmental sampling A sampling plan that combines judgmental sampling with systematic sampling. Figure 7.2 Example of a systematic sampling plan for collecting samples from a lake. Each solid dot represents a sample collected from within the sampling grid. systematic sampling Samples collected from the target population at regular intervals in time or space. 1400-CH07 9/8/99 4:03 PM Page 184 Chapter 7 Obtaining and Preparing Samples for Analysis 185 stratified sampling A sampling plan that divides the population into distinct strata from which random samples are collected. convenience sampling A sampling plan in which samples are collected because they are easily obtained. grab sample A single sample removed from the target population. ral distribution of pollutants is anticipated. For example, a plume of waste leaching from a landfill can reasonably be expected to move in the same di- rection as the flow of groundwater. The systematic–judgmental sampling plan shown in Figure 7.4 includes a rectangular grid for systematic sampling and linear transects extending the sampling along the plume’s suspected major and minor axes. 5 Stratified Sampling Another combination of the three primary approaches to sampling is judgmental–random, or stratified sampling. Many target populations are conveniently subdivided into distinct units, or strata. For example, in determining the concentration of particulate Pb in urban air, the target population can be subdivided by particle size. In this case samples can be collected in two ways. In a random sampling, differences in the strata are ignored, and individual samples are collected at random from the entire target population. In a stratified sampling the target population is divided into strata, and random samples are collected from within each stratum. Strata are analyzed separately, and their respective means are pooled to give an overall mean for the target population. The advantage of stratified sampling is that the composition of each stra- tum is often more homogeneous than that of the entire target population. When true, the sampling variance for each stratum is less than that when the target population is treated as a single unit. As a result, the overall sampling variance for stratified sampling is always at least as good as, and often better than, that obtained by simple random sampling. Convenience Sampling One additional method of sampling deserves brief mention. In convenience sampling, sample sites are selected using criteria other than minimizing sampling error and sampling variance. In a survey of groundwater quality, for example, samples can be collected by drilling wells at randomly selected sites, or by making use of existing wells. The latter method is usually the preferred choice. In this case, cost, expedience, and accessibility are the primary factors used in selecting sam- pling sites. 7 B.2 What Type of Sample to Collect After determining where to collect samples, the next step in designing a sampling plan is to decide what type of sample to collect. Three methods are commonly used to obtain samples: grab sampling, composite sampling, and in situ sampling. The most common type of sample is a grab sample, in which a portion of the target population is removed at a given time and location in space. A grab sample, there- fore, provides a “snapshot” of the target population. Grab sampling is easily adapted to any of the sampling schemes discussed in the previous section. If the tar- get population is fairly uniform in time and space, a set of grab samples collected at random can be used to establish its properties. A systematic sampling using grab samples can be used to characterize a target population whose composition varies over time or space. (a) (b) Figure 7.3 Effect of sampling frequency when monitoring periodic signals. In (a) the sampling frequency is 1.2 samples per period. The dashed line shows the apparent signal, while the solid line shows the true signal. In (b) a sampling frequency of five samples per period is sufficient to give an accurate estimation of the true signal. 1400-CH07 9/8/99 4:03 PM Page 185 A composite sample consists of a set of grab samples that are combined to form a single sample. After thoroughly mixing, the composite sample is analyzed. Because information is lost when individual samples are combined, it is normally desirable to analyze each grab sample separately. In some situations, however, there are advantages to working with composite samples. One such situation is in deter- mining a target population’s average composition over time or space. For example, wastewater treatment plants are required to monitor and report the average compo- sition of treated water released to the environment. One approach is to analyze a se- ries of individual grab samples, collected using a systematic sampling plan, and av- erage the results. Alternatively, the individual grab samples can be combined to form a single composite sample. Analyzing a single composite sample instead of many individual grab samples, provides an appreciable savings in time and cost. Composite sampling is also useful when a single sample cannot supply sufficient material for an analysis. For example, methods for determining PCBs in fish often require as much as 50 g of tissue, an amount that may be difficult to obtain from a single fish. Tissue samples from several fish can be combined and homogenized, and a 50-g portion of the composite sample taken for analysis. A significant disadvantage of grab samples and composite samples is the need to remove a portion of the target population for analysis. As a result, neither type of sam- ple can be used to continuously monitor a time-dependent change in the target popu- lation. In situ sampling, in which an analytical sensor is placed directly in the target population, allows continuous monitoring without removing individual grab samples. For example, the pH of a solution moving through an industrial production line can be continually monitored by immersing a pH electrode within the solution’s flow. 186 Modern Analytical Chemistry Figure 7.4 Systematic–judgmental sampling scheme for monitoring the leaching of pollutants from a landfill. Sites where samples are collected are represented by the solid dots. Direction of groundwater flow Landfill = Sample site Major axis of plume composite sample Several grab samples combined to form a single sample. in situ sampling Sampling done within the population without physically removing the sample. 1400-CH07 9/8/99 4:03 PM Page 186 Chapter 7 Obtaining and Preparing Samples for Analysis 187 *See Chapter 4 to review the properties of a binomial distribution. 7 B. 3 How Much Sample to Collect To minimize sampling errors, a randomly collected grab sample must be of an ap- propriate size. If the sample is too small its composition may differ substantially from that of the target population, resulting in a significant sampling error. Samples that are too large, however, may require more time and money to collect and ana- lyze, without providing a significant improvement in sampling error. As a starting point, let’s assume that our target population consists of two types of particles. Particles of type A contain analyte at a fixed concentration, and type B particles contain no analyte. If the two types of particles are randomly distributed, then a sample drawn from the population will follow the binomial distribution.* If we collect a sample containing n particles, the expected number of particles con- taining analyte, n A , is n A = np where p is the probability of selecting a particle of type A. The sampling standard deviation is 7.3 The relative standard deviation for sampling, s s,r , is obtained by dividing equation 7.3 by n A . Solving for n allows us to calculate the number of particles that must be sampled to obtain a desired sampling variance. 7.4 Note that the relative sampling variance is inversely proportional to the number of particles sampled. Increasing the number of particles in a sample, therefore, im- proves the sampling variance. EXAMPLE 7 . 4 Suppose you are to analyze a solid where the particles containing analyte represent only 1 · 10 –7 % of the population. How many particles must be collected to give a relative sampling variance of 1%? SOLUTION Since the particles of interest account for 1 · 10 –7 % of all particles in the population, the probability of selecting one of these particles is only 1 · 10 –9 . Substituting into equation 7.4 gives Thus, to obtain the desired sampling variance we need to collect 1 · 10 13 particles. n = ?× × ×=× ? ? 1110 110 1 001 110 9 92 13 () (. ) n p p s = ? × 11 2 s,r s np p np s,r = ?()1 snpp s =?()1 1400-CH07 9/8/99 4:03 PM Page 187 188 Modern Analytical Chemistry A sample containing 10 13 particles can be fairly large. Suppose this is equivalent to a mass of 80 g. Working with a sample this large is not practical; but does this mean we must work with a smaller sample and accept a larger relative sampling variance? Fortunately the answer is no. An important feature of equation 7.4 is that the relative sampling variance is a function of the number of particles but not their combined mass. We can reduce the needed mass by crushing and grinding the par- ticles to make them smaller. Our sample must still contain 10 13 particles, but since each particle is smaller their combined mass also is smaller. If we assume that a par- ticle is spherical, then its mass is proportional to the cube of its radius. Mass r 3 Decreasing a particle’s radius by a factor of 2, for example, decreases its mass by a fac- tor of 2 3 , or 8. Instead of an 80-g sample, a 10-g sample will now contain 10 13 particles. EXAMPLE 7 . 5 Assume that the sample of 10 13 particles from Example 7.4 weighs 80 g. By how much must you reduce the radius of the particles if you wish to work with a sample of 0.6 g? SOLUTION To reduce the sample from 80 g to 0.6 g you must change its mass by a factor of This can be accomplished by decreasing the radius of the particles by a factor of x 3 = 133 x = 5.1 Decreasing the radius by a factor of approximately 5 allows you to decrease the sample’s mass from 80 g to 0.6 g. Treating a population as though it contains only two types of particles is a use- ful exercise because it shows us that the relative sampling variance can be improved by collecting more particles of sample. Furthermore, we learned that the mass of sample needed can be reduced by decreasing particle size without affecting the rela- tive sampling variance. Both are important conclusions. Few populations, however, meet the conditions for a true binomial distribu- tion. Real populations normally contain more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, many well-mixed populations, in which the population’s composition is homogeneous on the scale at which we sample, approximate binomial sampling statistics. Under these conditions the following relationship between the mass of a randomly collected grab sample, m, and the percent relative standard deviation for sampling, R, is often valid. 6 mR 2 = K s 7.5 where K s is a sampling constant equal to the mass of sample producing a percent relative standard deviation for sampling of ±1%.* The sampling constant is evalu- 80 06 133 . = times *Problem 8 in the end-of-chapter problem set asks you to consider the relationship between equations 7.4 and 7.5. 1400-CH07 9/8/99 4:03 PM Page 188 Chapter 7 Obtaining and Preparing Samples for Analysis 189 ated by determining R using several samples of similar mass. Once K s is known, the mass of sample needed to achieve a desired relative standard deviation for sampling can be calculated. EXAMPLE 7 .6 The following data were obtained in a preliminary determination of the amount of inorganic ash in a breakfast cereal. Mass of Cereal Percentage Ash (g) (w/w) 0.9956 1.34 0.9981 1.29 1.0036 1.32 0.9994 1.26 1.0067 1.28 Determine K s and the amount of sample needed to give a relative standard deviation for sampling of ±2.0%. Predict the percent relative standard deviation and the absolute standard deviation if samples of 5 g are collected. SOLUTION To determine K s we need to know the average mass of the cereal samples and the relative standard deviation for the %(w/w) ash. The average mass of the five cereal samples is 1.0007 g. The average %(w/w) ash and the absolute standard deviation are, respectively, 1.298% and 0.03194. The percent relative standard deviation, therefore, is Thus K s = mR 2 = (1.0007 g)(2.46) 2 = 6.06 g The amount of sample needed to give a relative standard deviation of ±2%, therefore, is If we use 5.00-g samples, then the expected percent relative standard deviation is and the expected absolute standard deviation is s RX s == = 100 1 10 1 298 100 0 0143 (. )(. ) . R s X =× s 100 R K m == = s 606 500 110 . . .% g g m K R == = s 22 606 20 15 . (.) . g g R s X =× = × = s 100 0 03194 1 298 100 2 46 . . .% 1400-CH07 9/8/99 4:03 PM Page 189 190 Modern Analytical Chemistry When the target population is segregated, or stratified, equation 7.5 provides a poor estimate of the amount of sample needed to achieve a desired relative standard de- viation for sampling. A more appropriate relationship, which can be applied to both segregated and nonsegregated samples, has been proposed. 7 7.6 where n s is the number of samples to be analyzed, m is the mass of each sample, A is a homogeneity constant accounting for the random distribution of analyte in the target population, and B is a segregation constant accounting for the nonrandom distribution of analyte in the target population. Equation 7.6 shows that sampling variance due to the random distribution of analyte can be minimized by increasing either the mass of each sample or the total number of samples. Sampling errors due to the nonrandom distribution of the analyte, however, can only be minimized by increasing the total number of samples. Values for the homogeneity constant and heterogeneity constant are determined using two sets of samples that differ signifi- cantly in mass. EXAMPLE 7 . 7 To develop a sampling plan for the determination of PCBs in lake sediments, the following two experiments are conducted. First, 15 samples, each with a mass of 1.00 g, are analyzed, giving a sampling variance of 0.0183. In a second experiment, ten samples, each with a mass of 10.0 g, are analyzed, giving a sampling variance of 0.0069. If samples weighing 5.00 g are to be collected, how many are needed to give a sampling variance of 0.0100? If five samples are to be collected, how much should each sample weigh? SOLUTION Substituting known values for the two experiments into equation 7.6 gives the following pair of simultaneous equations Solving for A and B gives values of 0.228 and 0.0462, respectively. The number of 5.00-g samples is determined by solving for n, giving n = 9.2 ? 9 samples. When using five samples, the mass of each is given by the equation for which m is 60.0 g. 0 0100 0 228 5 0 0462 5 . . () . =+ m 0 0100 0 228 500 0 0462 . . (. ) . =+ nn 0 0069 10 0 10 10 . ( . )( ) =+ AB 0 0183 100 15 15 . ( . )( ) =+ AB s A mn B n s ss 2 =+ 1400-CH07 9/8/99 4:03 PM Page 190 Chapter 7 Obtaining and Preparing Samples for Analysis 191 7 B. 4 How Many Samples to Collect In the previous section we considered the amount of sample needed to minimize the sampling variance. Another important consideration is the number of samples required to achieve a desired maximum sampling error. If samples drawn from the target population are normally distributed, then the following equation describes the confidence interval for the sampling error where n s is the number of samples and s s is the sampling standard deviation. Rear- ranging and substituting e for the quantity (m – – X), gives the number of samples as 7.7 where s s 2 and e 2 are both expressed as absolute uncertainties or as relative uncertain- ties. Finding a solution to equation 7.7 is complicated by the fact that the value of t depends on n s . As shown in Example 7.8, equation 7.7 is solved iteratively. EXAMPLE 7 .8 In Example 7.6 we found that an analysis for the inorganic ash content of a breakfast cereal required a sample of 1.5 g to establish a relative standard deviation for sampling of ±2.0%. How many samples are needed to obtain a relative sampling error of no more than 0.80% at the 95% confidence level? SOLUTION Because the value of t depends on n s , and the value of n s is not yet known, we begin by letting n s = ￥ and use the associated value of t. From Appendix 1B, the value for t is 1.96. Substituting known values into equation 7.7 Letting n s = 24, the value of t from Appendix 1B is 2.075. Recalculating n s gives When n s = 27, the value of t is 2.066. Recalculating n s , we find Since two successive calculations give the same value for n s , an iterative solution has been found. Thus, 27 samples are needed to achieve the desired sampling error. Equation 7.7 only provides an estimate for the smallest number of samples ex- pected to produce the desired sampling error. The actual sampling error may be substantially higher if the standard deviation for the samples that are collected is significantly greater than the standard deviation due to sampling used to calculate n s . n s ==≈ (. )(.) (. ) . 2 066 2 0 080 26 7 27 22 2 n s ==≈ (. )(.) (. ) . 2 075 2 0 080 26 9 27 22 2 n s == (. )(.) (. ) 196 20 080 24 22 2 n ts e s s = 22 2 μ= ±X ts n s s 1400-CH07 9/8/99 4:03 PM Page 191 192 Modern Analytical Chemistry This is not an uncommon problem. For a target population with a relative sampling variance of 50 and a desired relative sampling error of ±5%, equation 7.7 predicts that ten samples are sufficient. In a simulation in which 1000 samples of size 10 were collected, however, only 57% of the samples resulted in sampling errors of less than ±5%. 8 By increasing the number of samples to 17 it was possible to ensure that the desired sampling error was achieved 95% of the time. 7 B. 5 Minimizing the Overall Variance A final consideration in developing a sampling plan is to minimize the overall vari- ance for the analysis. Equation 7.2 shows that the overall variance is a function of the variance due to the method and the variance due to sampling. As we have seen, we can improve the variance due to sampling by collecting more samples of proper size. Increasing the number of times we analyze each sample improves the variance due to the method. If s s 2 is significantly greater than s m 2 , then the method’s variance can be ignored and equation 7.7 used to estimate the number of samples to analyze. Analyzing any sample more than once will not improve the overall variance, since the variance due to the method is insignificant. If s m 2 is significantly greater than s s 2 , then we only need to collect and analyze a single sample. The number of replicate analyses, n r , needed to minimize the error due to the method is given by an equation similar to equation 7.7 Unfortunately, the simple situations just described are often the exception. In many cases, both the sampling variance and method variance are significant, and both multiple samples and replicate analyses of each sample are required. The over- all error in this circumstance is given by 7.8 Equation 7.8 does not have a unique solution because different combinations of n s and n r give the same overall error. The choice of how many samples to collect and how many times each sample should be analyzed is determined by other concerns, such as the cost of collecting and analyzing samples, and the amount of available sample. EXAMPLE 7 . 9 A certain analytical method has a relative sampling variance of 0.40% and a relative method variance of 0.070%. Evaluate the relative error (a = 0.05) if (a) you collect five samples, analyzing each twice; and, (b) you collect two samples, analyzing each five times. SOLUTION Both sampling strategies require a total of ten determinations. Using Appendix 1B, we find that the value of t is 2.26. Substituting into equation 7.8, we find that the relative error for the first sampling strategy is e =+ ? ? ? ? ? ? =226 040 5 0 070 52 067 12 . .. ()() .% / et s n s nn =+ ? ? ? ? ? ? s s m sr 22 12/ n ts e r m = 22 2 1400-CH07 9/8/99 4:03 PM Page 192 Chapter 7 Obtaining and Preparing Samples for Analysis 193 and that for the second sampling strategy is As expected, since the relative method variance is better than the relative sampling variance, a sampling strategy that favors the collection of more samples and few replicate analyses gives the better relative error. 7 C Implementing the Sampling Plan After a sampling plan has been developed, it is put into action. Implementing a sampling plan normally involves three steps: physically removing the sample from its target population, preserving the sample, and preparing the sample for analysis. Except for in situ sampling, the analysis of a sample occurs after removing it from the target population. Since sampling exposes the target population to potential contamination, the sampling device must be inert and clean. Once a sample is withdrawn from a target population, there is a danger that it may undergo a chemical or physical change. This is a serious problem since the properties of the sample will no longer be representative of the target population. For this reason, samples are often preserved before transporting them to the labora- tory for analysis. Even when samples are analyzed in the field, preservation may still be necessary. The initial sample is called the primary, or gross sample and may be a single increment drawn from the target population, or a composite of several increments. In many cases the gross sample cannot be analyzed without further treatment. Pro- cessing the gross sample may be used to reduce the sample’s particle size, to transfer the sample into a more readily analyzable form, or to improve its homogeneity. In the sections that follow, these three steps are considered for the sampling of liquids (including solutions), gases, and solids. 7 C.1 Solutions Typical examples of liquid samples include those drawn from containers of com- mercial solvents; beverages, such as milk or fruit juice; natural waters, including from lakes, streams, seawater, and rain; bodily fluids, such as blood and urine; and, suspensions, such as those found in many oral medications. Sample Collection Homogeneous solutions are easily sampled by siphoning, de- canting, or by using a pipet or syringe. Unfortunately, few solutions are truly homo- geneous. When the material to be sampled is of manageable size, manual shaking is often sufficient to ensure homogeneity. Samples may then be collected with a pipet, a syringe, or a bottle. The majority of solutions, however, cannot be sampled in this manner. To minimize the effect of heterogeneity, the method for collecting the gross sample must be adapted to the material being sampled. The environmental sampling of waters and wastewaters provides a good illus- tration of many of the methods used to sample solutions. The chemical composi- tion of surface waters, such as streams, rivers, lakes, estuaries, and oceans, is influ- enced by flow rate and depth. Rapidly flowing shallow streams and rivers, and shallow (<5 m) lakes are usually well mixed and show little stratification with e =+ ? ? ? ? ? ? =226 040 2 0 070 25 10 12 . .. ()() .% / gross sample The initial sample, collected from the target population without any processing. 1400-CH07 9/8/99 4:03 PM Page 193 depth. Grab samples are conveniently collected by submerging a capped bottle below the surface and removing the cap. The air–water interface, which may be en- riched with heavy metals 9 or contaminated with oil, is avoided when collecting the sample. After the sample bottle is filled, the cap is replaced and the bottle removed. Slowly moving streams and rivers, lakes deeper than 5 m, estuaries, and oceans may show substantial stratification. Grab samples from near the surface can be collected as described earlier, whereas samples at greater depths are collected with a weighted sample bottle that is lowered to the desired depth. Once it has reached the desired depth, the sample bottle is opened, allowed to fill, and closed before retrieving. Grab samples can be analyzed individually, giving information about changes in the analyte’s concentration with depth. Alternatively, the grab samples may be pooled to form a composite sample. Wells used for collecting groundwater samples must be purged before the sam- ple is collected, since the chemical composition of water in the well-casing and in the adjacent matrix may be significantly different from that of the surrounding groundwater. These differences may result from contaminants introduced when drilling the well, or differences in the groundwater’s redox potential when exposed to atmospheric oxygen. In general, wells are purged by pumping out a volume of water equivalent to several well-casing volumes, or until the water’s temperature, pH, or specific conductance are constant. Samples collected from municipal water supplies must also be purged since the chemical composition of water left standing in pipes may differ significantly from the treated water supply. Samples are collected at faucets after flushing the pipes for 2–3 min. Samples from municipal wastewater treatment plants and samples of industrial discharges often are collected as 24-h composites. Samples are obtained using an automatic sampler that periodically removes individual grab samples. The volume of each sample increment and the frequency of sampling may be constant or may vary in response to changes in flow rate. Sample containers for collecting solutions are made from glass or plastic. Con- tainers made from Kimax or Pyrex brand borosilicate glass have the advantage of being sterilizable, easy to clean, and inert to all solutions except those that are strongly alkaline. The disadvantages of glass containers are cost, weight, and the likelihood of breakage. Plastic containers are made from a variety of polymers, in- cluding polyethylene, polypropylene, polycarbonate, polyvinyl chloride, and Teflon (polytetrafluoroethylene). Plastic containers are lightweight, durable, and, except for those manufactured from Teflon, inexpensive. In most cases glass or plastic bot- tles may be used, although polyethylene bottles are generally preferred because of their lower cost. Glass containers are always used when collecting samples for the analysis of pesticides, oil and grease, and organics because these species often inter- act with plastic surfaces. Since glass surfaces easily adsorb metal ions, plastic bottles are preferred when collecting samples for the analysis of trace metals. In most cases the sample bottle has a wide mouth, making it easy to fill and re- move the sample. A narrow-mouth sample bottle is used when exposing the sample to the container cap or to the outside environment is undesirable. Unless exposure to plastic is a problem, caps for sample bottles are manufactured from polyethylene. When polyethylene must be avoided, the container cap includes an inert interior liner of neoprene or Teflon. Sample Preservation Once removed from its target population, a liquid sample’s chemical composition may change as a result of chemical, biological, or physical processes. Following its collection, samples are preserved by controlling the solu- 194 Modern Analytical Chemistry 1400-CH07 9/8/99 4:03 PM Page 194 tion’s pH and temperature, limiting its exposure to light or to the atmosphere, or by adding a chemical preservative. After preserving, samples may be safely stored for later analysis. The maximum holding time between preservation and analysis de- pends on the analyte’s stability and the effectiveness of sample preservation. Table 7.1 provides a list of sample preservation methods and maximum holding times for sev- eral analytes of importance in the analysis of water and wastewater. Sample Preparation Most analytical methods can be applied to analytes in a liquid or solution state. For this reason a gross sample of a liquid or solution does not need additional processing to bring it into a more suitable form for analysis. 7 C.2 Gases Typical examples of gaseous samples include automobile exhaust, emissions from industrial smokestacks, atmospheric gases, and compressed gases. Also included with gaseous samples are solid aerosol particulates. Sample Collection The simplest approach for collecting a gas sample is to fill a container, such as a stainless steel canister or a Tedlar/Teflon bag, with a portion of the gas. A pump is used to pull the gas into the container, and, after flushing the container for a predetermined time, the container is sealed. This method has the ad- vantage of collecting a more representative sample of the gas than other collection techniques. Disadvantages include the tendency for some gases to adsorb to the container’s walls, the presence of analytes at concentrations too low to detect with accuracy and precision, and the presence of reactive gases, such as ozone and nitro- gen oxides, that may change the sample’s chemical composition with time, or react with the container. When using a stainless steel canister many of these disadvan- tages can be overcome with cryogenic cooling, which changes the sample from a gaseous to a liquid state. Due to the difficulty of storing gases, most gas samples are collected using ei- ther a trap containing a solid sorbent or by filtering. Solid sorbents are used to col- lect volatile gases (vapor pressures more than approximately 10 –6 atm) and semi- volatile gases (vapor pressures between approximately 10 –6 atm and 10 –12 atm), and filtration is used to collect nonvolatile gases. Chapter 7 Obtaining and Preparing Samples for Analysis 195 Table 7 .1 Preservation Methods and Maximum Holding Times for Selected Water and Wastewater Parameters Parameter Preservation Maximum Holding Time ammonia cool to 4 °C; H 2 SO 4 to pH < 2 28 days chloride none required 28 days metals—Cr(VI) cool to 4 °C 24 h metals—Hg HNO 3 to pH < 2 28 days metals—all others HNO 3 to pH < 2 6 months nitrate none required 48 h organochlorine pesticides 1 mL 10 mg/mL HgCl 2 ; 7 days without extraction or addition of extracting 40 days with extraction solvent pH none required analyze immediately 1400-CH07 9/8/99 4:03 PM Page 195 Solid sorbent sampling is accomplished by passing the gas through a canister packed with sorbent particles. Typically 2–100 L of gas is sampled when collecting volatile compounds, and 2–500 m 3 when collecting semivolatile gases.* A variety of inorganic, organic polymer and carbon sorbents have been used. Inorganic sor- bents, such as silica gel, alumina, magnesium aluminum silicate, and molecular sieves, are efficient collectors for polar compounds. Their efficiency for collecting water, however, limits their sorption capacity for many organic compounds. Organic polymeric sorbents are manufactured using polymeric resins of 2,4-diphenyl-p-phenylene oxide or styrene-divinylbenzene for volatile compounds, or polyurethane foam for semivolatile compounds. These materials have a low affin- ity for water and are efficient collectors for all but the most highly volatile organic compounds and some low-molecular-weight alcohols and ketones. The adsorbing ability of carbon sorbents is superior to that of organic polymer resins. Thus, carbon sorbents can be used to collect those highly volatile organic compounds that cannot be collected by polymeric resins. The adsorbing ability of carbon sorbents may be a disadvantage, however, since the adsorbed compounds may be difficult to desorb. Nonvolatile compounds are normally present either as solid particulates or bound to solid particulates. Samples are collected by pulling large volumes of gas through a filtering unit where the particulates are collected on glass fiber filters. One of the most significant problems with sorbent sampling is the limited ca- pacity of the sorbent to retain gaseous analytes. If a sorbent’s capacity is exceeded before sampling is complete then a portion of the analyte will pass through the can- ister without being retained, making an accurate determination of its concentration impossible. For this reason it is not uncommon to place a second sorbent canister downstream from the first. If the analyte is not detected in the second canister, then it is safe to assume that the first canister’s capacity has not been exceeded. The vol- ume of gas that can be sampled before exceeding the sorbent’s capacity is called the breakthrough volume and is normally reported with units of m 3 /(g pack ), where g pack is the grams of sorbent. The short-term exposure of humans, animals, and plants to gaseous pollutants is more severe than that for pollutants in other matrices. Since the composition of atmospheric gases can show a substantial variation over a time, the continuous monitoring of atmospheric gases such as O 3 , CO, SO 2 , NH 3 , H 2 O 2 , and NO 2 by in situ sampling is important. 10 Sample Preservation and Preparation After collecting the gross sample there is generally little need for sample preservation or preparation. The chemical composi- tion of a gas sample is usually stable when it is collected using a solid sorbent, a fil- ter, or by cryogenic cooling. When using a solid sorbent, gaseous compounds may be removed before analysis by thermal desorption or by extracting with a suitable solvent. Alternatively, when the sorbent is selective for a single analyte, the increase in the sorbent’s mass can be used to determine the analyte’s concentration in the sample. 7 C. 3 Solids Typical examples of solid samples include large particulates, such as those found in ores; smaller particulates, such as soils and sediments; tablets, pellets, and capsules used in dispensing pharmaceutical products and animal feeds; sheet materials, such as polymers and rolled metals; and tissue samples from biological specimens. 196 Modern Analytical Chemistry breakthrough volume The volume of sample that can be passed through a solid sorbent before the analytes are no longer retained. *1 m 3 is equivalent to 1000 L. 1400-CH07 9/8/99 4:03 PM Page 196 Figure 7.5 Schematic diagram of grab sampler. When the sampler reaches the sediment, the jaws of the grab sampler are closed, collecting a sample of the sediment. Sample Collection Solids are usually heteroge- neous, and samples must be collected carefully if they are to be representative of the target popula- tion. As noted earlier, solids come in a variety of forms, each of which is sampled differently. Sediments from the bottom of streams, rivers, lakes, estuaries, and oceans are collected with a bot- tom grab sampler or with a corer. Grab samplers are equipped with a pair of “jaws” that close when they contact the sediment, scooping up sediment in the process (Figure 7.5). Their principal advantages are ease of use and the ability to collect a large sample. Disadvantages include the tendency to lose finer- grained sediment particles as water flows out of the sampler and the loss of spatial information, both lat- erally and with depth, due to mixing of the sample. An alternative method for sampling sediments uses a cylindrical coring device (Figure 7.6). The corer is dropped into the sediment, collecting a col- umn of sediment and the water in contact with the sediment. With the possible exception of sediment at the surface, which may experience mixing, sam- ples collected with a corer maintain their vertical profile. As a result, changes in the sediment’s com- position with depth are preserved. The main disad- vantage to a corer is that only a small surface area is sampled. For this reason sampling with a corer usu- ally requires more samples. Soil samples collected at depths of up to 30 cm are easily collected with scoops or shovels, although the sampling variance is generally high. A better method for obtaining soil samples near the surface is to use a soil punch. This thin-walled steel tube retains a core sample when it is pushed into the soil and removed. Soil samples collected at depths greater than 30 cm are obtained by digging a trench and collecting lateral samples with a soil punch. Alternatively, Chapter 7 Obtaining and Preparing Samples for Analysis 197 Open Closed Sedimentar y la y ers Sea bottom Figure 7.6 Schematic diagram of a piston corer in operation. The weight of the corer is sufficient to cause its penetration into the sediment, while the upward motion of the piston allows water pressure to help force the sediment column into the barrel of the corer. 1400-CH07 9/8/99 4:03 PM Page 197 Figure 7.8 Schematic diagram of a sample thief. Rotating the inner cylinder opens and closes the openings along the outer cylinder’s shaft. an auger may be used to drill a hole to the desired depth and the sample col- lected with a soil punch. The sampling of particulate material is often determined by the size of the par- ticles. Large particulate solids, such as coals and ores, can be sampled by randomly collecting samples with a shovel or by riffling. A riffle (Figure 7.7) is a trough con- taining an even number of compartments with adjacent compartments emptying on opposite sides of the riffle. Particulate material dumped into a riffle is divided into two parts. By repeatedly passing half of the separated material back through the riffle, a sample of any desired size may be collected. Smaller particulate materials, such as powders, are best collected with a sample thief, which allows material to be collected simultaneously from several locations (Figure 7.8). A typical sample thief consists of two tubes that are nestled together. Each tube has an identical set of slots aligned down their length. Before the sample thief is inserted into the material being sampled, the inner tube is rotated so that slots are closed. When the sample thief is in place, the inner tube is rotated to open the slots, allowing the powder to enter the sample thief through each slot. The inner tube is then rotated to the closed position and the sample thief withdrawn. When sampling a metal, it usually is necessary to obtain material from both the surface and the interior. When the metal is in the form of a sheet, random samples can be collected with a metal punch. Samples can be obtained from a metal wire by randomly cutting off pieces of an appropriate length. Larger pieces of metal, such as bars or bricks, are best sampled by sawing through the metal at randomly selected points and collecting the “sawdust” or by drilling through the metal and collecting the shavings. A surface coating can be sampled in situ or by dissolving the coating with an appropriate solvent. Sampling of biological tissue is done by removing the entire organ, which is then homogenized before smaller portions are taken for analysis. Alternatively, sev- eral small portions of tissue may be combined to form a composite sample. The composite sample is then homogenized and analyzed. Sample Preservation Without preservation, many solid samples are subject to changes in chemical composition due to the loss of volatile material, biodegrada- tion, and chemical reactivity (particularly redox reactions). Samples stored at re- duced temperatures are less prone to biodegradation and the loss of volatile mate- rial, but fracturing and phase separations may present problems. The loss of volatile material is minimized by ensuring that the sample completely fills its container without leaving a headspace where gases can collect. Samples collected from mate- rials that have not been exposed to O 2 are particularly susceptible to oxidation reac- tions. For example, the contact of air with anaerobic sediments must be prevented. Sample Preparation Unlike gases and liquids, which generally require little sample preparation, solid samples usually need some processing before analysis. There are two reasons for this. First, as discussed in Section 7B.3, sampling variance is a function of the number of particles sampled, not their combined mass. For extremely heteroge- neous populations consisting of large particulates, the gross sample may be too large to analyze. For example, a boxcar containing a load of a Ni-bearing ore with an average particle size of 5 mm may require a sample weighing one ton to obtain a reasonable sampling variance. Reducing the sample’s average particle size allows the same number of particles to be sampled with a smaller, more manageable combined mass. Second, the majority of analytical techniques, particularly those used for a quantitative analysis, require that the analyte be in solution. Solid samples, or at least the analytes in a solid sample, must be brought into solution. 198 Modern Analytical Chemistry Figure 7.7 Example of a four-unit riffle. A sample added through the top is divided into four piles, two on each of the riffle’s sides. 1400-CH07 9/8/99 4:03 PM Page 198 Figure 7.9 Illustration showing the method of coning and quartering as a means of reducing a gross sample for subsampling. (a) The gross sample is first piled into a cone and (b) flattened. Looking down from above, (c) the cone is divided into four quarters, (d) which are then separated. Reducing Particle Size A reduction in particle size is accomplished by a combina- tion of crushing and grinding the gross sample. The resulting particulates are then thoroughly mixed and divided into samples of smaller mass containing the appro- priate number of particles. The process seldom occurs in a single step. Instead, sam- ples are cycled through the process several times until a laboratory sample of de- sired mass is obtained. Crushing and grinding uses mechanical force to break larger particles into smaller ones. A variety of tools are used depending on the particle’s size and hard- ness. Large particles are crushed using jaw crushers capable of reducing particles to diameters of a few millimeters. Ball mills, disk mills, and mortars and pestles are used to further reduce particle size. Significant changes in composition may occur during crushing and grinding. Decreasing particle size increases available surface area. With more surface area there is a greater risk of losing volatile components, a problem made worse by the frictional heat accompanying the crushing and grinding. An increase in surface area also means that portions of the sample are freshly exposed to the atmosphere where oxidation may alter the sample’s composition. Other problems include contamina- tion from the mechanical abrasion of the materials used to crush and grind the sample, and differences in the ease with which particles are reduced in size. Softer particles are reduced in size more easily and may be lost as dust before the rest of the sample has been processed. This is a problem since the analyte’s distribution may not be uniform between particles of different size. To ensure that all particles are reduced to a uniform size, the sample is intermit- tently passed through a sieve. Processing of those particles not passing through the sieve continues until the entire sample is of uniform size. The sample is then mixed thoroughly to ensure homogeneity, and a secondary sample obtained with a riffle or by coning and quartering. The latter approach is outlined in Figure 7.9. The gross sample is piled into a cone, flattened, divided into four quarters, and two diagonally Chapter 7 Obtaining and Preparing Samples for Analysis 199 (a) (b) (c) (d) laboratory sample Sample taken into the lab for analysis after processing the gross sample. coning and quartering A process for reducing the size of a gross sample. 1400-CH07 9/8/99 4:03 PM Page 199 Figure 7.10 Schematic diagram of a microwave digestion vessel. opposed quarters are discarded. The remaining material is cycled through the process of coning and quartering until the desired amount of sample remains. Bringing Solid Samples into Solution If you are fortunate, the sample with which you are working will easily dissolve in a suitable solvent, requiring no more effort than gentle swirling and heating. Distilled water is usually the solvent of choice for inorganic salts, but an organic solvent, such as methanol, chloroform, or toluene, is used for organic materials. More often, one or more of the sample’s components re- sist simple dissolution. With samples that are difficult to dissolve, the first approach is usually to try di- gesting the sample with an acid or base. Table 7.2 lists the most commonly used acids and bases and summarizes their use. Digestion is commonly carried out in an open container, such as a beaker, using a hot plate as a source of heat. The chief ad- vantage of this approach is its low cost as it requires no special equipment. Volatile reaction products, however, are lost, leading to a determinate error if analyte is in- cluded among the volatile substances. Many digestions are now carried out in closed containers using microwave ra- diation as a source of energy for heating the solution. Vessels for microwave diges- tion are manufactured using Teflon (or some other fluoropolymer) or fused silica. Both materials are thermally stable, chemically resistant, transparent to microwave radiation, and capable of withstanding elevated pressures. A typical microwave di- gestion vessel is shown in Figure 7.10 and consists of the vessel body and cap, a tem- perature probe, and a pressure relief valve. Vessels are placed in a microwave oven (typically 6–12 vessels can be accommodated), and microwave energy is controlled by monitoring the temperature or pressure within the vessels. A microwave diges- tion has several important advantages over an open container digestion, including higher temperatures (200–300 °C) and pressures (40–100 bar). As a result, diges- tions requiring several hours in an open container may be accomplished in less than 200 Modern Analytical Chemistry Table 7 .2 Acids and Bases Used for Sample Digestion Solution (≈ %w/w) Uses and Properties HCl (37%) ? dissolves metals more easily reduced than H 2 (E o < 0) ? dissolves insoluble carbonates, sulfides, phosphates, fluorides, sulfates, and many oxides HNO 3 (70%) ? strong oxidizing agent ? dissolves most common metals except Al and Cr ? decomposes organics and biological samples (wet ashing) H 2 SO 4 (98%) ? dissolves many metals and alloys ? decomposes organics by oxidation and dehydration HF (50%) ? dissolves silicates forming volatile SiF 4 HClO 4 (70%) ? hot, concentrated solutions are strong oxidizing agents ? dissolves many metals and alloys ? decomposes organics (reactions with organics are often explosive, use only in specially equipped hoods with a blast shield and after prior decomposition with HNO 3 ) HCl:HNO 3 (3:1 v/v) ? also known as aqua regia ? dissolves Au and Pt NaOH ? dissolves Al and amphoteric oxides of Sn, Pb, Zn, and Cr Pressure relief valve Temperature probe Pressure probe Cap Vessel body 1400-CH07 9/8/99 4:03 PM Page 200 Chapter 7 Obtaining and Preparing Samples for Analysis 201 Table 7 . 3 Common Fluxes for Decomposing Inorganic Samples Melting Flux Temperature (°C) Crucible Typical Samples Na 2 CO 3 851 Pt silicates, oxides, phosphates, sulfides Li 2 B 4 O 7 930 LiBO 2 845 Pt, graphite aluminosilicates, carbonates NaOH 318 KOH 380 Au, Ag silicates, silicon carbide Na 2 O 2 — Ni silicates, chromium steel, Pt alloys K 2 S 2 O 7 300 Pt, porcelain oxides B 2 O 3 577 Pt silicates, oxides 30 min using microwave digestion. In addition, the closed container prevents the loss of volatile gases. Disadvantages include the inability to add reagents during di- gestion, limitations on the amount of sample that can be used (typically 1 g or less), and safety concerns due to the use of high pressures and corrosive reagents. Appli- cations include environmental and biological samples. Inorganic samples that resist decomposition by digestion with acids or bases often can be brought into solution by fusing with a large excess of an al- kali metal salt, called a flux. The sample and flux are mixed together in a cru- cible and heated till the substances fuse together in a molten state. The resulting melt is allowed to cool slowly to room temperature. Typically the melt dissolves readily in distilled water or dilute acid. Several common fluxes and their uses are listed in Table 7.3. Fusion works when other methods of decomposition do not because of the higher temperatures obtained and the high concentration of the reactive flux in the molten liquid. Disadvantages include a greater risk of contamination from the large quantity of flux and the crucible and the loss of volatile materials. Finally, organic materials may be decomposed by dry ashing. In this method the sample is placed in a suitable crucible and heated over a flame or in a furnace. Any carbon present in the sample is oxidized to CO 2 , and hydrogen, sulfur, and ni- trogen are removed as H 2 O, SO 2 and N 2 . These gases can be trapped and weighed to determine their content in the organic material. Often the goal of dry ashing is the removal of organic material, leaving behind an inorganic residue, or ash, that can be further analyzed. 7 D Separating the Analyte from Interferents When a method shows a high degree of selectivity for the analyte, the task of per- forming a quantitative, qualitative, or characterization analysis is simplified. For ex- ample, a quantitative analysis for glucose in honey is easier to accomplish if the method is selective for glucose, even in the presence of other reducing sugars, such as fructose. Unfortunately, analytical methods are rarely selective toward a single species. In the absence of interferents, the relationship between the sample’s signal, S samp , and the concentration of analyte, C A , is S samp = k A C A 7.9 1400-CH07 9/8/99 4:03 PM Page 201 202 Modern Analytical Chemistry *In equation 7.9, and the equations that follow, the concentration of analyte, C A , can be replaced by the moles of analyte, n A , when considering a total analysis technique. recovery The fraction of analyte or interferent remaining after a separation (R). where k A is the analyte’s sensitivity.* In the presence of an interferent, equation 7.9 becomes S samp = k A C A + k I C I 7.10 where k I and C I are the interferent’s sensitivity and concentration, respectively. A method’s selectivity is determined by the relative difference in its sensitivity toward the analyte and interferent. If k A is greater than k I , then the method is more selec- tive for the analyte. The method is more selective for the interferent if k I is greater than k A . Even if a method is more selective for an interferent, it can be used to deter- mine an analyte’s concentration if the interferent’s contribution to S samp is insignifi- cant. The selectivity coefficient, K A,I , was introduced in Chapter 3 as a means of characterizing a method’s selectivity. 7.11 Solving equation 7.11 for k I and substituting into equation 7.10 gives, after simplifying S samp = k A (C A + K A,I · C I ) 7.12 An interferent, therefore, will not pose a problem as long as the product of its con- centration and the selectivity coefficient is significantly smaller than the analyte’s concentration. K A,I · C I << C A When an interferent cannot be ignored, an accurate analysis must begin by separat- ing the analyte and interferent. 7 E General Theory of Separation Efficiency The goal of an analytical separation is to remove either the analyte or the interferent from the sample matrix. To achieve a separation there must be at least one signifi- cant difference between the chemical or physical properties of the analyte and inter- ferent. Relying on chemical or physical properties, however, presents a fundamental problem—a separation also requires selectivity. A separation that completely re- moves an interferent may result in the partial loss of analyte. Altering the separation to minimize the loss of analyte, however, may leave behind some of the interferent. A separation’s efficiency is influenced both by the failure to recover all the ana- lyte and the failure to remove all the interferent. We define the analyte’s recovery, R A , as where C A is the concentration of analyte remaining after the separation, and (C A ) o is the analyte’s initial concentration. A recovery of 1.00 means that none of the ana- lyte is lost during the separation. The recovery of the interferent, R I , is defined in the same manner 7.13 R C C I I Io = () R C C A A Ao = () K k k A,I I A = 1400-CH07 9/8/99 4:03 PM Page 202 Chapter 7 Obtaining and Preparing Samples for Analysis 203 where C I is the concentration of interferent remaining after the separation, and (C I ) o is the interferent’s initial concentration. The degree of separation is given by a separation factor, S I,A , which is the change in the ratio of interferent to analyte caused by the separation. 11 EXAMPLE 7 .10 An analysis to determine the concentration of Cu in an industrial plating bath uses a procedure for which Zn is an interferent. When a sample containing 128.6 ppm Cu is carried through a separation to remove Zn, the concentration of Cu remaining is 127.2 ppm. When a 134.9-ppm solution of Zn is carried through the separation, a concentration of 4.3 ppm remains. Calculate the recoveries for Cu and Zn and the separation factor. SOLUTION The recoveries for the analyte and interferent are and The separation factor is In an ideal separation R A = 1, R I = 0, and S I,A = 0. In general, the separation factor should be approximately 10 –7 for the quantitative analysis of a trace analyte in the presence of a macro interferent, and 10 –3 when the analyte and interferent are pres- ent in approximately equal amounts. Recoveries and separation factors are useful ways to evaluate the effectiveness of a separation. They do not, however, give a direct indication of the relative error introduced by failing to remove all interferents or failing to recover all the analyte. The relative error introduced by the separation, E, is defined as 7.14 where S * samp is the expected signal for an ideal separation when all the analyte is re- covered. S * samp = k A (C A ) o 7.15 Substituting equations 7.12 and 7.15 into 7.14 gives E kC K C kC kC = +×? AA A,I I AAo AAo ()() () E SS S = ? samp samp samp * * S R R Zn,Cu Zn Cu == = 0 032 0 9891 0 032 . . . R Zn == 43 134 9 0 032 3 2 . . ..% ppm ppm , or R Cu == 127 2 128 6 0 9891 98 91 . . ., .% ppm ppm or S CC CC R R I, A IA Io Ao I A == / ( ) /( ) separation factor A measure of the effectiveness of a separation at separating an analyte from an interferent (S I,A ). 1400-CH07 9/8/99 4:03 PM Page 203 204 Modern Analytical Chemistry which simplifies to 7.16 A more useful equation for the relative error is obtained by solving equation 7.13 for C I and substituting back into equation 7.16 7.17 The first term of equation 7.17 accounts for the incomplete recovery of analyte, and the second term accounts for the failure to remove all the interferent. EXAMPLE 7 .11 Following the separation outlined in Example 7.10, an analysis is to be carried out for the concentration of Cu in an industrial plating bath. The concentration ratio of Cu to Zn in the plating bath is 7:1. Analysis of standard solutions containing only Cu or Zn give the following standardization equations S Cu = 1250 · (ppm Cu) S Zn = 2310 · (ppm Zn) (a) What error is expected if no attempt is made to remove Zn before analyzing for Cu? (b) What is the error if the separation is carried out? (c) What is the maximum acceptable recovery for Zn if Cu is completely recovered and the error due to the separation must be no greater than 0.10%? SOLUTION (a) If the analysis is carried out without a separation, then R Cu and R Zn are equal to 1.000, and equation 7.17 simplifies to From equation 7.11 the selectivity coefficient is Although we do not know the actual concentrations of Zn or Cu in the sample, we do know that the concentration ratio (ppm Zn) o /(ppm Cu) o is 1/7. Thus (b) To calculate the error, we substitute the recoveries calculated in Example 7.10 into equation 7.17 E == ( . )( ) () ..% 185 1 7 0 264 26 4, or K k k Cu,Zn Zn Cu == = 2310 1250 185. E K = × Cu,Zn o o (ppm Zn) (ppm Cu) ER KC C R=?+ × × ? ? ? ? ? ? () () () A A,I I o Ao I 1 E CK CC C C C C C KC C R KC C = +×? =?+ × =?+ × AA,II Ao Ao A Ao Ao Ao A,I I Ao A A,I I Ao () () () () () () () () 1 1400-CH07 9/8/99 4:03 PM Page 204 Chapter 7 Obtaining and Preparing Samples for Analysis 205 Note that a negative determinate error introduced by failing to recover all the analyte is partially offset by a positive determinate error due to a failure to remove all the interferent. (c) To determine the maximum allowed recovery for Zn, we make appropriate substitutions into equation 7.17 and solve for R Zn , obtaining a recovery of 0.0038, or 0.38%. Thus, at least 99.62% of the Zn must be removed by the separation. 7 F Classifying Separation Techniques An analyte and an interferent can be separated if there is a significant difference in at least one of their chemical or physical properties. Table 7.4 provides a partial list of several separation techniques, classified by the chemical or physical property that is exploited. 7 F.1 Separations Based on Size The simplest physical property that can be exploited in a separation is size. The separation is accomplished using a porous medium through which only the analyte or inter- ferent can pass. Filtration, in which gravity, suction, or pressure is used to pass a sample through a porous filter is the most commonly encountered separation technique based on size. Particulate interferents can be separated from dis- solved analytes by filtration, using a filter whose pore size retains the interferent. This separation technique is important in the analysis of many natural waters, for which the presence of suspended solids may interfere in the analysis. Filtration also can be used to isolate analytes present as solid particulates from dissolved ions in the sample matrix. For example, this is a necessary step in gravimetry, in which the analyte is isolated as a precipi- tate. A more detailed description of the types of available filters is found in the discussion of precipitation gravimetry and particulate gravimetry in Chapter 8. 0 0010 1 000 1 185 1 7 .(.) ( . )( ) () =?+ × ? ? ? ? ? ? R Zn E =?+ × ? ? ? ? ? ? =? + =? ? (. ) ( . )( ) () . (. )(. ) ..% 0 9891 1 185 1 7 0 032 0 0109 0 0085 0 0024 0 24, or Table 7 . 4 Classification of Separation Techniques Basis of Separation Separation Technique size filtration dialysis size-exclusion chromatography mass and density centrifugation complex formation masking change in physical state distillation sublimation recrystallization change in chemical state precipitation ion exchange electrodeposition volatilization partitioning between phases extraction chromatography 1400-CH07 9/8/99 4:03 PM Page 205 Figure 7.11 Illustration of a dialysis membrane in action. In (a) the sample solution is placed in the dialysis tube and submerged in the solvent. (b) Smaller particles pass through the membrane, but larger particles remain within the dialysis tube. Another example of a separation technique based on size is dialysis, in which a semipermeable membrane is used to separate the analyte and interferent. Dialysis membranes are usually constructed from cellulose, with pore sizes of 1–5 nm. The sample is placed inside a bag or tube constructed from the membrane. The dialysis membrane and sample are then placed in a container filled with a solution whose composition differs from that of the sample. If the concentration of a particular species is not the same on the two sides of the membrane, the resulting concentra- tion gradient provides a driving force for its diffusion across the membrane. Al- though small particles may freely pass through the membrane, larger particles are unable to pass (Figure 7.11). Dialysis is frequently used to purify proteins, hormones, and enzymes. During kidney dialysis, metabolic waste products, such as urea, uric acid, and creatinine, are removed from blood by passing it over a dialysis membrane. Size-exclusion chromatography, which also is called gel permeation or molecular- exclusion chromatography, is a third example of a separation technique based on size. In this technique a column is packed with small, approximately 10-m m, porous particles of cross-linked dextrin or polyacrylamide. The pore size of the particles is controlled by the degree of cross-linking, with greater cross-linking resulting in smaller pore sizes. The sample to be separated is placed into a stream of solvent that is pumped through the column at a fixed flow rate. Particles too large to enter the pores are not retained and pass through the column at the same rate as the solvent. Those particles capable of entering into the pore structure take longer to pass through the column. Smaller particles, which penetrate more deeply into the pore structure, take the longest time to pass through the column. Size-exclusion chro- matography is widely used in the analysis of polymers and in biochemistry, where it is used for the separation of proteins. 7 F.2 Separations Based on Mass or Density If there is a difference in the mass or density of the analyte and interferent, then a separation using centrifugation may be possible. The sample, as a suspension, is placed in a centrifuge tube and spun at a high angular velocity (high numbers of revolutions per minute, rpm). Particles experiencing a greater centrifugal force have faster sedimentation rates and are preferentially pulled toward the bottom of the 206 Modern Analytical Chemistry dialysis A method of separation that uses a semi- permeable membrane. (a) (b) size-exclusion chromatography A separation method in which a mixture passes through a bed of porous particles, with smaller particles taking longer to pass through the bed due to their ability to move into the porous structure. 1400-CH07 9/8/99 4:03 PM Page 206 centrifuge tube. For particles of equal density the separation is based on mass, with heavier particles having greater sedimentation rates. When the particles are of equal mass, those with the highest density have the greatest sedimentation rate. Centrifugation is of particular importance as a separation technique in biochem- istry. As shown in Table 7.5, cellular components can be separated by centrifugation. 12 For example, lysosomes can be separated from other cellular components by repeated differential centrifugation, in which the sample is divided into a solid residue and a so- lution called the supernatant. After destroying the cell membranes, the solution is centrifuged at 15,000 · g (a centrifugal field strength that is 15,000 times that of the Earth’s gravitational field) for 20 min, leaving a residue of cell membranes and mitochondria. The supernatant is isolated by decanting from the residue and is centrifuged at 30,000 · g for 30 min, leaving a residue of lysosomes. An alternative approach to differential centrifugation is equilibrium– density–gradient centrifugation. The sample is either placed in a solution with a preformed density gradient or in a solution that, when centrifuged, forms a density gradient. For example, density gradients can be established with solutions of sucrose or CsCl. During centrifugation, the sample’s com- ponents undergo sedimentation at a rate determined by their centrifugal force. Because the solution’s density increases toward the bottom of the cen- trifuge tube, the sedimentation rate for each component decreases as it moves toward the bottom of the centrifuge tube. When a component reaches a posi- tion where its density is equal to that of the solution, the centrifugal force drops to zero and sedimentation stops. Each component, therefore, is isolated as a separate band positioned where the density of the component is equal to the density of the solution. For example, a mixture of proteins, RNA, and DNA can be separated in this way since their densities are different. A density gradient from 1.65 g/cm 3 to 1.80 g/cm 3 is established using CsCl. Proteins, with a density of less than 1.3 g/cm 3 experience no sedimentation, whereas RNA, with a density of greater than 1.8 g/cm 3 collects as a residue at the bottom of the centrifuge tube. The DNA, which has a density of approximately 1.7 g/cm 3 separates as a band near the middle of the centrifuge tube (Figure 7.12). 7 F. 3 Separations Based on Complexation Reactions (Masking) One of the most widely used techniques for preventing an interference is to bind the interferent as a soluble complex, preventing it from interfering in the analyte’s deter- mination. This process is known as masking. Technically, masking is not a separation Chapter 7 Obtaining and Preparing Samples for Analysis 207 Table 7 . 5 Conditions for the Separation of Selected Cellular Components by Centrifugation Centrifugal Force Time Components (· g) (min) eukaryotic cell 1000 5 cell membranes, nuclei 4000 10 mitochondria, bacterial cells 15,000 20 lysosomes, bacterial membranes 30,000 30 ribosomes 100,000 180 Source: Adapted from Zubay G. Biochemistry, 2nd ed. Macmillan: New York, 1988, p. 120. (a) (b) Protein DNA RNA Figure 7.12 Illustration showing separation by equilibrium–density–gradient centrifugation. The homogeneous mixture in (a) separates into three bands (b) after applying centrifugal force. masking A pseudo-separation method in which a species is prevented from participating in a chemical reaction by binding it with a masking agent in an unreactive complex. 1400-CH07 9/8/99 4:03 PM Page 207 208 Modern Analytical Chemistry Table 7 .6 Selected Masking Agents Masking Agent Species Which Can Be Masked CN – Ag, Au, Cd, Co, Cu, Fe, Hg, Mn, Ni, Pd, Pt, Zn SCN – Ag, Cd, Co, Cu, Fe, Ni, Pd, Pt, Zn NH 3 Ag, Co, Cu, Fe, Pd, Pt F – Al, Co, Cr, Mg, Mn, Sn, Zn S 2 O 3 2– Au, Cd, Co, Cu, Fe, Pb, Pd, Pt, Sb tartrate Al, Ba, Bi, Ca, Ce, Co, Cr, Cu, Fe, Hg, Mn, Pb, Pd, Pt, Sb, Sn, Zn oxalate Al, Fe, Mg, Mn, Sn thioglycolic acid Cu, Fe, Sn Source: Adapted from Meites, L. Handbook of Analytical Chemistry, McGraw-Hill: New York, 1963. technique because the analyte and interferent are never physically separated from each other. Masking can, however, be considered a pseudo-separation technique, and is in- cluded here for that reason. A wide variety of ions and molecules have been used as masking agents (Table 7.6), and, as a result, selectivity is usually not a problem. 13 EXAMPLE 7 .12 Suggest a masking agent for the analysis of Al in the presence of Fe. Repeat for the analysis of Fe when Al is an interferent. SOLUTION To find a suitable masking agent, we look for a species that binds with the interferent but does not bind with the analyte. Oxalate, for example, is an inappropriate choice because it binds with both Al and Fe. From Table 7.6 we find that thioglycolic acid is a selective masking agent for Fe in the presence of Al and that F – is a selective masking agent for Al in the presence of Fe. As shown in Example 7.13, a masking agent’s effectiveness can be judged by considering the equilibrium constants for the analytical and masking reactions. EXAMPLE 7 .1 3 Show that CN – is an appropriate masking agent for Ni 2+ in a method in which nickel’s complexation with EDTA is an interference. SOLUTION The relevant reactions and equilibrium constants from Appendix 3C are Ni 2+ +Y 4– fi NiY 2– K f = 4.2 · 10 18 Ni 2+ + 4CN – fi Ni(CN) 4 2– b 4 = 1.7 · 10 30 in which Y 4– is an abbreviation for ethylenediaminetetraacetic acid (EDTA). Cyanide is an appropriate masking agent because the formation constant for masking agent The reagent used to bind the species to be masked in an unreactive complex. 1400-CH07 9/8/99 4:03 PM Page 208 Ni(CN) 4 2– is greater than that for the Ni–EDTA complex. In fact, the equilibrium constant for the reaction in which EDTA displaces the masking agent Ni(CN) 4 2– +Y 4– t NiY 2– + 4CN – is very small, indicating that Ni(CN) 4 2– is relatively inert in the presence of EDTA. 7 F. 4 Separations Based on a Change of State Since an analyte and interferent are usually in the same phase, a separation often can be effected by inducing a change in one of their physical or chemical states. Changes in physical state that have been exploited for the purpose of a separation include liquid-to-gas and solid-to-gas phase transitions. Changes in chemical state involve one or more chemical reactions. Changes in Physical State When the analyte and interferent are mis- cible liquids, a separation based on distillation may be possible if their boiling points are significantly different. The progress of a distillation is outlined in Figure 7.13, which shows a plot of temperature versus the vapor-phase and liquid-phase composition of a mixture consisting of a low-boiling analyte and a high-boiling interferent. The initial mixture is indicated by the point labeled A. When this solution is brought to its boiling point, a vapor phase with the composition indi- cated by the point labeled B is in equilibrium with the original liquid phase. This equilibrium is indicated by the horizontal tie-line between points A and B. When the vapor phase at point B condenses, a new liquid phase with the same composition as the vapor phase (point C) results. The liquid phase at point C boils at a lower temperature, with an equilib- rium established with the vapor-phase composition indicated by point D. This process of repeated vaporization and condensation gradually separates the ana- lyte and interferent. Two examples of the equipment used for distillations are shown in Figure 7.14. The simple distillation apparatus shown in Figure 7.14a does not produce a very effi- cient separation and is useful only for separating a volatile liquid from nonvolatile liquids or for separating liquids with boiling points that differ by more than 150 °C. A more efficient separation is achieved by a fractional distillation (Figure 7.14b). Packing the distillation column with a high-surface-area material, such as a steel sponge or glass beads, provides more opportunity for the repeated process of vapor- ization and condensation necessary to effect a complete separation. When the sample is a solid, a separation of the analyte and interferent by subli- mation may be possible. The sample is heated at a temperature and pressure below its triple point where the solid vaporizes without passing through the liquid state. The vapor is then condensed to recover the purified solid. A good example of the use of sublimation is in the isolation of amino acids from fossil mollusk shells and deep-sea sediments. 14 K K == × × =× ? f β 4 18 30 12 42 10 17 10 25 10 . . . Chapter 7 Obtaining and Preparing Samples for Analysis 209 T emperature Mole % analyte 0 100 Mole % interferent Liquid Vapor 100 0 C B D A Figure 7.13 Boiling points versus composition diagram for a near-ideal solution, showing the progress of a distillation. 1400-CH07 9/8/99 4:03 PM Page 209 Another approach for purifying solids is recrystallization. The solid is dissolved in a minimum volume of solvent, for which the analyte’s solubility is significant when the solvent is hot, and minimal when the solvent is cold. The interferents must be less soluble in the hot solvent than the analyte, or present in much smaller amounts. A portion of the solvent is heated in an Erlenmeyer flask, and small amounts of sample are added until undissolved sample is visible. Additional heated solvent is added until the sample is again dissolved or until only insoluble impuri- ties remain. The process of adding sample and solvent is repeated until the entire sample has been added to the Erlenmeyer flask. If necessary, insoluble impurities are removed by filtering the heated solution. The solution is allowed to cool slowly, promoting the growth of large, pure crystals, and then cooled in an ice bath to min- imize solubility losses. The purified sample is isolated by filtration and rinsed to re- move soluble impurities. Finally, the sample is dried to remove any remaining traces of the solvent. Further purification, if necessary, can be accomplished by ad- ditional recrystallizations. Changes in Chemical State Distillation, sublimation, and recrystallization use a change in physical state as a means of separation. Chemical reactivity also can be used in a separation by effecting a change in the chemical state of the analyte or in- terferent. For example, SiO 2 can be separated from a sample by reacting with HF. The volatile SiF 4 that forms is easily removed by evaporation. In other cases distilla- tion may be used to remove a nonvolatile inorganic ion after chemically converting it to a more volatile form. For example, NH 4 + can be separated from a sample by mak- ing the solution basic, resulting in the formation of NH 3 . The ammonia that is produced can then be removed by distillation. Other examples are listed in Table 7.7. Other types of reactions can be used to chemically separate an analyte and interferent, including precipita- tion, electrodeposition, and ion exchange. Two impor- tant examples of the application of precipitation are the 210 Modern Analytical Chemistry Thermometer Distillation adaptor Distillation flask Condenser Receiving flask Thermometer Fractionating column Distillation adaptor Distillation flask Condenser Receiving flask Figure 7.14 Typical equipment for a (a) simple distillation; and a (b) fractional distillation. (a) (b) Table 7 . 7 Selected Examples of the Application of Distillation to the Separation of Inorganic Ions Analyte or Interferent Treatment a CO 3 2– CO 3 2– +2H 3 O + fi CO 2 +3H 2 O NH 4 + NH 4 + +OH – fi NH 3 +H 2 O SO 3 2– SO 3 2– +2H 3 O + fi SO 2 +3H 2 O S 2– S 2– +2H 3 O + fi H 2 S+2H 2 O a Underlined species is removed by distillation. 1400-CH07 9/8/99 4:04 PM Page 210 pH-dependent solubility of metal oxides and hydroxides, and the solubility of metal sulfides. Separations based on the pH-dependent solubility of oxides and hydroxides are usually accomplished using strong acids, strong bases, or NH 3 /NH 4 Cl buffers. Most metal oxides and hydroxides are soluble in hot concentrated HNO 3 , although a few oxides, such as WO 3 , SiO 2 , and SnO 2 remain insoluble even under these harsh con- ditions. In determining the amount of Cu in brass, for example, an interference from Sn is avoided by dissolving the sample with a strong acid. An insoluble residue of SnO 2 remains that can then be removed by filtration. Most metals will precipitate as the hydroxide in the presence of concentrated NaOH. Metals forming amphoteric hydroxides, however, remain soluble in concen- trated NaOH due to the formation of higher-order hydroxo-complexes. For exam- ple, Zn 2 + and Al 3 + will not precipitate in concentrated NaOH due to the formation of Zn(OH) 3 – and Al(OH) 4 – . The solubility of Al 3 + in concentrated NaOH is used to isolate aluminum from impure bauxite, an ore of Al 2 O 3 . The ore is powdered and placed in a solution of concentrated NaOH where the Al 2 O 3 dissolves to form Al(OH) 4 – . Other oxides that may be present in the ore, such as Fe 2 O 3 and SiO 2 , re- main insoluble. After filtering, the filtrate is acidified to recover the aluminum as a precipitate of Al(OH) 3 . The pH of an NH 3 /NH 4 Cl buffer (pK a = 9.24) is sufficient to ensure the precip- itation of most metals as the hydroxide. The alkaline earths and alkaline metals, however, will not precipitate at this pH. In addition, metal ions that form soluble complexes with NH 3 , such as Cu 2+ , Zn 2+ , Ni 2+ , and Co 2+ , also will not precipitate under these conditions. Historically, the use of S 2– as a precipitating reagent is one of the earliest exam- ples of a separation technique. In Fresenius’s 1881 text, A System of Instruction in Quantitative Chemical Analysis, 15 sulfide is frequently used as a means for separat- ing metal ions from the remainder of the sample matrix. The importance of sulfide as a precipitating reagent for separations is due to two factors: most metal ions, ex- cept for the alkaline earths and alkaline metals, form insoluble sulfides; and the sol- ubilities of these metal sulfides show a substantial variation. Since the concentration of S 2– is pH-dependent, control of pH was used to determine which metal ions would precipitate. For example, in Fresenius’s gravimetric procedure for the deter- mination of Ni in ore samples (see Figure 1.1 in Chapter 1 for a schematic diagram of this procedure), sulfide is used three times as a means of separating Co 2+ and Ni 2+ from Cu 2+ and, to a lesser extent from Pb 2+ . 7 F. 5 Separations Based on a Partitioning Between Phases The most important class of separation techniques is based on the selective parti- tioning of the analyte or interferent between two immiscible phases. When a phase containing a solute, S, is brought into contact with a second phase, the solute parti- tions itself between the two phases. S phase 1 t S phase 2 7.18 The equilibrium constant for reaction 7.18 is called the distribution constant, or partition coefficient. If K D is sufficiently large, then the solute will move from phase 1 to phase 2. The solute will remain in phase 1, K S S D phase 2 phase 1 = [] [] Chapter 7 Obtaining and Preparing Samples for Analysis 211 partition coefficient An equilibrium constant describing the distribution of a solute between two phases; only one form of the solute is used in defining the partition coefficient (K D ). 1400-CH07 9/8/99 4:04 PM Page 211 Figure 7.17 Solid-phase extraction cartridges: (a) disk cartridge; (b) column cartridge. however, if the partition coefficient is sufficiently small. If a phase containing two solutes is brought into contact with a second phase, and K D is favorable for only one of the solutes, then a separation of the solutes may be possible. The physical states of the two phases are identified when describing the separation process, with the phase containing the sample listed first. For example, when the sample is in a liquid phase and the second phase is a solid, the separation involves liquid–solid partitioning. Extraction Between Two Phases When the sample is initially present in one of the phases, the separation is known as an extraction. In a simple extraction the sample is extracted one or more times with portions of the second phase. Simple extrac- tions are particularly useful for separations in which only one component has a fa- vorable distribution ratio. Several important separation techniques are based on simple extractions, including liquid–liquid, liquid–solid, solid–liquid, and gas–solid extractions. Liquid—Liquid Extractions Liquid–liquid extractions are usually accomplished with a separatory funnel (Figure 7.15). The two liquids are placed in the separa- tory funnel and shaken to increase the surface area between the phases. When the extraction is complete, the liquids are allowed to separate, with the denser phase settling to the bottom of the separatory funnel. Liquid–liquid extractions also may be carried out in the sample container by adding the extracting solvent when the sample is collected. Pesticides in water, for example, may be preserved for longer periods by extracting into a small volume of hexane added to the sample in the field. Liquid–liquid microextractions, in which the extracting phase is a 1-m L drop suspended from a microsyringe (Figure 7.16) also have been described. 16 Be- cause of its importance, a more thorough discussion of liquid–liquid extraction is given in Section 7G. Solid-Phase Extractions In a solid-phase extraction the sample is passed through a cartridge containing solid particulates that serve as the adsorbent material. For liq- uid samples the solid adsorbent is isolated in either a disk cartridge or a column (Figure 7.17). The choice of adsorbent is determined by the properties of the species being retained and the matrix in which it is found. Representative solid adsorbents 212 Modern Analytical Chemistry Phase 2 Phase 1 Syringe needle Microliter drop of extraction solvent Direction of solvent flow Direction of solvent flow Porous retainer Porous retainer Solid adsorbent (a) (b) Figure 7.16 Schematic of a liquid–liquid microextraction showing syringe needle with attached 1-m L droplet. Figure 7.15 Separatory funnel for use in a liquid–liquid extraction. extraction The process by which a solute is transferred from one phase to a new phase. 1400-CH07 9/8/99 4:04 PM Page 212 are listed in Table 7.8. For example, sedatives, such as secobarbital and phenobarbi- tal, can be isolated from serum by a solid-phase extraction using a C-18 solid adsor- bent. Typically a 500-m L sample of serum is passed through the cartridge, with the sedatives being retained by a liquid–solid extraction. The cartridge is then washed with distilled water to remove any residual traces of the serum’s matrix. Finally, the retained sedatives are eluted from the cartridge by a solid–liquid extraction using 500 m L of acetone. For many analyses, solid–phase extractions are replacing liquid–liquid extractions due to their ease of use, faster extraction times, decreased volumes of solvent, and their superior ability to concentrate the analytes. The last advantage is discussed in more detail in the final section of this chapter. Solid-phase microextractions also have been developed. In one approach, a fused silica fiber is placed inside a syringe needle. The fiber, which is coated with a thin organic film, such as poly(dimethyl siloxane), is lowered into the sample by de- pressing a plunger and exposed to the sample for a predetermined time. The fiber is then withdrawn into the needle and transferred to a gas chromatograph for analysis. 17 In gas–solid extractions the sample is passed through a container packed with a solid adsorbent. One example of the application of gas–solid extraction is in the analysis of organic compounds for carbon and hydrogen. The sample is combusted in a flowing stream of O 2 , and the gaseous combustion products are passed through a series of solid-phase adsorbents that remove the CO 2 and H 2 O. Continuous Extractions An extraction is still feasible even when the component of interest has an unfavorable partition coefficient, provided that all other components in the sample have significantly smaller partition coefficients. Because the partition Chapter 7 Obtaining and Preparing Samples for Analysis 213 Table 7 .8 Selected Adsorbents for Solid-Phase Extraction of Liquid Samples Adsorbent Surface Structure Properties and Uses silica —O—Si—O—Si—O— ? retains low-to-moderate polarity species || from organic matrices HO OH ? fat-soluble vitamins, steroids alumina —O—Al—O—Al—O— ? retains hydrophilic species from organic || matrices HO OH cyanopropyl —C 3 H 6 CN ? retains wide variety of species from aqueous and organic matrices ? pesticides, hydrophobic peptides diol —CH 2 —CH 2 — ? retains wide variety of species from || aqueous and organic matrices OH OH ? proteins, peptides, fungicides octadecyl (C-18) —C 18 H 37 ? retains hydrophobic species from aqueous matrices ? caffeine, sedatives, polyaromatic hydrocarbons, carbohydrates, pesticides octyl (C-8) —C 8 H 17 ? similar to C-18 styrene ? wide variety of organic species from divinylbenzene aqueous matrices ? polyaromatic hydrocarbons 1400-CH07 9/8/99 4:04 PM Page 213 Figure 7.18 Schematic diagram of a Soxhlet extractor. coefficient is unfavorable, a simple extraction will not be quantitative. Instead, the extraction is accomplished by continuously passing the extracting phase through the sample until a quantitative extraction is achieved. Many continuous extractions involving solid samples are carried out with a Soxhlet extractor (Figure 7.18). The extracting solvent is placed in the lower reser- voir and heated to its boiling point. Solvent in the vapor phase moves upward through the tube on the left side of the apparatus to the condenser where it con- denses back to the liquid state. The solvent then passes through the sample, which is held in a porous cellulose filter thimble, collecting in the upper reservoir. When the volume of solvent in the upper reservoir reaches the upper bend of the return tube, the solvent and any extracted components are siphoned back to the lower reservoir. Over time, the concentration of the extracted component in the lower reservoir increases. Soxhlet extractions have been replaced in some applications by microwave- assisted extractions. 18 The process is the same as that described earlier for microwave digestion. The sample is placed in a sealed digestion vessel along with the liquid ex- traction phase, and a microwave oven is used to heat the extraction mixture. Using a sealed digestion vessel allows the extraction to take place at a higher temperature and pressure, thereby reducing the amount of time needed for a quantitative extrac- tion. In a Soxhlet extraction the temperature is limited by the solvent’s boiling point at atmospheric pressure. For example, when acetone is the solvent, a Soxhlet extrac- tion is limited to 56 °C. With a microwave-assisted extraction, however, a tempera- ture of over 150 °C can be obtained when using acetone as the solvent. Two other examples of a continuous extraction deserve mention. Volatile or- ganic compounds (VOCs) can be quantitatively removed from liquid samples by a liquid–gas extraction. As shown in Figure 7.19, the VOCs are removed by passing an inert purging gas, such as He, through the sample. The He removes the VOCs, which are then carried by the He to a tube where they are collected on a solid adsor- bent. When the extraction is complete, the VOCs can then be removed from the trap for analysis by rapidly heating the tube while flushing with He. This technique is known as a purge and trap. Recoveries for analytes using a purge and trap may not be reproducible, requiring the use of internal standards for quantitative work. 214 Modern Analytical Chemistry Water-cooled condenser Extraction thimble Upper reservoir Lower reservoir Return tube Sample purge and trap A technique for separating volatile analytes from liquid samples in which the analytes are subsequently trapped on a solid adsorbent. Sample Primary adsorbent trap Secondary adsorbent trap Purge gas Figure 7.19 Schematic diagram of a purge-and-trap system. Analyte is collected in the primary adsorption trap. The secondary adsorption trap is monitored for evidence of breakthrough. 1400-CH07 9/8/99 4:04 PM Page 214 Chapter 7 Obtaining and Preparing Samples for Analysis 215 supercritical fluid A state of matter where a substance is held at a temperature and pressure that exceeds its critical temperature and pressure. Continuous extractions also can be accomplished with supercritical fluids. 19 When a substance is heated above its critical temperature and pressure, it forms a supercritical fluid whose properties are between those of a gas and a liquid. Super- critical fluids are better solvents than gases, making them a better reagent for ex- tractions. In addition, the viscosity of a supercritical fluid is significantly less than that of a liquid solvent, allowing it to pass more readily through particulate samples. One example of a supercritical extraction is the determination of total petroleum hydrocarbons (TPHs) in soils, sediments, and sludges with supercritical CO 2 . Ap- proximately 3 g of sample is placed in a 10-mL stainless steel cartridge, and super- critical CO 2 , at a pressure of 340 atm and a temperature of 80 °C, is passed through the cartridge for 30 min at flow rate of 1–2 mL/min. The petroleum hydrocarbons are collected by passing the effluent from the cartridge through 3 mL of tetra- chloroethylene at room temperature. At this temperature the CO 2 reverts to the gas phase and is released to the atmosphere. 20 Chromatographic Separations In an extraction, the sample is initially present in one phase, and the component of interest is extracted into a second phase. Separa- tions can also be accomplished by continuously passing one sample-free phase, called the mobile phase, over a second sample-free phase that remains fixed or sta- tionary. The sample is then injected or placed into the mobile phase. As the sam- ple’s components move with the mobile phase, they partition themselves between the mobile and stationary phases. Those components having the largest partition coefficients are more likely to move into the stationary phase, taking longer to pass through the system. This is the basis of all chromatographic separation techniques. As currently practiced, modern chromatography provides a means both of separat- ing analytes and interferents and of performing a qualitative or quantitative analysis of the analyte. For this reason a more thorough treatment of chromatography is found in Chapter 12. 7 G Liquid—Liquid Extractions A liquid–liquid extraction is one of the most important separation techniques used in environmental, clinical, and industrial laboratories. Two examples from envi- ronmental analysis serve to illustrate its importance. Public drinking water supplies are routinely monitored for trihalomethanes (CHCl 3 , CHBrCl 2 , CHBr 2 Cl, and CHBr 3 ) because of their known or suspected carcinogeneity. Before their analysis by gas chromatography, trihalomethanes are separated from their aqueous matrix by a liquid–liquid extraction using pentane. 21 A liquid–liquid extraction is also used in screening orange juice for the presence of organophosphorous pesticides. A sample of orange juice is mixed with acetonitrite and filtered. Any organophospho- rous pesticides that might be present in the filtrate are extracted with petroleum ether before a gas chromatographic analysis. 22 In a simple liquid–liquid extraction the solute is partitioned between two im- miscible phases. In most cases one of the phases is aqueous, and the other phase is an organic solvent such as diethyl ether or chloroform. Because the phases are im- miscible, they form two layers, with the denser phase on the bottom. The solute is initially present in one phase, but after extraction it is present in both phases. The efficiency of a liquid–liquid extraction is determined by the equilibrium constant for the solute’s partitioning between the two phases. Extraction efficiency is also in- fluenced by any secondary reactions involving the solute. Examples of secondary re- actions include acid–base and complexation equilibria. 1400-CH07 9/8/99 4:04 PM Page 215 Figure 7.20 Scheme for a simple liquid–liquid extraction without any secondary reactions. 7 G.1 Partition Coefficients and Distribution Ratios Earlier we learned that the partitioning of a solute between two phases is described by a partition coefficient. If the solute is initially in an aqueous phase and is ex- tracted into an organic phase* S aq t S org the partition coefficient is A large value for K D indicates that the extraction of the solute into the organic phase is favorable. In evaluating the efficiency of an extraction, however, we must consider the solute’s total concentration in each phase. We define the distribution ratio, D, to be the ratio of the solute’s total concentration in each phase. When the solute exists in only one form in each phase, then the partition coefficient and the distribution ratio are identical. If, however, the solute exists in more than one form in either phase, then K D and D usually have different values. For example, if the solute exists in two forms in the aqueous phase, A and B, only one of which, A, partitions itself between the two phases, then This distinction between K D and D is important. The partition coefficient is an equilibrium constant and has a fixed value for the solute’s partitioning between the two phases. The value of the distribution ratio, however, changes with solution con- ditions if the relative amounts of forms A and B change. If we know the equilibrium reactions taking place within each phase and between the phases, we can derive an algebraic relationship between K D and D. 7 G.2 Liquid—Liquid Extraction with No Secondary Reactions In the simplest form of liquid–liquid extraction, the only reaction affecting extrac- tion efficiency, is the partitioning of the solute between the two phases (Figure 7.20). In this case the distribution ratio and the partition coefficient are equal. 7.19 Conservation of mass requires that the moles of solute initially present in one phase equal the combined moles of solute in the aqueous and organic phases after the extraction; thus (Moles aq) 0 = (moles aq) 1 + (moles org) 1 7.20 D S S S S == [] [] [] [] org tot aq tot org aq D S SS K S S = + ≤= [] [] [] [] [] org A aq A aq B D org A aq A D S S = [] [] org tot aq tot K S S D org aq = [] [] 216 Modern Analytical Chemistry *Although the following treatment assumes that the solute is initially present in the aqueous phase, the resulting equations for the distribution of the solute between the two phases are independent of which phase originally contains the solute. distribution ratio A ratio expressing the total concentration of solute in one phase relative to a second phase; all forms of the solute are considered in defining the distribution ratio (D). Organic Aqueous S org S aq 1400-CH07 9/8/99 4:04 PM Page 216 Chapter 7 Obtaining and Preparing Samples for Analysis 217 where the subscript indicates the extraction number. The concentration of S in the aqueous phase after the extraction is 7.21 whereas the solute’s concentration in the organic phase is 7.22 where V aq and V org are the volumes of the aqueous and organic phases. Solving equation 7.20 for (moles org) 1 and substituting into equation 7.22 leave us with 7.23 Substituting equations 7.21 and 7.23 into equation 7.19, we obtain Rearranging and solving for the fraction of solute remaining in the aqueous phase after one extraction, (q aq ) 1 , gives 7.24 The fraction present in the organic phase after one extraction, (q org ) 1 , is Example 7.14 shows how equation 7.24 is used to calculate the efficiency of a simple liquid–liquid extraction. EXAMPLE 7 .1 4 A solute, S, has a K D between water and chloroform of 5.00. A 50.00-mL sample of a 0.050 M aqueous solution of the solute is extracted with 15.00 mL of chloroform. (a) What is the extraction efficiency for this separation? (b) What is the solute’s final concentration in each phase? (c) What volume of chloroform is needed to extract 99.9% of the solute? SOLUTION For a simple liquid–liquid extraction, the distribution ratio, D, and the partition coefficient, K D , are identical. (a) The fraction of solute remaining in the aqueous phase after the extraction is given by equation 7.24 The fraction of solute present in the organic phase is, therefore, 0.600. Extraction efficiency is the percentage of solute successfully transferred from its initial phase to the extracting phase. The extraction efficiency is, therefore, 60.0%. () . . .q aq mL (5.00)(15.00 mL) mL 1 50 00 50 00 0 400= + = () ( ( ()qq DV DV V org 1 0 aq org org aq moles org) moles org) 11 1==?= + () ( ( q V DV V aq 1 0 aq org aq moles aq) moles aq) 1 == + D V V VV V = ? = ?[( ( ]/ (/ (( ( moles aq) moles aq) moles aq) moles aq) moles aq) moles aq) 0 1 org 1aq 0aq 1aq org1 [] (( S V org 01 org moles aq) moles aq) = ? [] ( S V org 1 org moles org) = []S V aq 1 aq (moles aq) = 1400-CH07 9/8/99 4:04 PM Page 217 Colorplate 4 shows an example of a liquid—liquid extraction. 218 Modern Analytical Chemistry (b) The moles of solute present in the aqueous phase before the extraction is Since 40.0% of the solute remains in the aqueous phase, and 60.0% has been extracted into the organic phase, the moles of solute in the two phases after extraction are (Moles aq) 1 = (moles aq) 0 · (q aq ) 1 = 0.0025 mol · (0.400) = 0.0010 mol (Moles org) 1 = (moles aq) 0 – (moles aq) 1 = 0.0025 mol – 0.0010 mol = 0.0015 mol The solute’s concentration in each phase is (c) To extract 99.9% of the solute (q aq ) 1 must be 0.001. Solving equation 7.24 for V org , and making appropriate substitutions for (q aq ) 1 and V aq gives Clearly, a single extraction is not reasonable under these conditions. In Example 7.14 a single extraction results in an extraction efficiency of only 60%. If a second extraction is carried out, the fraction of solute remaining in the aqueous phase, (q aq ) 2 , is given by If the volumes of the aqueous and organic layers are the same for both extractions, then the cumulative fraction of solute remaining in the aqueous layer after two ex- tractions, (Q aq ) 2 , is In general, for a series of n identical extractions, the fraction of analyte remaining in the aqueous phase after the last extraction is 7.25 EXAMPLE 7 .1 5 For the extraction described in Example 7.14, determine (a) the extrac- tion efficiency for two extractions and for three extractions; and (b) the number of extractions required to ensure that 99.9% of the solute is extracted. ()Q V DV V n n aq aq org aq = + ? ? ? ? ? ? () ( ( ()()Qqq V DV V aq 2 0 aq aq aq org aq moles aq) moles aq) 212 2 === + ? ? ? ? ? ? () ( ( q V DV V aq 2 1 aq org aq moles aq) moles aq) 2 == + V VqV qD org aq aq aq aq mL mL mL= ? = ? = () () .(.)(.) (. )(. ) 1 1 50 00 0 001 50 00 0 001 5 00 9990 [] (. . .S V org 1 org moles org) mol L M 1 0 0015 0 01500 010=== [] (. . .S V aq 1 aq moles aq) mol L M 1 0 0010 0 05000 0 020=== ([] . ..Moles aq) mol L L mol 0aq aq =×= × =SV 0 0 050 0 05000 0 0025 1400-CH07 9/8/99 4:04 PM Page 218 Figure 7.21 Plot of extraction efficiency versus number of extractions for the liquid–liquid extraction scheme in Figure 7.20. SOLUTION (a) The fraction of solute remaining in the aqueous phase after two and three extractions is Thus, the extraction efficiencies are 84.0% with two extractions and 93.6% with three extractions. (b) To determine the minimum number of extractions for an efficiency of 99.9%, we set (Q aq ) n to 0.001 and solve for n in equation 7.25 Taking the log of both sides log(0.001) = nlog(0.400) and solving for n gives n = 7.54 Thus, a minimum of eight extractions is necessary. An important observation from Examples 7.14 and 7.15 is that an extraction efficiency of 99.9% can be obtained with less solvent when using multiple extrac- tions. Obtaining this extraction efficiency with one extraction requires 9990 mL of the organic solvent. Eight extractions using separate 15-mL portions of the or- ganic solvent, however, requires only 120 mL. Although extraction efficiency in- creases dramatically with the first few multiple extractions, the effect quickly di- minishes as the number of extractions is increased (Figure 7.21). In most cases there is little gain in extraction efficiency after five or six extractions. In Example 7.15 five extractions are needed to reach an extraction efficiency of 99%, and an additional three extractions are required to obtain the extra 0.9% increase in ex- traction efficiency. 7 G. 3 Liquid—Liquid Extractions Involving Acid—Base Equilibria In a simple liquid–liquid extraction the distribution ratio and the partition coeffi- cient are identical. As a result, the distribution ratio is unaffected by any change in the composition of the aqueous or organic phase. If the solute also partici- pates in a single-phase equilibrium reaction, then the distribution ratio and the partition coefficient may not be the same. For example, Figure 7.22 shows the equilibria occurring when extracting an aqueous solution contain- ing a molecular weak acid, HA, with an organic phase in which ionic species are not soluble. In this case the partition coefficient and the distribution ratio are 7.26 K D org aq HA HA = [] [] 0 001 50 00 50 00 0 400. . ). (. )= + ? ? ? ? ? ? = mL (5.00)(15.00 mL mL n n () . ( . )( . ) . .Q aq mL mL mL 3 3 50 00 500 1500 5000 0 064= + ? ? ? ? ? ? = () . . .Q aq mL (5.00)(15.00 mL) mL 2 2 50 00 50 00 0 160= + ? ? ? ? ? ? = Chapter 7 Obtaining and Preparing Samples for Analysis 219 Extraction efficienc y Number of extractions 0 0 10 20 30 40 50 60 70 80 90 100 510 Organic Aqueous HA K D K a HA + H 2 O H 3 O + + A – Figure 7.22 Scheme for the liquid–liquid extraction of a molecular weak acid. 1400-CH07 9/8/99 4:04 PM Page 219 220 Modern Analytical Chemistry K a 3aq + aq aq HO A HA = ? [][] [] 7.27 Since the position of an acid–base equilibrium depends on the pH, the distribution ratio must also be pH-dependent. To derive an equation for D showing this depen- dency, we begin with the acid dissociation constant for HA. 7.28 Solving equation 7.28 for [A aq – ] and substituting into equation 7.27 gives Factoring [HA aq ] from the denominator and substituting equation 7.26 gives, after simplifying, the sought-after relationship between the distribution ratio and the pH of the aqueous solution 7.29 The value for D given by equation 7.29 can be used in equation 7.25 to determine extraction efficiency. EXAMPLE 7 .16 An acidic solute, HA, has an acid dissociation constant of 1.00 · 10 –5 , and a partition coefficient between water and benzene of 3.00. Calculate the extraction efficiency when 50.00 mL of a 0.025 M aqueous solution of HA buffered to a pH of 3.00, is extracted with 50.00 mL of benzene. Repeat for cases in which the pH of the aqueous solution is buffered to 5.00 and 7.00. SOLUTION When the pH is 3.00, the [H 3 O aq + ] is 1.00 · 10 –3 , and the distribution ratio for the extraction is D = × ×+× = ? ?? (. )(. ) .. . 3 00 1 00 10 1 00 10 1 00 10 297 3 35 D K K = + D3aq + 3aq + a HO HO [] [] D K K = + D a3aq + HO1 ( /[ ]) D K = + [] [ ]{ ( /[ ])} HA HA H O org aq a 3 aq + 1 D K = + [] []([]/[ ]) HA HA HA H O org aq a aq 3 aq + [] [] [] A K aq aaq 3aq HA HO ? + = D == + ? [] [] [] [][] HA HA HA HA A org tot aq tot org aq aq 1400-CH07 9/8/99 4:04 PM Page 220 The fraction of solute remaining in the aqueous phase is The extraction efficiency, therefore, is almost 75%. When the same calculation is carried out at a pH of 5.00, the extraction efficiency is 60%, but the extraction efficiency is only 3% at a pH of 7.00. As expected, extraction efficiency is better at more acidic pHs when HA is the predominate species in the aqueous phase. A graph of extraction efficiency versus pH for this system is shown in Figure 7.23. Note that the extraction efficiency is greatest for pHs more acidic than the weak acid’s pK a and decreases substantially at pHs more basic than the pK A . A ladder diagram for HA is superimposed on the graph to help illustrate this effect. The same approach can be used to derive an equation for the distri- bution ratio when the solute is a molecular weak base, B, (Figure 7.24). The resulting distribution ratio is where K a is the acid dissociation constant for the weak base’s conjugate weak acid. 7 G. 4 Liquid—Liquid Extractions Involving Metal Chelators One of the most common applications of a liquid–liquid extraction is the selective extraction of metal ions using a chelating agent. Unfor- tunately, many chelating agents have a limited solubility in water or are subject to hydrolysis or air oxidation in aqueous solutions. For these reasons the chelating agent is added to the organic solvent in- stead of the aqueous phase. The chelating agent is extracted into the aqueous phase, where it reacts to form a stable metal–ligand complex with the metal ion. The metal–ligand complex is then extracted into the organic phase. A summary of the relevant equilibria is shown in Figure 7.25. If the ligand’s concentration is much greater than the metal ion’s concentration, the distribution ratio is given as* 7.30 where C L is the initial concentration of ligand in the organic phase be- fore the extraction. The distribution ratio calculated using equation 7.30 can be substituted back into equation 7.25 to determine the extraction efficiency. As shown in Example 7.17, the extraction efficiency for metal ions shows a marked pH dependency. D KKC KKC nn nnn = + β β D,c a L D,L 3aq + a L HO[] D KK K = + Da a3aq + HO[] () . ). .Q aq mL (2.97)(50.00 mL mL 1 50 00 50 00 0 252= + = Chapter 7 Obtaining and Preparing Samples for Analysis 221 Extraction efficienc y pH 135 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 7 9 11 13 Organic Aqueous B K D K b B + H 2 OOH – + HB + Organic Aqueous HL HL + H 2 O L – + + H 3 O + K D,HL K a K D,c n M n+ ML n ML n b Figure 7.23 Plot of extraction efficiency versus pH of the aqueous phase for the liquid–liquid extraction of the molecular weak acid in Example 7.16. Figure 7.25 Scheme for the liquid–liquid extraction of a metal ion by a metal chelator. Figure 7.24 Scheme for the liquid–liquid extraction of a molecular weak base. *The derivation of equation 7.31 is considered in problem 33 in the end-of-chapter problem set. 1400-CH07 9/8/99 4:04 PM Page 221 222 Modern Analytical Chemistry EXAMPLE 7 .1 7 A divalent metal ion, M 2+ , is to be extracted from an aqueous solution into an organic solvent using a chelating agent, HL, dissolved in the organic solvent. The partition coefficients for the chelating agent, K D,L , and the metal–ligand complex, K D,c , are 1.0 · 10 4 and 7.0 · 10 4 , respectively. The acid dissociation constant for the chelating agent, K a , is 5.0 · 10 –5 , and the formation constant for the metal–ligand complex, b , is 2.5 · 10 16 . Calculate the extraction efficiency when 100.0 mL of a 1.0 · 10 –6 M aqueous solution of M 2+ , buffered to a pH of 1.00, is extracted with 10.00 mL of an organic solvent that is 0.1 mM in the chelating agent. Repeat the calculation at a pH of 3.00. SOLUTION At a pH of 1.00 ([H 3 O aq + ] = 0.10), the distribution ratio for the extraction is and the fraction of metal ion remaining in the aqueous phase is Thus, at a pH of 1.00, only 0.40% of the metal is extracted. Changing the pH to 3.00, however, gives an extraction efficiency of 97.8%. A plot of extraction efficiency versus the pH of the aqueous phase is shown in Figure 7.26. () . ). .Q aq mL (0.0438)(10.00 mL mL 1 100 0 100 0 0 996= + = D = ××× × ×+××× = ?? ?? (. )(. )(. )(. ) (. )(. ) (. )(. )(. ) . 25 10 70 10 50 10 10 10 1010 010 2510 5010 1010 0 0438 16 4 5 2 4 2 42 2 16 52 42 Extraction efficienc y pH 123 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 4567 Figure 7.26 Typical plot of extraction efficiency versus pH for the liquid–liquid extraction of a metal ion by metal chelator. One of the advantages of using a chelating agent is the high degree of selec- tivity that it brings to the extraction of metal ions. As shown in Example 7.17 and Figure 7.26, the extraction efficiency for a divalent cation increases from approximately 0%–100% over a range of only 2 pH units. Furthermore, a chelating agent’s metal–ligand formation constant varies substantially between metal ions. As a result, significant differences arise in the pH range over which different metal ions experience an increase in extraction efficiency from 0% to 100% (Figure 7.27). 1400-CH07 9/8/99 4:04 PM Page 222 Figure 7.27 Plot of extraction efficiency for selected metals using dithizone in CCl 4 . The metal ions are: (a) Cu 2+ ; (b) Co 2+ ; (c) Ni 2+ ; (d) Sn 2+ ; (e) Pb 2+ ; and (f) Cd 2+ . Chapter 7 Obtaining and Preparing Samples for Analysis 223 P e r cent e xtracted pH –4 –2 0 (a) (b) (d) (c) (e) (f) 0 10 20 30 40 50 60 70 80 90 100 2468 1210 14 EXAMPLE 7 .18 Using Figure 7.27, explain how an aqueous mixture of Cu 2+ , Pb 2+ , and Cd 2+ can be separated by extraction with dithizone in CCl 4 . SOLUTION From Figure 7.27 we see that a quantitative separation of Cu 2+ from Pb 2+ and Cd 2+ can be accomplished if the aqueous phase is buffered to a pH of less than 5.5. After the extraction is complete, the pH can be buffered to approximately 9.5, allowing the selective extraction of Pb 2+ . Liquid–liquid extractions using ammonium pyrrolidine dithiocarbamate (APDC) as a metal chelating agent are commonly encountered in the analysis of metal ions in aqueous samples. The sample and APDC are mixed together, and the resulting metal–ligand complexes are extracted into methyl isobutyl ketone before analysis. 7 H Separation Versus Preconcentration Two frequently encountered analytical problems are: (1) the presence of matrix components interfering with the analysis of the analyte; and (2) the presence of an- alytes at concentrations too small to analyze accurately. We have seen how a separa- tion can be used to solve the former problem. Interestingly, separation techniques can often be used to solve the second problem as well. For separations in which a complete recovery of the analyte is desired, it may be possible to transfer the analyte in a manner that increases its concentration. This step in an analytical procedure is known as a preconcentration. Two examples from the analysis of water samples illustrate how a separation and preconcentration can be accomplished simultaneously. In the gas chromato- graphic analysis for organophosphorous pesticides in environmental waters, the an- alytes in a 1000-mL sample may be separated from their aqueous matrix by a solid- phase extraction using 15 mL of ethyl acetate. 23 After the extraction, the analytes are present in the ethyl acetate at a concentration that is 67 times greater than that in preconcentration The process of increasing an analyte’s concentration before its analysis. 1400-CH07 9/8/99 4:04 PM Page 223 the original sample (if the extraction is 100% efficient). The preconcentration of metal ions is accomplished by a liquid–liquid extraction with a metal chelator. For example, before their analysis by atomic absorption spectrophotometry, metal ions in aqueous samples can be concentrated by extraction into methyl isobutyl ketone (MIBK) using ammonium pyrrolidine dithiocarbamate (APDC) as a chelating agent. Typically, a 100-mL sample is treated with 1 mL of APDC, and extracted with ten mL of MIBK. The result is a ten-fold increase in the concentration of the metal ions. This procedure can be adjusted to increase the concentrations of the metal ions by as much as a factor of 40. 224 Modern Analytical Chemistry 7 I KEY TERMS breakthrough volume (p. 196) composite sample (p. 186) coning and quartering (p. 199) convenience sampling (p. 185) dialysis (p. 206) distribution ratio (p. 216) extraction (p. 212) grab sample (p. 185) gross sample (p. 193) in situ sampling (p. 186) judgmental sampling (p. 184) laboratory sample (p. 199) masking (p. 207) masking agent (p. 208) Nyquist theorem (p. 184) partition coefficient (p. 211) preconcentration (p. 223) purge and trap (p. 214) random sample (p. 183) recovery (p. 202) sampling plan (p. 182) separation factor (p. 203) size-exclusion chromatography (p. 206) stratified sampling (p. 185) systematic–judgmental sampling (p. 184) systematic sampling (p. 184) supercritical fluid (p. 215) An analysis requires a sample, and how we acquire the sample is critical. To be useful, the samples we collect must accurately repre- sent their target population. Just as important, our sampling plan must provide a sufficient number of samples of appropriate size so that the variance due to sampling does not limit the precision of our analysis. A complete sampling plan requires several considerations, including the type of sampling (random, judgmental, systematic, systematic–judgmental, stratified, or convenience); whether to col- lect grab samples, composite samples, or in situ samples; whether the population is homogeneous or heterogeneous; the appropriate size for each sample; and, the number of samples to collect. Removing a sample from its population may induce a change in its composition due to a chemical or physical process. For this reason, samples are collected in inert containers and are often pre- served at the time of collection. When the analytical method’s selectivity is insufficient, it may be necessary to separate the analyte from potential interferents. Such separations can take advantage of physical properties, such as size, mass or density, or chemical proper- ties. Important examples of chemical separations include mask- ing, distillation, and extractions. In a solid-phase extraction the analytes are first extracted from their solution matrix into a solid adsorbent. After wash- ing to remove impurities, the analytes are removed from the adsorbent with a suitable solvent. Alternatively, the extraction can be carried out using a Soxhlet extractor. In a liquid–liquid extraction, the analyte (or interferent) is ex- tracted from one liquid phase into a second, immiscible liquid phase. When the analyte is involved in secondary equilibrium re- actions, it is often possible to improve selectivity by carefully ad- justing the composition of one or both phases. 7 J SUMMARY 1400-CH07 9/8/99 4:04 PM Page 224 Chapter 7 Obtaining and Preparing Samples for Analysis 225 Bauer, C. F. “Sampling Error Lecture Demonstration,” J. Chem. Educ. 1985, 62, 253. This short paper describes a demonstration suitable for use in the classroom. Two populations of corks are sampled to determine the concentration of labeled corks. The exercise demonstrates how increasing the number of particles sampled improves the standard deviation due to sampling. Clement, R. E. “Environmental Sampling for Trace Analysis,” Anal. Chem. 1992, 64, 1076A–1081A. Sampling of a large population (n = 900) of colored candies (M&M’s work well) is used to demonstrate the importance of sample size in determining the concentration of species at several different concentration levels. This experiment is similar to the preceding one described by Bauer but incorporates several analytes. Guy, R. D.; Ramaley, L.; Wentzell, P. D. “An Experiment in the Sampling of Solids for Chemical Analysis,” J. Chem. Educ. 1998, 75, 1028–1033. This experiment uses the molybdenum-blue method to determine the concentration of phosphate in a phosphate/sodium chloride mixture. Flow-injection analysis is used to increase the speed of analysis, allowing students to collect a sufficient amount of data in a single laboratory period. The overall variance of the analysis is partitioned into components due to the method, to the preparation of samples, and to sample collection. The validity of equation 7.5 is also evaluated. Kratochvil, B.; Reid, R. S.; Harris, W. E. “Sampling Error in a Particulate Mixture,” J. Chem. Educ. 1980, 57, 518–520. In this experiment the overall variance for the analysis of potassium hydrogen phthalate (KHP) in a mixture of KHP and sucrose is partitioned into that due to sampling and that due to the analytical method (an acid–base titration). By having individuals analyze samples with different % w/w KHP, the relationship between sampling error and concentration of analyte can be explored. Lochmuler, C. “Atomic Spectroscopy—Determination of Calcium and Magnesium in Sand with a Statistical Treatment of Measurements” published on the web at http://www.chem.duke.edu/~clochmul/exp4/exp4.html. This experiment introduces random sampling. The experiment’s overall variance is divided into that due to the instrument, that due to sample preparation, and that due to sampling. 7 K Suggested EXPERIMENTS The following set of experiments introduce students to the important effect of sampling on the quality of analytical results. Each experiment is annotated with a brief description of the principles that it emphasizes. Experiments The following experiments describe homemade sampling devices for collecting samples in the field. Delumyea, R. D.; McCleary, D. L. “A Device to Collect Sediment Cores,” J. Chem. Educ. 1993, 70, 172–173. Directions are provided for preparing and using a simple coring device using PVC pipe. This experiment also details a procedure for determining the weight percent of organic material in sediments as a function of depth. Rockwell, D. M.; Hansen, T. “Sampling and Analyzing Air Pollution,” J. Chem. Educ. 1994, 71, 318–322. Two simple air samplers are described as well as their use for determining particulates in air. Saxena, S.; Upadhyay, R.; Upadhyay, P. “A Simple and Low- Cost Air Sampler,” J. Chem. Educ. 1996, 73, 787–788. This experiment describes the construction of an air sampler using an aquarium pump, a flow meter, a filter holder, and bottles that serve as traps for analytes. Applications include the determinations of SO 2 , NO 2 , HCHO, and suspended particulate matter. Shooter, D. “Nitrogen Dioxide and Its Determination in the Atmosphere,” J. Chem. Educ. 1993, 70, A133–A140. This paper describes the construction and use of a diffusion tube for sampling NO 2 from the atmosphere. Examples of its use include the determination of NO 2 concentrations at various heights above ground level in an urban environment and through a tree’s leaf canopy. —Continued 1400-CH07 9/8/99 4:04 PM Page 225 226 Modern Analytical Chemistry Continued from page 225 Experiments The following experiments introduce students to the importance of sample preparation and methods for extracting analytes from their matrix. Each experiment includes a brief description of the sample and analyte, as well as the method of analysis used to measure the analyte’s concentration. Dunn, J. G.; Phillips, D. N.; von Bronswijk, W. “An Exercise to Illustrate the Importance of Sample Preparation in Chemical Analysis,” J. Chem. Educ. 1997, 74, 1188–1190. Ore samples are analyzed for %w/w Ni. A jaw crusher is used to break the original ore sample into smaller pieces that are then sieved into 5 size fractions. A portion of each fraction is reduced in size using a disk mill and samples taken for analysis by coning and quartering. The effect of particle size on the determination of %w/w Ni is evaluated. “Extract-Clean SPE Sample Preparation Guide Volume 1,” Bulletin No. 83, Alltech Associates, Inc. Deerfield, IL. This publication provides several examples of the use of solid-phase extractions for separating analytes from their matrices. Some of the examples included are caffeine from coffee, polyaromatic hydrocarbons from water, parabens from cosmetics, chlorinated pesticides from water, and steroids from hydrocortisone creams. Extracted analytes may be determined quantitatively by gas (GC) or liquid chromatography (LC). Freeman, R. G.; McCurdy, D. L. “Using Microwave Sample Decomposition in Undergraduate Analytical Chemistry,” J. Chem. Educ. 1998, 75, 1033–1032. Although experimental details are not included, the application of microwave digestion is discussed for the determination of Na in food samples by flame atomic emission spectroscopy, and the Kjeldahl analysis of nitrogen is detailed. Snow, N. H.; Dunn, M.; Patel, S. “Determination of Crude Fat in Food Products by Supercritical Fluid Extraction and Gravimetric Analysis,” J. Chem. Educ. 1997, 74, 1108–1111. Supercritical CO 2 is used to extract fat from candy bars. Samples are placed in an extraction vessel and its weight determined before and after extraction. The %w/w fat content is determined by difference. The volume of CO 2 needed to effect a complete extraction is determined experimentally. Variations in results for different samples illustrate the importance of sampling. Yang, M. J.; Orton, M. L.; Pawliszyn, J. “Quantitative Determination of Caffeine in Beverages Using a Combined SPME-GC/MS Method,” J. Chem. Educ. 1997, 74, 1130–1132. Caffeine is extracted from beverages by a solid-phase microextraction using an uncoated fused silica fiber. The fiber is suspended in the sample for 5 min and the sample stirred to assist the mass transfer of analyte to the fiber. Immediately after removing the fiber from the sample it is transferred to the gas chromatograph’s injection port where the analyte is thermally desorbed. Quantitation is accomplished by using a 13 C 3 caffeine solution as an internal standard. 1. Because of the risk of lead poisoning, the exposure of children to lead-based paint is a significant public health concern. The first step in the quantitative analysis of lead in dried paint chips is to dissolve the sample. Corl evaluated several dissolution techniques. 24 In this study, samples of paint were collected and pulverized with a Pyrex mortar and pestle. Replicate portions of the powdered paint were then taken for analysis. Results for an unknown paint sample and for a standard reference material, in which dissolution was accomplished by a 4–6-h digestion with HNO 3 on a hot plate, are shown in the following table. %w/w Pb %w/w Pb Replicate in Unknown in Standard 1 5.09 11.48 2 6.29 11.62 3 6.64 11.47 4 4.63 11.86 (a) Determine the overall variance, the variance due to the method, and the variance due to sampling. (b) What percentage of the overall variance is due to sampling? How might the variance due to sampling be decreased? 7 L PROBLEMS 1400-CH07 9/8/99 4:04 PM Page 226 2. A shipment of 100 barrels of an organic solvent is to be evaluated by collecting and analyzing single samples from 10 of the barrels. A random number table is used to determine the barrels to be sampled. From which barrels should the samples be drawn if the first barrel is given by the twelfth entry in the random number table in Appendix 1E, with subsequent barrels given by every third entry? 3. The concentration of dissolved O 2 in a lake shows a daily cycle due to the effect of photosynthesis and a yearly cycle due to seasonal changes in temperature. Suggest an appropriate systematic sampling plan for monitoring the daily changes in dissolved O 2 . Suggest an appropriate systematic sampling plan for monitoring the yearly changes in dissolved O 2 . 4. The following data were collected during a preliminary study of the pH of an industrial wastewater stream Time Time (h) pH (h) pH 0.5 4.4 9.0 5.7 1.0 4.8 9.5 5.5 1.5 5.2 10.0 6.5 2.0 5.2 10.5 6.0 2.5 5.6 11.0 5.8 3.0 5.4 11.5 6.0 3.5 5.4 12.0 5.6 4.0 4.4 12.5 5.6 4.5 4.8 13.0 5.4 5.0 4.8 13.5 4.9 5.5 4.2 14.0 5.2 6.0 4.2 14.5 4.4 6.5 3.8 15.0 4.0 7.0 4.0 15.5 4.5 7.5 4.0 16.0 4.0 8.0 3.9 16.5 5.0 8.5 4.7 17.0 5.0 Construct a graph of pH as a function of time, and suggest an appropriate sampling frequency for a long-term monitoring program. 5. Suppose you have been asked to monitor the daily fluctuations in atmospheric ozone levels in the downtown area of a city to determine the relationship between daily traffic patterns and ozone levels. (a) Describe the sampling plan you would choose (random, systematic, judgmental, systematic–judgmental, or stratified). (b) Would you choose to collect and analyze a series of grab samples or form a single composite sample? (c) How would your answers to these questions change if the purpose of your work is to determine if the average daily ozone level exceeds a threshold value? 6. The best sampling plan for collecting samples depends on whether the population is homogeneous or heterogeneous. (a) Define homogeneous and heterogeneous. (b) If you collect and analyze a single sample, can you determine if the population is homogeneous or heterogeneous? 7. Examine equation 7.6 for sampling a heterogeneous population. (a) Explain why the contribution of heterogeneity to the overall sampling variance can be minimized by increasing the number of samples, but not the mass of the individual samples. (b) Explain why the contribution of homogeneity to the overall sampling variance can be minimized by increasing both the number of samples and the mass of individual samples. 8. Show that equation 7.5 can be derived from equation 7.4. Assume that the particles are spherical with a radius of r and a density of d. 9. The sampling constant for the radioisotope 24 Na in a sample of homogenized human liver has been reported as approximately 35 g. 25 (a) What is the expected relative standard deviation for sampling if 1.0-g samples are analyzed? (b) How many 1.0-g samples need to be analyzed to obtain a maximum sampling error of ±5% at the 95% confidence level? 10. Engels and Ingamells reported the following results for the %w/w K 2 O in a mixture of amphibolite and orthoclase. 26 0.247 0.300 0.236 0.258 0.304 0.330 0.247 0.275 0.212 0.311 0.258 0.187 Each of the 12 samples had a nominal weight of 0.1 g. Determine the approximate value for K s , and the mass of sample needed to achieve a percent relative standard deviation of 2%. 11. The following data have been reported for the determination of KH 2 PO 4 in a mixture of KH 2 PO 4 and NaCl. 27 Nominal Mass Actual Mass %w/w KH 2 PO 4 (g) (g) 0.10 0.1039 0.085 0.1015 1.078 0.1012 0.413 0.1010 1.248 0.1060 0.654 0.0997 0.507 0.25 0.2515 0.847 0.2465 0.598 0.2770 0.431 0.2460 0.842 0.2485 0.964 0.2590 1.178 0.50 0.5084 1.009 0.4954 0.947 0.5286 0.618 0.5232 0.744 0.4965 0.572 0.4995 0.709 Chapter 7 Obtaining and Preparing Samples for Analysis 227 1400-CH07 9/8/99 4:04 PM Page 227 1.00 1.027 0.696 0.987 0.843 0.991 0.535 0.998 0.750 0.997 0.711 1.001 0.639 2.50 2.496 0.766 2.504 0.769 2.496 0.682 2.496 0.609 2.557 0.589 2.509 0.617 (a) Prepare a graph of %w/w KH 2 PO 4 versus actual sample mass, and discuss how this graph is consistent with your understanding of factors affecting sampling variance. (b) For each nominal mass, calculate the percent relative standard deviation for the analysis. The value of K s for this analysis has been estimated as 350. For each nominal mass, use K s to determine the percent relative standard deviation due to sampling. Considering these two calculations, what conclusion can you make about the importance of indeterminate sampling errors for this analysis? (c) For each nominal mass, convert the percent relative standard deviation to an absolute standard deviation. Plot points on your graph corresponding to ±1 absolute standard deviation about the overall average %w/w KH 2 PO 4 . Draw smooth curves through these two sets of points. Considering these results, does the sample appear to be homogeneous on the scale at which it is sampled? 12. In this problem you will collect and analyze data in a simulation of the sampling process. Obtain a pack of M&M’s or other similar candy. Obtain a sample of five candies, and count the number that are red. Report the result of your analysis as % red. Return the candies to the bag, mix thoroughly, and repeat the analysis for a total of 20 determinations. Calculate the mean and standard deviation for your data. Remove all candies, and determine the true % red for the population. Sampling in this exercise should follow binomial statistics. Calculate the expected mean value and expected standard deviation, and compare to your experimental results. 13. The following two experiments were conducted in developing a sampling plan for determining the concentration of the herbicide diaquat in soil. First, eight samples, each with a mass of 1.50 g, were analyzed, giving a sampling variance of 0.013. In the second experiment, eight samples, each with a mass of 3.00 g, were analyzed, giving a sampling variance of 0.0071. (a) If samples of 1.00 g are to be collected, how many are needed to give a sampling variance of 0.0100? (b) If five samples are to be collected, how much should each sample weigh? 14. Determine the error (a = 0.05) for the following situations. In each case, assume that the variance for a single determination is 0.0025 and that the variance for collecting a single sample is 0.050. (a) Nine samples are collected, each of which is analyzed once. (b) One sample is collected and analyzed nine times. (c) Five samples are collected, each of which is analyzed three times. 15. Which of the sampling schemes in problem 14 is best if you wish to have an overall error of less than ±0.25, and the cost of collecting a single sample is 1 (arbitrary units), and the cost of analyzing a single sample is 10? Which is best if the cost of collecting a single sample is 7, and the cost of analyzing a single sample is 3? 16. Maw, Witry, and Emond evaluated a microwave digestion method for Hg against the standard open-vessel digestion method. 28 The standard method requires a 2-h digestion and is operator-intensive, whereas the microwave digestion is complete in approximately 0.5 h and requires little monitoring by the operator. Samples of baghouse dust from air-pollution-control equipment were collected from a hazardous waste incinerator and digested in triplicate by each method before determining the concentration of Hg in parts per million (ppm). Results are summarized in the following table. Parts per Million Hg Following Microwave Digestion Sample Replicate 1 Replicate 2 Replicate 3 1 7.12 7.66 7.17 2 16.1 15.7 15.6 3 4.89 4.62 4.28 4 9.64 9.03 8.44 5 6.76 7.22 7.50 6 6.19 6.61 7.61 7 9.44 9.56 10.7 8 30.8 29.0 26.2 Parts per million Hg Following Standard Digestion Sample Replicate 1 Replicate 2 Replicate 3 1 5.50 5.54 5.40 2 13.1 12.8 13.0 3 5.39 5.12 5.36 4 6.59 6.52 7.20 5 6.20 6.03 5.77 6 6.25 5.65 5.61 7 15.0 13.9 14.0 8 20.4 16.1 20.0 Evaluate whether the microwave digestion method yields acceptable results in comparison with the standard digestion method. 228 Modern Analytical Chemistry 1400-CH07 9/8/99 4:04 PM Page 228 17. Simpson, Apte, and Batley investigated methods for preserving water samples collected from anoxic (O 2 -poor) environments containing high concentrations of dissolved sulfide. 29 They found that preserving the water samples by adding HNO 3 (which is a common preservation method for aerobic samples) gave significant negative determinate errors when analyzing for Cu 2+ . When samples were preserved by adding H 2 O 2 followed by HNO 3 , the concentration of Cu 2+ was determined without a determinate error. Explain their observations. 18. In a particular analysis the selectivity coefficient, K A,I , is 0.816. When a standard sample known to contain an analyte-to- interferent ratio of 5:1 is carried through the analysis, the error in determining the analyte is +6.3%. (a) Determine the apparent recovery for the analyte if R I = 0. (b) Determine the apparent recovery for the interferent if R A =1. 19. The amount of Co in an ore sample is to be determined using a procedure for which Fe is an interferent. To evaluate the procedure’s accuracy, a standard sample of ore known to have a Co/Fe ratio of 10.2:1 is analyzed. When pure samples of Co and Fe are taken through the procedure, the following calibration relationships are obtained S Co = 0.786 · number of grams Co S Fe = 0.699 · number of grams Fe When 278.3 mg of Co is taken through the separation step, 275.9 mg is recovered. Only 3.6 mg of Fe is recovered when a 184.9-mg sample of Fe is carried through the separation step. Calculate (a) the recoveries for Co and Fe; (b) the separation factor; (c) the error if no attempt is made to separate the Co and Fe before the analysis; (d) the error if the separation step is carried out; and (e) the maximum recovery for Fe if all the Co is recovered and the maximum allowed error is 0.05%. 20. The amount of calcium in a sample of urine was determined by a method for which magnesium is an interferent. The selectivity coefficient, K Ca,Mg , for the method is 0.843. When a sample with a Mg/Ca ratio of 0.50 was carried through the procedure, an error of –3.7% was obtained. The error was +5.5% when a sample with a Mg/Ca ratio of 2.0 was used. (a) Determine the recoveries for Ca and Mg. (b) What is the expected error for a urine sample in which the Mg/Ca ratio is 10.0? 21. Show that F – is an effective masking agent in preventing a reaction of Al 3+ with EDTA. Assume that the only significant forms of fluoride and EDTA are F – and Y 4– . 22. Cyanide is frequently used as a masking agent for metal ions. The effectiveness of CN – as a masking agent is generally better in more basic solutions. Explain the reason for this pH dependency. 23. Explain how an aqueous sample consisting of Cu 2+ , Sn 4+ , Pb 2+ , and Zn 2+ can be separated into its component parts by adjusting the pH of the solution. 24. A solute, S, has a distribution ratio between water and ether of 7.5. Calculate the extraction efficiency if a 50.0-mL aqueous sample of S is extracted using 50.0 mL of ether as (a) a single portion of 50.0 mL; (b) two portions, each of 25.0 mL; (c) four portions, each of 12.5 mL; and (d) five portions, each of 10.0 mL. Assume that the solute is not involved in any secondary equilibria. 25. What volume of ether is needed to extract 99.9% of the solute in problem 24 when using (a) one extraction; (b) two extractions; (c) four extractions; and (d) five extractions. 26. What must a solute’s distribution ratio be if 99% of the solute in a 50.0-mL sample is to be extracted with a single 50.0-mL portion of an organic solvent? Repeat for the case where two 25.0-mL portions of the organic solvent are used. 27. A weak acid, HA, with a K a of 1.0 · 10 –5 , has a partition coefficient, K D , between water and an organic solvent of 1200. What restrictions on the sample’s pH are necessary to ensure that a minimum of 99.9% of the weak acid is extracted in a single step from 50.0 mL of water using 50.0 mL of the organic solvent? 28. For problem 27, how many extractions will be necessary if the pH of the sample cannot be decreased below 7.0? 29. A weak base, with a K b of 1.0 · 10 –3 , has a partition coefficient, K D , between water and an organic solvent of 500. What restrictions on the pH of the sample are necessary to ensure that a minimum of 99.9% of the weak base is extracted from 50.0 mL of water using two 25.00-mL portions of the organic solvent? 30. A sample contains a weak acid analyte, HA, and a weak acid interferent, HB. The acid dissociation constants and partition coefficients for the weak acids are as follows: K a,HA = 1.0 · 10 –3 , K a,HB = 1.0 · 10 –7 , K D,HA = K D,HB = 500. (a) Calculate the extraction efficiency for HA and HB when 50.0 mL of sample, buffered to a pH of 7.0, is extracted with 50.0 mL of the organic solvent. (b) Which phase is enriched in the analyte? (c) What are the recoveries for the analyte and interferent in this phase? (d) What is the separation factor? (e) A quantitative analysis is conducted on the contents of the phase enriched in analyte. What is the expected relative error if the selectivity coefficient, K HA,HB , is 0.500 and the initial ratio of HB/HA was 10.0? 31. The relevant equilibria for the extraction of I 2 from an aqueous solution of KI into an organic phase are shown in the following diagram. Chapter 7 Obtaining and Preparing Samples for Analysis 229 Organic Aqueous I 2 K D K f I 2 + I – I 3 – (a) Will the extraction efficiency for I 2 be greater for higher or lower concentrations of I – ? (b) Derive an expression for the distribution ratio for this extraction. 1400-CH07 9/8/99 4:04 PM Page 229 32. The relevant equilibria for extracting a neutral metal–ligand complex from an aqueous solution into an organic phase are shown in the following diagram. Distribution Ratio for pH Hg 2+ Pb 2+ Zn 2+ 1 3.3 0.0 0.0 2 10.0 0.43 0.0 3 32.3 999 0.0 4 32.3 9999 0.0 5 19.0 9999 0.18 6 4.0 9999 0.33 7 1.0 9999 0.82 8 0.54 9999 1.50 9 0.15 9999 2.57 10 0.05 9999 2.57 (a) Suppose that you have 50.0 mL of an aqueous solution containing Hg 2+ , Pb 2+ , and Zn 2+ . Describe how you would go about effecting a separation in which these metal ions are extracted into separate portions of the organic solvent. (b) Under the conditions you have selected for extracting Hg 2+ , what percent of the Hg 2+ remains in the aqueous phase after three extractions with 50.0 mL each of the organic solvent? (c) Under the conditions you have chosen for extracting Pb 2+ , what is the minimum volume of organic solvent needed to extract 99.5% of the Pb 2+ in a single extraction? (d) Under the conditions you have chosen for extracting Zn 2+ , how many extractions are needed to remove 99.5% of the Zn 2+ if each extraction is to use 25.0 mL of organic solvent? 230 Modern Analytical Chemistry Organic Aqueous ML 2 K D b 2 M 2+ + 2L – ML 2 (a) Derive an expression for the distribution ratio for this extraction. (b) Calculate the extraction efficiency when 50.0 mL of an aqueous solution that is 0.15 mM in M 2+ and 0.12 M in L – is extracted with 25.0 mL of the organic phase. Assume that K D is 10.3 and b 2 is 560. 33. Derive equation 7.30. 34. The following information is available for the extraction of Cu 2+ into CCl 4 with dithizone: K D,c =7· 10 4 ; b =5· 10 22 ; K a,HL =3· 10 –5 ; K D,HL = 1.1 · 10 4 ; n = 2. What is the extraction efficiency if 100 mL of an aqueous solution of 1.0 · 10 –7 M Cu 2+ that is 1 M in HCl is extracted with 10 mL of CCl 4 containing 4.0 · 10 –4 M HL? 35. Cupferron is a ligand whose strong affinity for metal ions makes it useful as a chelating agent in liquid–liquid extractions. The following distribution ratios are known for the extraction of Hg 2+ , Pb 2+ , and Zn 2+ from aqueous solutions to an organic solvent. The following paper provides a general introduction to the terminology used in describing sampling. Majors, R. E. “Nomenclature for Sampling in Analytical Chemistry.” LC?GC 1992, 10, 500–506. Further information on the statistics of sampling is covered in the following papers. Kratochvil, B.; Goewie, C. E.; Taylor, J. K. “Sampling Theory for Environmental Analysis,” Trends Anal. Chem. 1986, 5, 253–256. Kratochvil, B.; Taylor, J. K. “Sampling for Chemical Analysis,” Anal. Chem. 1981, 53, 924A–938A. The following sources may be consulted for further details regarding the collection of environmental samples. The paper by Benoit and colleagues provides a good discussion of how easily samples can be contaminated during collection and preservation. Barceló, D.; Hennion, M. C. “Sampling of Polar Pesticides from Water Matrices,” Anal. Chim. Acta 1997, 338, 3–18. Batley, G. E.; Gardner, D. “Sampling and Storage of Natural Waters for Trace Metal Analysis”, Wat. Res. 1977, 11, 745-756. Benoit, G.; Hunter, K. S.; Rozan, T. F. “Sources of Trace Metal Contamination Artifacts During Collection, Handling, and Analysis of Freshwaters,” Anal. Chem. 1997, 69, 1006–1011. Keith, L. H., ed. Principles of Environmental Sampling, American Chemical Society: Washington, DC, 1988. Keith, L. H. Environmental Sampling and Analysis—A Practical Guide, Lewis Publishers, Boca Raton, FL, 1991. The following sources provide additional information on preparing samples for analysis, including the separation of analytes and interferents. Anderson, R. Sample Pretreatment and Separation, Wiley: Chichester, 1987. Baiulescu, G. E.; Dumitrescu, P.; Zuaravescu, P. G. Sampling. Ellis Horwood: New York, 1991. Compton, T. R. Direct Preconcentration Techniques. Oxford Science Publications: Oxford, 1993. Compton, T. R. Complex-Formation Preconcentration Techniques. Oxford Science Publications: Oxford, 1993. Gy, P. M. Sampling of Particulate Materials: Theory and Practice; Elsevier: Amsterdam, 1979. 7 M SUGGESTED READINGS 1400-CH07 9/8/99 4:04 PM Page 230 Gy, P. M. Sampling of Heterogeneous and Dynamic Materials: Theories of Heterogeneity, Sampling and Homogenizing. Elsevier: Amsterdam, 1992. Gy, P. ed. Sampling for Analytical Purposes. Wiley: New York, 1998. Karger, B. L.; Snyder, L. R.; Harvath, C. An Introduction to Separation Science, Wiley-Interscience: New York, 1973. Kingston, H. M.; Haswell, S. J., eds. Microwave-Enhanced Chemistry: Fundamentals, Sample Preparation, and Applications. American Chemical Society: Washington, DC, 1997. Majors, R. E.; Raynie, D. E. “Sample Preparation and Solid-Phase Extraction,” LC?GC 1997, 15, 1106–1117. Miller, J. M. Separation Methods in Chemical Analysis, Wiley- Interscience: New York, 1975. Morrison, G. H.; Freiser, H. Solvent Extraction in Analytical Chemistry, John Wiley and Sons: New York, 1957. Pawliszyn, J. Solid-Phase Microextraction: Theory and Practice, Wiley: New York, 1997. Smith, R.; James, G. V. The Sampling of Bulk Materials. Royal Society of Chemistry: London, 1981. Sulcek, Z.; Povondra, P. Methods of Decomposition in Inorganic Analysis. CRC Press: Boca Raton, FL, 1989. Thurman, E. M.; Mills, M. S. Solid-Phase Extraction: Principles and Practice, Wiley: New York, 1998. A web-site dedicated to sample preparation, which contains useful information about acid digestion and microwave digestion, is found at http://www.sampleprep.duq.edu/sampleprep/ Chapter 7 Obtaining and Preparing Samples for Analysis 231 1. Youden, Y. J. J. Assoc. Off. Anal. Chem. 1981, 50, 1007–1013. 2. Fricke, G. H.; Mischler, P. G.; Staffieri, F. P.; et al. Anal. Chem. 1987, 59, 1213–1217. 3. (a) Cohen, R. D. J. Chem. Educ. 1991, 68, 902–903; (b) Cohen, R. D. J. Chem. Educ. 1992, 69, 200–203. 4. Keith, L. H. Environ. Sci. Technol. 1990, 24, 610–617. 5. Flatman, G. T.; Englund, E. J.; Yfantis, A. A. In Keith, L. H., ed. Principles of Environmental Sampling. American Chemical Society: Washington, DC, 1988, 73–84. 6. Ingamells; C. O.; Switzer, P. Talanta 1973, 20, 547–568. 7. Viseman, J. Mat. Res. Stds. 1969, 9(11), 8–13. 8. Blackwood, L. G. Environ. Sci. Technol. 1991, 25, 1366–1367. 9. Duce, R. A.; Quinn, J. G.; Olney, C. E. et al. Science, 1972, 176, 161–163. 10. Tanner, R. L. In Keith, L. H., ed. Principles of Environmental Sampling. American Chemical Society: Washington, DC, 1988, 275–286. 11. (a) Sandell, E. B. Colorimetric Determination of Traces of Metals, Interscience Publishers: New York, 1950, pp. 19–20; (b) Sandell, E. B. Anal. Chem. 1968, 40, 834–835. 12. Zubay, G. Biochemistry, 2nd ed. Macmillan: New York, 1988, p. 120. 13. Meites, L. Handbook of Analytical Chemistry, McGraw-Hill: New York, 1963. 14. Glavin, D. P.; Bada, J. L. Anal. Chem. 1998, 70, 3119–3122. 15. Fresenius, C. R. A System of Instruction in Quantitative Chemical Analysis. John Wiley and Sons: New York, 1881. 16. Jeannot, M. A.; Cantwell, F. F. Anal. Chem. 1997, 69, 235–239. 17. Zhang, A.; Yang, M. J.; Pawliszyn, J. Anal. Chem. 1994, 66, 844A–853A. 18. Renoe, B. W. Am. Lab. August 1994, 34–40. 19. McNally, M. E. Anal. Chem. 1995, 67, 308A–315A. 20. “TPH Extraction by SFE”, ISCO, Inc., Lincoln, NE, Revised Nov. 1992. 21. “The Analysis of Trihalomethanes in Drinking Water by Liquid Extraction”; US Environmental Protection Agency, Environmental Monitoring and Support Laboratory, Cincinnati, OH, 9 Sept. 1977. 22. Official Methods of Analysis, 11th ed., Association of Official Analytical Chemists, Washington, DC, 1970, p. 475. 23. Aguilar, C.; Borrull, F.; Marcé, R. M. LC?GC 1996, 14, 1048–1054. 24. Corl, W. E. Spectroscopy 1991, 6(8), 40–43. 25. Kratochvil, B.; Taylor, J. K. Anal. Chem. 1981, 53, 924A–938A. 26. Engels, J. C.; Ingamells, C. O. Geochim. Cosmochim. Acta 1970, 34, 1007–1017. 27. Guy, R. D.; Ramaley, L.; Wentzell, P. D. J. Chem. Educ. 1998, 75, 1028–1033. 28. Maw, R.; Witry, L.; Emond, T. Spectroscopy 1994, 9, 39–41. 29. Simpson, S. L.; Apte, S. C.; Batley, G. E. Anal. Chem. 1998, 70, 4202–4205. 7 N REFERENCES 1400-CH07 9/8/99 4:04 PM Page 231 232 Chapter 8 Gravimetric Methods of Analysis Gravimetry encompasses all techniques in which we measure mass or a change in mass. When you step on a scale after exercising you are making, in a sense, a gravimetric determination of your mass. Measuring mass is the most fundamental of all analytical measurements, and gravimetry is unquestionably the oldest analytical technique. 1400-CH08 9/9/99 2:17 PM Page 232 Chapter 8 Gravimetric Methods of Analysis 233 8A Overview of Gravimetry Before we look more closely at specific gravimetric methods and their applications, let’s take a moment to develop a broad survey of gravimetry. Later, as you read through the sections of this chapter discussing different gravimetric methods, this survey will help you focus on their similarities. It is usually easier to understand a new method of analysis when you can see its relationship to other similar methods. 8A.1 Using Mass as a Signal At the beginning of this chapter we indicated that in gravimetry we measure mass or a change in mass. This suggests that there are at least two ways to use mass as an analytical signal. We can, of course, measure an analyte’s mass directly by placing it on a balance and recording its mass. For example, suppose you are to determine the total suspended solids in water released from a sewage-treatment facility. Sus- pended solids are just that; solid matter that has yet to settle out of its solution ma- trix. The analysis is easy. You collect a sample and pass it through a preweighed fil- ter that retains the suspended solids. After drying to remove any residual moisture, you weigh the filter. The difference between the filter’s original mass and final mass gives the mass of suspended solids. We call this a direct analysis because the analyte itself is the object being weighed. What if the analyte is an aqueous ion, such as Pb 2+ ? In this case we cannot iso- late the analyte by filtration because the Pb 2+ is dissolved in the solution’s matrix. We can still measure the analyte’s mass, however, by chemically converting it to a solid form. If we suspend a pair of Pt electrodes in our solution and apply a suffi- ciently positive potential between them for a long enough time, we can force the reaction Pb 2+ (aq)+4H 2 O(l) t PbO 2 (s)+H 2 (g)+2H 3 O + (aq) to go to completion. The Pb 2+ ion in solution oxidizes to PbO 2 and deposits on the Pt electrode serving as the anode. If we weigh the Pt anode before and after applying the potential, the difference in the two measurements gives the mass of PbO 2 and, from the reaction’s stoichiometry, the mass of Pb 2+ . This also is a direct analysis be- cause the material being weighed contains the analyte. Sometimes it is easier to remove the analyte and use a change in mass as the analytical signal. Imagine how you would determine a food’s moisture content by a direct analysis. One possibility is to heat a sample of the food to a temperature at which the water in the sample vaporizes. If we capture the vapor in a preweighed absorbent trap, then the change in the absorbent’s mass provides a di- rect determination of the amount of water in the sample. An easier approach, however, is to weigh the sample of food before and after heating, using the change in its mass as an indication of the amount of water originally present. We call this an indirect analysis since we determine the analyte by a signal representing its disappearance. The indirect determination of moisture content in foods is done by difference. The sample’s initial mass includes the water, whereas the final mass is measured after removing the water. We can also determine an analyte indirectly without its ever being weighed. Again, as with the determination of Pb 2+ as PbO 2 (s), we take advantage of the analyte’s chemistry. For example, phosphite, PO 3 3– , reduces Hg 2+ to Hg 2 2+ . In the presence of Cl – a solid precipitate of Hg 2 Cl 2 forms. 2HgCl 2 (aq)+PO 3 3– (aq)+3H 2 O(l) t Hg 2 Cl 2 (s)+2H 3 O + (aq) + 2Cl – (aq)+ PO 4 3– (aq) gravimetry Any method in which the signal is a mass or change in mass. 1400-CH08 9/9/99 2:17 PM Page 233 If HgCl 2 is added in excess, each mole of PO 3 3– produces one mole of Hg 2 Cl 2 . The precipitate’s mass, therefore, provides an indirect measurement of the mass of PO 3 3– present in the original sample. Summarizing, we can determine an analyte gravimetrically by directly deter- mining its mass, or the mass of a compound containing the analyte. Alternatively, we can determine an analyte indirectly by measuring a change in mass due to its loss, or the mass of a compound formed as the result of a reaction involving the analyte. 8A.2 Types of Gravimetric Methods In the previous section we used four examples to illustrate the different ways that mass can serve as an analytical signal. These examples also illustrate the four gravi- metric methods of analysis. When the signal is the mass of a precipitate, we call the method precipitation gravimetry. The indirect determination of PO 3 3– by precipi- tating Hg 2 Cl 2 is a representative example, as is the direct determination of Cl – by precipitating AgCl. In electrogravimetry the analyte is deposited as a solid film on one electrode in an electrochemical cell. The oxidation of Pb 2+ , and its deposition as PbO 2 on a Pt anode is one example of electrogravimetry. Reduction also may be used in electro- gravimetry. The electrodeposition of Cu on a Pt cathode, for example, provides a direct analysis for Cu 2+ . When thermal or chemical energy is used to remove a volatile species, we call the method volatilization gravimetry. In determining the moisture content of food, thermal energy vaporizes the H 2 O. The amount of carbon in an organic com- pound may be determined by using the chemical energy of combustion to convert C to CO 2 . Finally, in particulate gravimetry the analyte is determined following its re- moval from the sample matrix by filtration or extraction. The determination of sus- pended solids is one example of particulate gravimetry. 8A. 3 Conservation of Mass An accurate gravimetric analysis requires that the mass of analyte present in a sam- ple be proportional to the mass or change in mass serving as the analytical signal. For all gravimetric methods this proportionality involves a conservation of mass. For gravimetric methods involving a chemical reaction, the analyte should partici- pate in only one set of reactions, the stoichiometry of which indicates how the pre- cipitate’s mass relates to the analyte’s mass. Thus, for the analysis of Pb 2+ and PO 3 3– described earlier, we can write the following conservation equations Moles Pb 2+ = moles PbO 2 Moles PO 3 3– = moles Hg 2 Cl 2 Removing the analyte from its matrix by filtration or extraction must be complete. When true, the analyte’s mass can always be found from the analytical signal; thus, for the determination of suspended solids we know that Filter’s final mass – filter’s initial mass = g suspended solid whereas for the determination of the moisture content we have Sample’s initial mass – sample’s final mass = g H 2 O 234 Modern Analytical Chemistry precipitation gravimetry A gravimetric method in which the signal is the mass of a precipitate. electrogravimetry A gravimetric method in which the signal is the mass of an electrodeposit on the cathode or anode in an electrochemical cell. volatilization gravimetry A gravimetric method in which the loss of a volatile species gives rise to the signal. particulate gravimetry A gravimetric method in which the mass of a particulate analyte is determined following its separation from its matrix. 1400-CH08 9/9/99 2:17 PM Page 234 Chapter 8 Gravimetric Methods of Analysis 235 precipitant A reagent that causes the precipitation of a soluble species. Specific details, including worked examples, are found in the sections of this chapter covering individual gravimetric methods. 8A. 4 Why Gravimetry Is Important Except for particulate gravimetry, which is the most trivial form of gravimetry, it is entirely possible that you will never use gravimetry after you are finished with this course. Why, then, is familiarity with gravimetry still important? The answer is that gravimetry is one of only a small number of techniques whose measurements re- quire only base SI units, such as mass and moles, and defined constants, such as Avogadro’s number and the mass of 12 C.* The result of an analysis must ultimately be traceable to methods, such as gravimetry, that can be related to fundamental physical properties. 1 Most analysts never use gravimetry to validate their methods. Verifying a method by analyzing a standard reference material, however, is com- mon. Estimating the composition of these materials often involves a gravimetric analysis. 2 8B Precipitation Gravimetry Precipitation gravimetry is based on the formation of an insoluble compound fol- lowing the addition of a precipitating reagent, or precipitant, to a solution of the analyte. In most methods the precipitate is the product of a simple metathesis reac- tion between the analyte and precipitant; however, any reaction generating a pre- cipitate can potentially serve as a gravimetric method. Most precipitation gravimet- ric methods were developed in the nineteenth century as a means for analyzing ores. Many of these methods continue to serve as standard methods of analysis. 8B.1 Theory and Practice A precipitation gravimetric analysis must have several important attributes. First, the precipitate must be of low solubility, high purity, and of known composition if its mass is to accurately reflect the analyte’s mass. Second, the precipitate must be in a form that is easy to separate from the reaction mixture. The theoretical and exper- imental details of precipitation gravimetry are reviewed in this section. Solubility Considerations An accurate precipitation gravimetric method requires that the precipitate’s solubility be minimal. Many total analysis techniques can rou- tinely be performed with an accuracy of better than ±0.1%. To obtain this level of accuracy, the isolated precipitate must account for at least 99.9% of the analyte. By extending this requirement to 99.99% we ensure that accuracy is not limited by the precipitate’s solubility. Solubility losses are minimized by carefully controlling the composition of the solution in which the precipitate forms. This, in turn, requires an understanding of the relevant equilibrium reactions affecting the precipitate’s solubility. For example, Ag + can be determined gravimetrically by adding Cl – as a precipitant, forming a precipitate of AgCl. Ag + (aq)+Cl – (aq) t AgCl(s) 8.1 *Two other techniques that depend only on base SI units are coulometry and isotope-dilution mass spectrometry. Coulometry is discussed in Chapter 11. Isotope-dilution mass spectroscopy is beyond the scope of an introductory text, however, the list of suggested readings includes a useful reference. 1400-CH08 9/9/99 2:17 PM Page 235 Figure 8.1 Solubility of AgCl as a function of pCl. The dashed line shows the predicted S AgCl , assuming that only reaction 8.1 and equation 8.2 affect the solubility of AgCl. The solid line is calculated using equation 8.7, and includes the effect of reactions 8.3–8.5. A ladder diagram for the AgCl complexation equilibria is superimposed on the pCl axis. If this is the only reaction considered, we would falsely conclude that the precipi- tate’s solubility, S AgCl , is given by 8.2 and that solubility losses may be minimized by adding a large excess of Cl – . In fact, as shown in Figure 8.1, adding a large excess of Cl – eventually increases the precipi- tate’s solubility. To understand why AgCl shows a more complex solubility relationship than that suggested by equation 8.2, we must recognize that Ag + also forms a series of soluble chloro-complexes Ag + (aq)+Cl – (aq) t K 1 AgCl(aq) 8.3 Ag + (aq) +2Cl – (aq) t b 2 AgCl 2 – (aq) 8.4 Ag + (aq) + 3Cl – (aq) t b 3 AgCl 3 2– (aq) 8.5 The solubility of AgCl, therefore, is the sum of the equilibrium concentrations for all soluble forms of Ag + . S AgCl = [Ag + ] + [AgCl(aq)] + [AgCl 2 – ] + [AgCl 3 2– ] 8.6 Substituting the equilibrium constant expressions for reactions 8.3–8.5 into equation 8.6 defines the solubility of AgCl in terms of the equilibrium concentration of Cl – . 8.7 Equation 8.7 explains the solubility curve for AgCl shown in Figure 8.1. As Cl – is added to a solution of Ag + , the solubility of AgCl initially decreases because of re- action 8.1. Note that under these conditions, the final three terms in equation 8.7 are small, and that equation 8.1 is sufficient to describe the solubility of AgCl. In- creasing the concentration of chloride, however, leads to an increase in the solubil- ity of AgCl due to the soluble chloro-complexes formed in reactions 8.3–8.5.* S K KK K K AgCl sp sp sp sp Cl Cl Cl=+ + + ? ?? [] [] [] 12 3 2 ββ S K AgCl + sp Ag Cl == ? [] [] 236 Modern Analytical Chemistry 654 –7.00 –6.00 –5.00 –4.00 –3.00 –2.00 –1.00 0.00 3 pCl log (Solubility) 210 Ag + (aq) AgCl (aq) AgCl 2 – (aq) *Also shown in Figure 8.1 is a ladder diagram for this system. Note that the increase in solubility begins when the higher-order soluble complexes, AgCl 2 – and AgCl 3 2– , become the dominant species. 1400-CH08 9/9/99 2:17 PM Page 236 Chapter 8 Gravimetric Methods of Analysis 237 Figure 8.2 (a) Ladder diagram for phosphate; (b) Solubility diagram for Ca 3 (PO 4 ) 2 showing the predominate form of phosphate for each segment of the solubility curve. Clearly the equilibrium concentration of chloride is an important parameter if the concentration of silver is to be determined gravimetrically by precipitating AgCl. In particular, a large excess of chloride must be avoided. Another important parameter that may affect a precipitate’s solubility is the pH of the solution in which the precipitate forms. For example, hydroxide precipitates, such as Fe(OH) 3 , are more soluble at lower pH levels at which the concentration of OH – is small. The effect of pH on solubility is not limited to hydroxide precipitates, but also affects precipitates containing basic or acidic ions. The solubility of Ca 3 (PO 4 ) 2 is pH-dependent because phosphate is a weak base. The following four reactions, therefore, govern the solubility of Ca 3 (PO 4 ) 2 . Ca 3 (PO 4 ) 2 (s) t K sp 3Ca 2+ (aq) + 2PO 4 3– (aq) 8.8 PO 4 3– (aq)+H 2 O(l) t K b1 HPO 4 2– (aq)+OH – (aq) 8.9 HPO 4 2– (aq)+H 2 O(l) t K b2 H 2 PO 4 – (aq)+OH – (aq) 8.10 H 2 PO 4 – (aq)+H 2 O(l) t K b3 H 3 PO 4 (aq)+OH – (aq) 8.11 Depending on the solution’s pH, the predominate phosphate species is either PO 4 3– , HPO 4 2– , H 2 PO 4 – , or H 3 PO 4 . The ladder diagram for phosphate, shown in Figure 8.2a, provides a convenient way to evaluate the pH-dependent solubility of phos- phate precipitates. When the pH is greater than 12.4, the predominate phosphate species is PO 4 3– , and the solubility of Ca 3 (PO 4 ) 2 will be at its minimum because only reaction 8.8 occurs to an appreciable extent (see Figure 8.2b). As the solution becomes more acidic, the solubility of Ca 3 (PO 4 ) 2 increases due to the contributions of reactions 8.9–8.11. Solubility can often be decreased by using a nonaqueous solvent. A precip- itate’s solubility is generally greater in aqueous solutions because of the ability of water molecules to stabilize ions through solvation. The poorer solvating ability of nonaqueous solvents, even those that are polar, leads to a smaller solu- bility product. For example, PbSO 4 has a K sp of 1.6 · 10 –8 in H 2 O, whereas in a 50:50 mixture of H 2 O/ethanol the K sp at 2.6 · 10 –12 is four orders of magnitude smaller. pH H 3 PO 4 H 2 PO 4 – H 2 PO 4 – HPO 4 2– HPO 4 2– pK a3 = 12.35 pK a2 = 7.20 pK a1 = 2.15 PO 4 3– 14 12 10 8 6 4 2 0 246 pH 8 1012140 H 3 PO 4 H 2 PO 4 – log(solubility) Ca 3 (PO 4 ) 2 HPO 4 2– PO 4 3– (a) (b) 1400-CH08 9/9/99 2:17 PM Page 237 Figure 8.3 Schematic model of AgCl showing difference between bulk and surface atoms of silver. Silver and chloride ions are not shown to scale. Avoiding Impurities Precipitation gravimetry is based on a known stoichiometry between the analyte’s mass and the mass of a precipitate. It follows, therefore, that the precipitate must be free from impurities. Since precipitation typically occurs in a solution rich in dissolved solids, the initial precipitate is often impure. Any impu- rities present in the precipitate’s matrix must be removed before obtaining its weight. The greatest source of impurities results from chemical and physical interac- tions occurring at the precipitate’s surface. A precipitate is generally crystalline, even if only on a microscopic scale, with a well-defined lattice structure of cations and anions. Those cations and anions at the surface of the precipitate carry, respec- tively, a positive or a negative charge as a result of their incomplete coordination spheres. In a precipitate of AgCl, for example, each Ag + ion in the bulk of the pre- cipitate is bound to six Cl – ions. Silver ions at the surface, however, are bound to no more than five Cl – ions, and carry a partial positive charge (Figure 8.3). Precipitate particles grow in size because of the electrostatic attraction between charged ions on the surface of the precipitate and oppositely charged ions in solu- tion. Ions common to the precipitate are chemically adsorbed, extending the crystal lattice. Other ions may be physically adsorbed and, unless displaced, are incorpo- rated into the crystal lattice as a coprecipitated impurity. Physically adsorbed ions are less strongly attracted to the surface and can be displaced by chemically ad- sorbed ions. One common type of impurity is an inclusion. Potential interfering ions whose size and charge are similar to a lattice ion may substitute into the lattice structure by chemical adsorption, provided that the interferent precipitates with the same crystal structure (Figure 8.4a). The probability of forming an inclusion is greatest when the interfering ion is present at substantially higher concentrations than the dissolved lattice ion. The presence of an inclusion does not decrease the amount of analyte that precipitates, provided that the precipitant is added in sufficient excess. Thus, the precipitate’s mass is always larger than expected. Inclusions are difficult to remove since the included material is chemically part of the crystal lattice. The only way to remove included material is through reprecip- itation. After isolating the precipitate from the supernatant solution, it is dissolved 238 Modern Analytical Chemistry Cl – Ag + Bulk silver ion surrounded by 6 chloride ions Surface silver ion surrounded by 4 chloride ions inclusion A coprecipitated impurity in which the interfering ion occupies a lattice site in the precipitate. 1400-CH08 9/9/99 2:17 PM Page 238 in a small portion of a suitable solvent at an elevated temperature. The solution is then cooled to re-form the precipitate. Since the concentration ratio of interferent to analyte is lower in the new solution than in the original supernatant solution, the mass percent of included material in the precipitate decreases. This process of re- precipitation is repeated as needed to completely remove the inclusion. Potential solubility losses of the analyte, however, cannot be ignored. Thus, reprecipitation requires a precipitate of low solubility, and a solvent for which there is a significant difference in the precipitate’s solubility as a function of temperature. Occlusions, which are a second type of coprecipitated impurity, occur when physically adsorbed interfering ions become trapped within the growing precipitate. Occlusions form in two ways. The most common mechanism occurs when physically adsorbed ions are surrounded by additional precipitate before they can be desorbed or displaced (see Figure 8.4a). In this case the precipitate’s mass is always greater than expected. Occlusions also form when rapid precipitation traps a pocket of solution within the growing precipitate (Figure 8.4b). Since the trapped solution contains dis- solved solids, the precipitate’s mass normally increases. The mass of the precipitate may be less than expected, however, if the occluded material consists primarily of the analyte in a lower-molecular-weight form from that of the precipitate. Occlusions are minimized by maintaining the precipitate in equilibrium with its supernatant solution for an extended time. This process is called digestion and may be carried out at room temperature or at an elevated temperature. During di- gestion, the dynamic nature of the solubility–precipitation equilibrium, in which the precipitate dissolves and re-forms, ensures that occluded material is eventually exposed to the supernatant solution. Since the rate of dissolution and reprecipita- tion are slow, the chance of forming new occlusions is minimal. After precipitation is complete the surface continues to attract ions from solu- tion (Figure 8.4c). These surface adsorbates, which may be chemically or physically adsorbed, constitute a third type of coprecipitated impurity. Surface adsorption is minimized by decreasing the precipitate’s available surface area. One benefit of di- gestion is that it also increases the average size of precipitate particles. This is not surprising since the probability that a particle will dissolve is inversely proportional to its size. During digestion larger particles of precipitate increase in size at the ex- pense of smaller particles. One consequence of forming fewer particles of larger size is an overall decrease in the precipitate’s surface area. Surface adsorbates also may be removed by washing the precipitate. Potential solubility losses, however, cannot be ignored. Inclusions, occlusions, and surface adsorbates are called coprecipitates because they represent soluble species that are brought into solid form along with the de- sired precipitate. Another source of impurities occurs when other species in solu- tion precipitate under the conditions of the analysis. Solution conditions necessary to minimize the solubility of a desired precipitate may lead to the formation of an additional precipitate that interferes in the analysis. For example, the precipitation of nickel dimethylgloxime requires a pH that is slightly basic. Under these condi- tions, however, any Fe 3+ that might be present precipitates as Fe(OH) 3 . Finally, since most precipitants are not selective toward a single analyte, there is always a risk that the precipitant will react, sequentially, with more than one species. The formation of these additional precipitates can usually be minimized by carefully controlling solution conditions. Interferents forming precipitates that are less soluble than the analyte may be precipitated and removed by filtration, leaving the analyte behind in solution. Alternatively, either the analyte or the interferent can be masked using a suitable complexing agent, preventing its precipitation. Chapter 8 Gravimetric Methods of Analysis 239 Figure 8.4 Example of coprecipitation: (a) schematic of a chemically adsorbed inclusion or a physically adsorbed occlusion in a crystal lattice, where C and A represent the cation–anion pair comprising the analyte and the precipitant, and is the impurity; (b) schematic of an occlusion by entrapment of supernatant solution; (c) surface adsorption of excess C. CACACACACACACACACACA ACACACACACACACACACAC CACACACAMACACACACACA ACACACACACACAMACACAC CACACACACACACACACACA AMACACACACACACACACAC CACACACACACACACACACA (a) CACACACACACACACACACA ACACACACACACACACACAC CACACACACACACACACACA ACACACACACACACACACAC CACACACACACACACACACA ACACACACACACACACACAC CACACACACACACACACACA (b) CA ACC CC C C CACACACAC C C ACACACACACACAC CACACACACACACACACACAC (c) occlusion A coprecipitated impurity trapped within a precipitate as it forms. digestion The process by which a precipitate is given time to form larger, purer particles. adsorbate A coprecipitated impurity that adsorbs to the surface of a precipitate. M 1400-CH08 9/9/99 2:17 PM Page 239 Both of the above-mentioned approaches are illustrated in Fresenius’s analyti- cal method for determining Ni and Co in ores containing Pb 2+ , Cu 2+ , and Fe 3+ as potential interfering ions (see Figure 1.1 in Chapter 1). The ore is dissolved in a so- lution containing H 2 SO 4 , selectively precipitating Pb 2+ as PbSO 4 . After filtering, the supernatant solution is treated with H 2 S. Because the solution is strongly acidic, however, only CuS precipitates. After removing the CuS by filtration, the solution is made basic with ammonia until Fe(OH) 3 precipitates. Cobalt and nickel, which form soluble amine complexes, remain in solution. In some situations the rate at which a precipitate forms can be used to separate an analyte from a potential interferent. For example, due to similarities in their chemistry, a gravimetric analysis for Ca 2+ may be adversely affected by the presence of Mg 2+ . Precipitates of Ca(OH) 2 , however, form more rapidly than precipitates of Mg(OH) 2 . If Ca(OH) 2 is filtered before Mg(OH) 2 begins to precipitate, then a quantitative analysis for Ca 2+ is feasible. Finally, in some cases it is easier to isolate and weigh both the analyte and the interferent. After recording its weight, the mixed precipitate is treated to convert at least one of the two precipitates to a new chemical form. This new mixed precipitate is also isolated and weighed. For example, a mixture containing Ca 2+ and Mg 2+ can be analyzed for both cations by first isolating a mixed precipitate of CaCO 3 and MgCO 3 . After weighing, the mixed precipitate is heated, converting it to a mixture of CaO and MgO. Thus Grams of mixed precipitate 1 = grams CaCO 3 + grams MgCO 3 Grams of mixed precipitate 2 = grams CaO + grams MgO Although these equations contain four unknowns (grams CaCO 3 , grams MgCO 3 , grams CaO, and grams MgO), the stoichiometric relationships between CaCO 3 and CaO Moles CaCO 3 = moles CaO and between MgCO 3 and MgO Moles MgCO 3 = moles MgO provide enough additional information to determine the amounts of both Ca 2+ and Mg 2+ in the sample.* Controlling Particle Size Following precipitation and digestion, the precipitate must be separated from the supernatant solution and freed of any remaining impu- rities, including residual solvent. These tasks are accomplished by filtering, rinsing, and drying the precipitate. The size of the precipitate’s particles determines the ease and success of filtration. Smaller, colloidal particles are difficult to filter because they may readily pass through the pores of the filtering device. Large, crystalline particles, however, are easily filtered. By carefully controlling the precipitation reaction we can significantly increase a precipitate’s average particle size. Precipitation consists of two distinct events: nu- cleation, or the initial formation of smaller stable particles of precipitate, and the subsequent growth of these particles. Larger particles form when the rate of particle growth exceeds the rate of nucleation. 240 Modern Analytical Chemistry *Example 8.2 shows how to solve this type of problem. 1400-CH08 9/9/99 2:17 PM Page 240 Chapter 8 Gravimetric Methods of Analysis 241 A solute’s relative supersaturation, RSS, can be expressed as 8.12 where Q is the solute’s actual concentration, S is the solute’s expected concentra- tion at equilibrium, and Q – S is a measure of the solute’s supersaturation when precipitation begins. 3 A large, positive value of RSS indicates that a solution is highly supersaturated. Such solutions are unstable and show high rates of nucle- ation, producing a precipitate consisting of numerous small particles. When RSS is small, precipitation is more likely to occur by particle growth than by nucleation. Examining equation 8.12 shows that we can minimize RSS by either decreasing the solute’s concentration or increasing the precipitate’s solubility. A precipitate’s solubility usually increases at higher temperatures, and adjusting pH may affect a precipitate’s solubility if it contains an acidic or basic anion. Temperature and pH, therefore, are useful ways to increase the value of S. Conducting the precipitation in a dilute solution of analyte, or adding the precipitant slowly and with vigorous stir- ring are ways to decrease the value of Q. There are, however, practical limitations to minimizing RSS. Precipitates that are extremely insoluble, such as Fe(OH) 3 and PbS, have such small solubilities that a large RSS cannot be avoided. Such solutes inevitably form small particles. In addi- tion, conditions that yield a small RSS may lead to a relatively stable supersaturated solution that requires a long time to fully precipitate. For example, almost a month is required to form a visible precipitate of BaSO 4 under conditions in which the ini- tial RSS is 5. 4 An increase in the time required to form a visible precipitate under conditions of low RSS is a consequence of both a slow rate of nucleation and a steady decrease in RSS as the precipitate forms. One solution to the latter problem is to chemically generate the precipitant in solution as the product of a slow chemical reaction. This maintains the RSS at an effectively constant level. The precipitate initially forms under conditions of low RSS, leading to the nucleation of a limited number of parti- cles. As additional precipitant is created, nucleation is eventually superseded by par- ticle growth. This process is called homogeneous precipitation. 5 Two general methods are used for homogeneous precipitation. If the precipi- tate’s solubility is pH-dependent, then the analyte and precipitant can be mixed under conditions in which precipitation does not occur. The pH is then raised or lowered as needed by chemically generating OH – or H 3 O + . For example, the hydrol- ysis of urea can be used as a source of OH – . CO(NH 2 ) 2 (aq)+H 2 O(l) t CO 2 (g) + 2NH 3 (aq) NH 3 (aq)+H 2 O(l) t NH 4 + (aq)+OH – (aq) The hydrolysis of urea is strongly temperature-dependent, with the rate being negli- gible at room temperature. The rate of hydrolysis, and thus the rate of precipitate formation, can be controlled by adjusting the solution’s temperature. Precipitates of BaCrO 4 , for example, have been produced in this manner. In the second method of homogeneous precipitation, the precipitant itself is generated by a chemical reaction. For example, Ba 2+ can be homogeneously precipi- tated as BaSO 4 by hydrolyzing sulphamic acid to produce SO 4 2– . NH 2 SO 3 H(aq)+2H 2 O(l) t NH 4 + (aq)+H 3 O + (aq)+SO 4 2– (aq) RSS QS S = ? relative supersaturation A measure of the extent to which a solution, or a localized region of solution, contains more dissolved solute than that expected at equilibrium (RSS). homogeneous precipitation A precipitation in which the precipitant is generated in situ by a chemical reaction. Color Plate 5 shows the difference between a precipitate formed by direct precipitation and a precipitate formed by a homogeneous precipitation. 1400-CH08 9/9/99 2:17 PM Page 241 Figure 8.5 Schematic model of the solid–solution interface at a particle of AgCl in a solution containing excess AgNO 3 . Homogeneous precipitation affords the dual advantages of producing large particles of precipitate that are relatively free from impurities. These advantages, however, may be offset by increasing the time needed to produce the precipitate, and a tendency for the precipitate to deposit as a thin film on the container’s walls. The latter problem is particularly severe for hydroxide precipitates generated using urea. An additional method for increasing particle size deserves mention. When a precipitate’s particles are electrically neutral, they tend to coagulate into larger par- ticles. Surface adsorption of excess lattice ions, however, provides the precipitate’s particles with a net positive or negative surface charge. Electrostatic repulsion be- tween the particles prevents them from coagulating into larger particles. Consider, for instance, the precipitation of AgCl from a solution of AgNO 3 , using NaCl as a precipitant. Early in the precipitation, when NaCl is the limiting reagent, excess Ag + ions chemically adsorb to the AgCl particles, forming a posi- tively charged primary adsorption layer (Figure 8.5). Anions in solution, in this case NO 3 – and OH – , are attracted toward the surface, forming a negatively charged sec- ondary adsorption layer that balances the surface’s positive charge. The solution outside the secondary adsorption layer remains electrically neutral. Coagulation cannot occur if the secondary adsorption layer is too thick because the individual particles of AgCl are unable to approach one another closely enough. Coagulation can be induced in two ways: by increasing the concentration of the ions responsible for the secondary adsorption layer or by heating the solution. One way to induce coagulation is to add an inert electrolyte, which increases the concen- tration of ions in the secondary adsorption layer. With more ions available, the thickness of the secondary absorption layer decreases. Particles of precipitate may now approach one another more closely, allowing the precipitate to coagulate. The amount of electrolyte needed to cause spontaneous coagulation is called the critical coagulation concentration. 242 Modern Analytical Chemistry AgCl ClAgCl AgClAgCl ClAgClAgCl AgClAgClAgCl ClAgClAgClAg AgClAgClAgCl ClAgClAgCl AgClAgClAg ClAgClAgCl AgClAgCl OH – OH – OH – OH – NO 3 – Ag + Ag + Ag + Ag + Ag + Ag + Ag + Ag + Ag + Ag + Ag + NO 3 – NO 3 – NO 3 – Cl – NO 3 – NO 3 – NO 3 – NO 3 – NO 3 – NO 3 – NO 3 – NO 3 – Precipitate Primary adsorption layer Secondary adsorption layer Bulk solution coagulation The process of smaller particles of precipitate clumping together to form larger particles. 1400-CH08 9/9/99 2:17 PM Page 242 Chapter 8 Gravimetric Methods of Analysis 243 Figure 8.6 Proper procedure for filtering solids using filter paper. The filter paper circle in (a) is folded in half (b), and folded in half again (c). The filter paper is parted (d), and a small corner is torn off (e). The filter paper is opened up into a cone and placed in the funnel (f). Note that the torn corner is placed to the outside. Heating the solution and precipitate provides a second way to induce coagula- tion. As the temperature increases, the number of ions in the primary adsorption layer decreases, lowering the precipitate’s surface charge. In addition, increasing the particle’s kinetic energy may be sufficient to overcome the electrostatic repulsion preventing coagulation at lower temperatures. Filtering the Precipitate After precipitation and digestion are complete, the precip- itate is separated from solution by filtration using either filter paper or a filtering cru- cible. The most common filtering medium is cellulose-based filter paper, which is classified according to its filtering speed, its size, and its ash content on ignition. Fil- tering speed is a function of the paper’s pore size, which determines the particle sizes retained by the filter. Filter paper is rated as fast (retains particles > 20–25 m m), medium fast (retains particles > 16 m m), medium (retains particles > 8 m m), and slow (retains particles > 2–3 m m). The proper choice of filtering speed is important. If the filtering speed is too fast, the precipitate may pass through the filter paper re- sulting in a loss of precipitate. On the other hand, the filter paper can become clogged when using a filter paper that is too slow. Filter paper is hygroscopic and is not easily dried to a constant weight. As a re- sult, in a quantitative procedure the filter paper must be removed before weighing the precipitate. This is accomplished by carefully igniting the filter paper. Following ignition, a residue of noncombustible inorganic ash remains that contributes a posi- tive determinate error to the precipitate’s final mass. For quantitative analytical pro- cedures a low-ash filter paper must be used. This grade of filter paper is pretreated by washing with a mixture of HCl and HF to remove inorganic materials. Filter paper classed as quantitative has an ash content of less than 0.010% w/w. Qualita- tive filter paper typically has a maximum ash content of 0.06% w/w. Filtering is accomplished by folding the filter paper into a cone, which is then placed in a long-stem funnel (Figure 8.6). A seal between the filter cone and the (a) (d) Tear (e) (f) (c)(b) 1400-CH08 9/9/99 2:17 PM Page 243 Figure 8.7 Proper procedure for transferring the supernatant to the filter paper cone. funnel is formed by dampening the paper with water and pressing the paper to the wall of the funnel. When properly prepared, the stem of the funnel will fill with the solution being filtered, increasing the rate of filtration. Filtration is accomplished by the force of gravity. The precipitate is transferred to the filter in several steps (Figure 8.7). The first step is to decant the majority of the supernatant through the filter paper without transferring the precipitate. This is done to prevent the filter paper from becoming clogged at the beginning of the filtration process. Initial rinsing of the precipitate is done in the beaker in which the precipitation was performed. These rinsings are also decanted through the filter paper. Finally, the precipitate is transferred onto the filter paper using a stream of rinse solution. Any precipitate clinging to the walls of the beaker is transferred using a rubber policeman (which is simply a flexible rubber spatula attached to the end of a glass stirring rod). An alternative method for filtering the precipitate is a filtering crucible (Fig- ure 8.8). The most common is a fritted glass crucible containing a porous glass disk filter. Fritted glass crucibles are classified by their porosity: coarse (retaining particles > 40–60 m m), medium (retaining particles > 10–15 m m), and fine (re- taining particles > 4–5.5 m m). Another type of filtering crucible is the Gooch cru- cible, a porcelain crucible with a perforated bottom. A glass fiber mat is placed in the crucible to retain the precipitate, which is transferred to the crucible in the same manner described for filter paper. The supernatant is drawn through the crucible with the assistance of suction from a vacuum aspirator or pump. Rinsing the Precipitate Filtering removes most of the supernatant solution. Resid- ual traces of the supernatant, however, must be removed to avoid a source of deter- minate error. Rinsing the precipitate to remove this residual material must be done carefully to avoid significant losses of the precipitate. Of greatest concern is the po- tential for solubility losses. Usually the rinsing medium is selected to ensure that solubility losses are negligible. In many cases this simply involves the use of cold solvents or rinse solutions containing organic solvents such as ethanol. Precipitates containing acidic or basic ions may experience solubility losses if the rinse solution’s pH is not appropriately adjusted. When coagulation plays an important role in de- 244 Modern Analytical Chemistry Vent to Vacuum Trap Suction flask Rubber hose Rubber adapter Crucible Figure 8.8 Procedure for filtering through a filtering crucible. The trap is used to prevent water from a water aspirator from backwashing into the suction flask. supernatant The solution that remains after a precipitate forms. 1400-CH08 9/9/99 2:17 PM Page 244 termining particle size, a volatile inert electrolyte is often added to the rinse water to prevent the precipitate from reverting into smaller particles that may not be re- tained by the filtering device. This process of reverting to smaller particles is called peptization. The volatile electrolyte is removed when drying the precipitate. When rinsing a precipitate there is a trade-off between introducing positive de- terminate errors due to ionic impurities from the precipitating solution and intro- ducing negative determinate errors from solubility losses. In general, solubility losses are minimized by using several small portions of the rinse solution instead of a single large volume. Testing the used rinse solution for the presence of impurities is another way to ensure that the precipitate is not overrinsed. This can be done by testing for the presence of a targeted solution ion and rinsing until the ion is no longer detected in a freshly collected sample of the rinse solution. For example, when Cl – is known to be a residual impurity, its presence can be tested for by adding a small amount of AgNO 3 to the collected rinse solution. A white precipitate of AgCl indicates that Cl – is present and additional rinsing is necessary. Additional rinsing is not needed, however, if adding AgNO 3 does not produce a precipitate. Drying the Precipitate Finally, after separating the precipitate from its super- natant solution the precipitate is dried to remove any residual traces of rinse solu- tion and any volatile impurities. The temperature and method of drying depend on the method of filtration, and the precipitate’s desired chemical form. A temperature of 110 °C is usually sufficient when removing water and other easily volatilized im- purities. A conventional laboratory oven is sufficient for this purpose. Higher tem- peratures require the use of a muffle furnace, or a Bunsen or Meker burner, and are necessary when the precipitate must be thermally decomposed before weighing or when using filter paper. To ensure that drying is complete the precipitate is repeat- edly dried and weighed until a constant weight is obtained. Filter paper’s ability to absorb moisture makes its removal necessary before weighing the precipitate. This is accomplished by folding the filter paper over the precipitate and transferring both the filter paper and the precipitate to a porcelain or platinum crucible. Gentle heating is used to first dry and then to char the filter paper. Once the paper begins to char, the temperature is slowly increased. Although the paper will often show traces of smoke, it is not allowed to catch fire as any pre- cipitate retained by soot particles will be lost. After the paper is completely charred the temperature is slowly raised to a higher temperature. At this stage any carbon left after charring is oxidized to CO 2 . Fritted glass crucibles cannot withstand high temperatures and, therefore, should only be dried in an oven at temperatures below 200 °C. The glass fiber mats used in Gooch crucibles can be heated to a maximum temperature of approxi- mately 500 °C. Composition of Final Precipitate The quantitative application of precipitation gravimetry, which is based on a conservation of mass, requires that the final precipi- tate have a well-defined composition. Precipitates containing volatile ions or sub- stantial amounts of hydrated water are usually dried at a temperature that is suffi- cient to completely remove the volatile species. For example, one standard gravimetric method for the determination of magnesium involves the precipitation of MgNH 4 PO 4 6H 2 O. Unfortunately, this precipitate is difficult to dry at lower temperatures without losing an inconsistent amount of hydrated water and ammo- nia. Instead, the precipitate is dried at temperatures above 1000 °C, where it decom- poses to magnesium pyrophosphate, Mg 2 P 2 O 7 . Chapter 8 Gravimetric Methods of Analysis 245 peptization The reverse of coagulation in which a coagulated precipitate reverts to smaller particles. 1400-CH08 9/9/99 2:17 PM Page 245 246 Modern Analytical Chemistry Representative Methods An additional problem is encountered when the isolated solid is non- stoichiometric. For example, precipitating Mn 2+ as Mn(OH) 2 , followed by heating to produce the oxide, frequently produces a solid with a stoichiometry of MnO x , where x varies between 1 and 2. In this case the nonstoichiometric product results from the formation of a mixture of several oxides that differ in the oxidation state of manganese. Other nonstoichiometric compounds form as a result of lattice de- fects in the crystal structure. 6 Representative Method The best way to appreciate the importance of the theoreti- cal and practical details discussed in the previous section is to carefully examine the procedure for a typical precipitation gravimetric method. Although each method has its own unique considerations, the determination of Mg 2+ in water and waste- water by precipitating MgNH 4 PO 4 6H 2 O and isolating Mg 2 P 2 O 7 provides an in- structive example of a typical procedure. Method 8.1 Determination of Mg 2+ in Water and Wastewater 7 Description of Method. Magnesium is precipitated as MgNH 4 PO 4 6H 2 O using (NH 4 ) 2 HPO 4 as the precipitant. The precipitate’s solubility in neutral solutions (0.0065 g/100 mL in pure water at 10 °C) is relatively high, but it is much less soluble in the presence of dilute ammonia (0.0003 g/100 mL in 0.6 M NH 3 ). The precipitant is not very selective, so a preliminary separation of Mg 2+ from potential interferents is necessary. Calcium, which is the most significant interferent, is usually removed by its prior precipitation as the oxalate. The presence of excess ammonium salts from the precipitant or the addition of too much ammonia can lead to the formation of Mg(NH 4 ) 4 (PO 4 ) 2 , which is subsequently isolated as Mg(PO 3 ) 2 after drying. The precipitate is isolated by filtration using a rinse solution of dilute ammonia. After filtering, the precipitate is converted to Mg 2 P 2 O 7 and weighed. Procedure. Transfer a sample containing no more than 60 mg of Mg 2+ into a 600-mL beaker. Add 2–3 drops of methyl red indicator, and, if necessary, adjust the volume to 150 mL. Acidify the solution with 6 M HCl, and add 10 mL of 30% w/v (NH 4 ) 2 HPO 4 . After cooling, add concentrated NH 3 dropwise, and while constantly stirring, until the methyl red indicator turns yellow (pH > 6.3). After stirring for 5 min, add 5 mL of concentrated NH 3 , and continue stirring for an additional 10 min. Allow the resulting solution and precipitate to stand overnight. Isolate the precipitate by filtration, rinsing with 5% v/v NH 3 . Dissolve the precipitate in 50 mL of 10% v/v HCl, and precipitate a second time following the same procedure. After filtering, carefully remove the filter paper by charring. Heat the precipitate at 500 °C until the residue is white, and then bring the precipitate to constant weight at 1100 °C. Questions 1. Why does the procedure call for a sample containing no more than 60 mg of Mg 2+ ? A sample containing 60 mg of Mg 2+ will generate approximately 600 mg, or 0.6 g, of MgNH 4 PO 4 6H 2 O. This is a substantial amount of precipitate to work with during the filtration step. Large quantities of precipitate may be difficult to filter and difficult to adequately rinse free of impurities. —Continued 1400-CH08 9/9/99 2:17 PM Page 246 Chapter 8 Gravimetric Methods of Analysis 247 2. Why is the solution acidified with HCl before the precipitant is added? The HCl is added to ensure that MgNH 4 PO 4 6H 2 O does not precipitate when the precipitant is initially added. Because PO 4 3– is a weak base, the precipitate is soluble in a strongly acidic solution. If the precipitant is added under neutral or basic conditions (high RSS), the resulting precipitate will consist of smaller, less pure particles. Increasing the pH by adding base allows the precipitate of MgNH 4 PO 4 6H 2 O to form under more favorable (low RSS) conditions. 3. Why is the acid–base indicator methyl red added to the solution? The indicator’s color change, which occurs at a pH of approximately 6.3, indicates when sufficient NH 3 has been added to neutralize the HCl added at the beginning of the procedure. The amount of NH 3 added is crucial to this procedure. If insufficient NH 3 is added, the precipitate’s solubility increases, leading to a negative determinate error. If too much NH 3 is added, the precipitate may contain traces of Mg(NH 4 ) 4 (PO 4 ) 2 , which, on ignition, forms Mg(PO 3 ) 2 . This increases the mass of the ignited precipitate, giving a positive determinate error. Once enough NH 3 has been added to neutralize the HCl, additional NH 3 is added to quantitatively precipitate MgNH 4 PO 4 6H 2 O. 4. Explain why the formation of Mg(PO 3 ) 2 in place of Mg 2 P 2 O 7 increases the mass of precipitate. The desired final precipitate, Mg 2 P 2 O 7 , contains two moles of Mg, and the impurity, Mg(PO 3 ) 2 , contains only one mole of Mg. Conservation of mass, therefore, requires that two moles of Mg(PO 3 ) 2 must form in place of each mole of Mg 2 P 2 O 7 . One mole of Mg 2 P 2 O 7 weights 222.6 g. Two moles of Mg(PO 3 ) 2 weigh 364.5 g. Any replacement of Mg 2 P 2 O 7 with Mg(PO 3 ) 2 must increase the precipitate’s mass. 5. What additional steps in the procedure, beyond those discussed in questions 2 and 3, are taken to improve the precipitate’s purity? Two additional steps in the procedure help form a precipitate that is free of impurities: digestion and reprecipitation. 6. Why is the precipitate rinsed with a solution of 5% v/v NH 3 ? This is done for the same reason that precipitation is carried out in an ammonical solution; using dilute ammonia minimizes solubility losses when rinsing the precipitate. 8B.2 Quantitative Applications Although not in common use, precipitation gravimetry still provides a reliable means for assessing the accuracy of other methods of analysis or for verifying the composition of standard reference materials. In this section we review the general application of precipitation gravimetry to the analysis of inorganic and organic compounds. Inorganic Analysis The most important precipitants for inorganic cations are chromate, the halides, hydroxide, oxalate, sulfate, sulfide, and phosphate. A sum- mary of selected methods, grouped by precipitant, is shown in Table 8.1. Many in- organic anions can be determined using the same reactions by reversing the analyte 1400-CH08 9/9/99 2:17 PM Page 247 and precipitant. For example, chromate can be determined by adding BaCl 2 and precipitating BaCrO 4 . Methods for other selected inorganic anions are summarized in Table 8.2. Methods for the homogeneous generation of precipitants are shown in Table 8.3. The majority of inorganic precipitants show poor selectivity. Most organic pre- cipitants, however, are selective for one or two inorganic ions. Several common or- ganic precipitants are listed in Table 8.4. Precipitation gravimetry continues to be listed as a standard method for the analysis of Mg 2+ and SO 4 2– in water and wastewater analysis. A description of the procedure for Mg 2+ was discussed earlier in Method 8.1. Sulfate is analyzed by pre- cipitating BaSO 4 , using BaCl 2 as the precipitant. Precipitation is carried out in an 248 Modern Analytical Chemistry Table 8.1 Selected Gravimetric Method for Inorganic Cations Based on Precipitation Analyte Precipitant Precipitate Formed Precipitate Weighed Ba 2+ (NH 4 ) 2 CrO 4 BaCrO 4 BaCrO 4 Pb 2+ K 2 CrO 4 PbCrO 4 PbCrO 4 Ag + HCl AgCl AgCl Hg 2 2+ HCl Hg 2 Cl 2 Hg 2 Cl 2 Al 3+ NH 3 Al(OH) 3 Al 2 O 3 Be 2+ NH 3 Be(OH) 2 BeO Fe 3+ NH 3 Fe(OH) 3 Fe 2 O 3 Ca 2+ (NH 4 ) 2 C 2 O 4 CaC 2 O 4 CaCO 3 or CaO Sb 3+ H 2 SSb 2 S 3 Sb 2 S 3 As 3+ H 2 SAs 2 S 3 As 2 S 3 Hg 2+ H 2 S HgS HgS Ba 2+ H 2 SO 4 BaSO 4 BaSO 4 Pb 2+ H 2 SO 4 PbSO 4 PbSO 4 Sr 2+ H 2 SO 4 SrSO 4 SrSO 4 Be 2+ (NH 4 ) 2 HPO 4 NH 4 BePO 4 Be 2 P 2 O 7 Mg 2+ (NH 4 ) 2 HPO 4 NH 4 MgPO 4 Mg 2 P 2 O 7 Sr 2+ KH 2 PO 4 SrHPO 4 Sr 2 P 2 O 7 Zn 2+ (NH 4 ) 2 HPO 4 NH 4 ZnPO 4 Zn 2 P 2 O 7 Table 8.2 Selected Gravimetric Methods for Inorganic Anions Based on Precipitation Analyte Precipitant Precipitate Formed Precipitate Weighed CN – AgNO 3 AgCN AgCN I – AgNO 3 AgI AgI Br – AgNO 3 AgBr AgBr Cl – AgNO 3 AgCl AgCl ClO 3 – FeSO 4 /AgNO 3 AgCl AgCl SCN – SO 2 /CuSO 4 CuSCN CuSCN SO 4 2– BaCl 2 BaSO 4 BaSO 4 1400-CH08 9/9/99 2:18 PM Page 248 Chapter 8 Gravimetric Methods of Analysis 249 Table 8. 3 Reactions for the Homogeneous Preparation of Selected Inorganic Precipitants Precipitant Reaction OH – (NH 2 ) 2 CO+3H 2 O t 2NH 4 + +CO 2 + 2OH – SO 4 2– NH 2 HSO 3 +2H 2 O t NH 4 + +H 3 O + +SO 4 2– S 2– CH 3 CSNH 2 +H 2 O t CH 3 CONH 2 +H 2 S IO 3 – HOCH 2 CH 2 OH+IO 4 – t 2HCHO + H 2 O+IO 3 – PO 4 2– (CH 3 O) 3 PO+3H 2 O t 3CH 3 OH+H 3 PO 4 C 2 O 4 2– (C 2 H 5 ) 2 C 2 O 4 +2H 2 O t 2C 2 H 5 OH+H 2 C 2 O 4 CO 3 2– Cl 3 CCOOH + 2OH – t CHCl 3 +CO 3 2– +H 2 O Table 8. 4 Selected Gravimetric Methods for Inorganic Cations Based on Precipitation with Organic Precipitants Analyte Precipitant Structure Precipitate Formed Precipitate Weighed Ni 2+ dimethylgloxime Ni(C 4 H 7 O 2 N 2 ) 2 Ni(C 4 H 7 O 2 N 2 ) 2 Fe 3+ cupferron Fe(C 6 H 5 N 2 O 2 ) 3 Fe 2 O 3 Cu 2+ cupron CuC 14 H 11 O 2 N CuC 14 H 11 O 2 N Co 2+ 1-nitroso-2-naphthol Co(C 10 H 6 O 2 N) 3 Co or CoSO 4 K + sodium tetraphenylborate K[B(C 6 H 5 ) 4 ] K[B(C 6 H 5 ) 4 ] NO 3 – nitron C 20 H 16 N 4 HNO 3 C 20 H 16 N 4 HNO 3 NN OH OH O – NH 4 + NO N H 5 C 6 C 6 H 5 N OH OH OH NO Na[B(C 6 H 5 ) 4 ] NC 6 H 5 N + N C 6 H 5 C 6 H 5 – N 1400-CH08 9/9/99 2:18 PM Page 249 acidic solution (acidified to pH 4.5–5.0 with HCl) to prevent the possible precipita- tion of BaCO 3 or Ba 3 (PO 4 ) 2 and performed near the solution’s boiling point. The precipitate is digested at 80–90 °C for at least 2 h. Ashless filter paper pulp is added to the precipitate to aid in filtration. After filtering, the precipitate is ignited to con- stant weight at 800 °C. Alternatively, the precipitate can be filtered through a fine- porosity fritted glass crucible (without adding filter paper pulp) and dried to con- stant weight at 105 °C. This procedure is subject to a variety of errors, including occlusions of Ba(NO 3 ) 2 , BaCl 2 , and alkali sulfates. Organic Analysis Several organic functional groups or heteroatoms can be deter- mined using gravimetric precipitation methods; examples are outlined in Table 8.5. Note that the procedures for the alkoxy and alkimide functional groups are exam- ples of indirect analyses. Quantitative Calculations In precipitation gravimetry the relationship between the analyte and the precipitate is determined by the stoichiometry of the relevant reactions. As discussed in Section 2C, gravimetric calculations can be simplified by applying the principle of conservation of mass. The following example demonstrates the application of this approach to the direct analysis of a single analyte. 250 Modern Analytical Chemistry Table 8. 5 Selected Gravimetric Methods for the Analysis of Organic Functional Groups and Heteroatoms Based on Precipitation Analyte Treatment Precipitant Precipitate Organic halides Oxidation with HNO 3 in presence of Ag + AgNO 3 AgX R-X X = Cl, Br, I Organic Halides Combustion in O 2 (with Pt catalyst) in presence of Ag + AgNO 3 AgX R-X X = Cl, Br, I Organic sulfur Oxidation with HNO 3 in presence of Ba 2+ BaCl 2 BaSO 4 Organic sulfur Combustion in O 2 (with Pt catalyst) to produce SO 2 and SO 3 , which are collected in dilute H 2 O 2 BaCl 2 BaSO 4 Alkoxy groups Reaction with HI to produce RI AgNO 3 AgI R’-OR R=CH 3 or C 2 H 5 or Alkimide group Reaction with HI to produce RI AgNO 3 AgI R=CH 3 , C 2 H 5 N may be 1°, 2°, or 3° NR O CORR' 1400-CH08 9/9/99 2:18 PM Page 250 Chapter 8 Gravimetric Methods of Analysis 251 EXAMPLE 8.1 An ore containing magnetite, Fe 3 O 4 , was analyzed by dissolving a 1.5419-g sample in concentrated HCl, giving a mixture of Fe 2+ and Fe 3+ . After adding HNO 3 to oxidize any Fe 2+ to Fe 3+ , the resulting solution was diluted with water and the Fe 3+ precipitated as Fe(OH) 3 by adding NH 3 . After filtering and rinsing, the residue was ignited, giving 0.8525 g of pure Fe 2 O 3 . Calculate the %w/w Fe 3 O 4 in the sample. SOLUTION This is an example of a direct analysis since the iron in the analyte, Fe 3 O 4 , is part of the isolated precipitate, Fe 2 O 3 . Applying a conservation of mass to Fe, we write 3 · moles Fe 3 O 4 =2· moles Fe 2 O 3 Using formula weights, FW, to convert from moles to grams in the preceding equation leaves us with which can be solved for grams of Fe 3 O 4 and %w/w Fe 3 O 4 in the sample. As discussed earlier, the simultaneous analysis of samples containing two analytes requires the isolation of two precipitates. As shown in Example 8.2, conservation of mass can be used to write separate stoichiometric equations for each precipitate. These equations can then be solved simultaneously for both analytes. EXAMPLE 8.2 A 0.6113-g sample of Dow metal, containing aluminum, magnesium, and other metals, was dissolved and treated to prevent interferences by the other metals. The aluminum and magnesium were precipitated with 8-hydroxyquinoline. After filtering and drying, the mixture of Al(C 9 H 6 NO) 3 and Mg(C 9 H 6 NO) 2 was found to weigh 7.8154 g. The mixture of dried precipitates was then ignited, converting the precipitate to a mixture of Al 2 O 3 and MgO. The weight of this mixed solid was found to be 1.0022 g. Calculate the %w/w Al and %w/w Mg in the alloy. SOLUTION This is an example of a direct analysis in which the two analytes are determined without a prior separation. The weight of the original precipitate and the ignited precipitate are given by the following two equations g Al(C 9 H 6 NO) 3 + g Mg(C 9 H 6 NO) 2 = 7.8154 g Al 2 O 3 + g MgO = 1.0022 g Fe O g sample g 1.5419 g w/w Fe 34 ×= ×=100 0 82405 100 53 44 . .% 2 3 2 0 8525 0 82405 ×× × = ×× × = g Fe O FW Fe O FW Fe O g 231.54 g/mol 3 159.69 g/mol g Fe O 23 34 23 34 . . 32× = ×g Fe O FW Fe O g Fe O FW Fe O 34 34 23 23 1400-CH08 9/9/99 2:18 PM Page 251 containing four unknown terms. To solve this pair of equations, we must find two additional equations relating the four unknowns to one another. These additional equations describe the stoichiometric relationships between the two compounds containing aluminum and the two compounds containing magnesium and are based on the conservation of Al and Mg. Thus, for Al we have 2 · moles Al 2 O 3 = moles Al(C 9 H 6 NO) 3 Converting from moles to grams and solving yields an equation relating the grams of Al 2 O 3 to the grams of Al(C 9 H 6 NO) 3 For Mg we have Moles MgO = moles Mg(C 9 H 6 NO) 2 Substituting the equations for g MgO and g Al 2 O 3 into the equation for the combined weights of MgO and Al 2 O 3 leaves us with two equations and two unknowns. g Al(C 9 H 6 NO) 3 + g Mg(C 9 H 6 NO) 2 = 7.8154 0.11096 · g Al(C 9 H 6 NO) 3 + 0.12893 · g Mg(C 9 H 6 NO) 2 = 1.0022 Multiplying the first equation by 0.11096 and subtracting the second equation gives –0.01797 · g Mg(C 9 H 6 NO) 2 = –0.1350 which can be solved for the mass of Mg(C 9 H 6 NO) 2 . g Mg(C 9 H 6 NO) 2 = 7.5125 g The mass of Al(C 9 H 6 NO) 3 can then be calculated using the known combined mass of the two original precipitates. 7.8154 g – g Mg(C 9 H 6 NO) 2 = 7.8154 g – 7.5125 g = 0.3029 g Al(C 9 H 6 NO) 3 Using the conservation of Mg and Al, the %w/w Mg and %w/w Al in the sample can now be determined as in Example 8.1, where AW is an atomic weight. ) ./ / . g MgO FW MgO g Mg(C H NO) FW Mg(C H NO g MgO g Mg(C H NO) FW MgO FW Mg(C H NO) g Mg(C H NO) g mol 312.61 g mol g Mg(C H NO) 96 2 96 96 2 96 2 96 2 96 2 = = × = × =× 2 40 304 0 12893 . . . 2 2 101 96 2 459 45 0 11096 × = = × × = × × =× g Al O FW Al O g Al(C H NO) FW Al(C H NO) g Al O g Al(C H NO) FW Al O FW Al(C H NO) g Al(C H NO) g/mol g/mol g Al(C H NO) 23 23 96 3 96 3 23 96 3 23 96 3 96 3 96 3 252 Modern Analytical Chemistry 1400-CH08 9/9/99 2:18 PM Page 252 Chapter 8 Gravimetric Methods of Analysis 253 g Mg AW Mg g Mg(C H NO) FW Mg(C H NO) g Mg g Mg(C H NO) AW Mg FW Mg(C H NO) g 24.305 g/mol 312.61 g/mol g %Mg = g Mg g sample g 0.6113 g w/w Mg 96 2 96 2 96 2 96 2 = = × = × = × ×= 7 5125 0 5841 100 0 5841 100 95 55 . . . .% g Al AW Al g Al(C H NO) FW Al(C H NO) g Al g Al(C H NO) AW Al FW Al(C H NO) .3029 g 26.982 g /mol 459.45 g /mol g %Al g Al g sample g 0.6113 g w/w Al 96 3 96 3 96 3 96 3 = = × = × = =× ×= 0 0 0178 100 0 0178 100 2 91 . . .% In an indirect analysis the precipitate does not contain the analyte, but is the product of a reaction involving the analyte. Despite the additional complexity, a stoichiometric relationship between the analyte and the precipitate can be written by applying the conservation principles discussed in Section 2C. EXAMPLE 8. 3 An impure sample of Na 3 PO 3 weighing 0.1392 g was dissolved in 25 mL of water. A solution containing 50 mL of 3% w/v mercury(II) chloride, 20 mL of 10% w/v sodium acetate and 5 mL of glacial acetic acid was then prepared. The solution containing the phosphite was added dropwise to the second solution, oxidizing PO 3 3– to PO 4 3– and precipitating Hg 2 Cl 2 . After digesting, filtering, and rinsing, the precipitated Hg 2 Cl 2 was found to weigh 0.4320 g. Report the purity of the original sample as %w/w Na 3 PO 3 . SOLUTION This is an example of an indirect analysis since the isolated precipitate, Hg 2 Cl 2 , does not contain the analyte, Na 3 PO 3 . The stoichiometry of the redox reaction in which Na 3 PO 3 is a reactant and Hg 2 Cl 2 is a product was given earlier in the chapter. It also can be determined by using the conservation of electrons. Phosphorus has an oxidation state of +3 in PO 3 3– and +5 in PO 4 3– ; thus, oxidizing PO 3 3– to PO 4 3– requires two electrons. The formation of Hg 2 Cl 2 by reduction of HgCl 2 requires 2 electrons as the oxidation state of each mercury changes from +2 to +1. Since the oxidation of PO 3 3– and the formation of Hg 2 Cl 2 both require two electrons, we have Moles Na 3 PO 3 = moles Hg 2 Cl 2 Converting from moles to grams leaves us with g Na PO FW Na PO g Hg Cl FW Hg Cl 33 33 22 22 = Moles Mg = moles Mg(C 9 H 6 NO) 2 Moles Al = moles Al(C 9 H 6 NO) 3 1400-CH08 9/9/99 2:18 PM Page 253 which can be solved for g Na 3 PO 3 and %w/w Na 3 PO 3 in the sample. 8B. 3 Qualitative Applications Precipitation gravimetry can also be applied to the identification of inorganic and organic analytes, using precipitants such as those outlined in Tables 8.1, 8.2, 8.4, and 8.5. Since this does not require quantitative measurements, the analytical signal is simply the observation that a precipitate has formed. Although qualitative appli- cations of precipitation gravimetry have been largely replaced by spectroscopic methods of analysis, they continue to find application in spot testing for the pres- ence of specific analytes. 8 8B. 4 Evaluating Precipitation Gravimetry Scale of Operation The scale of operation for precipitation gravimetry is governed by the sensitivity of the balance and the availability of sample. To achieve an accu- racy of ±0.1% using an analytical balance with a sensitivity of ±0.1 mg, the precipi- tate must weigh at least 100 mg. As a consequence, precipitation gravimetry is usu- ally limited to major or minor analytes, and macro or meso samples (see Figure 3.6 in Chapter 3). The analysis of trace level analytes or micro samples usually requires a microanalytical balance. Accuracy For macro–major samples, relative errors of 0.1–0.2% are routinely achieved. The principal limitations are solubility losses, impurities in the precipitate, and the loss of precipitate during handling. When it is difficult to obtain a precipitate free from impurities, an empirical relationship between the precipitate’s mass and the mass of the analyte can be determined by an appropriate standardization. Precision The relative precision of precipitation gravimetry depends on the amount of sample and precipitate involved. For smaller amounts of sample or precipitate, relative precisions of 1–2 ppt are routinely obtained. When working with larger amounts of sample or precipitate, the relative precision can be ex- tended to several parts per million. Few quantitative techniques can achieve this level of precision. Sensitivity For any precipitation gravimetric method, we can write the following general equation relating the signal (grams of precipitate) to the absolute amount of analyte in the sample Grams precipitate = k · grams of analyte 8.13 where k, the method’s sensitivity, is determined by the stoichiometry between the precipitate and the analyte. Note that equation 8.13 assumes that a blank has been used to correct the signal for the reagent’s contribution to the precipitate’s mass. g Na PO g sample g g % w/w Na PO 33 33 ×= ×=100 0 1354 0 1392 100 97 27 . . . g Na PO g Hg Cl FW Na PO FW Hg Cl g 147.94 g /mol 472.09 g /mol g 33 22 3 3 22 = × = × = 0 4320 0 1354 . . 254 Modern Analytical Chemistry 1400-CH08 9/9/99 2:18 PM Page 254 Chapter 8 Gravimetric Methods of Analysis 255 Consider, for example, the determination of Fe as Fe 2 O 3 . Using a conservation of mass for Fe we write 2 · moles Fe 2 O 3 = moles Fe Converting moles to grams and rearranging yields an equation in the form of 8.13 where k is equal to 8.14 As can be seen from equation 8.14, we may improve a method’s sensitivity in two ways. The most obvious way is to increase the ratio of the precipitate’s molar mass to that of the analyte. In other words, it is desirable to form a precipitate with as large a formula weight as possible. A less obvious way to improve the calibration sensitivity is indicated by the term of 1/2 in equation 8.14, which accounts for the stoichiometry between the analyte and precipitate. Sensitivity also may be improved by forming precipitates containing fewer units of the analyte. Selectivity Due to the chemical nature of the precipitation process, precipitants are usually not selective for a single analyte. For example, silver is not a selective precipitant for chloride because it also forms precipitates with bromide and iodide. Consequently, interferents are often a serious problem that must be considered if accurate results are to be obtained. Time, Cost, and Equipment Precipitation gravimetric procedures are time-intensive and rarely practical when analyzing a large number of samples. However, since much of the time invested in precipitation gravimetry does not require an analyst’s immedi- ate supervision, it may be a practical alternative when working with only a few sam- ples. Equipment needs are few (beakers, filtering devices, ovens or burners, and bal- ances), inexpensive, routinely available in most laboratories, and easy to maintain. 8C Volatilization Gravimetry A second approach to gravimetry is to thermally or chemically decompose a solid sample. The volatile products of the decomposition reaction may be trapped and weighed to provide quantitative information. Alternatively, the residue remaining when decomposition is complete may be weighed. In thermogravimetry, which is one form of volatilization gravimetry, the sample’s mass is continuously monitored while the applied temperature is slowly increased. 8C.1 Theory and Practice Whether the analysis is direct or indirect, volatilization gravimetry requires that the products of the decomposition reaction be known. This requirement is rarely a problem for organic compounds for which volatilization is usually accomplished by combustion and the products are gases such as CO 2 , H 2 O, and N 2 . For inorganic compounds, however, the identity of the volatilization products may depend on the temperature at which the decomposition is conducted. k =× 1 2 FW Fe O AW Fe 23 g Fe O FW Fe O AW Fe g Fe 23 23 =× × 1 2 thermogravimetry A form of volatilization gravimetry in which the change in a sample’s mass is monitored while it is heated. 1400-CH08 9/9/99 2:18 PM Page 255 Figure 8.9 Thermogram for CaC 2 O 4 H 2 O. 256 Modern Analytical Chemistry Mass (mg) Temperature (°C) 0 0.00 5.00 10.00 15.00 20.00 25.00 100 200 300 400 500 600 700 800 900 1000 thermogram A graph showing change in mass as a function of applied temperature. Thermogravimetry The products of a thermal decomposition can be deduced by monitoring the sample’s mass as a function of applied temperature. (Figure 8.9). The loss of a volatile gas on thermal decomposition is indicated by a step in the thermogram. As shown in Example 8.4, the change in mass at each step in a ther- mogram can be used to identify both the volatilized species and the solid residue. EXAMPLE 8. 4 The thermogram in Figure 8.9 shows the change in mass for a sample of calcium oxalate monohydrate, CaC 2 O 4 H 2 O. The original sample weighed 24.60 mg and was heated from room temperature to 1000 °C at a rate of 5 °C min. The following changes in mass and corresponding temperature ranges were observed: Loss of 3.03 mg from 100–250 °C Loss of 4.72 mg from 400–500 °C Loss of 7.41 mg from 700–850 °C Determine the identities of the volatilization products and the solid residue at each step of the thermal decomposition. SOLUTION The loss of 3.03 mg from 100–250 °C corresponds to a 12.32% decrease in the original sample’s mass. In terms of CaC 2 O 4 H 2 O, this corresponds to a loss of 18.00 g/mol. 0.1232 · 146.11 g/mol = 18.00 g/mol 303 24 60 100 12 32 . . .% mg mg ×= 1400-CH08 9/9/99 2:18 PM Page 256 The product’s molar mass, coupled with the temperature range, suggests that this represents the loss of H 2 O. The residue is CaC 2 O 4. The loss of 4.72 mg from 400–500 °C represents a 19.19% decrease in the original mass of 24.60 g, or a loss of 0.1919 · 146.11 g/mol = 28.04 g/mol This loss is consistent with CO as the volatile product, leaving a residue of CaCO 3 . Finally, the loss of 7.41 mg from 700–850 °C is a 30.12% decrease in the original mass of 24.60 g. This is equivalent to a loss of 0.3012 · 146.11 g/mol = 44.01 g/mol suggesting the loss of CO 2 . The final residue is CaO. Once the products of thermal decomposition have been determined, an analyt- ical procedure can be developed. For example, the thermogram in Figure 8.9 shows that a precipitate of CaC 2 O 4 H 2 O must be heated at temperatures above 250 °C, but below 400 °C if it is to be isolated as CaC 2 O 4 . Alternatively, by heating the sam- ple to 1000 °C, the precipitate can be isolated as CaO. Knowing the identity of the volatilization products also makes it possible to design an analytical method in which one or more of the gases are trapped. Thus, a sample of CaC 2 O 4 H 2 O could be analyzed by heating to 1000 °C and passing the volatilized gases through a trap that selectively retains H 2 O, CO, or CO 2 . Equipment Depending on the method, the equipment for volatilization gravime- try may be simple or complex. In the simplest experimental design, the weight of a solid residue is determined following either thermal decomposition at a fixed tem- perature or combustion. Thermal decomposition or combustion is accomplished using a Bunsen or Meker burner, a laboratory oven or a muffle furnace, with the volatile products vented to the atmosphere. The weight of the sample and the solid residue are determined using an analytical balance. Constant-temperature decomposition or combustion, followed by trapping and weighing the volatilized gases, requires more specialized equipment. Decom- position of the sample is conducted in a closed container, and the volatilized gases are carried by a purge-gas stream through one or more selective absorbent traps. In a thermogravimetric analysis, the sample is placed in a small weighing boat attached to one arm of a specially designed electromagnetic balance and placed inside an electric furnace. The temperature of the electric furnace is slowly increased at a fixed rate of a few degrees per minute, and the sample’s weight is monitored. Representative Method Although each volatilization gravimetric procedure has its own unique characteristics, the following indirect method for the determination of Si in ores and alloys by formation of volatile SiF 4 provides an instructive example of a typical procedure. Chapter 8 Gravimetric Methods of Analysis 257 1400-CH08 9/9/99 2:18 PM Page 257 258 Modern Analytical Chemistry Representative Methods Method 8.2 Determination of Si in Ores and Alloys 9 Description of Method. Silicon is determined by dissolving the sample in acid. Dehydration of the resulting solution precipitates silicon as SiO 2 . Because a variety of other insoluble oxides also form, the precipitate’s mass does not provide a direct measure of the amount of silicon in the sample. Treating the solid residue with HF results in the formation of volatile SiF 4 . The decrease in mass following the loss of SiF 4 provides an indirect measure of the amount of silicon in the original sample. Procedure. Transfer a sample of between 0.5 and 5 g to a platinum crucible along with an excess of Na 2 CO 3 , and heat until a melt is formed. After cooling, dissolve the residue in dilute HCl. Dehydrating silicon to SiO 2 is accomplished by evaporating the solution to dryness on a steam bath and heating the residue for 1 hour at 110 °C. Moisten the residue with HCl, and repeat the dehydration. Remove any acid-soluble materials from the residue by adding 50 mL of water and 5 mL of concentrated HCl. Bring to a boil, and filter through #40 filter paper. Wash the residue with hot 2% v/v HCl followed by hot water. Evaporate the filtrate to dryness twice, and, following the same procedure, treat to remove any acid–soluble materials. Combine the two precipitates, and dry and ignite to a constant weight at 1200 °C. After cooling, add 2 drops of 50% v/v H 2 SO 4 and 10 mL of HF. Remove the volatile SiF 4 by evaporating to dryness on a hot plate. Finally, bring the residue to constant weight by igniting at 1200 °C. Questions 1. According to the procedure, the sample should weigh between 0.5 and 5 g. On what basis should a decision on the amount of sample be made? In this procedure the critical measurement is the decrease in mass following the volatilization of SiF 4 . The reaction that occurs is SiO 2 (s) + 4HF(aq) fi SiF 4 (g)+2H 2 O(l) The water and any excess HF are removed during the final ignition and do not contribute to the change in mass. The loss in weight, therefore, is equal to the grams of SiO 2 present after the dehydration step. For every 0.1 g of Si in the original sample, a weight loss of 0.21 g is expected. The amount of sample used is determined by how much Si is present. If a sample is 50% w/w Si, a 0.5-g sample will give a respectable weight loss of 0.53 g. A 0.5-g sample that is only 5% w/w Si, however, will give a weight loss of only 0.053 g. In this case, a larger sample is needed. 2. Why are acid-soluble materials removed before the dehydrated residue is treated with HF? Any acid-soluble materials present in the sample will react with HF or H 2 SO 4 . If the products of these reactions are volatile or decompose at the ignition temperature of 1200 °C, then the change in weight will not be due solely to the volatilization of SiF 4 . The result is a positive determinate error. 3. Why is H 2 SO 4 added with the HF? Many samples containing silicon also contain aluminum and iron. After dehydration, these metals are present as Al 2 O 3 and Fe 2 O 3 . These oxides are potential interferents since they also are capable of forming volatile fluorides. In the presence of H 2 SO 4 , however, aluminum and iron form nonvolatile sulfates. These sulfates decompose back to their respective oxides when ignited to 1200 °C. As a result, the change in weight after treating with HF and H 2 SO 4 is due only to the loss of SiF 4 . 1400-CH08 9/9/99 2:18 PM Page 258 8C.2 Quantitative Applications Unlike precipitation gravimetry, which is rarely used as a standard method of analy- sis, gravimetric methods based on volatilization reactions continue to play an im- portant role in chemical analysis. Several important examples are discussed in the following sections. Inorganic Analysis Determining the inorganic ash content of organic materials, such as polymers and paper, is an example of a direct volatilization gravimetric analysis. The sample is weighed, placed in an appropriate crucible, and the organic material is carefully removed by combustion. The crucible containing the residue is then heated to a constant weight using either a burner or an oven. Another example of volatilization gravimetry is the determination of dissolved solids in water and wastewater. In this method a sample of the water is transferred to a weighed dish and dried to a constant weight at either 103–105 °C, or at 180 °C. Samples dried at the lower temperature retain some occluded water and lose some carbonate as CO 2 . The loss of organic material, however, is minimal. At the higher temperature, the residue is free from occluded water, but losses of carbonate are greater. In addition, some chloride, nitrate, and organic material are lost through thermal decomposition. The residue remaining after drying at either temperature can be ignited to constant weight at 500 °C. The loss in weight on ignition provides an indirect measure of the amount of volatile solids in the sample, and the weight of the remaining residue gives the amount of fixed solids. Indirect analyses based on the weight of the residue remaining after volatiliza- tion are commonly used in determining moisture in a variety of products and in de- termining silica in water, wastewater, and rocks. Moisture is determined by drying a preweighed sample with an infrared lamp or in a low-temperature oven. The differ- ence between the original weight and the weight after drying equals the mass of water lost. The determination of silicon is commonly encountered in metallurgical and mining laboratories responsible for the analysis of ores, slags, and alloys. The volatilization gravimetric method, which is appropriate for samples containing high concentrations of silicon, was described earlier in Method 8.2. As a final example, the determination of carbon in steels and other metal alloys can be determined by heating the sample. The carbon is converted to CO 2 , which is collected in an appropriate absorbent trap, providing a direct measure of the amount of C in the original sample. Organic Analysis The most important application of volatilization gravimetry to the analysis of organic materials is an elemental analysis. When burned in a stream of pure O 2 , many elements, such as carbon and hydrogen, are released as gaseous combustion products, such as CO 2 and H 2 O. The combustion products are passed through preweighed tubes containing appropriate absorbents. The increase in the mass of these tubes provides a direct indication of the mass percent of carbon and hydrogen in the organic material. Alkaline metals and earths in organic materials can be determined by adding H 2 SO 4 to the sample before combustion. Following combustion, the metal remains behind as a solid residue of metal sulfate. Silver, gold, and platinum can be deter- mined by burning the organic sample, leaving a metallic residue of Ag, Au, or Pt. Other metals are determined by adding HNO 3 before combustion, leaving a residue of the metal oxide. Chapter 8 Gravimetric Methods of Analysis 259 1400-CH08 9/9/99 2:18 PM Page 259 260 Modern Analytical Chemistry Volatilization gravimetry is also used to determine biomass in water and waste- water. Biomass is a water quality index, providing an indication of the total mass of organisms contained within a sample of water. A known volume of the sample is passed through a preweighed 0.45-m m membrane filter or a glass-fiber filter and dried at 105 °C for 24 h. The residue’s mass provides a direct measure of biomass. If samples are known to contain a substantial amount of dissolved inorganic solids, the residue can be ignited at 500 °C for 1 h, thereby removing all organic materials. The resulting residue is wetted with distilled water to rehydrate any clay minerals and dried to a constant weight at 105 °C. The difference in weight before and after ignition provides an indirect measure of biomass. Quantitative Calculations When needed, the relationship between the analyte and the analytical signal is given by the stoichiometry of any relevant reactions. Calcula- tions are simplified, however, by applying the principle of conservation of mass. The most frequently encountered example of a direct volatilization gravimetric analysis is the determination of a compound’s elemental composition. EXAMPLE 8. 5 A 101.3-mg sample of an organic compound known to contain Cl is burned in pure O 2 and the combustion gases collected in absorbent tubes. The tube used to trap CO 2 increases in mass by 167.6 mg, and the tube for trapping H 2 O shows a 13.7-mg increase. A second sample of 121.8 mg is treated with concentrated HNO 3 producing Cl 2 , which subsequently reacts with Ag + , forming 262.7 mg of AgCl. Determine the compound’s composition, as well as its empirical formula. SOLUTION Applying the principle of conservation of mass to carbon, we write Moles C = moles CO 2 Converting from moles to grams, and rearranging to solve for milligrams of carbon gives Thus, the %w/w C in the sample is The calculation is then repeated for hydrogen Moles H = 2 · moles H 2 O mg C mg sample mg 101.3 mg w/w C×= ×=100 45 74 100 45 15 . .% g CO 2 AW C 1000 mg/g FW CO 2 g 12.011 g/mol 1000 mg/g 44.011 g/mol mg C ×× = ×× = 0 1676 45 74 . . 2 g H O AW H 1000 mg/g FW H O 2 g 1.008 g/mol 1000 mg/g 18.015 g/mol mg H mg 101.3 mg 2 2 ××× = ×× × = ×= ×= 0 0137 1 533 100 1 533 100 1 51 . . . .% mg H mg sample w/w H 1400-CH08 9/9/99 2:18 PM Page 260 Chapter 8 Gravimetric Methods of Analysis 261 and for chlorine Moles Cl = moles AgCl Adding together the weight percents for C, H, and Cl gives a total of 100.01%. The compound, therefore, contains only these three elements. To determine the compound’s empirical formula, we assume that we have 1 g of the compound, giving 0.4515 g of C, 0.0151 g of H, and 0.5335 g of Cl. Expressing each element in moles gives 0.0376 mol C, 0.0150 mol H, and 0.0150 mol Cl. Hydrogen and chlorine are present in a 1:1 molar ratio. The molar ratio of C to moles of H or Cl is Thus, the simplest, or empirical, formula for the compound is C 5 H 2 Cl 2 . In an indirect volatilization gravimetric analysis, the change in the sample’s weight is proportional to the amount of analyte. Note that in the following example it is not necessary to apply the conservation of mass to relate the analytical signal to the analyte. EXAMPLE 8.6 A sample of slag from a blast furnace is analyzed for SiO 2 by decomposing a 0.5003-g sample with HCl, leaving a residue with a mass of 0.1414 g. After treating with HF and H 2 SO 4 and evaporating the volatile SiF 4 , a residue with a mass of 0.0183 g remains. Determine the %w/w SiO 2 in the sample. SOLUTION In this procedure the difference in the residue’s mass before and after volatilizing SiF 4 gives the mass of SiO 2 in the sample. Thus the sample contained 0.1414 g – 0.0183 g = 0.1231 g SiO 2 The %w/w SiO 2 , therefore, is Finally, in some quantitative applications it is necessary to compare the result for a sample with a similar result obtained using a standard. g SiO g sample g g w/w SiO 2 2 ×= ×=100 0 1231 0 5003 100 24 61 . . .% Moles C Moles H moles C moles Cl ===≈ 0 0376 0 0150 251 25 . . .. mg Cl mg sample mg 121.8 mg w/w Cl×= ×=100 64 98 100 53 35 . .% g AgCl AW Cl 1000 mg/g FW AgCl g g/mol 1000 mg/g 143.32 g/mol mg Cl ×× = ×× = 0 2627 35 453 64 98 .. . 1400-CH08 9/9/99 2:18 PM Page 261 262 Modern Analytical Chemistry EXAMPLE 8. 7 A 26.23-mg sample of MgC 2 O 4 ?H 2 O and inert materials is heated to constant weight at 1200 °C, leaving a residue weighing 20.98 mg. A sample of pure MgC 2 O 4 ?H 2 O, when treated in the same fashion, undergoes a 69.08% change in its mass. Determine the %w/w MgC 2 O 4 ?H 2 O in the sample. SOLUTION The change in mass when analyzing the mixture is 5.25 mg, thus the grams of MgC 2 O 4 ?H 2 O in the sample is The %w/w MgC 2 O 4 ?H 2 O, therefore, is 8C. 3 Evaluating Volatilization Gravimetry The scale of operation, accuracy, and precision of gravimetric volatilization methods are similar to that described in Section 8B.4 for precipitation gravime- try. The sensitivity for a direct analysis is fixed by the analyte’s chemical form following combustion or volatilization. For an indirect analysis, however, sensi- tivity can be improved by carefully choosing the conditions for combustion or volatilization so that the change in mass is as large as possible. For example, the thermogram in Figure 8.9 shows that an indirect analysis for CaC 2 O 4 H 2 O based on the weight of the residue following ignition at 1000 °C will be more sensitive than if the ignition was done 300 °C. Selectivity does not present a problem for direct volatilization gravimetric methods in which the analyte is a gaseous product retained in an absorbent trap. A direct analysis based on the residue’s weight following combustion or volatilization is possible when the residue only contains the analyte of interest. As noted earlier, indirect analyses are only feasible when the residue’s change in mass results from the loss of a sin- gle volatile product containing the analyte. Volatilization gravimetric methods are time- and labor-intensive. Equipment needs are few except when combustion gases must be trapped or for a thermogravi- metric analysis, which requires specialized equipment. 8D Particulate Gravimetry Gravimetric methods based on precipitation or volatilization reactions require that the analyte, or some other species in the sample, participate in a chemical reaction producing a change in physical state. For example, in direct precipitation gravime- try, a soluble analyte is converted to an insoluble form that precipitates from solu- tion. In some situations, however, the analyte is already present in a form that may be readily separated from its liquid, gas, or solid matrix. When such a separation is possible, the analyte’s mass can be directly determined with an appropriate balance. In this section the application of particulate gravimetry is briefly considered. mg MgC O H O mg sample mg 26.23 mg w/w MgC O H O 24 2 24 2 ? ? ×= ×=100 760 100 29 0 . .% 525 760.. mg lost 100 mg MgC O H O 69.08 mg lost mg MgC O H O 24 2 24 2 ×= ? ? 1400-CH08 9/9/99 2:18 PM Page 262 Chapter 8 Gravimetric Methods of Analysis 263 Figure 8.10 Four possible mechanisms for solid-state extraction: (a) adsorption onto a solid substrate; (b) absorption into a thin polymer or chemical film coated on a solid substrate; (c) metal–ligand complexation in which the ligand is covalently bound to the solid substrate; and (d) antibody–antigen binding in which the receptor is covalently bound to the solid substrate. 8D.1 Theory and Practice Two approaches have been used to separate the analyte from its matrix in particu- late gravimetry. The most common approach is filtration, in which solid particu- lates are separated from their gas, liquid, or solid matrix. A second approach uses a liquid-phase or solid-phase extraction. Filtration Liquid samples are filtered by pulling the liquid through an appropriate filtering medium, either by gravity or by applying suction from a vacuum pump or aspirator. The choice of filtering medium is dictated primarily by the size of the solid particles and the sample’s matrix. Filters are constructed from a variety of ma- terials, including cellulose fibers, glass fibers, cellulose nitrate, and polytetrafluo- roethylene (PTFE). Particle retention depends on the size of the filter’s pores. Cellu- lose fiber filters, commonly referred to as filter paper, range in pore size from 30 m m to 2–3 m m. Glass fiber filters, constructed from chemically inert borosilicate glass, range in pore size from 2.5 m m to 0.3 m m. Membrane filters, which are made from a variety of materials, including cellulose nitrate and PTFE, are available with pore sizes from 5.0 m m to 0.1 m m. Solid aerosol particulates in gas samples are filtered using either a single or multiple stage. In a single-stage system the gas is passed through a single filter, re- taining particles larger than the filter’s pore size. When sampling a gas line, the filter is placed directly in line. Atmospheric gases are sampled with a high-volume sam- pler that uses a vacuum pump to pull air through the filter at a rate of approxi- mately 75 m 3 /h. In either case, the filtering medium used for liquid samples also can be used for gas samples. In a multiple-stage system, a series of filtering units is used to separate the particles by size. Solid samples are separated by particle size using one or more sieves. By select- ing several sieves of different mesh size, particulates with a narrow size range can be isolated from the solid matrix. Sieves are available in a variety of mesh sizes, ranging from approximately 25 mm to 40 m m. Extraction Filtering limits particulate gravimetry to solid particulate analytes that are easily separated from their matrix. Particulate gravimetry can be extended to the analysis of gas-phase analytes, solutes, and poorly filterable solids if the analyte can be extracted from its matrix with a suitable solvent. After extraction, the solvent can be evaporated and the mass of the extracted analyte determined. Alternatively, the analyte can be determined indirectly by measuring the change in a sample’s mass after extracting the analyte. Solid-phase extractions, such as those described in Chapter 7, also may be used. More recently, methods for particulate gravimetry have been developed in which the analyte is separated by adsorption onto a metal surface, by absorption into a thin polymer or chemical film coated on a solid support, or by chemically binding to a suitable receptor covalently bound to a solid support (Figure 8.10). Ad- sorption, absorption, and binding occur at the interface between the metal surface, the thin film, or the receptor, and the solution containing the analyte. Conse- quently, the amount of analyte extracted is minuscule, and the resulting change in mass is too small to detect with a conventional balance. This problem is overcome by using a quartz crystal microbalance as a support. The measurement of mass using a quartz crystal microbalance is based on the piezoelectric effect. 10 When a piezoelectric material, such as a quartz crystal, experi- ences a mechanical stress, it generates an electrical potential whose magnitude is proportional to the applied stress. Conversely, when an alternating electrical field is A A A A A A A L (c) (d) (a) (b) A L A L A A A 1400-CH08 9/9/99 2:18 PM Page 263 applied across a quartz crystal, an oscillatory vibrational motion is induced in the crystal. Every quartz crystal vibrates at a characteristic resonant frequency that is a function of the crystal’s properties, including the mass per unit area of any material coated on the crystal’s surface. The change in mass following adsorption, absorption, or binding of the analyte, therefore, can be determined by monitoring the change in the quartz crystal’s characteristic resonant frequency. The exact relationship between the change in frequency and mass is determined by a calibration curve. 8D.2 Quantitative Applications Particulate gravimetry is commonly encountered in the environmental analysis of water, air, and soil samples. The analysis for suspended solids in water samples, for example, is accomplished by filtering an appropriate volume of a well-mixed sample through a glass fiber filter and drying the filter to constant weight at 103–105 °C. Microbiological testing of water also is accomplished by particulate gravimetry. For example, in the analysis for coliform bacteria an appropriate volume of sample is passed through a sterilized 0.45-m m membrane filter. The filter is then placed on a sterilized absorbent pad saturated with a culturing medium and incubated for 22–24 h at 35 ±0.5 °C. Coliform bacteria are identified by the presence of individual bacterial colonies that form during the incubation period. As with qualitative appli- cations of precipitation gravimetry, the signal in this case is a visual observation rather than a measurement of mass. Total airborne particulates are determined using a high-volume air sampler equipped with either cellulose fiber or glass fiber filters. Samples taken from urban environments require approximately 1 h of sampling time, but samples from rural environments require substantially longer times. Grain size distributions for sediments and soils are used to determine the amount of sand, silt, and clay present in a sample. For example, a grain size of 2 mm serves as a boundary between gravel and sand. Grain size boundaries for sand–silt and silt–clay are given as 1/16 mm and 1/256 mm, respectively. Several standard methods for the quantitative analysis of food samples are based on measuring the sample’s mass following a selective solvent extraction. For example, the crude fat content in chocolate can be determined by extracting with ether for 16 h in a Soxhlet extractor. After the extraction is complete, the ether is al- lowed to evaporate, and the residue is weighed after drying at 100 °C. This analysis has also been accomplished indirectly by weighing a sample before and after ex- tracting with supercritical CO 2 . Quartz crystal microbalances equipped with thin-film polymer or chemical coatings have found numerous quantitative applications in environmental analysis. Methods have been reported for the analysis of a variety of gaseous pollutants, in- cluding ammonia, hydrogen sulfide, ozone, sulfur dioxide, and mercury. 10 Bio- chemical particulate gravimetric sensors also have been developed. For example, a piezoelectric immunosensor has been developed that shows a high selectivity for human serum albumin and is capable of detecting microgram quantities. 11 Quantitative Calculations The result of a quantitative analysis by particulate gravimetry is just the ratio, using appropriate units, of the amount of analyte to the amount of sample. 264 Modern Analytical Chemistry 1400-CH08 9/9/99 2:18 PM Page 264 Chapter 8 Gravimetric Methods of Analysis 265 EXAMPLE 8.8 A 200.0-mL sample of water was filtered through a preweighed glass fiber filter. After drying to constant weight at 105 °C, the filter was found to have increased in mass by 48.2 mg. Determine the total suspended solids for the sample in parts per million. SOLUTION Parts per million is the same as milligrams of analyte per liter of solution; thus, the total suspended solids for the sample is 8D. 3 Evaluating Particulate Gravimetry The scale of operation and detection limit for particulate gravimetry can be ex- tended beyond that of other gravimetric methods by increasing the size of the sample taken for analysis. This is usually impossible for other gravimetric meth- ods because of the difficulty of manipulating a larger sample through the indi- vidual steps of the analysis. With particulate gravimetry, however, the part of the sample that is not analyte is removed when filtering or extracting. Conse- quently, particulate gravimetry is easily extended to the analysis of trace-level analytes. Except for methods relying on a quartz crystal microbalance, particulate gravimetry uses the same balances as other gravimetric methods and is capable of achieving similar levels of accuracy and precision. Since particulate gravime- try is defined in terms of the mass of the particle itself, the sensitivity of the analysis is given by the balance’s sensitivity. Selectivity, on the other hand, is de- termined by either the filter’s pore size or the properties of the extracting phase. Particulate gravimetric methods based on filtration are generally less time-, labor-, and capital-intensive than other gravimetric methods since they require only a filtration step. 48 2 241 . mg 0.2000 L ppm= 8E KEY TERMS adsorbate (p. 239) coagulation (p. 242) digestion (p. 239) electrogravimetry (p. 234) gravimetry (p. 233) homogeneous precipitation (p. 241) inclusion (p. 238) occlusion (p. 239) particulate gravimetry (p. 234) peptization (p. 245) precipitant (p. 235) precipitation gravimetry (p. 234) relative supersaturation (p. 241) supernatant (p. 244) thermogram (p. 256) thermogravimetry (p. 255) volatilization gravimetry (p. 234) 1400-CH08 9/9/99 2:18 PM Page 265 266 Modern Analytical Chemistry In a gravimetric analysis a measurement of mass or change in mass provides quantitative information about the amount of ana- lyte in a sample. The most common form of gravimetry uses a pre- cipitation reaction to generate a product whose mass is propor- tional to the analyte. In many cases the precipitate includes the analyte; however, an indirect analysis in which the analyte causes the precipitation of another compound also is possible. Precipita- tion gravimetric procedures must be carefully controlled to pro- duce precipitates that are easily filterable, free from impurities, and of known stoichiometry. In volatilization gravimetry, thermal or chemical energy is used to decompose the sample containing the analyte. The mass of residue remaining after decomposition, the mass of volatile prod- ucts collected with a suitable trap, or a change in mass due to the loss of volatile material are all gravimetric measurements. When the analyte is already present in a particulate form that is easily separated from its matrix, then a particulate gravimetric analysis may be feasible. Examples include the determination of dissolved solids and the determination of fat in foods. 8F SUMMARY Burrows, H. D.; Ellis, H. A.; Odilora, C. A. “The Dehydrochlorination of PVC,” J. Chem. Educ. 1995, 72, 448–450. This experiment describes a simple gravimetric procedure for determining the %w/w Cl in samples of poly(vinyl chloride). Carmosini, N.; Ghoreshy, S.; Koether, M. C. “The Gravimetric Analysis of Nickel Using a Microwave Oven,” J. Chem. Educ. 1997, 74, 986–987. A procedure for using a microwave oven to digest samples of Ni ore and to dry precipitates of nickel–dimethylgloxime is described in this experiment. Harris, T. M. “Revitalizing the Gravimetric Determination in Quantitative Analysis Laboratory,” J. Chem. Educ. 1995, 72, 355–356. This experiment investigates the accuracy of a gravimetric analysis when modifying a standard procedure. The gravimetric procedure is the determination of Ba 2+ as BaSO 4 . Modifications that are investigated include the addition of a potential interferent (Ca 2+ ), changing the pH at which precipitation occurs, changing the rate at which the precipitant is added, changing the conditions for digesting the precipitate, and changing the procedure for filtering and drying the precipitate. Errors introduced by modifying the standard procedure can be explained by considering the process by which precipitation occurs. Henrickson, C. H.; Robinson, P. R. “Gravimetric Determination of Calcium as CaC 2 O 4 H 2 O,” J. Chem. Educ. 1979, 56, 341–342. A procedure is provided for the analysis of calcium in samples of CaCO 3 . Precipitation is done homogeneously using urea and acidified ammonium oxalate. By acidifying the ammonium oxalate, the oxalate is introduced as oxalic acid and does not precipitate the Ca 2+ . Heating the solution hydrolyzes the urea, forming NH 3 . As the NH 3 neutralizes the acid in solution, oxalate acid is converted to oxalate and CaC 2 O 4 H 2 O precipitates. The acid–base indicator methyl red is used to signal the completion of the precipitation. Snow, N. H.; Dunn, M.; Patel, S. “Determination of Crude Fat in Food Products by Supercritical Fluid Extraction and Gravimetric Analysis,” J. Chem. Educ. 1997, 74, 1108–1111. The %w/w fat in candy bars is determined by an indirect particulate gravimetric analysis. Supercritical CO 2 is used to extract the fat from the sample, and the change in the sample’s weight is used to determine the fat content. Thompson, R. Q.; Ghadiali, M. “Microwave Drying of Precipitates for Gravimetric Analysis,” J. Chem. Educ. 1993, 70, 170–171. This article describes conditions for using a household microwave oven to dry precipitates for the determination of Cl – as AgCl, the determination of SO 4 2– as BaSO 4 , and the determination of Ca 2+ as CaC 2 O 4 H 2 O. 8G Suggested EXPERIMENTS A number of gravimetric methods, such as the determination of Cl – in a soluble salt, have been part of the “standard” repertoire of experiments for introductory courses in analytical chemistry. Listed here are additional experiments that may be used to provide practical examples of gravimetry. Experiments 1400-CH08 9/9/99 2:18 PM Page 266 Chapter 8 Gravimetric Methods of Analysis 267 1. Starting with the equilibrium constant expressions for reactions 8.1, and 8.3–8.5, verify that equation 8.7 is correct. 2. Equation 8.7 shows how the solubility of AgCl varies as a function of the equilibrium concentration of Cl – . Derive a similar equation to describe the solubility of AgCl as a function of the equilibrium concentration of Ag + . Graph the resulting solubility function and compare it with that shown in Figure 8.1. 3. Derive a solubility diagram (solubility versus pH) for Zn(OH) 2 that takes into account the following soluble zinc hydroxide complexes: Zn(OH) + , Zn(OH) 3 – , Zn(OH) 4 2– . What is the optimum pH for the quantitative precipitation of Zn(OH) 2 ? 4. For what pH range will the following precipitates have their lowest solubility? a. CaC 2 O 4 b. PbCrO 4 c. BaSO 4 d. SrCO 3 e. ZnS 5. When solutions of 1.5 M KNO 3 and 1.5 M HClO 4 are mixed, a white precipitate of KClO 4 is formed. If traces of MnO 4 – are present, an inclusion of KMnO 4 is possible. Impure precipitates of KClO 4 are colored purple by the included KMnO 4 . Following are the descriptions and results for two experiments in which KClO 4 is precipitated in the presence of MnO 4 – . Explain why the two experiments lead to different results (see Color Plate 6). Experiment 1. Place 1 mL of 1.5 M KNO 3 in a test tube, add 3 drops of 0.1 M KMnO 4 , and swirl to mix. Add 1 mL of 1.5 M HClO 4 dropwise, agitating the solution between drops. Destroy the excess KMnO 4 by adding 0.1 M NaHSO 3 dropwise. The resulting precipitate of KClO 4 has an intense purple color. Experiment 2. Place 1 mL of 1.5 M HClO 4 in a test tube, add 3 drops of 0.1 M KMnO 4 , and swirl to mix. Add 1 mL of 1.5 M KNO 3 dropwise, agitating the solution between drops. Destroy the excess KMnO 4 by adding 0.1 M NaHSO 3 dropwise. The resulting precipitate of KClO 4 is pale purple or white in color. 6. When solutions of Ba(SCN) 2 and MgSO 4 are mixed, a precipitate of BaSO 4 forms. Following are the descriptions and results for several experiments in which only the concentrations of Ba(SCN) 2 and MgSO 4 are different. Explain why these experiments produce different results. Experiment 1. When equal volumes of 3.5 M Ba(SCN) 2 and 3.5 M MgSO 4 are mixed, a gelatinous precipitate immediately forms. Experiment 2. When equal volumes of 1.5 M Ba(SCN) 2 and 1.5 M MgSO 4 are mixed, a curdy precipitate immediately forms. Individual particles of BaSO 4 can be seen as points under magnification at 1500· . Experiment 3. When equal volumes of 0.0005 M Ba(SCN) 2 and 0.0005 M MgSO 4 are mixed, the complete precipitation of BaSO 4 requires 2–3 h. Individual crystals of BaSO 4 obtain lengths of approximately 0.005 mm. 7. Aluminum can be determined gravimetrically by precipitating as Al(OH) 3 and isolating as Al 2 O 3 . A sample containing approximately 0.1 g of Al is dissolved in 200 mL of H 2 O and 5 g of NH 4 Cl and a few drops of methyl red indicator is added (methyl red is red at pH levels below 4 and yellow at pH levels above 6). The solution is heated to boiling, and 1:1 NH 3 is added dropwise till the indicator turns yellow, precipitating Al(OH) 3 . The precipitate is held at the solution’s boiling point for several minutes, filtered, and washed with a hot solution of 2%, w/v NH 4 NO 3 . The precipitate is then ignited at 1000–1100 °C, forming Al 2 O 3 . (a) Cite two ways in which this procedure has been designed to encourage the formation of larger particles of precipitate. (b) The ignition step must be carried out carefully to ensure that Al(OH) 3 is quantitatively converted to Al 2 O 3 . What effect would an incomplete conversion have on the reported %w/w Al? (c) What role do NH 4 Cl and methyl red indicator play in this procedure? (d) An alternative procedure involves isolating and weighing the precipitate as the 8- hydroxyquinolate, Al(C 9 H 6 ON) 3 . Why might this be a more advantageous form of Al for a gravimetric analysis? 8. Calcium is determined gravimetrically by precipitating it as CaC 2 O 4 H 2 O, followed by isolating the precipitate as CaCO 3 . The sample to be analyzed is dissolved in 10 mL of water and 15 mL of 6 M HCl. After dissolution, the resulting solution is heated to boiling, and a warm solution of excess ammonium oxalate is added. The solution is maintained at 80 °C, and 6 M NH 3 is added dropwise, with stirring, until the solution is faintly alkaline. The resulting precipitate and solution are removed from the heat and allowed to stand for at least 1 h. After testing the solution for completeness of precipitation, the sample is filtered, washed with 0.1% w/v ammonium oxalate, and dried at 100–120 °C for 1 h. The precipitate is then transferred to a muffle furnace where it is converted to CaCO 3 by drying at 500 ± 25 °C until a constant weight. (a) Why is the precipitate of CaC 2 O 4 H 2 O converted to CaCO 3 ? (b) In the final step, if the sample is heated at too high of a temperature, some CaCO 3 may be converted to CaO. What effect would this have on the reported %w/w Ca? (c) Why is the precipitant, (NH 4 ) 2 C 2 O 4 , added to a hot, acidic solution rather than to a cold, alkaline solution? 8H PROBLEMS 1400-CH08 9/9/99 2:18 PM Page 267 9. Iron can be determined gravimetrically by precipitating as Fe(OH) 3 and igniting to Fe 2 O 3 . The sample to be analyzed is weighed and transferred to a 400-mL beaker where it is dissolved in 50 mL of H 2 O and 10 mL of 6 M HCl. Any Fe 2+ that is present is oxidized to Fe 3+ with 1–2 mL of concentrated HNO 3 . After boiling to remove the oxides of nitrogen, the solution is diluted to 200 mL, brought to boiling, and Fe(OH) 3 is precipitated by slowly adding 1:1 NH 3 until the odor of NH 3 is detected. The solution is boiled for an additional minute, and the precipitate is allowed to settle to the bottom of the beaker. The precipitate is then filtered and washed with several portions of hot 1% w/v NH 4 NO 3 until no Cl – is found in the wash water. Finally, the precipitate is ignited to constant weight at 500–550 °C, and weighed as Fe 2 O 3 . (a) If the ignition is not carried out under oxidizing conditions (plenty of O 2 present), the final product will contain some Fe 3 O 4 . What effect would this have on the reported %w/w Fe? (b) The precipitate is washed with a dilute solution of NH 4 NO 3 . Why is NH 4 NO 3 added to the wash water? (c) Why does the procedure call for adding NH 3 until the odor of ammonia is detected? (d) Describe how you might test the filtrate for Cl – . 10. Sinha and Shome described a gravimetric method for molybdenum in which it is precipitated with n- benzoylphenylhydroxylamine, C 13 H 11 NO 2 , giving a precipitate of MoO 2 (C 13 H 10 NO 2 ) 2 . 12 The precipitate is weighed after igniting to MoO 3 . As part of their study they determined the optimum conditions for the analysis. Samples containing a known amount of Mo were taken through the procedure while varying the temperature, the amount of precipitant added, and the pH of the solution. The solution volume was held constant at 300 mL for all experiments. A summary of their results are shown in the following table g Mo Temperature g mL 10 M taken (°C) precipitant HCl g MoO 3 0.0770 30 0.20 0.9 0.0675 0.0770 30 0.30 0.9 0.1014 0.0770 30 0.35 0.9 0.1140 0.0770 30 0.42 0.9 0.1155 0.0770 30 0.42 0.3 0.1150 0.0770 30 0.42 18.0 0.1152 0.0770 30 0.42 48.0 0.1160 0.0770 30 0.42 75.0 0.1159 0.0770 50 0.42 0.9 0.1156 0.0770 75 0.42 0.9 0.1158 0.0770 80 0.42 0.9 0.1129 Considering these results, discuss the optimum conditions for the determination of Mo by this method. Express your results for the precipitant as the minimum concentration in excess, as %w/v, needed to ensure a quantitative precipitation. 11. A sample of an impure iron ore is believed to be approximately 55% w/w Fe. The amount of Fe in the sample is to be determined gravimetrically by isolating it as Fe 2 O 3 . How many grams of sample should be taken to ensure that approximately 1 g of Fe 2 O 3 will be isolated? 12. The concentration of arsenic in an insecticide can be determined gravimetrically by precipitating it as MgNH 4 AsO 4 . The precipitate is ignited and weighed as Mg 2 As 2 O 7 . Determine the %w/w As 2 O 3 in a 1.627-g sample of insecticide that yields 106.5 mg of Mg 2 As 2 O 7 . 13. After preparing a sample of alum, K 2 SO 4 ? Al 2 (SO 4 ) 3 ? 24H 2 O, a student determined its purity gravimetrically. A 1.2931-g sample was dissolved and the aluminum precipitated as Al(OH) 3 . The precipitate was collected by filtration, washed, and ignited to Al 2 O 3 , yielding 0.1357 g. What is the purity of the alum preparation? 14. To determine the amount of iron in a dietary supplement, a random sample of 15 tablets weighing a total of 20.505 g was ground into a fine powder. A 3.116-g sample of the powdered tablets was dissolved and treated to precipitate the iron as Fe(OH) 3 . The precipitate was collected, rinsed, and ignited to a constant weight as Fe 2 O 3 , yielding 0.355 g. Report the iron content of the dietary supplement as g FeSO 4 7H 2 O per tablet. 15. A 1.4639-g sample of limestone was analyzed for Fe, Ca, and Mg. The iron was determined as Fe 2 O 3, yielding 0.0357 g. Calcium was isolated as CaSO 4 , yielding a precipitate of 1.4058 g, and Mg was isolated as 0.0672 g of Mg 2 P 2 O 7 . Report the amount of Fe, Ca, and Mg in the limestone sample as %w/w Fe 2 O 3 , %w/w CaO, and %w/w MgO. 16. The number of ethoxy groups (CH 3 CH 2 O–) in an organic compound can be determined by the following sequence of reactions R(OCH 2 CH 3 ) x + xHI fi R(OH) x + xCH 3 CH 2 I CH 3 CH 2 I+Ag + +H 2 O fi AgI(s)+CH 3 CH 2 OH A 36.92-mg sample of an organic compound with an approximate molecular weight of 176 was treated in this fashion, yielding 0.1478 g of AgI. How many ethoxy groups are there in each molecule? 17. A 516.7-mg sample containing a mixture of K 2 SO 4 and (NH 4 ) 2 SO 4 was dissolved in water and treated with BaCl 2 , precipitating the SO 4 2– as BaSO 4 . The resulting precipitate was isolated by filtration, rinsed free of impurities, and dried to a constant weight, yielding 863.5 mg of BaSO 4 . What is the %w/w K 2 SO 4 in the sample? 18. The amount of iron and manganese in an alloy can be determined by precipitating the metals with 8- hydroxyquinoline, C 9 H 7 NO. After weighing the mixed precipitate, the precipitate is dissolved and the amount of 8-hydroxyquinoline determined by another method. In a typical analysis, a 127.3-mg sample of an alloy containing iron, manganese, and other metals was dissolved in acid and 268 Modern Analytical Chemistry 1400-CH08 9/9/99 2:18 PM Page 268 treated with appropriate masking agents to prevent an interference from other metals. The iron and manganese were precipitated and isolated as Fe(C 9 H 6 NO) 3 and Mn(C 9 H 6 NO) 2 , yielding a total mass of 867.8 mg. The amount of 8-hydroxyquinolate in the mixed precipitate was determined to be 5.276 mmol. Calculate the %w/w Fe and %w/w Mn in the alloy. 19. A 0.8612-g sample of a mixture consisting of NaBr, NaI, and NaNO 3 was analyzed by adding AgNO 3 to precipitate the Br – and I – , yielding a 1.0186-g mixture of AgBr and AgI. The precipitate was then heated in a stream of Cl 2 , converting it to 0.7125 g of AgCl. Calculate the %w/w NaNO 3 in the sample. 20. The earliest determinations of elemental atomic weights were accomplished gravimetrically. In determining the atomic weight of manganese, a carefully purified sample of MnBr 2 weighing 7.16539 g was dissolved and the Br – precipitated as AgBr, yielding 12.53112 g. What is the atomic weight for Mn if the atomic weights for Ag and Br are taken to be 107.868 and 79.904, respectively? 21. While working as a laboratory assistant you prepared 0.4 M solutions of AgNO 3 , Pb(NO 3 ) 2 , BaCl 2 , KI, and Na 2 SO 4 . Unfortunately, you became distracted and forgot to label the solutions before leaving the laboratory. Realizing your error, you labeled the solutions A–E and performed all possible binary mixings of the five solutions. The following results were obtained where NR means no reaction was observed, W means a white precipitate formed, and Y means a yellow precipitate formed. 23. Two methods have been proposed for the analysis of sulfur in impure samples of pyrite, FeS 2 . Sulfur can be determined in a direct analysis by oxidizing it to SO 4 2– and precipitating as BaSO 4 . An indirect analysis is also possible if the iron is precipitated as Fe(OH) 3 and isolated as Fe 2 O 3 . Which of these methods will provide a more sensitive determination for sulfur? What other factors should be considered in deciding between these methods? 24. A sample of impure pyrite known to be approximately 90–95% w/w FeS 2 is to be analyzed by oxidizing the sulfur to SO 4 2– and precipitating as BaSO 4 . How many grams of the sample must be taken if a minimum of 1 g of BaSO 4 is desired? 25. A series of samples consisting of any possible combination of KCl, NaCl, and NH 4 Cl is to be analyzed by adding AgNO 3 to precipitate AgCl. What is the minimum volume of 5% w/v AgNO 3 necessary to completely precipitate the chloride in any 0.5-g sample? 26. When precipitation does not result in a “stoichiometric” precipitate, a gravimetric analysis is still feasible if the stoichiometric ratio between analyte and precipitate can be established experimentally. Consider, for example, the precipitation gravimetric analysis of Pb as PbCrO 4 . (a) For each gram of Pb in a sample, how many grams of PbCrO 4 are expected to form? (b) In a study of this procedure, Grote found that 1.568 g of PbCrO 4 formed per gram of Pb. 14 What is the apparent stoichiometry between Pb and PbCrO 4 ? (c) Does failing to account for the actual stoichiometry lead to a positive or negative determinate error for the analysis? 27. Determine the uncertainty for the gravimetric analysis described in Example 8.1. (a) How does your result compare with the expected accuracy of 0.1–0.2% for precipitation gravimetry? (b) What sources of error might account for any discrepancy between the most probable measurement error and the expected accuracy? 28. A 38.63-mg sample of potassium ozonide, KO 3 , was heated to 70 °C for 1 h, undergoing a weight loss of 7.10 mg. Write a balanced chemical reaction describing this decomposition reaction. A 29.6-mg sample of impure KO 3 experiences a 4.86-mg weight loss when treated under similar condition. What is the %w/w KO 3 in the sample? 29. The water content of an 875.4-mg sample of cheese was determined with a moisture analyzer. What is the %w/w H 2 O in the cheese if the final mass was found to be 545.8 mg? 30. Method 8.2 describes a procedure for determining Si in ores and alloys. In this analysis a weight loss of 0.21 g corresponds to 0.1 g of Si. Show that this relationship is correct. 31. The iron content of an organometallic compound was determined by treating a 0.4873-g sample with HNO 3 and heating to volatilize the organic material. After ignition, the residue of Fe 2 O 3 was found to weigh 0.2091 g. (a) What is the %w/w Fe in this compound? The carbon and hydrogen in a second sample of the compound were determined by a Chapter 8 Gravimetric Methods of Analysis 269 AB A B C D NR Y Y NR W NR W W NR W CDE Identify solutions A–E. 22. A solid sample is known to consist of approximately equal amounts of two or more of the following soluble salts. AgNO 3 ZnCl 2 K 2 CO 3 MgSO 4 Ba(C 2 H 3 O 2 ) 2 NH 4 NO 3 A sample of the solid, sufficient to give at least 0.04 mol of any single salt, was added to 100 mL of water, yielding a white precipitate and a clear solution. The precipitate was collected and rinsed with water. When a portion of the precipitate was placed in dilute HNO 3 , it completely dissolved, leaving a colorless solution. A second portion of the precipitate was placed in dilute HCl, yielding a precipitate and a clear solution. Finally, the filtrate from the original precipitate was treated with excess NH 3 , yielding a white precipitate. Indicate which of the soluble salts must be present in the sample, which must be absent and those for which there is insufficient information to make this determination. 13 1400-CH08 9/9/99 2:18 PM Page 269 combustion analysis. When a 0.5123-g sample was carried through the analysis, 1.2119 g of CO 2 and 0.2482 g of H 2 O were collected. (b) What are the %w/w C and %w/w H in this compound? (c) What is the compound’s empirical formula? 32. A polymer’s ash content is determined by placing a weighed sample in a Pt crucible that has been previously brought to a constant weight. The polymer is melted under gentle heating from a Bunsen burner until the volatile vapor ignites. The polymer is allowed to burn until only a noncombustible residue remains. The residue is then brought to constant weight at 800 °C in a muffle furnace. The following data were collected during the analysis of two samples of a polymer resin: g crucible + g crucible + Polymer A g crucible polymer ash replicate 1 19.1458 21.2287 19.7717 replicate 2 15.9193 17.9522 16.5310 replicate 3 15.6992 17.6660 16.2909 g crucible + g crucible + Polymer B g crucible polymer ash replicate 1 19.1457 21.0693 19.7187 replicate 2 15.6991 17.8273 16.3327 replicate 3 15.9196 17.9037 16.5110 (a) Determine the average and standard deviation for the %w/w ash of each polymer resin. (b) Is there any evidence at a = 0.05 for a significant difference between the two polymer resins? 33. In the presence of water vapor the surface of zirconia, ZrO 2 , chemically adsorbs H 2 O, forming surface hydroxyls, ZrOH (additional water is physically adsorbed as H 2 O). When heated above 200 °C the surface hydroxyls convert to H 2 O(g), with one molecule of water released for every two surface hydroxyls. Below 200 °C only physically absorbed water is lost. Nawrocki and coworkers 15 used thermogravimetry to determine the density of surface hydroxyls on a sample of zirconia that was heated to 700 °C and allowed to cool in a desiccator containing humid N 2 , finding that 0.006 g H 2 O was lost between 200 °C and 900 °C for every gram of dehydroxylated ZrO 2 . Given that the zirconia sample had a surface area of 33 m 2 /g and that one molecule of H 2 O forms two surface hydroxyls, calculate the density of surface hydroxyls in micromoles per square meter. 34. Suppose that you have a mixture of CaC 2 O 4 , MgC 2 O 4 , and some inert impurities, and that you wish to determine the %w/w CaC 2 O 4 in the sample. One way to accomplish this analysis is by volatilization gravimetry. The following information is available to you as the result of a thermogravimetric study: CaC 2 O 4 converts to CaCO 3 at a temperature of 500 °C; CaCO 3 converts to CaO at a temperature of 900 °C; and MgC 2 O 4 converts to MgO at a temperature of 500 °C. (a) Describe how you would carry out this analysis, paying particular attention to the measurements that you need to make. (b) How would you calculate the %w/w CaC 2 O 4 in the sample? 35. The concentration of airborne particulates in an industrial workplace was determined by pulling the air through a single- stage air sampler equipped with a glass fiber filter. The air was sampled for 20 min at a rate of 75 m 3 /h. At the end of the sampling period the glass fiber filter was found to have increased in mass by 345.2 mg. What is the concentration of particulates in the air sample in milligrams per cubic meter and in milligrams per liter? 36. The fat content of potato chips can be determined indirectly by weighing a sample before and after extracting the fat with supercritical CO 2 . The following data were obtained for the analysis of one sample of potato chips. 16 Sample Initial Sample Final Sample Number Weight (g) Weight (g) 1 1.1661 0.9253 2 1.1723 0.9252 3 1.2525 0.9850 4 1.2280 0.9562 5 1.2837 1.0119 (a) Determine the average fat content (%w/w) for this sample of potato chips. (b) This sample of potato chips is known to have a fat content of 22.7% w/w. Is there any evidence for a determinate error in the data at a = 0.05? 37. Delumyea and McCleary report data for the determination of %w/w organic material in sediment samples collected at different depths from a cove on the St. Johns River in Jacksonville, FL. 17 After collecting the sediment core it was sectioned into a set of 2-cm increments. Each increment was placed in 50 mL of deionized water and the slurry filtered through a piece of preweighed filter paper. The filter paper and sediment increment were then placed in a preweighed evaporating dish and dried in an oven at 270 Modern Analytical Chemistry 1400-CH08 9/9/99 2:18 PM Page 270 110 °C. After weighing, the filter paper and sediment were transferred to a muffle furnace where the filter paper and any organic material were removed by ashing. Finally, the inorganic residue remaining after ashing was weighed. Using the following data, determine the %w/w organic material as a function of the average depth for each increment. Weight of Weight of Weight Weight Paper Evaporating After After Depth Filter Dish Drying Ashing (cm) (g) (g) (g) (g) 0–2 1.590 43.21 52.10 49.49 2–4 1.745 40.62 48.83 46.00 4–6 1.619 41.23 52.86 47.84 6–8 1.611 42.10 50.59 47.13 8–10 1.658 43.62 51.88 47.53 10–12 1.628 43.24 49.45 45.31 12–14 1.633 43.08 47.92 44.20 14–16 1.630 43.96 58.31 55.53 16–18 1.636 43.36 54.37 52.75 38. Yao and associates recently described a method for the quantitative analysis of thiourea based on its reaction with I 2 . 18 CS(NH 2 ) 2 +4I 2 +6H 2 O fi (NH 4 ) 2 SO 4 +8 HI+CO 2 The procedure calls for placing a 100-m L aqueous sample containing the thiourea in a 60-mL separatory funnel along with 10 mL of a pH 7 buffer and 10 mL of a 12 m M solution of I 2 in CCl 4 . The contents of the separatory funnel are shaken, and the organic and aqueous layers are allowed to separate. The organic layer, containing the excess I 2 , is transferred to the surface of a piezoelectric crystal on which a thin layer of Au has been deposited. After allowing the I 2 to adsorb to the Au, the CCl 4 is removed and the crystal’s frequency shift is measured. The following data are reported for a series of thiourea standards. [thiourea] (M) D f (Hz) 3.00 · 10 –7 74.6 5.00 · 10 –7 120 7.00 · 10 –7 159 9.00 · 10 –7 205 15.00 · 10 –7 327 25.00 · 10 –7 543 35.00 · 10 –7 789 50.00 · 10 –7 1089 (a) Characterize this method with respect to the scale of operation shown in Figure 3.6 of Chapter 3. (b) Using a regression analysis, determine the relationship between the crystal’s frequency shift and the concentration of thiourea. (c) A sample containing an unknown amount of thiourea is taken through the procedure and gives a D f of 176 Hz. What is the molar concentration of thiourea in the sample? (d) What is the 95% confidence interval for the concentration of thiourea in this sample assuming one replicate? Chapter 8 Gravimetric Methods of Analysis 271 The following resources provide a general history of gravimetry. Beck, C. M. “Classical Analysis: A Look at the Past, Present, and Future,” Anal. Chem. 1991, 63, 993A–1003A. Laitinen, H. A.; Ewing, G. W., eds. A History of Analytical Chemistry. The Division of Analytical Chemistry of the American Chemical Society: Washington, DC, 1977, pp. 10–24. Sources providing additional examples of inorganic and organic gravimetric methods include the following texts. Bassett, J.; Denney, R. C.; Jeffery, G. H.; et al. Vogel’s Textbook of Quantitative Inorganic Analysis, 4th ed. Longman: London, 1981. Erdey, L. Gravimetric Analysis, Pergamon: Oxford, 1965. Steymark, A. Quantitative Organic Microanalysis, The Blakiston Co.: New York, 1951. The following text provides more information on thermogravimetry. Wendlandt, W. W. Thermal Methods of Analysis, 2nd ed. Wiley: New York, 1986. For a review of isotope dilution mass spectrometry see the following article. Fassett, J. D.; Paulsen, P. J. “Isotope Dilution Mass Spectrometry for Accurate Elemental Analysis,” Anal. Chem. 1989, 61, 643A–649A. 8I SUGGESTED READINGS 1400-CH08 9/9/99 2:18 PM Page 271 272 Modern Analytical Chemistry 1. Valcárcel, M.; Rios, A. Analyst 1995, 120, 2291–2297. 2. (a) Moody, J. R.; Epstein, M. S. Spectrochim. Acta 1991, 46B, 1571–1575. (b) Epstein, M. S. Spectrochim. Acta 1991, 46B, 1583–1591. 3. Von Weimarn, P. P. Chem. Revs. 1925, 2, 217. 4. Bassett, J.; Denney, R. C.; Jeffery, G. H.; et al. Vogel’s Textbook of Quantitative Inorganic Analysis, 4th ed. Longman: London, 1981, p. 408. 5. Gordon, L; Salutsky, M. L.; Willard, H. H. Precipitation from Homogeneous Solution, Wiley: New York, 1959. 6. Ward, R., ed. Non-Stoichiometric Compounds (Ad. Chem. Ser. 39); American Chemical Society: Washington, DC, 1963. 7. Method 3500-Mg D as published in Standard Methods for the Examination of Water and Wastewater, 18th ed. American Public Health Association: Washington, DC, 1992, pp. 3–73 to 3–74. 8. Jungreis, E. Spot Test Analysis, 2nd ed. Wiley: New York, 1997. 9. Young, R. S. Chemical Analysis in Extractive Metallurgy, Griffen: London, 1971, pp. 302–304. 10. (a) Ward, M. D.; Buttry, D. A. Science 1990, 249, 1000–1007; (b) Grate, J. W.; Martin, S. J.; White, R. M. Anal. Chem. 1993, 65, 940A–948A; (c) Grate, J. W.; Martin, S. J.; White, R. M. Anal. Chem. 1993, 65, 987A–996A. 11. Muratsugu, M.; Ohta, F.; Miya, Y. et al. Anal. Chem. 1993, 65, 2933–2937. 12. Sinha, S. K.; Shome, S. C. Anal. Chim. Acta 1960, 24, 33–36. 13. Adapted from Sorum, C. H.; Lagowski, J. J. Introduction to Semimicro Qualitative Analysis, 5th ed. Prentice-Hall: Englewood Cliffs, N. J., 1977, p. 285. 14. Grote, Z. Anal. Chem. 1941, 122, 395. 15. Nawrocki, J.; Carr, P. W.; Annen, M. J.; et al. Anal. Chim. Acta 1996, 327, 261–266. 16. Data taken from the pamphlet Fat Determination by SFE, ISCO, Inc.: Lincoln, NE. 17. Delumyea, R. D.; McCleary, D. L. J. Chem. Educ. 1993, 70, 172–173. 18. Yao, S. F.; He, F. J.; Nie, L. H. Anal. Chim. Acta 1992, 268, 311–314. 8J REFERENCES 1400-CH08 9/9/99 2:18 PM Page 272 Chapter 9 273 Titrimetric Methods of Analysis Titrimetry, in which we measure the volume of a reagent reacting stoichiometrically with the analyte, first appeared as an analytical method in the early eighteenth century. Unlike gravimetry, titrimetry initially did not receive wide acceptance as an analytical technique. Many prominent late-nineteenth century analytical chemists preferred gravimetry over titrimetry and few of the standard texts from that era include titrimetric methods. By the early twentieth century, however, titrimetry began to replace gravimetry as the most commonly used analytical method. Interestingly, precipitation gravimetry developed in the absence of a theory of precipitation. The relationship between the precipitate s mass and the mass of analyte, called a gravimetric factor, was determined experimentally by taking known masses of analyte (an external standardization). Gravimetric factors could not be calculated using the precipitation reaction s stoichiometry because chemical formulas and atomic weights were not yet available! Unlike gravimetry, the growth and acceptance of titrimetry required a deeper understanding of stoichiometry, thermodynamics, and chemical equilibria. By the early twentieth century the accuracy and precision of titrimetric methods were comparable to that of gravimetry, establishing titrimetry as an accepted analytical technique. 1400-CH09 9/9/99 2:12 PM Page 273 274 Modern Analytical Chemistry 9 A Overview of Titrimetry Titrimetric methods are classified into four groups based on the type of reaction in- volved. These groups are acid–base titrations, in which an acidic or basic titrant re- acts with an analyte that is a base or an acid; complexometric titrations involving a metal–ligand complexation reaction; redox titrations, where the titrant is an oxidiz- ing or reducing agent; and precipitation titrations, in which the analyte and titrant react to form a precipitate. Despite the difference in chemistry, all titrations share several common features, providing the focus for this section. 9 A.1 Equivalence Points and End Points For a titration to be accurate we must add a stoichiometrically equivalent amount of titrant to a solution containing the analyte. We call this stoichiometric mixture the equivalence point. Unlike precipitation gravimetry, where the precipitant is added in excess, determining the exact volume of titrant needed to reach the equiv- alence point is essential. The product of the equivalence point volume, V eq , and the titrant’s concentration, C T , gives the moles of titrant reacting with the analyte. Moles titrant = V eq · C T Knowing the stoichiometry of the titration reaction(s), we can calculate the moles of analyte. Unfortunately, in most titrations we usually have no obvious indication that the equivalence point has been reached. Instead, we stop adding titrant when we reach an end point of our choosing. Often this end point is indicated by a change in the color of a substance added to the solution containing the analyte. Such sub- stances are known as indicators. The difference between the end point volume and the equivalence point volume is a determinate method error, often called the titra- tion error. If the end point and equivalence point volumes coincide closely, then the titration error is insignificant and can be safely ignored. Clearly, selecting an ap- propriate end point is critical if a titrimetric method is to give accurate results. 9 A.2 Volume as a Signal* Almost any chemical reaction can serve as a titrimetric method provided that three conditions are met. The first condition is that all reactions involving the titrant and analyte must be of known stoichiometry. If this is not the case, then the moles of titrant used in reaching the end point cannot tell us how much ana- lyte is in our sample. Second, the titration reaction must occur rapidly. If we add titrant at a rate that is faster than the reaction’s rate, then the end point will ex- ceed the equivalence point by a significant amount. Finally, a suitable method must be available for determining the end point with an acceptable level of accu- racy. These are significant limitations and, for this reason, several titration strate- gies are commonly used. A simple example of a titration is an analysis for Ag + using thiocyanate, SCN – , as a titrant. Ag + (aq) + SCN – (aq) t AgSCN(s) equivalence point The point in a titration where stoichiometrically equivalent amounts of analyte and titrant react. end point The point in a titration where we stop adding titrant. indicator A colored compound whose change in color signals the end point of a titration. titration error The determinate error in a titration due to the difference between the end point and the equivalence point. titrant The reagent added to a solution containing the analyte and whose volume is the signal. *Instead of measuring the titrant’s volume we also can measure its mass. Since the titrant’s density is a measure of its mass per unit volume, the mass of titrant and volume of titrant are proportional. titrimetry Any method in which volume is the signal. 1400-CH09 9/9/99 2:12 PM Page 274 This reaction occurs quickly and is of known stoichiometry. A titrant of SCN – is easily prepared using KSCN. To indicate the titration’s end point we add a small amount of Fe 3+ to the solution containing the analyte. The formation of the red- colored Fe(SCN) 2+ complex signals the end point. This is an example of a direct titration since the titrant reacts with the analyte. If the titration reaction is too slow, a suitable indicator is not available, or there is no useful direct titration reaction, then an indirect analysis may be possible. Sup- pose you wish to determine the concentration of formaldehyde, H 2 CO, in an aque- ous solution. The oxidation of H 2 CO by I 3 – H 2 CO(aq) + 3OH – (aq)+I 3 – (aq) t HCO 2 – (aq)+3I – (aq)+2H 2 O(l) is a useful reaction, except that it is too slow for a direct titration. If we add a known amount of I 3 – , such that it is in excess, we can allow the reaction to go to comple- tion. The I 3 – remaining can then be titrated with thiosulfate, S 2 O 3 2– . I 3 – (aq)+2S 2 O 3 2– (aq) t S 4 O 6 2– (aq)+3I – (aq) This type of titration is called a back titration. Calcium ion plays an important role in many aqueous environmental systems. A useful direct analysis takes advantage of its reaction with the ligand ethylenedi- aminetetraacetic acid (EDTA), which we will represent as Y 4– . Ca 2+ (aq)+Y 4– (aq) t CaY 2– (aq) Unfortunately, it often happens that there is no suitable indicator for this direct titration. Reacting Ca 2+ with an excess of the Mg 2+ –EDTA complex Ca 2+ (aq) + MgY 2– (aq) t CaY 2– (aq)+Mg 2+ (aq) releases an equivalent amount of Mg 2+ . Titrating the released Mg 2+ with EDTA Mg 2+ (aq)+Y 4– (aq) t MgY 2– (aq) gives a suitable end point. The amount of Mg 2+ titrated provides an indirect mea- sure of the amount of Ca 2+ in the original sample. Since the analyte displaces a species that is then titrated, we call this a displacement titration. When a suitable reaction involving the analyte does not exist it may be possible to generate a species that is easily titrated. For example, the sulfur content of coal can be determined by using a combustion reaction to convert sulfur to sulfur dioxide. S(s)+O 2 (g) fi SO 2 (g) Passing the SO 2 through an aqueous solution of hydrogen peroxide, H 2 O 2 , SO 2 (g)+H 2 O 2 (aq) fi H 2 SO 4 (aq) produces sulfuric acid, which we can titrate with NaOH, H 2 SO 4 (aq) + 2OH – (aq) t SO 4 2– (aq)+2H 2 O(l) providing an indirect determination of sulfur. 9 A. 3 Titration Curves To find the end point we monitor some property of the titration reaction that has a well-defined value at the equivalence point. For example, the equivalence point for a titration of HCl with NaOH occurs at a pH of 7.0. We can find the end point, Chapter 9 Titrimetric Methods of Analysis 275 back titration A titration in which a reagent is added to a solution containing the analyte, and the excess reagent remaining after its reaction with the analyte is determined by a titration. displacement titration A titration in which the analyte displaces a species, usually from a complex, and the amount of the displaced species is determined by a titration. 1400-CH09 9/9/99 2:12 PM Page 275 Figure 9.1 Acid–base titration curve for 25.0 mL of 0.100 M HCl with 0.100 M NaOH. therefore, by monitoring the pH with a pH electrode or by adding an indicator that changes color at a pH of 7.0. Suppose that the only available indicator changes color at a pH of 6.8. Is this end point close enough to the equivalence point that the titration error may be safely ignored? To answer this question we need to know how the pH changes dur- ing the titration. A titration curve provides us with a visual picture of how a property, such as pH, changes as we add titrant (Figure 9.1). We can measure this titration curve ex- perimentally by suspending a pH electrode in the solution containing the analyte, monitoring the pH as titrant is added. As we will see later, we can also calculate the expected titration curve by considering the reactions responsible for the change in pH. However we arrive at the titration curve, we may use it to evaluate an indica- tor’s likely titration error. For example, the titration curve in Figure 9.1 shows us that an end point pH of 6.8 produces a small titration error. Stopping the titration at an end point pH of 11.6, on the other hand, gives an unacceptably large titration error. The titration curve in Figure 9.1 is not unique to an acid–base titration. Any titration curve that follows the change in concentration of a species in the titration reaction (plotted logarithmically) as a function of the volume of titrant has the same general sigmoidal shape. Several additional examples are shown in Figure 9.2. Concentration is not the only property that may be used to construct a titration curve. Other parameters, such as temperature or the absorbance of light, may be used if they show a significant change in value at the equivalence point. Many titra- tion reactions, for example, are exothermic. As the titrant and analyte react, the temperature of the system steadily increases. Once the titration is complete, further additions of titrant do not produce as exothermic a response, and the change in temperature levels off. A typical titration curve of temperature versus volume of titrant is shown in Figure 9.3. The titration curve contains two linear segments, the intersection of which marks the equivalence point. 276 Modern Analytical Chemistry pH Volume NaOH (mL) 0.00 0.00 4.00 6.00 8.00 Equivalence point 10.00 12.00 14.00 10.00 20.00 30.00 50.00 2.00 40.00 titration curve A graph showing the progress of a titration as a function of the volume of titrant added. 1400-CH09 9/9/99 2:12 PM Page 276 Figure 9.2 Examples of titration curves for (a) a complexation titration, (b) a redox titration, and (c) a precipitation titration. 9 A. 4 The Buret The only essential piece of equipment for an acid–base titration is a means for deliv- ering the titrant to the solution containing the analyte. The most common method for delivering the titrant is a buret (Figure 9.4). A buret is a long, narrow tube with graduated markings, and a stopcock for dispensing the titrant. Using a buret with a small internal diameter provides a better defined meniscus, making it easier to read the buret’s volume precisely. Burets are available in a variety of sizes and tolerances Chapter 9 Titrimetric Methods of Analysis 277 pCd Volume of titrant (mL) 0.00 (a) 0.0 4.0 6.0 8.0 10.0 16.0 18.0 10.00 20.00 30.00 50.00 2.0 40.00 14.0 12.0 pCl Volume of titrant (mL) 0.00 (c) 0.0 2.0 4.0 8.0 10.00 20.00 30.00 50.0040.00 6.0 P otential (V) Volume of titrant (mL) 0 (b) 0.000 0.400 0.600 0.800 1.000 1.600 1.800 20 40 60 100 0.200 80 1.400 1.200 T emperature ( ° C) Volume of titrant (mL) Equivalence point Figure 9.3 Example of a thermometric titration curve. buret Volumetric glassware used to deliver variable, but known volumes of solution. 1400-CH09 9/9/99 2:12 PM Page 277 Figure 9.4 Volumetric buret showing a portion of its graduated scale. 278 Modern Analytical Chemistry 0 1 2 Table 9 .1 Specifications for Volumetric Burets Volume Subdivision Tolerance (mL) Class a (mL) (mL) 5 A 0.01 ±0.01 B 0.01 ±0.02 10 A 0.02 ±0.02 B 0.02 ±0.04 25 A 0.1 ±0.03 B 0.1 ±0.06 50 A 0.1 ±0.05 B 0.1 ±0.10 100 A 0.2 ±0.10 B 0.2 ±0.20 a Specifications for class A and class B glassware are taken from American Society for Testing Materials (ASTM) E288, E542, and E694 standards. (Table 9.1), with the choice of buret determined by the demands of the analysis. The accuracy obtainable with a buret can be improved by calibrating it over several intermediate ranges of volumes using the same method described in Chapter 5 for calibrating pipets. In this manner, the volume of titrant delivered can be corrected for any variations in the buret’s internal diameter. Titrations may be automated using a pump to deliver the titrant at a constant flow rate, and a solenoid valve to control the flow (Figure 9.5). The volume of titrant delivered is determined by multiplying the flow rate by the elapsed time. Au- tomated titrations offer the additional advantage of using a microcomputer for data storage and analysis. 9 B Titrations Based on Acid—Base Reactions The earliest acid–base titrations involved the determination of the acidity or alka- linity of solutions, and the purity of carbonates and alkaline earth oxides. Before 1800, acid–base titrations were conducted using H 2 SO 4 , HCl, and HNO 3 as acidic titrants, and K 2 CO 3 and Na 2 CO 3 as basic titrants. End points were determined using visual indicators such as litmus, which is red in acidic solutions and blue in basic solutions, or by observing the cessation of CO 2 effervescence when neutraliz- ing CO 3 2– . The accuracy of an acid–base titration was limited by the usefulness of the indicator and by the lack of a strong base titrant for the analysis of weak acids. The utility of acid–base titrimetry improved when NaOH was first introduced as a strong base titrant in 1846. In addition, progress in synthesizing organic dyes led to the development of many new indicators. Phenolphthalein was first synthe- sized by Bayer in 1871 and used as a visual indicator for acid–base titrations in 1877. Other indicators, such as methyl orange, soon followed. Despite the increas- ing availability of indicators, the absence of a theory of acid–base reactivity made se- lecting a proper indicator difficult. Developments in equilibrium theory in the late nineteenth century led to sig- nificant improvements in the theoretical understanding of acid–base chemistry and, acid–base titration A titration in which the reaction between the analyte and titrant is an acid–base reaction. 1400-CH09 9/9/99 2:12 PM Page 278 Figure 9.5 Typical instrumentation for performing an automatic titration. Courtesy of Fisher Scientific. in turn, of acid–base titrimetry. S?renson’s establishment of the pH scale in 1909 provided a rigorous means for comparing visual indicators. The determination of acid–base dissociation constants made the calculation of theoretical titration curves possible, as outlined by Bjerrum in 1914. For the first time a rational method ex- isted for selecting visual indicators, establishing acid–base titrimetry as a useful al- ternative to gravimetry. 9 B.1 Acid—Base Titration Curves In the overview to this chapter we noted that the experimentally determined end point should coincide with the titration’s equivalence point. For an acid–base titra- tion, the equivalence point is characterized by a pH level that is a function of the acid–base strengths and concentrations of the analyte and titrant. The pH at the end point, however, may or may not correspond to the pH at the equivalence point. To understand the relationship between end points and equivalence points we must know how the pH changes during a titration. In this section we will learn how to construct titration curves for several important types of acid–base titrations. Our Chapter 9 Titrimetric Methods of Analysis 279 1400-CH09 9/9/99 2:12 PM Page 279 approach will make use of the equilibrium calculations described in Chapter 6. We also will learn how to sketch a good approximation to any titration curve using only a limited number of simple calculations. Titrating Strong Acids and Strong Bases For our first titration curve let’s consider the titration of 50.0 mL of 0.100 M HCl with 0.200 M NaOH. For the reaction of a strong base with a strong acid the only equilibrium reaction of importance is H 3 O + (aq)+OH – (aq) t 2H 2 O(l) 9.1 The first task in constructing the titration curve is to calculate the volume of NaOH needed to reach the equivalence point. At the equivalence point we know from re- action 9.1 that Moles HCl = moles NaOH or M a V a = M b V b where the subscript ‘a’ indicates the acid, HCl, and the subscript ‘b’ indicates the base, NaOH. The volume of NaOH needed to reach the equivalence point, there- fore, is Before the equivalence point, HCl is present in excess and the pH is determined by the concentration of excess HCl. Initially the solution is 0.100 M in HCl, which, since HCl is a strong acid, means that the pH is pH = –log[H 3 O + ] = –log[HCl] = –log(0.100) = 1.00 The equilibrium constant for reaction 9.1 is (K w ) –1 , or 1.00 · 10 14 . Since this is such a large value we can treat reaction 9.1 as though it goes to completion. After adding 10.0 mL of NaOH, therefore, the concentration of excess HCl is giving a pH of 1.30. At the equivalence point the moles of HCl and the moles of NaOH are equal. Since neither the acid nor the base is in excess, the pH is determined by the dissoci- ation of water. K w = 1.00 · 10 –14 =[H 3 O + ][OH – ]=[H 3 O + ] 2 [H 3 O + ] = 1.00 · 10 –7 M Thus, the pH at the equivalence point is 7.00. Finally, for volumes of NaOH greater than the equivalence point volume, the pH is determined by the concentration of excess OH – . For example, after adding 30.0 mL of titrant the concentration of OH – is [ ( . )( . ) ( )( . ) .. . HCl] moles excess HCl total volume M mL .200 M mL mL mL M aa bb ab == ? + = ? + = MV MV VV 0 100 50 0 0 10 0 50 0 10 0 0 050 VV MV M eq b aa b M mL M) mL== = = (. )( . ) (. . 0 100 50 0 0 200 25 0 280 Modern Analytical Chemistry 1400-CH09 9/9/99 2:12 PM Page 280 To find the concentration of H 3 O + , we use the K w expression giving a pH of 12.10. Table 9.2 and Figure 9.1 show additional results for this titra- tion curve. Calculating the titration curve for the titration of a strong base with a strong acid is handled in the same manner, except that the strong base is in excess before the equivalence point and the strong acid is in excess after the equivalence point. Titrating a Weak Acid with a Strong Base For this example let’s consider the titra- tion of 50.0 mL of 0.100 M acetic acid, CH 3 COOH, with 0.100 M NaOH. Again, we start by calculating the volume of NaOH needed to reach the equivalence point; thus Moles CH 3 COOH = moles NaOH M a V a = M b V b VV MV M eq b aa b M mL M mL== = = (. )( . ) (. ) . 0 100 50 0 0 100 50 0 [] [] . . .HO OH 3 w + ? ? ? == × =× K 100 10 0 0125 800 10 14 13 [] (. )( . ) (. )( . ) .. . OH moles excess NaOH total volume M mL M mL mL mL M bb aa ab ? == ? + = ? + = MV MV VV 0 200 30 0 0 100 50 0 50 0 30 0 0 0125 Chapter 9 Titrimetric Methods of Analysis 281 Table 9 .2 Data for Titration of 50.00 mL of 0.100 M HCl with 0.0500 M NaOH Volume (mL) of Titrant pH 0.00 1.00 5.00 1.14 10.00 1.30 15.00 1.51 20.00 1.85 22.00 2.08 24.00 2.57 25.00 7.00 26.00 11.42 28.00 11.89 30.00 12.50 35.00 12.37 40.00 12.52 45.00 12.62 50.00 12.70 1400-CH09 9/9/99 2:12 PM Page 281 Before adding any NaOH the pH is that for a solution of 0.100 M acetic acid. Since acetic acid is a weak acid, we calculate the pH using the method outlined in Chapter 6. CH 3 COOH(aq)+H 2 O(l) t H 3 O + (aq)+CH 3 COO – (aq) x =[H 3 O + ] = 1.32 · 10 –3 At the beginning of the titration the pH is 2.88. Adding NaOH converts a portion of the acetic acid to its conjugate base. CH 3 COOH(aq)+OH – (aq) t H 2 O(l)+CH 3 COO – (aq) 9.2 Any solution containing comparable amounts of a weak acid, HA, and its conjugate weak base, A – , is a buffer. As we learned in Chapter 6, we can calculate the pH of a buffer using the Henderson–Hasselbalch equation. The equilibrium constant for reaction 9.2 is large (K = K a /K w = 1.75 · 10 9 ), so we can treat the reaction as one that goes to completion. Before the equivalence point, the concentration of unreacted acetic acid is and the concentration of acetate is For example, after adding 10.0 mL of NaOH the concentrations of CH 3 COOH and CH 3 COO – are giving a pH of A similar calculation shows that the pH after adding 20.0 mL of NaOH is 4.58. At the equivalence point, the moles of acetic acid initially present and the moles of NaOH added are identical. Since their reaction effectively proceeds to comple- tion, the predominate ion in solution is CH 3 COO – , which is a weak base. To calcu- late the pH we first determine the concentration of CH 3 COO – . The pH is then calculated as shown in Chapter 6 for a weak base. [] (. )( . ) .. .CH COO moles CH COOH total volume M mL mL mL M 3 3 ? == + = 0 100 50 0 50 0 50 0 0 0500 pH =+ =476 0 0167 0 0667 416. log . . . [] (. )( . ) .. .CH COO M mL mL mL M 3 ? = + = 0 100 10 0 50 0 10 0 0 0167 [ (. )( . ) (. )( . ) .. .CH COOH] M mL M mL mL mL M 3 = ? + = 0 100 50 0 0 100 10 0 50 0 10 0 0 0667 [CH COO ] moles NaOH added total volume 3 bb ab ? == + MV VV [CH COOH] moles unreacted CH COOH total volume 3 3aab ab == ? + MV MV VV pH p A HA] a =+ ? K log [] [ K xx x a == ? =× ? ? [][ ] [ ()() . . H O CH COO CH COOH] 3 + 3 3 0 100 175 10 5 282 Modern Analytical Chemistry 1400-CH09 9/9/99 2:12 PM Page 282 CH 3 COO – (aq)+H 2 O(l) t OH – (aq)+CH 3 COOH(aq) x = [OH – ] = 5.34 · 10 –6 M The concentration of H 3 O + , therefore, is 1.87 · 10 –9 , or a pH of 8.73. After the equivalence point NaOH is present in excess, and the pH is deter- mined in the same manner as in the titration of a strong acid with a strong base. For example, after adding 60.0 mL of NaOH, the concentration of OH – is giving a pH of 11.96. Table 9.3 and Figure 9.6 show additional results for this titra- tion. The calculations for the titration of a weak base with a strong acid are handled in a similar manner except that the initial pH is determined by the weak base, the pH at the equivalence point by its conjugate weak acid, and the pH after the equiva- lence point by the concentration of excess strong acid. [] (. )( . ) (. )( . ) .. .OH M mL M mL mL mL M ? = ? + = 0 100 60 0 0 100 50 0 50 0 60 0 0 00909 K xx x b 3 3 OH CH COOH] [CH COO == ? =× ? ? ? [][ ] ()() . . 0 0500 571 10 10 Chapter 9 Titrimetric Methods of Analysis 283 pH Volume NaOH (mL) 0 0.00 4.00 6.00 8.00 10.00 12.00 14.00 20 40 60 100 2.00 80 Table 9 . 3 Data for Titration of 50.0 mL of 0.100 M Acetic Acid with 0.100 M NaOH Volume of NaOH (mL) pH 0.00 2.88 5.00 3.81 10.00 4.16 15.00 4.39 20.00 4.58 25.00 4.76 30.00 4.94 35.00 5.13 40.00 5.36 45.00 5.71 48.00 6.14 50.00 8.73 52.00 11.29 55.00 11.68 60.00 11.96 65.00 12.12 70.00 12.22 75.00 12.30 80.00 12.36 85.00 12.41 90.00 12.46 95.00 12.49 100.00 12.52 Figure 9.6 Titration curve for 50.0 mL of 0.100 M acetic acid (pK a = 4.76) with 0.100 M NaOH. 1400-CH09 9/9/99 2:12 PM Page 283 284 Modern Analytical Chemistry The approach that we have worked out for the titration of a monoprotic weak acid with a strong base can be extended to reactions involving multiprotic acids or bases and mixtures of acids or bases. As the complexity of the titration increases, however, the necessary calculations become more time-consuming. Not surpris- ingly, a variety of algebraic 1 and computer spreadsheet 2 approaches have been de- scribed to aid in constructing titration curves. Sketching an Acid—Base Titration Curve To evaluate the relationship between an equivalence point and an end point, we only need to construct a reasonable approx- imation to the titration curve. In this section we demonstrate a simple method for sketching any acid–base titration curve. Our goal is to sketch the titration curve quickly, using as few calculations as possible. To quickly sketch a titration curve we take advantage of the following observa- tion. Except for the initial pH and the pH at the equivalence point, the pH at any point of a titration curve is determined by either an excess of strong acid or strong base, or by a buffer consisting of a weak acid and its conjugate weak base. As we have seen in the preceding sections, calculating the pH of a solution containing ex- cess strong acid or strong base is straightforward. We can easily calculate the pH of a buffer using the Henderson–Hasselbalch equa- tion. We can avoid this calculation, however, if we make the following assumption. You may recall that in Chapter 6 we stated that a buffer operates over a pH range ex- tending approximately – 1 pH units on either side of the buffer’s pK a . The pH is at the lower end of this range, pH = pK a – 1, when the weak acid’s concentration is approxi- mately ten times greater than that of its conjugate weak base. Conversely, the buffer’s pH is at its upper limit, pH = pK a + 1, when the concentration of weak acid is ten times less than that of its conjugate weak base. When titrating a weak acid or weak base, therefore, the buffer region spans a range of volumes from approximately 10% of the equivalence point volume to approximately 90% of the equivalence point volume.* Our strategy for quickly sketching a titration curve is simple. We begin by draw- ing our axes, placing pH on the y-axis and volume of titrant on the x-axis. After calcu- lating the volume of titrant needed to reach the equivalence point, we draw a vertical line that intersects the x-axis at this volume. Next, we determine the pH for two vol- umes before the equivalence point and for two volumes after the equivalence point. To save time we only calculate pH values when the pH is determined by excess strong acid or strong base. For weak acids or bases we use the limits of their buffer region to esti- mate the two points. Straight lines are drawn through each pair of points, with each line intersecting the vertical line representing the equivalence point volume. Finally, a smooth curve is drawn connecting the three straight-line segments. Example 9.1 illus- trates this approach for the titration of a weak acid with a strong base. EXAMPLE 9 .1 Sketch the titration curve for the titration of 50.0 mL of 0.100 M acetic acid with 0.100 M NaOH. This is the same titration for which we previously calculated the titration curve (Table 9.3 and Figure 9.6). SOLUTION We begin by drawing the axes for the titration curve (Figure 9.7a). We have already shown that the volume of NaOH needed to reach the equivalence point is 50 mL, so we draw a vertical line intersecting the x-axis at this volume (Figure 9.7b). *Question 4 in the end-of-chapter problems asks you to consider why these pH limits correspond to approximately 10% and 90% of the equivalence point volume. 1400-CH09 9/9/99 2:12 PM Page 284 Chapter 9 Titrimetric Methods of Analysis 285 pH Volume of titrant Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 (a) pH Volume of titrant (b) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 pH Volume of titrant (c) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 pH Volume of titrant (d) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 pH Volume of titrant (e) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 pH Volume of titrant (f) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 Figure 9.7 How to sketch an acid–base titration curve; see text for explanation. 1400-CH09 9/9/99 2:12 PM Page 285 Figure 9.8 Sketches of titration curves for (a) 50.00 mL of 0.0500 M diprotic weak acid (pK a1 =3, pK a2 = 7) with 0.100 M strong base; and (b) 50.00 mL of a mixture of weak acids consisting of 0.075 M HA (pK a,HA = 3) and 0.025 M HB (pK a,HB = 7) with 0.100 M strong base. The points used to sketch the titration curves are indicated by the dots (?). Equivalence points are indicated by the arrows. Before the equivalence point the titrant is the limiting reagent, and the pH is controlled by a buffer consisting of unreacted acetic acid and its conjugate weak base, acetate. The pH limits for the buffer region are plotted by superimposing the ladder diagram for acetic acid on the y-axis (Figure 9.7c) and adding the appropriate points at 10% (5.0 mL) and 90% (45.0 mL) of the equivalence point volume. After the equivalence point the pH is controlled by the concentration of excess NaOH. Again, we have already done this calculation. Using values from Table 9.3, we plot two additional points. An approximate sketch of the titration curve is completed by drawing separate straight lines through the two points in the buffer region and the two points in the excess titrant region (Figure 9.7e). Finally, a smooth curve is drawn connecting the three straight-line segments (Figure 9.7f). This approach can be used to sketch titration curves for other acid–base titra- tions including those involving polyprotic weak acids and bases or mixtures of weak acids and bases (Figure 9.8). Figure 9.8a, for example, shows the titration curve when titrating a diprotic weak acid, H 2 A, with a strong base. Since the analyte is 286 Modern Analytical Chemistry pH Volume of titrant (a) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 pH Volume of titrant (b) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 12.0 50.00 500 100 200150 100.00 2.0 14.0 1400-CH09 9/9/99 2:12 PM Page 286 Figure 9.9 Titration curves for 50.00 mL of 0.100 M weak acid with 0.100 M strong base. The pK a s of the weak acids are (a) 1, (b) 3, (c) 5, (d) 7, (e) 9, (f) 11. diprotic there are two equivalence points, each requiring the same volume of titrant. Before the first equivalence point the pH is controlled by a buffer consisting of H 2 A and HA – , and the HA – /A 2– buffer determines the pH between the two equiv- alence points. After the second equivalence point, the pH reflects the concentration of the excess strong base titrant. Figure 9.8b shows a titration curve for a mixture consisting of two weak acids: HA and HB. Again, there are two equivalence points. In this case, however, the equivalence points do not require the same volume of titrant because the concen- tration of HA is greater than that for HB. Since HA is the stronger of the two weak acids, it reacts first; thus, the pH before the first equivalence point is controlled by the HA/A – buffer. Between the two equivalence points the pH reflects the titration of HB and is determined by the HB/B – buffer. Finally, after the second equivalence point, the excess strong base titrant is responsible for the pH. 9 B.2 Selecting and Evaluating the End Point Earlier we made an important distinction between an end point and an equivalence point. The difference between these two terms is important and deserves repeating. The equivalence point occurs when stoichiometrically equal amounts of analyte and titrant react. For example, if the analyte is a triprotic weak acid, a titration with NaOH will have three equivalence points corresponding to the addition of one, two, and three moles of OH – for each mole of the weak acid. An equivalence point, therefore, is a theoretical not an experimental value. An end point for a titration is determined experimentally and represents the analyst’s best estimate of the corresponding equivalence point. Any difference be- tween an equivalence point and its end point is a source of determinate error. As we shall see, it is even possible that an equivalence point will not have an associated end point. Where Is the Equivalence Point? We have already learned how to calculate the equivalence point for the titration of a strong acid with a strong base, and for the titration of a weak acid with a strong base. We also have learned to sketch a titra- tion curve with a minimum of calculations. Can we also locate the equivalence point without performing any calculations? The answer, as you may have guessed, is often yes! It has been shown 3 that for most acid–base titrations the inflection point, which corresponds to the greatest slope in the titration curve, very nearly coincides with the equivalence point. The inflection point actually precedes the equivalence point, with the error approaching 0.1% for weak acids or weak bases with dissocia- tion constants smaller than 10 –9 , or for very dilute solutions. Equivalence points de- termined in this fashion are indicated on the titration curves in Figure 9.8. The principal limitation to using a titration curve to locate the equivalence point is that an inflection point must be present. Sometimes, however, an inflection point may be missing or difficult to detect. Figure 9.9, for example, demonstrates the influence of the acid dissociation constant, K a , on the titration curve for a weak acid with a strong base titrant. The inflection point is visible, even if barely so, for acid dissociation constants larger than 10 –9 , but is missing when K a is 10 –11 . Another situation in which an inflection point may be missing or difficult to detect occurs when the analyte is a multiprotic weak acid or base whose successive dissociation constants are similar in magnitude. To see why this is true let’s con- sider the titration of a diprotic weak acid, H 2 A, with NaOH. During the titration the following two reactions occur. Chapter 9 Titrimetric Methods of Analysis 287 pH Volume of titrant 0.00 (f) (e) (d) (c) (b) (a) 0.0 4.0 6.0 8.0 10.0 12.0 20.00 60.00 2.0 40.00 14.0 1400-CH09 9/9/99 2:12 PM Page 287 Figure 9.10 Titration curves for (a) maleic acid, pK a1 = 1.91, pK a2 = 6.33; (b) malonic acid, pK a1 = 2.85, pK a2 = 5.70; (c) succinic acid, pK a1 = 4.21, pK a2 = 5.64. Titration curves are for 50.00 mL of 0.0500 M acid with 0.100 M strong base. Equivalence points for all three titrations occur at 25.00 and 50.00 mL of titrant. H 2 A(aq)+OH – (aq) fi HA – (aq)+H 2 O(l) 9.3 HA – (aq)+OH – (aq) fi A 2– (aq)+H 2 O(l) 9.4 Two distinct inflection points are seen if reaction 9.3 is essentially complete before reaction 9.4 begins. Figure 9.10 shows titration curves for three diprotic weak acids. The titra- tion curve for maleic acid, for which K a1 is approximately 20,000 times larger than K a2 , shows two very distinct inflection points. Malonic acid, on the other hand, has acid dissociation constants that differ by a factor of approximately 690. Although malonic acid’s titration curve shows two inflection points, the first is not as distinct as that for maleic acid. Finally, the titration curve for suc- cinic acid, for which the two K a values differ by a factor of only 27, has only a single inflection point corresponding to the neutralization of HC 4 H 4 O 4 – to C 4 H 4 O 4 2– . In general, separate inflection points are seen when successive acid dissociation constants differ by a factor of at least 500 (a D pK a of at least 2.7). Finding the End Point with a Visual Indicator One interesting group of weak acids and bases are derivatives of organic dyes. Because such com- pounds have at least one conjugate acid–base species that is highly colored, their titration results in a change in both pH and color. This change in color can serve as a useful means for determining the end point of a titration, pro- vided that it occurs at the titration’s equivalence point. The pH at which an acid–base indicator changes color is determined by its acid dissociation constant. For an indicator that is a monoprotic weak acid, HIn, the following dissociation reaction occurs HIn(aq)+H 2 O(l) t H 3 O + (aq)+In – (aq) for which the equilibrium constant is 9.5 Taking the negative log of each side of equation 9.5, and rearranging to solve for pH gives a familiar equation. 9.6 The two forms of the indicator, HIn and In – , have different colors. The color of a solution containing an indicator, therefore, continuously changes as the concen- tration of HIn decreases and the concentration of In – increases. If we assume that both HIn and In – can be detected with equal ease, then the transition between the two colors reaches its midpoint when their concentrations are identical or when the pH is equal to the indicator’s pK a . The equivalence point and the end point coin- cide, therefore, if an indicator is selected whose pK a is equal to the pH at the equiva- lence point, and the titration is continued until the indicator’s color is exactly halfway between that for HIn and In – . Unfortunately, the exact pH at the equiva- lence point is rarely known. In addition, detecting the point where the concentra- tions of HIn and In – are equal may be difficult if the change in color is subtle. We can establish a range of pHs over which the average analyst will observe a change in color if we assume that a solution of the indicator is the color of HIn whenever its concentration is ten times more than that of In – , and the color of In – pH p In HIn a =+ ? K log [] [] K a 3 In H O HIn = ?+ [][ ] [] 288 Modern Analytical Chemistry pH Volume of titrant (a) 0.00 0.0 4.0 6.0 8.0 10.0 12.0 40.00 80.00 2.0 60.0020.00 14.0 Maleic acid pH Volume of titrant (b) 0.00 0.0 4.0 6.0 8.0 10.0 12.0 40.00 80.00 2.0 60.0020.00 14.0 Malonic acid pH Volume of titrant (c) 0.00 0.0 4.0 6.0 8.0 10.0 12.0 40.00 80.00 2.0 60.0020.00 14.0 Succinic acid 1400-CH09 9/9/99 2:12 PM Page 288 Figure 9.11 Ladder diagram showing the range of pH levels over which a typical acid–base indicator changes color. whenever the concentration of HIn is ten times less than that of In – . Substi- tuting these inequalities into equation 9.6 shows that an indicator changes color over a pH range of – 1 units on either side of its pK a (Figure 9.11). Thus, the indicator will be the color of HIn when the pH is less than pK a – 1, and the color of In – for pHs greater than pK a +1. The pH range of an indicator does not have to be equally distributed on either side of the indicator’s pK a . For some indicators only the weak acid or weak base is col- ored. For other indicators both the weak acid and weak base are colored, but one form may be easier to see. In either case, the pH range is skewed toward those pH lev- els for which the less colored form of the indicator is present in higher concentration. A list of several common acid–base indicators, along with their pK a s, color changes, and pH ranges, is provided in the top portion of Table 9.4. In some cases, pH p p aa =+ =+KKlog 10 1 1 pH p p aa =+ =?KKlog 1 10 1 Chapter 9 Titrimetric Methods of Analysis 289 In – HIn pH = pK a,HIn + 1 pH = pK a,HIn – 1 pH = pK a,HIn Indicator’s color transition region pH Table 9 . 4 Properties of Selected Indicators, Mixed Indicators, and Screened Indicators for Acid–Base Titrations Indicator Acid Color Base Color pH Range pK a cresol red red yellow 0.2–1.8 — thymol blue red yellow 1.2–2.8 1.7 bromophenol blue yellow blue 3.0–4.6 4.1 methyl orange red orange 3.1–4.4 3.7 Congo red blue red 3.0–5.0 — bromocresol green yellow blue 3.8–5.4 4.7 methyl red red yellow 4.2–6.3 5.0 bromocresol purple yellow purple 5.2–6.8 6.1 litmus red blue 5.0–8.0 — bromothymol blue yellow blue 6.0–7.6 7.1 phenol red yellow red 6.8–8.4 7.8 cresol red yellow red 7.2–8.8 8.2 thymol blue yellow blue 8.0–9.6 8.9 phenolphthalein colorless red 8.3–10.0 9.6 alizarin yellow R yellow orange/red 10.1–12.0 — Mixed Indicator Acid Color Base Color pH Range bromocresol green and methyl orange orange blue-green 3.5–4.3 bromocresol green and chlorophenol red yellow-green blue-violet 5.4–6.2 bromothymol blue and phenol red yellow violet 7.2–7.6 Screened Indicator Acid Color Base Color pH Range dimethyl yellow and methylene blue blue-violet green 3.2–3.4 methyl red and methylene blue red-violet green 5.2–5.6 neutral red and methylene blue violet-blue green 6.8–7.3 1400-CH09 9/9/99 2:12 PM Page 289 Figure 9.12 Titration curve for 50.00 mL of 0.100 M CH 3 COOH with 0.100 M NaOH showing the range of pHs and volumes of titrant over which the indicators bromothymol blue and phenolphthalein are expected to change color. mixed indicators, which are a mixture of two or more acid–base indicators, provide a narrower range of pHs over which the color change occurs. A few examples of such mixed indicators are included in the middle portion of Table 9.4. Adding a neutral screening dye, such as methylene blue, also has been found to narrow the pH range over which an indicator changes color (lower portion of Table 9.4). In this case, the neutral dye provides a gray color at the midpoint of the indicator’s color transition. The relatively broad range of pHs over which any indicator changes color places additional limitations on the feasibility of a titration. To minimize a determi- nate titration error, an indicator’s entire color transition must lie within the sharp transition in pH occurring near the equivalence point. Thus, in Figure 9.12 we see that phenolphthalein is an appropriate indicator for the titration of 0.1 M acetic acid with 0.1 M NaOH. Bromothymol blue, on the other hand, is an inappropriate indicator since its change in color begins before the initial sharp rise in pH and, as a result, spans a relatively large range of volumes. The early change in color increases the probability of obtaining inaccurate results, and the range of possible end point volumes increases the probability of obtaining imprecise results. The need for the indicator’s color transition to occur in the sharply rising por- tion of the titration curve justifies our earlier statement that not every equivalence point has an end point. For example, trying to use a visual indicator to find the first equivalence point in the titration of succinic acid (see Figure 9.10c) is pointless since any difference between the equivalence point and the end point leads to a large titration error. Finding the End Point by Monitoring pH An alternative approach to finding a titration’s end point is to monitor the titration reaction with a suitable sensor whose signal changes as a function of the analyte’s concentration. Plotting the data gives us the resulting titration curve. The end point may then be determined from the titration curve with only a minimal error. The most obvious sensor for an acid–base titration is a pH electrode.* For ex- ample, Table 9.5 lists values for the pH and volume of titrant obtained during the titration of a weak acid with NaOH. The resulting titration curve, which is called a potentiometric titration curve, is shown in Figure 9.13a. The simplest method for finding the end point is to visually locate the inflection point of the titration curve. This is also the least accurate method, particularly if the titration curve’s slope at the equivalence point is small. 290 Modern Analytical Chemistry pH Volume of titrant 0.00 0.0 4.0 6.0 8.0 10.0 12.0 20.00 70.00 2.0 30.00 40.00 50.00 60.0010.00 14.0 Phenolphthalein Bromothymol blue *See Chapter 11 for more details about pH electrodes. 1400-CH09 9/9/99 2:12 PM Page 290 Chapter 9 Titrimetric Methods of Analysis 291 Table 9 . 5 Data for the Titration of a Weak Acid with 0.100 M NaOH Normal Titration First Derivative Second Derivative Volume Volume Volume (mL) pH (mL) (mL) 0.00 2.89 1.00 0.815 2.00 4.52 2.00 –0.273 3.00 0.270 4.00 5.06 5.00 –0.033 7.00 0.138 10.00 5.89 9.00 –0.002 11.00 0.130 12.00 6.15 12.00 0.055 13.00 0.240 14.00 6.63 13.75 0.140 14.50 0.450 15.00 7.08 14.88 1.19 15.25 1.34 15.50 7.75 15.40 17.2 15.55 6.50 15.60 8.40 15.60 24.0 15.65 8.90 15.70 9.29 15.70 –11.0 15.75 7.80 15.80 10.07 15.83 –32.8 15.90 2.90 16.00 10.65 16.20 –3.68 16.50 0.690 17.00 11.34 17.00 –0.400 17.50 0.290 18.00 11.63 18.25 –0.070 19.00 0.185 20.00 12.00 20.50 –0.027 22.00 0.103 24.00 12.41 ? ? 2 pH 2 V ? ? pH V Another method for finding the end point is to plot the first or second deriva- tive of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as D pH/D V, where D pH is the change in pH between successive additions of titrant. For exam- ple, the initial point in the first derivative titration curve for the data in Table 9.5 is and is plotted at the average of the two volumes (1.00 mL). The remaining data for the first derivative titration curve are shown in Table 9.5 and plotted in Figure 9.13b. ? ? pH V = ? ? = 452 289 200 000 0 815 .. .. . 1400-CH09 9/9/99 2:12 PM Page 291 Figure 9.13 Titration curves for a weak acid with 0.100 M NaOH—(a) normal titration curve; (b) first derivative titration curve; (c) second derivative titration curve; (d) Gran plot. The second derivative of a titration curve may be more useful than the first de- rivative, since the end point is indicated by its intersection with the volume axis. The second derivative is approximated as D (D pH/D V)/D V, or D 2 pH/D V 2 . For the titration data in Table 9.5, the initial point in the second derivative titration curve is and is plotted as the average of the two volumes (2.00 mL). The remainder of the data for the second derivative titration curve are shown in Table 9.5 and plotted in Figure 9.13c. Derivative methods are particularly well suited for locating end points in multi- protic and multicomponent systems, in which the use of separate visual indicators for each end point is impractical. The precision with which the end point may be located also makes derivative methods attractive for the analysis of samples with poorly defined normal titration curves. Derivative methods work well only when sufficient data are recorded during the sharp rise in pH occurring near the equivalence point. This is usually not a problem when the titration is conducted with an automatic titrator, particularly when operated under computer control. Manual titrations, however, often contain only a few data points in the equivalence point region, due to the limited range of volumes over which the transition in pH occurs. Manual titrations are, however, information-rich during the more gently rising portions of the titration curve be- fore and after the equivalence point. Consider again the titration of a monoprotic weak acid, HA, with a strong base. At any point during the titration the weak acid is in equilibrium with H 3 O + and A – HA(aq)+H 2 O(l) t H 3 O + (aq)+A – ? ? 2 0 270 0 815 300 100 0 273 pH 2 V = ? ? =? .. .. . 292 Modern Analytical Chemistry pH Volume of titrant (mL) 0 0 4 6 8 10 12 10 25 2 15 205 14 ? pH/ ? V Volume of titrant (mL) 0 0 4 6 8 10 25 2 15 205 10 ? 2 pH/ ? V 2 Volume of titrant (mL) 2 –40 10 0 20 30 10 22 –10 –20 –30 14 16 18 201264 8 40 V b [H + ] · 10 7 Volume of titrant (mL) 10 0 80 100 120 12 16 60 40 20 13 14 1511 140 (a) (b) (c) (d) 1400-CH09 9/9/99 2:12 PM Page 292 for which Before the equivalence point, and for volumes of titrant in the titration curve’s buffer region, the concentrations of HA and A – are given by the following equations. Substituting these equations into the K a expression for HA, and rearranging gives Finally, recognizing that the equivalence point volume is leaves us with the following equation. V b · [H 3 O + ]=K a · V eq – K a · V b For volumes of titrant before the equivalence point, a plot of V b · [H 3 O + ] versus V b is a straight line with an x-intercept equal to the volume of titrant at the end point and a slope equal to –K a .* Results for the data in Table 9.5 are shown in Table 9.6 and plotted in Figure 9.13d. Plots such as this, which convert a portion of a titration curve into a straight line, are called Gran plots. Finding the End Point by Monitoring Temperature The reaction between an acid and a base is exothermic. Heat generated by the reaction increases the temperature of the titration mixture. The progress of the titration, therefore, can be followed by monitoring the change in temperature. An idealized thermometric titration curve (Figure 9.14a) consists of three distinct linear regions. Before adding titrant, any change in temperature is due to the cooling or warming of the solution containing the analyte. Adding titrant ini- tiates the exothermic acid–base reaction, resulting in an increase in temperature. This part of a thermometric titration curve is called the titration branch. The V MV M eq aa b = KMV KMV MV KMV M KV V aaa abb 3 bb aaa b ab 3 b HO HO ?= ?= + + [] [] K MV V MV MV V V MV MV MV a 3bbab aa bb a b 3bb aa bb HO V HO = + ?+ = ? + + [ ]( )/( ) ( )/( ) [] []A moles OH added total volume bb ab ? ? == + MV VV [HA] = moles HA moles OH added total volume aa bb ab ? = ? + ? MV MV VV K a 3 HO A HA] = +? [][] [ Chapter 9 Titrimetric Methods of Analysis 293 *Values of K a determined by this method may be in substantial error if the effect of activity is not considered. Gran plot A linearized form of a titration curve. 1400-CH09 9/9/99 2:12 PM Page 293 temperature continues to rise with each addition of titrant until the equivalence point is reached. After the equivalence point, any change in temperature is due to the difference between the temperatures of the analytical solution and the titrant, and the enthalpy of dilution for the excess titrant. Actual thermometric titration curves (Figure 9.14b) frequently show curvature at the intersection of the titration branch and the excess titrant branch due to the incompleteness of the neutraliza- tion reaction, or excessive dilution of the analyte during the titration. The latter problem is minimized by using a titrant that is 10–100 times more concentrated than the analyte, although this results in a very small end point volume and a larger relative error. The end point is indicated by the intersection of the titration branch and the excess titrant branch. In the idealized thermometric titration curve (see Figure 9.14a) the end point is easily located. When the intersection between the two branches is curved, the end point can be found by extrapolation (Figure 9.14b). Although not commonly used, thermometric titrations have one distinct ad- vantage over methods based on the direct or indirect monitoring of pH. As dis- cussed earlier, visual indicators and potentiometric titration curves are limited by the magnitude of the relevant equilibrium constants. For example, the titra- tion of boric acid, H 3 BO 3 , for which K a is 5.8 · 10 –10 , yields a poorly defined equivalence point (Figure 9.15a). The enthalpy of neutralization for boric acid with NaOH, however, is only 23% less than that for a strong acid (–42.7 kJ/mol 294 Modern Analytical Chemistry Table 9 .6 Gran Plot Treatment of the Data in Table 9.5 Volume (mL) V b · [H 3 O + ] · 10 7 2.00 604 4.00 348 10.00 129 12.00 85.0 14.00 33.8 15.00 12.5 15.55 2.76 15.60 0.621 15.70 0.0805 15.80 0.0135 Figure 9.14 Thermometric titration curves—(a) ideal; (b) showing curvature at the intersection of the titration and excess titrant branches. Equivalence points are indicated by the dots (?). T emperature V titr 0 Titration branch Excess titration branch T emperature V titr 0 (a) (b) pH Volume of titrant 0.00 0.0 4.0 6.0 8.0 10.0 12.0 2.00 6.00 2.0 3.00 4.00 5.001.00 14.0 T emperature Volume of titrant –1 24.980 25.020 25.060 162 3 4 50 25.100 Figure 9.15 Titration curves for 50.00 mL of 0.0100 M H 3 BO 3 with 0.100 M NaOH determined by monitoring (a) pH, and (b) temperature. (a) (b) 1400-CH09 9/9/99 2:12 PM Page 294 for H 3 BO 3 versus –55.6 kJ/mol for HCl), resulting in a favorable thermometric titration curve (Figure 9.15b). 9 B. 3 Titrations in Nonaqueous Solvents Thus far we have assumed that the acid and base are in an aqueous solution. In- deed, water is the most common solvent in acid–base titrimetry. When considering the utility of a titration, however, the solvent’s influence cannot be ignored. The dissociation, or autoprotolysis constant for a solvent, SH, relates the con- centration of the protonated solvent, SH 2 + , to that of the deprotonated solvent, S – . For amphoteric solvents, which can act as both proton donors and proton accep- tors, the autoprotolysis reaction is 2SH t SH 2 + +S – with an equilibrium constant of K s = [SH 2 + ][S – ] You should recognize that K w is just the specific form of K s for water. The pH of a solution is now seen to be a general statement about the relative abundance of pro- tonated solvent pH = –log[SH 2 + ] where the pH of a neutral solvent is given as Perhaps the most obvious limitation imposed by K s is the change in pH during a titration. To see why this is so, let’s consider the titration of a 50 mL solution of 10 –4 M strong acid with equimolar strong base. Before the equivalence point, the pH is determined by the untitrated strong acid, whereas after the equivalence point the concentration of excess strong base determines the pH. In an aqueous solution the concentration of H 3 O + when the titration is 90% complete is corresponding to a pH of 5.3. When the titration is 110% complete, the concentra- tion of OH – is or a pOH of 5.3. The pH, therefore, is pH=pK w – pOH = 14.0 – 5.3 = 8.7 The change in pH when the titration passes from 90% to 110% completion is D pH = 8.7 – 5.3 = 3.4 [] (( . OH M)(55 mL) M)(50 mL) M bb aa ab ? ?? ? = ? + = ×?× + =× MV MV VV 110 110 50 55 48 10 44 6 [] (( . HO M)(50 mL) M)(45 mL) M 3 + aa bb ab = ? + = ×?× + =× ?? ? MV MV VV 110 110 50 45 53 10 44 6 pH p neut s = 1 2 K Chapter 9 Titrimetric Methods of Analysis 295 1400-CH09 9/9/99 2:12 PM Page 295 If the same titration is carried out in a nonaqueous solvent with a K s of 1.0 · 10 –20 , the pH when the titration is 90% complete is still 5.3. However, the pH when the titration is 110% complete is now pH=pK s – pOH = 20.0 – 5.3 = 14.7 In this case the change in pH of D pH = 14.7 – 5.3 = 9.4 is significantly greater than that obtained when the titration is carried out in water. Figure 9.16 shows the titration curves in both the aque- ous and nonaqueous solvents. Nonaqueous solvents also may be used to increase the change in pH when titrating weak acids or bases (Fig- ure 9.17). Another parameter affecting the feasibility of a titration is the dis- sociation constant of the acid or base being titrated. Again, the solvent plays an important role. In the Br?nsted–Lowry view of acid–base be- havior, the strength of an acid or base is a relative measure of the ease with which a proton is transferred from the acid to the solvent, or from the solvent to the base. For example, the strongest acid that can exist in water is H 3 O + . The acids HCl and HNO 3 are considered strong because they are better proton donors than H 3 O + . Strong acids essentially donate all their protons to H 2 O, “leveling” their acid strength to that of H 3 O + . In a different solvent HCl and HNO 3 may not behave as strong acids. When acetic acid, which is a weak acid, is placed in water, the dis- sociation reaction CH 3 COOH(aq)+H 2 O(l) t H 3 O + (aq)+CH 3 COO – (aq) does not proceed to a significant extent because acetate is a stronger base than water and the hydronium ion is a stronger acid than acetic acid. If acetic acid is placed in a solvent that is a stronger base than water, such as ammonia, then the reaction CH 3 COOH + NH 3 t NH 4 + +CH 3 COO – proceeds to a greater extent. In fact, HCl and CH 3 COOH are both strong acids in ammonia. All other things being equal, the strength of a weak acid increases if it is placed in a solvent that is more basic than water, whereas the strength of a weak base in- creases if it is placed in a solvent that is more acidic than water. In some cases, how- ever, the opposite effect is observed. For example, the pK b for ammonia is 4.76 in water and 6.40 in the more acidic glacial acetic acid. In contradiction to our expec- tations, ammonia is a weaker base in the more acidic solvent. A full description of the solvent’s effect on a weak acid’s pK a or on the pK b of a weak base is beyond the scope of this text. You should be aware, however, that titrations that are not feasible in water may be feasible in a different solvent. 9 B. 4 Representative Method Although each acid–base titrimetric method has its own unique considerations, the following description of the determination of protein in bread provides an instruc- tive example of a typical procedure. 296 Modern Analytical Chemistry pH Volume of titrant 0.00 0.0 10.0 15.0 100.00 5.0 20.00 40.00 (b) (a) 60.00 80.00 20.0 pH Volume of titrant 0.00 0.0 10.0 15.0 100.00 5.0 20.00 40.00 (b) (a) 60.00 80.00 20.0 Figure 9.16 Titration curves for 50.00 mL of 10 –4 M HCl with 10 –4 M NaOH in (a) water, K w =l· 10 –14 , and (b) nonaqueous solvent, K s =1· 10 –20 . Figure 9.17 Titration curves for 50.00 mL of 0.100 M weak acid (pK a = 11) with 0.100 M NaOH in (a) water, K w =1· 10 –14 ; and (b) nonaqueous solvent, K s =1· 10 –20 . The titration curve in (b) assumes that the change in solvent has no effect on the acid dissociation constant of the weak acid. leveling Acids that are better proton donors than the solvent are leveled to the acid strength of the protonated solvent; bases that are better proton acceptors than the solvent are leveled to the base strength of the deprotonated solvent. 1400-CH09 9/9/99 2:12 PM Page 296 Chapter 9 Titrimetric Methods of Analysis 297 Representative Methods —Continued Method 9 .1 Determination of Protein in Bread 4 Description of the Method. This quantitative method of analysis for proteins is based on a determination of the %w/w N in the sample. Since different cereal proteins have similar amounts of nitrogen, the experimentally determined %w/w N is multiplied by a factor of 5.7 to give the %w/w protein in the sample (on average there are 5.7 g of cereal protein for every gram of nitrogen). As described here, nitrogen is determined by the Kjeldahl method. The protein in a sample of bread is oxidized in hot concentrated H 2 SO 4 , converting the nitrogen to NH 4 + . After making the solution alkaline, converting NH 4 + to NH 3 , the ammonia is distilled into a flask containing a known amount of standard strong acid. Finally, the excess strong acid is determined by a back titration with a standard strong base titrant. Procedure. Transfer a 2.0-g sample of bread, which has previously been air dried and ground into a powder, to a suitable digestion flask, along with 0.7 g of HgO as a catalyst, 10 g of K 2 SO 4 , and 25 mL of concentrated H 2 SO 4 . Bring the solution to a boil, and continue boiling until the solution turns clear, and for at least an additional 30 min. After cooling to below room temperature, add 200 mL of H 2 O and 25 mL of 4% w/v K 2 S to remove the Hg 2+ catalyst. Add a few Zn granules to serve as boiling stones, and 25 g of NaOH. Quickly connect the flask to a distillation apparatus, and distill the NH 3 into a collecting flask containing a known amount of standardized HCl. The tip of the condenser should be placed below the surface of the strong acid. After the distillation is complete, titrate the excess strong acid with a standard solution of NaOH, using methyl red as a visual indicator. Questions 1. Oxidizing the protein converts the nitrogen to NH 4 + . Why is the amount of nitrogen not determined by titrating the NH 4 + with a strong base? There are two reasons for not titrating the ammonium ion. First, NH 4 + is a very weak acid (K a = 5.7 · 10 –10 ) that yields a poorly defined end point when titrated with a strong base. Second, even if the end point can be determined with acceptable accuracy and precision, the procedure calls for adding a substantial amount of H 2 SO 4 . After the oxidation is complete, the amount of excess H 2 SO 4 will be much greater than the amount of NH 4 + that is produced. The presence of two acids that differ greatly in concentration makes for a difficult analysis. If the titrant’s concentration is similar to that of H 2 SO 4 , then the equivalence point volume for the titration of NH 4 + may be too small to measure reliably. On the other hand, if the concentration of the titrant is similar to that of NH 4 + , the volume needed to neutralize the H 2 SO 4 will be unreasonably large. 2. Ammonia is a volatile compound as evidenced by the strong smell of even dilute solutions. This volatility presents a possible source of determinate error. Will this determinate error be negative or positive? The conversion of N to NH 3 follows the following pathway N fi NH 4 + NH 4 + fi NH 3 Any loss of NH 3 is loss of analyte and a negative determinate error. The photo in Colorplate 8a shows the indicator s color change for this titration. 1400-CH09 9/9/99 2:12 PM Page 297 9 B. 5 Quantitative Applications Although many quantitative applications of acid–base titrimetry have been replaced by other analytical methods, there are several important applications that continue to be listed as standard methods. In this section we review the general application of acid–base titrimetry to the analysis of inorganic and organic compounds, with an emphasis on selected applications in environmental and clinical analysis. First, however, we discuss the selection and standardization of acidic and basic titrants. Selecting and Standardizing a Titrant Most common acid–base titrants are not readily available as primary standards and must be standardized before they can be used in a quantitative analysis. Standardization is accomplished by titrating a known amount of an appropriate acidic or basic primary standard. The majority of titrations involving basic analytes, whether conducted in aque- ous or nonaqueous solvents, use HCl, HClO 4 , or H 2 SO 4 as the titrant. Solutions of these titrants are usually prepared by diluting a commercially available concentrated stock solution and are stable for extended periods of time. Since the concentrations of concentrated acids are known only approximately,* the titrant’s concentration is determined by standardizing against one of the primary standard weak bases listed in Table 9.7. The most common strong base for titrating acidic analytes in aqueous solutions is NaOH. Sodium hydroxide is available both as a solid and as an approximately 50% w/v solution. Solutions of NaOH may be standardized against any of the pri- mary weak acid standards listed in Table 9.7. The standardization of NaOH, how- ever, is complicated by potential contamination from the following reaction be- tween CO 2 and OH – . CO 2 (g) + 2OH – (aq) fi CO 3 2– (aq)+H 2 O(l) 9.7 When CO 2 is present, the volume of NaOH used in the titration is greater than that needed to neutralize the primary standard because some OH – reacts with the CO 2 . 298 Modern Analytical Chemistry 3. Discuss the steps taken in this procedure to minimize this determinate error. Three specific steps are taken to minimize the loss of ammonia: (1) the solution is cooled to below room temperature before adding NaOH; (2) the digestion flask is quickly connected to the distillation apparatus after adding NaOH; and (3) the condenser tip of the distillation apparatus is placed below the surface of the HCl to ensure that the ammonia will react with the HCl before it can be lost through volatilization. 4. How does K 2 S remove Hg 2+ , and why is this important? Adding sulfide precipitates the Hg 2+ as HgS. This is important because NH 3 forms stable complexes with many metal ions, including Hg 2+ . Any NH 3 that is complexed with Hg 2+ will not be collected by distillation, providing another source of determinate error. Continued from page 297 *The nominal concentrations are 12.1 M HCl, 11.7 M HClO 4 , and 18.0 M H 2 SO 4 . 1400-CH09 9/9/99 2:12 PM Page 298 The calculated concentration of OH – , therefore, is too small. This is not a problem when titrations involving NaOH are restricted to an end point pH less than 6. Below this pH any CO 3 2– produced in reaction 9.7 reacts with H 3 O + to form car- bonic acid. CO 3 2– (aq)+2H 3 O + (aq) fi H 2 CO 3 (aq)+2H 2 O(l) 9.8 Combining reactions 9.7 and 9.8 gives an overall reaction of CO 2 (g)+H 2 O(l) fi H 2 CO 3 (aq) which does not include OH – . Under these conditions the presence of CO 2 does not affect the quantity of OH – used in the titration and, therefore, is not a source of de- terminate error. For pHs between 6 and 10, however, the neutralization of CO 3 2– requires only one proton CO 3 2– (aq)+H 3 O + (aq) fi HCO 3 – (aq)+H 2 O(l) and the net reaction between CO 2 and OH – is CO 2 (g)+OH – (aq) fi HCO 3 – (aq) Under these conditions some OH – is consumed in neutralizing CO 2 . The result is a determinate error in the titrant’s concentration. If the titrant is used to analyze an analyte that has the same end point pH as the primary standard used during stan- dardization, the determinate errors in the standardization and the analysis cancel, and accurate results may still be obtained. Solid NaOH is always contaminated with carbonate due to its contact with the atmosphere and cannot be used to prepare carbonate-free solutions of NaOH. So- lutions of carbonate-free NaOH can be prepared from 50% w/v NaOH since Na 2 CO 3 is very insoluble in concentrated NaOH. When CO 2 is absorbed, Na 2 CO 3 Chapter 9 Titrimetric Methods of Analysis 299 Table 9 . 7 Selected Primary Standards for the Standardization of Strong Acid and Strong Base Titrants Standardization of Acidic Titrants Primary Standard Titration Reaction Comment Na 2 CO 3 Na 2 CO 3 +2H 3 O + fi H 2 CO 3 + 2Na + +2H 2 Oa TRIS (HOCH 2 ) 3 CNH 2 +H 3 O + fi (HOCH 2 ) 3 CNH 3 + +H 2 Ob Na 2 B 4 O 7 Na 2 B 4 O 7 +2H 3 O + +3H 2 O fi 2Na + +4H 3 BO 3 Standardization of Basic Titrants Primary Standard Titration Reaction Comment KHC 8 H 4 O 4 KHC 8 H 4 O 4 +OH – fi K + +C 8 H 4 O 4 2– +H 2 Oc C 6 H 5 COOH C 6 H 5 COOH + OH – fi C 6 H 5 COO – +H 2 Od KH(IO 3 ) 2 KH(IO 3 ) 2 +OH – fi K + + 2IO 3 – +H 2 O a The end point for this titration is improved by titrating to the second equivalence point, boiling the solution to expel CO 2 , and retitrating to the second equivalence point. In this case the reaction is Na 2 CO 3 +2H 3 O + fi CO 2 + 2Na + +3H 2 O b TRIS stands for tris-(hydroxymethyl)aminomethane. c KHC 8 H 4 O 4 is also known as potassium hydrogen phthalate, or KHP. d Due to its poor solubility in water, benzoic acid is dissolved in a small amount of ethanol before being diluted with water. 1400-CH09 9/9/99 2:12 PM Page 299 precipitates and settles to the bottom of the container, allowing access to the carbonate-free NaOH. Dilution must be done with water that is free from dissolved CO 2 . Briefly boiling the water expels CO 2 and, after cooling, it may be used to pre- pare carbonate-free solutions of NaOH. Provided that contact with the atmosphere is minimized, solutions of carbonate-free NaOH are relatively stable when stored in polyethylene bottles. Standard solutions of sodium hydroxide should not be stored in glass bottles because NaOH reacts with glass to form silicate. Inorganic Analysis Acid–base titrimetry is a standard method for the quantitative analysis of many inorganic acids and bases. Standard solutions of NaOH can be used in the analysis of inorganic acids such as H 3 PO 4 or H 3 AsO 4 , whereas standard solutions of HCl can be used for the analysis of inorganic bases such as Na 2 CO 3 . Inorganic acids and bases too weak to be analyzed by an aqueous acid–base titration can be analyzed by adjusting the solvent or by an indirect analysis. For ex- ample, the accuracy in titrating boric acid, H 3 BO 3 , with NaOH is limited by boric acid’s small acid dissociation constant of 5.8 · 10 –10 . The acid strength of boric acid, however, increases when mannitol is added to the solution because it forms a com- plex with the borate ion. The increase in K a to approximately 1.5 · 10 –4 results in a sharper end point and a more accurate titration. Similarly, the analysis of ammo- nium salts is limited by the small acid dissociation constant of 5.7 · 10 –10 for NH 4 + . In this case, NH 4 + can be converted to NH 3 by neutralizing with strong base. The NH 3 , for which K b is 1.8 · 10 –5 , is then removed by distillation and titrated with a standard strong acid titrant. Inorganic analytes that are neutral in aqueous solutions may still be analyzed if they can be converted to an acid or base. For example, NO 3 – can be quantitatively analyzed by reducing it to NH 3 in a strongly alkaline solution using Devarda’s alloy, a mixture of 50% w/w Cu, 45% w/w Al, and 5% w/w Zn. 3NO 3 – (aq) + 8Al(s) + 5OH – (aq)+2H 2 O(l) fi 8AlO 2 – (aq) + 3NH 3 (aq) The NH 3 is removed by distillation and titrated with HCl. Alternatively, NO 3 – can be titrated as a weak base in an acidic nonaqueous solvent such as anhydrous acetic acid, using HClO 4 as a titrant. Acid–base titrimetry continues to be listed as the standard method for the de- termination of alkalinity, acidity, and free CO 2 in water and wastewater analysis. Al- kalinity is a measure of the acid-neutralizing capacity of a water sample and is as- sumed to arise principally from OH – , HCO 3 – , and CO 3 2– , although other weak bases, such as phosphate, may contribute to the overall alkalinity. Total alkalinity is determined by titrating with a standard solution of HCl or H 2 SO 4 to a fixed end point at a pH of 4.5, or to the bromocresol green end point. Alkalinity is reported as milligrams CaCO 3 per liter. When the sources of alkalinity are limited to OH – , HCO 3 – , and CO 3 2– , titra- tions to both a pH of 4.5 (bromocresol green end point) and a pH of 8.3 (phe- nolphthalein or metacresol purple end point) can be used to determine which species are present, as well as their respective concentrations. Titration curves for OH – , HCO 3 – , and CO 3 2– are shown in Figure 9.18. For a solution containing only OH – alkalinity, the volumes of strong acid needed to reach the two end points are identical. If a solution contains only HCO 3 – alkalinity, the volume of strong acid needed to reach the end point at a pH of 8.3 is zero, whereas that for the pH 4.5 end point is greater than zero. When the only source of alkalinity is CO 3 2– , the volume of strong acid needed to reach the end point at a pH of 4.5 is exactly twice that needed to reach the end point at a pH of 8.3. 300 Modern Analytical Chemistry alkalinity A measure of a water’s ability to neutralize acid. 1400-CH09 9/9/99 2:12 PM Page 300 Figure 9.18 Titration curves for (a) 50.00 mL of 0.100 M NaOH with 0.100 M HCl; (b) 50.00 mL of 0.100 M Na 2 CO 3 with 0.100 M HCl; and (c) 50.00 mL of 0.100 M NaHCO 3 with M HCl. The dashed lines indicate the pH 8.3 and pH 4.5 end points. Mixtures of OH – and CO 3 2– , or HCO 3 – and CO 3 2– alkalinities also are possible. Consider, for example, a mixture of OH – and CO 3 2– . The volume of strong acid needed to titrate OH – will be the same whether we titrate to the pH 8.3 or pH 4.5 end point. Titrating CO 3 2– to the end point at a pH of 4.5, however, requires twice as much strong acid as when titrating to the pH 8.3 end point. Consequently, when titrating a mixture of these two ions, the volume of strong acid needed to reach the pH 4.5 end point is less than twice that needed to reach the end point at a pH of 8.3. For a mixture of HCO 3 – and CO 3 2– , similar reasoning shows that the volume of strong acid needed to reach the end point at a pH of 4.5 is more than twice that need to reach the pH 8.3 end point. Solutions containing OH – and HCO 3 – alkalini- ties are unstable with respect to the formation of CO 3 2– and do not exist. Table 9.8 summarizes the relationship between the sources of alkalinity and the volume of titrant needed to reach the two end points. Acidity is a measure of a water sample’s capacity for neutralizing base and is conveniently divided into strong acid and weak acid acidity. Strong acid acidity is due to the presence of inorganic acids, such as HCl, HNO 3 , and H 2 SO 4 , and is com- monly found in industrial effluents and acid mine drainage. Weak acid acidity is usually dominated by the formation of H 2 CO 3 from dissolved CO 2 , but also Chapter 9 Titrimetric Methods of Analysis 301 pH Volume of titrant 0.00 0.0 4.0 6.0 8.0 10.0 12.0 70.00 2.0 10.00 20.00 30.00 40.00 50.00 60.00 14.0 pH Volume of titrant 0.00 0.0 4.0 6.0 8.0 10.0 12.0 140.00 2.0 20.00 40.00 60.00 80.00 100.00 120.00 14.0 0.00 70.0010.00 20.00 30.00 40.00 50.00 60.00 pH Volume of titrant 0.0 4.0 6.0 8.0 10.0 12.0 2.0 14.0 (a) (b) (c) acidity A measure of a water’s ability to neutralize base. 1400-CH09 9/9/99 2:12 PM Page 301 includes contributions from hydrolyzable metal ions such as Fe 3+ , Al 3+ , and Mn 2+ . In addition, weak acid acidity may include a contribution from organic acids. Acidity is determined by titrating with a standard solution of NaOH to fixed end points at pH 3.7 and pH 8.3. These end points are located potentiometrically, using a pH meter, or by using an appropriate indicator (bromophenol blue for pH 3.7, and metacresol purple or phenolphthalein for pH 8.3). Titrating to a pH of 3.7 provides a measure of strong acid acidity,* and titrating to a pH of 8.3 provides a measure of total acidity. Weak acid acidity is given indirectly as the difference be- tween the total and strong acid acidities. Results are expressed as the milligrams of CaCO 3 per liter that could be neutralized by the water sample’s acidity. An alterna- tive approach for determining strong and weak acidity is to obtain a potentiometric titration curve and use Gran plot methodology to determine the two equivalence points. This approach has been used, for example, in determining the forms of acid- ity in atmospheric aerosols. 5 Water in contact with either the atmosphere or carbonate-bearing sediments contains dissolved or free CO 2 that exists in equilibrium with gaseous CO 2 and the aqueous carbonate species H 2 CO 3 , HCO 3 – , and CO 3 2– . The concentration of free CO 2 is determined by titrating with a standard solution of NaOH to the phenol- phthalein end point, or to a pH of 8.3, with results reported as milligrams CO 2 per liter. This analysis is essentially the same as that for the determination of total acid- ity, and can only be applied to water samples that do not contain any strong acid acidity. Organic Analysis The use of acid–base titrimetry for the analysis of organic com- pounds continues to play an important role in pharmaceutical, biochemical, agri- cultural, and environmental laboratories. Perhaps the most widely employed acid–base titration is the Kjeldahl analysis for organic nitrogen, described earlier in Method 9.1. This method continues to be used in the analysis of caffeine and sac- charin in pharmaceutical products, as well as for the analysis of proteins, fertilizers, sludges, and sediments. Any nitrogen present in the –3 oxidation state is quantita- tively oxidized to NH 4 + . Some aromatic heterocyclic compounds, such as pyridine, are difficult to oxidize. A catalyst, such as HgO, is used to ensure that oxidation is complete. Nitrogen in an oxidation state other than –3, such as nitro- and azo- nitrogens, is often oxidized to N 2 , resulting in a negative determinate error. Adding a reducing agent, such as salicylic acid, reduces the nitrogen to a –3 oxidation state, 302 Modern Analytical Chemistry *This is sometimes referred to as “methyl orange acidity” since, at one time, methyl orange was the traditional indicator of choice. Table 9 .8 Relationship Between End Point Volumes and Sources of Alkalinity Source of Alkalinity Relationship Between End Point Volumes OH – V pH 4.5 = V pH 8.3 CO 3 2– V pH 4.5 =2· V pH 8.3 HCO 3 – V pH 8.3 = 0; V pH 4.5 > 0 OH – and CO 3 2– V pH 4.5 < 2 · V pH 8.3 CO 3 2– and HCO 3 – V pH 4.5 > 2 · V pH 8.3 Kjeldahl analysis An acid–base titrimetric method for determining the amount of nitrogen in organic compounds. 1400-CH09 9/9/99 2:12 PM Page 302 eliminating this source of error. Other examples of elemental analyses based on the conversion of the element to an acid or base are outlined in Table 9.9. Several organic functional groups have weak acid or weak base properties that allow their direct determination by an acid–base titration. Carboxylic (—COOH), sulfonic (—SO 3 H), and phenolic (—C 6 H 5 OH) functional groups are weak acids that can be successfully titrated in either aqueous or nonaqueous solvents. Sodium hydroxide is the titrant of choice for aqueous solutions. Nonaqueous titrations are often carried out in a basic solvent, such as ethylenediamine, using tetrabutylam- monium hydroxide, (C 4 H 9 ) 4 NOH, as the titrant. Aliphatic and aromatic amines are weak bases that can be titrated using HCl in aqueous solution or HClO 4 in glacial acetic acid. Other functional groups can be analyzed indirectly by use of a func- tional group reaction that produces or consumes an acid or base. Examples are shown in Table 9.10. Many pharmaceutical compounds are weak acids or bases that can be analyzed by an aqueous or nonaqueous acid–base titration; examples include salicylic acid, phenobarbital, caffeine, and sulfanilamide. Amino acids and proteins can be ana- lyzed in glacial acetic acid, using HClO 4 as the titrant. For example, a procedure for determining the amount of nutritionally available protein has been developed that is based on an acid–base titration of lysine residues. 6 Chapter 9 Titrimetric Methods of Analysis 303 Table 9 . 9 Selected Elemental Analyses Based on Acid–Base Titrimetry Element Liberated as Reaction Producing Acid or Base to Be Titrated a Titration NNH 3 (g)NH 3 (g)+H 3 O + (aq) fi NH 4 + (aq)+H 2 O(l) excess H 3 O + with strong base SSO 2 (g)SO 2 (g)+H 2 O 2 (aq) fi H 2 SO 4 (aq)H 2 SO 4 with strong base CC 2 (g)C 2 (g)+Ba(OH) 2 (aq) fi BaCO 3 (s)+H 2 O(l) excess Ba(OH) 2 with strong acid Cl HCl(g) HCl(g)+H 2 O(l) fi H 3 O + (aq)+Cl – (aq 3 O + with strong base F SiF 4 (g) 3SiF 4 (g)+2H 2 O(l) fi 2H 2 SiF 6 (aq) + SiO 2 (s)H 2 SiF 6 with strong base a The acid or base that is eventually titrated is indicated in bold. Table 9 .10 Selected Acid–Base Titrimetric Procedures for Organic Functional Groups Based on the Production or Consumption of Acid or Base Functional Group Reaction Producing Acid or Base to Be Titrated a Titration ester RCOOR’(aq)+OH – (aq) fi RCOO – (aq) + HOR’(aq) excess OH – with strong acid alcohol b [1] (CH 3 CO) 2 O + ROH fi CH 3 COOR + CH 3 COOH CH 3 COOH with strong base; ROH is determined [2] (CH 3 CO) 2 O+H 2 O fi 2CH 3 COOH from the difference in the amount of titrant needed to react with a blank consisting only of acetic anhydride, and the amount reacting with the sample. a The acid or base that is eventually titrated is indicated in bold. b The acetylation reaction, [1], is carried out in pyridine to avoid the hydrolysis of acetic anhydride by water. After the acetylation is complete, water is added to convert the remaining acetic anhydride to acetic acid, [2]. R C O NH OH HCI 22 — () ()aq aq+→ ? R C NOH H O 22 — () () ()aq aq++HCI l carbonyl HCI with strong base 1400-CH09 9/9/99 2:12 PM Page 303 304 Modern Analytical Chemistry Quantitative Calculations In acid–base titrimetry the quantitative relationship be- tween the analyte and the titrant is determined by the stoichiometry of the relevant reactions. As outlined in Section 2C, stoichiometric calculations may be simplified by focusing on appropriate conservation principles. In an acid–base reaction the number of protons transferred between the acid and base is conserved; thus The following example demonstrates the application of this approach in the direct analysis of a single analyte. EXAMPLE 9 .2 A 50.00-mL sample of a citrus drink requires 17.62 mL of 0.04166 M NaOH to reach the phenolphthalein end point (Figure 9.19a). Express the sample’s acidity in terms of grams of citric acid, C 6 H 8 O 7 , per 100 mL. SOLUTION Since citric acid is a triprotic weak acid, we must first decide to which equivalence point the titration has been carried. The three acid dissociation constants are pK a1 = 3.13 pK a2 = 4.76 pK a3 = 6.40 moles of H donated mole acid moles acid moles of H accepted mole base moles base + ×= × + 0.00 30.005.00 10.00 15.00 20.00 25.00 pH Volume of NaOH 0.0 4.0 6.0 8.0 10.0 12.0 2.0 14.0 Cit 3– HCit 2– H 2 Cit – H 3 Cit pH = 3.13 pH = 4.76 pH = 6.40 pH Figure 9.19 (a) Titration curve for 50.00 mL of a 0.00489 M solution of citric acid, using 0.04166 M NaOH as the titrant; (b) ladder diagram for citric acid. (a) (b) 1400-CH09 9/9/99 2:12 PM Page 304 Chapter 9 Titrimetric Methods of Analysis 305 The phenolphthalein end point is basic, occurring at a pH of approximately 8.3 and can be reached only if the titration proceeds to the third equivalence point (Figure 9.19b); thus, we write 3 · moles citric acid = moles NaOH Making appropriate substitutions for the moles of citric acid and moles of NaOH gives the following equation which can be solved for the grams of citric acid Since this is the grams of citric acid in a 50.00-mL sample, the concentration of citric acid in the citrus drink is 0.09402 g/100 mL. In an indirect analysis the analyte participates in one or more preliminary reac- tions that produce or consume acid or base. Despite the additional complexity, the stoichiometry between the analyte and the amount of acid or base produced or con- sumed may be established by applying the conservation principles outlined in Sec- tion 2C. Example 9.3 illustrates the application of an indirect analysis in which an acid is produced. EXAMPLE 9 . 3 The purity of a pharmaceutical preparation of sulfanilamide, C 6 H 4 N 2 O 2 S, can be determined by oxidizing the sulfur to SO 2 and bubbling the SO 2 through H 2 O 2 to produce H 2 SO 4 . The acid is then titrated with a standard solution of NaOH to the bromothymol blue end point, where both of sulfuric acid’s acidic protons have been neutralized. Calculate the purity of the preparation, given that a 0.5136-g sample required 48.13 mL of 0.1251 M NaOH. SOLUTION Conservation of protons for the titration reaction requires that 2 · moles H 2 SO 4 = moles NaOH Since all the sulfur in H 2 SO 4 comes from sulfanilamide, we use a conservation of mass on sulfur to establish the following stoichiometric relationship. Moles C 6 H 4 N 2 O 2 S = moles H 2 SO 4 Combining the two conservation equations gives a single equation relating the moles of analyte to the moles of titrant. 2 · moles C 6 H 4 N 2 O 2 S = moles NaOH Making appropriate substitutions for the moles of sulfanilamide and moles of NaOH gives 2 × =× g sulfanilamide FW sulfanilamide bb MV MV bb FW citric acid M)(0.01762 L)(192.13 g/mol) 3 g citric acid ×× = = 3 0 04166 0 04701 (. . 3 × =× g citric acid FW citric acid bb MV 1400-CH09 9/9/99 2:12 PM Page 305 306 Modern Analytical Chemistry which can be solved for the grams of sulfanilamide Thus, the purity of the preparation is This approach is easily extended to back titrations, as shown in the following example. EXAMPLE 9 . 4 The amount of protein in a sample of cheese is determined by a Kjeldahl analysis for nitrogen. After digesting a 0.9814-g sample of cheese, the nitrogen is oxidized to NH 4 + , converted to NH 3 with NaOH, and distilled into a collection flask containing 50.00 mL of 0.1047 M HCl. The excess HCl is then back titrated with 0.1183 M NaOH, requiring 22.84 mL to reach the bromothymol blue end point. Report the %w/w protein in the cheese given that there is 6.38 g of protein for every gram of nitrogen in most dairy products. SOLUTION In this procedure, the HCl reacts with two different bases; thus Moles HCl = moles HCl reacting with NH 3 + moles HCl reacting with NaOH Conservation of protons requires that Moles HCl reacting with NH 3 = moles NH 3 Moles HCl reacting with NaOH = moles NaOH A conservation of mass on nitrogen gives the following equation. Moles NH 3 = moles N Combining all four equations gives a final stoichiometric equation of Moles HCl = moles N + moles NaOH Making appropriate substitutions for the moles of HCl, N, and NaOH gives which we solve for the grams of nitrogen. g N=(M a · V a – M b · V b ) · AW N (0.1047 M · 0.05000 L – 0.1183 M · 0.02284 L) · 14.01 g/mol = 0.03549 g N MV MV aa bb g N AW N ×= + × g sulfanilamide g sample g 0.5136 g w/w sulfanilamide×= ×=100 0 5063 100 98 58 . .% MV bb FW sulfanilamide 2 M)(0.04813 L)(168.18 g/mol) 2 g sulfanilamide ×× = = (. . 0 1251 0 5063 1400-CH09 9/9/99 2:12 PM Page 306 Chapter 9 Titrimetric Methods of Analysis 307 The mass of protein, therefore, is and the %w/w protein is Earlier we noted that an acid–base titration may be used to analyze a mixture of acids or bases by titrating to more than one equivalence point. The concentration of each analyte is determined by accounting for its contribution to the volume of titrant needed to reach the equivalence points. EXAMPLE 9 . 5 The alkalinity of natural waters is usually controlled by OH – , CO 3 2– , and HCO 3 – , which may be present singularly or in combination. Titrating a 100.0-mL sample to a pH of 8.3 requires 18.67 mL of a 0.02812 M solution of HCl. A second 100.0-mL aliquot requires 48.12 mL of the same titrant to reach a pH of 4.5. Identify the sources of alkalinity and their concentrations in parts per million. SOLUTION Since the volume of titrant needed to reach a pH of 4.5 is more than twice that needed to reach a pH of 8.3, we know, from Table 9.8, that the alkalinity of the sample is controlled by CO 3 2– and HCO 3 2– . Titrating to a pH of 8.3 neutralizes CO 3 2– to HCO 3 – , but does not lead to a reaction of the titrant with HCO 3 – (see Figure 9.14b). Thus Moles HCl to pH 8.3 = moles CO 3 2– or Solving for the grams of carbonate gives g CO 3 2– = M a · V a, pH 8.3 · FW CO 3 2– 0.02812 M · 0.01867 L · 60.01 g/mol = 0.03151 g CO 3 2– The concentration of CO 3 2– , therefore, is Titrating to the second end point at pH 4.5 neutralizes CO 3 2– to H 2 CO 3 , and HCO 3 – to H 2 CO 3 (see Figures 9.18b,c). The conservation of protons, therefore, requires that Moles HCl to pH 4.5 = 2 · moles CO 3 2– + moles HCO 3 – or mg CO liter mg L ppm CO 3 2 3 2 ? ? == 31 51 0 1000 315 1 . . . MV a a,pH 8.3 3 2 3 2 g CO FW CO ×= ? ? g protein g sample g 0.9814 g w/w protein×= ×=100 0 2264 100 23 1 . .% 638 0 03549 0 2264 . .. g protein g N g N g protein×= 1400-CH09 9/9/99 2:12 PM Page 307 Solving for the grams of bicarbonate gives The concentration of HCO 3 – , therefore, is 9 B.6 Qualitative Applications We have already come across one example of the qualitative application of acid–base titrimetry in assigning the forms of alkalinity in waters (see Example 9.5). This approach is easily extended to other systems. For example, the composition of solutions containing one or two of the following species H 3 PO 4 H 2 PO 4 – HPO 4 2– PO 4 3– NaOH HCl can be determined by titrating with either a strong acid or a strong base to the methyl orange and phenolphthalein end points. As outlined in Table 9.11, each species or mixture of species has a unique relationship between the volumes of titrant needed to reach these two end points. mg HCO liter mg 0.1000 L ppm HCO 3 3 ? ? == 18 49 184 9 . . 0 02812 0 04812 2 0 03151 61 02 0 01849.. . .. M L g 60.01 g/mol g/mol g HCO 3 2 ×? ×? ? ? ? ? ? ×= ? g HCO g CO FW CO FW HCO 3 a a, pH 4.5 3 2 3 2 3 ? ? ? ? =× ? × ? ? ? ? ? ? ×MV 2 MV a a, pH 4.5 3 2 3 2 3 3 g CO FW CO g HCO FW HCO ×= × + ? ? ? ? 2 308 Modern Analytical Chemistry Table 9 .11 Relationship Between End Point Volumes for Solutions of Phosphate Species with HCl and NaOH Solution Relationship Between End Point Relationship Between End Point Composition Volumes with Strong Base Titrant a Volumes with Strong Acid Titrant a H 3 PO 4 V PH =2· V MO — b H 2 PO 4 – V PH > 0; V MO =0 — HPO 4 2– — V MO > 0; V PH =0 PO 4 3– — V MO =2· V PH HCl V PH = V MO — NaOH — V MO = V PH HCl and H 3 PO 4 V PH < 2 · V MO — H 3 PO 4 and H 2 PO 4 – V PH > 2 · V MO — H 2 PO 4 – and HPO 4 2– V PH > 0; V MO =0 V MO > 0; V PH =0 HPO 4 2– and PO 4 3– — V MO > 2 · V PH PO 4 3– and NaOH — V MO < 2 · V PH a V MO and V PH are, respectively, the volume of titrant needed to reach the methyl orange and phenolphthalein end points. b When no information is given, the volume of titrant needed to reach either end point is zero. 1400-CH09 9/9/99 2:12 PM Page 308 Chapter 9 Titrimetric Methods of Analysis 309 *See Section 2B.2 for a review of chemical equivalents and equivalent weights. 9 B. 7 Characterization Applications Two useful characterization applications involving acid–base titrimetry are the de- termination of equivalent weight, and the determination of acid–base dissociation constants. Equivalent Weights Acid–base titrations can be used to characterize the chemical and physical properties of matter. One simple example is the determination of the equivalent weight* of acids and bases. In this method, an accurately weighed sample of a pure acid or base is titrated to a well-defined equivalence point using a mono- protic strong acid or strong base. If we assume that the titration involves the trans- fer of n protons, then the moles of titrant needed to reach the equivalence point is given as Moles titrant = n · moles analyte and the formula weight is Since the actual number of protons transferred between the analyte and titrant is uncertain, we define the analyte’s equivalent weight (EW) as the apparent formula weight when n = 1. The true formula weight, therefore, is an integer multiple of the calculated equivalent weight. FW = n · EW Thus, if we titrate a monoprotic weak acid with a strong base, the EW and FW are identical. If the weak acid is diprotic, however, and we titrate to its second equiva- lence point, the FW will be twice as large as the EW. EXAMPLE 9 .6 A 0.2521-g sample of an unknown weak acid is titrated with a 0.1005 M solution of NaOH, requiring 42.68 mL to reach the phenolphthalein end point. Determine the compound’s equivalent weight. Which of the following compounds is most likely to be the unknown weak acid? ascorbic acid C 6 H 8 O 6 FW = 176.1 monoprotic malonic acid C 3 H 4 O 4 FW = 104.1 diprotic succinic acid C 4 H 6 O 4 FW = 118.1 diprotic citric acid C 6 H 8 O 7 FW = 192.1 triprotic SOLUTION The moles of NaOH needed to reach the end point is M b · V b = 0.1005 M · 0.04268 L = 4.289 · 10 –3 mol NaOH giving an equivalent weight of EW g analyte moles titrant g mol g/mol== × = ? 0 2521 4 289 10 58 78 3 . . . FW g analyte moles analyte g analyte moles titrant ==×n 1400-CH09 9/9/99 2:13 PM Page 309 Figure 9.20 Estimating the pK a for a weak acid from its titration curve with a strong base. The possible formula weights for the unknown weak acid are for n = 1: FW = EW = 58.78 g/mol for n =2: FW=2· EW = 117.6 g/mol for n =3: FW=3· EW = 176.3 g/mol If the weak acid is monoprotic, then the FW must be 58.78 g/mol, eliminating ascorbic acid as a possibility. If the weak acid is diprotic, then the FW may be either 58.78 g/mol or 117.6 g/mol, depending on whether the titration was to the first or second equivalence point. Succinic acid, with a formula weight of 118.1 g/mol is a possibility, but malonic acid is not. If the analyte is a triprotic weak acid, then its FW must be 58.78 g/mol, 117.6 g/mol, or 176.3 g/mol. None of these values is close to the formula weight for citric acid, eliminating it as a possibility. Only succinic acid provides a possible match. Equilibrium Constants Another application of acid–base titrimetry is the determi- nation of equilibrium constants. Consider, for example, the titration of a weak acid, HA, with a strong base. The dissociation constant for the weak acid is 9.9 When the concentrations of HA and A – are equal, equation 9.9 reduces to K a =[H 3 O + ], or pH = pK a . Thus, the pK a for a weak acid can be determined by measuring the pH for a solution in which half of the weak acid has been neutralized. On a titration curve, the point of half-neutralization is approximated by the volume of titrant that is half of that needed to reach the equivalence point. As shown in Fig- ure 9.20, an estimate of the weak acid’s pK a can be obtained directly from the titra- tion curve. This method provides a reasonable estimate of the pK a , provided that the weak acid is neither too strong nor too weak. These limitations are easily appreciated by considering two limiting cases. For the first case let’s assume that the acid is strong enough that it is more than 50% dissociated before the titration begins. As a result the concentration of HA before the equivalence point is always less than the con- centration of A – , and there is no point along the titration curve where [HA] = [A – ]. At the other extreme, if the acid is too weak, the equilibrium constant for the titra- tion reaction HA(aq)+OH – (aq) t H 2 O(l)+A – (aq) may be so small that less than 50% of HA will have reacted at the equivalence point. In this case the concentration of HA before the equivalence point is always greater K a 3 AHO HA] = ?+ [][ ] [ 310 Modern Analytical Chemistry 0.00 70.0010.00 pK a V eq pt 2 · V eq pt 20.00 30.00 40.00 50.00 pH Volume of titrant 0.0 4.0 6.0 8.0 10.0 12.0 2.0 60.00 14.0 1400-CH09 9/9/99 2:13 PM Page 310 than that of A – . Determining the pK a by the half-equivalence point method overes- timates its value if the acid is too strong and underestimates its value if the acid is too weak. A second approach for determining the pK a of an acid is to replot the titration curve in a linear form as a Gran plot. For example, earlier we learned that the titra- tion of a weak acid with a strong base can be plotted in a linear form using the fol- lowing equation V b · [H 3 O + ]=K a · V eq – K a · V b Plotting V b · [H 3 O + ] versus V b , for volumes less than the equivalence point volume yields a straight line with a slope of –K a . Other linearizations have been developed that use all the points on a titration curve 7 or require no assumptions. 8 This ap- proach to determining acidity constants has been used to study the acid–base prop- erties of humic acids, which are naturally occurring, large-molecular-weight organic acids with multiple acidic sites. In one study, a sample of humic acid was found to have six titratable sites, three of which were identified as carboxylic acids, two of which were believed to be secondary or tertiary amines, and one of which was iden- tified as a phenolic group. 9 9 B.8 Evaluation of Acid—Base Titrimetry Scale of Operation In an acid–base titration the volume of titrant needed to reach the equivalence point is proportional to the absolute amount of analyte present in the analytical solution. Nevertheless, the change in pH at the equiv- alence point, and thus the utility of an acid–base titration, is a function of the analyte’s concentration in the solution being titrated. When the sample is available as a solution, the smallest concentration of analyte that can be readily analyzed is approximately 10 –3 M (Figure 9.21). If, for example, the analyte has a gram formula weight of 120 g/mol, then the lower concentration limit is 120 ppm. When the analyte is a solid, it must first be placed into solution, the volume of which must be sufficient to allow the titration’s end point to be monitored using a visual indicator or a suitable probe. If we assume a minimum volume of 25 mL, and a lower concentration limit of 120 ppm, then a sample containing at least 3 mg of analyte is re- quired. Acid–base titrations involving solid or solution samples, therefore, are generally limited to major and minor analytes (see Figure 3.6 in Chapter 3). The analysis of gases can be extended to trace analytes by pulling a large vol- ume of the gas through a suitable collection solution. Efforts have been made to develop methods for conducting acid–base titrations on a much smaller scale. In one experimental design, samples of 20–100 m L were held by capillary action between a flat-surface pH electrode and a stainless steel rod. 10 The titrant was added by using the oscillations of a piezoelectric ceramic device to move an angled glass rod in and out of a tube connected to a reservoir containing the titrant (see Figure 9.22). Each time the glass tube was withdrawn an approximately 2-nL microdroplet of titrant was released. The microdroplets were allowed to fall onto the steel rod containing the sample, with mixing accomplished by spinning the rod at 120 rpm. A total of 450 microdroplets, with a combined volume of 0.81–0.84 m L, was dispensed between each pH measurement. In this fashion a titration curve was constructed. This method was used to titrate solutions of 0.1 M HCl and 0.1 M CH 3 COOH with 0.1 M NaOH. Absolute errors ranged from a minimum of +0.1% to a maximum of –4.1%, with relative Chapter 9 Titrimetric Methods of Analysis 311 0 100 (a) (b) (c) (d) (a) (b) (c) (d) 20 40 60 80 pH Volume of NaOH (mL) 0.00 4.00 6.00 8.00 10.00 12.00 2.00 14.00 Piezoelectric ceramic Sample pH meter Titrant Figure 9.21 Titration curves for (a) 10 –1 M HCl, (b) 10 –2 M HCl, (c) 10 –3 M HCl, and (d) 10 –4 M HCl. In each case the titrant is an equimolar solution of NaOH. Figure 9.22 Experimental design for a microdroplet titration apparatus. 1400-CH09 9/9/99 2:13 PM Page 311 Figure 9.23 (a) Experimental set-up for a diffusional microtitration; (b) close-up showing the tip of the diffusional microburet in contact with the drop of sample. standard deviations from 0.15% to 4.7%. The smallest volume of sample that was successfully titrated was 20 m L. More recently, a method has been described in which the acid–base titration is conducted within a single drop of solution. 11 The titrant is added using a microbu- ret fashioned from a glass capillary micropipet (Figure 9.23). The microburet has a 1–2 m m tip filled with an agar gel membrane. The tip of the microburet is placed within a drop of the sample solution, which is suspended in heptane, and the titrant is allowed to diffuse into the sample. The titration is followed visually using a col- ored indicator, and the time needed to reach the end point is measured. The rate of the titrant’s diffusion from the microburet must be determined by calibration. Once calibrated, the end point time can be converted to an end point volume. Sam- ples usually consisted of picoliter volumes (10 –12 L), with the smallest sample being 0.7 pL. The precision of the titrations was usually about 2%. Titrations conducted with microliter or picoliter sample volumes require a smaller absolute amount of analyte. For example, diffusional titrations have been successfully conducted on as little as 29 femtomoles (10 –15 mol) of nitric acid. Nev- ertheless, the analyte must still be present in the sample at a major or minor level for the titration to be performed accurately and precisely. Accuracy When working with macro–major and macro–minor samples, acid–base titrations can be accomplished with relative errors of 0.1–0.2%. The prin- cipal limitation to accuracy is the difference between the end point and the equiva- lence point. Precision The relative precision of an acid–base titration depends primarily on the precision with which the end point volume can be measured and the precision of the end point signal. Under optimum conditions, an acid–base titration can be ac- complished with a relative precision of 0.1–0.2%. The relative precision can be im- proved by using the largest volume buret that is feasible and ensuring that most of its capacity is used to reach the end point. Smaller volume burets are used when the cost of reagents or waste disposal is of concern or when the titration must be com- pleted quickly to avoid competing chemical reactions. Automatic titrators are par- ticularly useful for titrations requiring small volumes of titrant, since the precision with which the volume can be measured is significantly better (typically about – 0.05% of the buret’s volume). The precision of the end point signal depends on the method used to locate the end point and the shape of the titration curve. With a visual indicator, the precision of the end point signal is usually between – 0.03 mL and 0.10 mL. End points deter- mined by direct monitoring often can be determined with a greater precision. Sensitivity For an acid–base titration we can write the following general analytical equation Volume of titrant = k · moles of analyte where k, the sensitivity, is determined by the stoichiometric relationship between analyte and titrant. Note that this equation assumes that a blank has been analyzed to correct the signal for the volume of titrant reacting with the reagents. Consider, for example, the determination of sulfurous acid, H 2 SO 3 , by titrating with NaOH to the first equivalence point. Using the conservation of protons, we write Moles NaOH = moles H 2 SO 3 312 Modern Analytical Chemistry Sample (a) (b) Heptane Agar gel membrane Diffusional microburet 1400-CH09 9/9/99 2:13 PM Page 312 Substituting the molarity and volume of titrant for moles, and rearranging gives 9.10 where k is equivalent to There are two ways in which the sensitivity can be increased. The first, and most ob- vious, is to decrease the concentration of the titrant, since it is inversely propor- tional to the sensitivity, k. The second method, which only applies if the analyte is multiprotic, is to titrate to a later equivalence point. When H 2 SO 3 is titrated to the second equivalence point, for instance, equation 9.10 becomes where k is now equal to In practice, however, any improvement in the sensitivity of an acid–base titration due to an increase in k is offset by a decrease in the precision of the equivalence point volume when the buret needs to be refilled. Consequently, standard analytical procedures for acid–base titrimetry are usually written to ensure that titrations re- quire 60–100% of the buret’s volume. Selectivity Acid–base titrants are not selective. A strong base titrant, for example, will neutralize any acid, regardless of strength. Selectivity, therefore, is determined by the relative acid or base strengths of the analyte and the interferent. Two limiting situations must be considered. First, if the analyte is the stronger acid or base, then the titrant will begin reacting with the analyte before reacting with the interferent. The feasibility of the analysis depends on whether the titrant’s reaction with the in- terferent affects the accurate location of the analyte’s equivalence point. If the acid dissociation constants are substantially different, the end point for the analyte can be accurately determined (Figure 9.24a). Conversely, if the acid dissociation con- stants for the analyte and interferent are similar, then an accurate end point for the analyte may not be found (Figure 9.24b). In the latter case, a quantitative analysis for the analyte is not possible. In the second limiting situation the analyte is a weaker acid or base than the in- terferent. In this case the volume of titrant needed to reach the analyte’s equivalence point is determined by the concentration of both the analyte and the interferent. To account for the contribution from the interferent, an equivalence point for the in- terferent must be present. Again, if the acid dissociation constants for the analyte and interferent are significantly different, the analyte’s determination is possible. If, however, the acid dissociation constants are similar, only a single equivalence point is found, and the analyte’s and interferent’s contributions to the equivalence point volume cannot be separated. Time, Cost, and Equipment Acid–base titrations require less time than most gravi- metric procedures, but more time than many instrumental methods of analysis, particularly when analyzing many samples. With the availability of instruments for k M = 2 b V M b b 23 moles H SO=× ×2 1 k M = 1 b V M b b 23 moles H SO=× 1 Chapter 9 Titrimetric Methods of Analysis 313 1400-CH09 9/9/99 2:13 PM Page 313 Figure 9.24 Titration curves for a 50.00 mL mixture of 0.100 M HA and 0.100 M HB with 0.100 M NaOH, where (a) pK a,HA = 3 and pK a,HB =8; and (b) pK a,HA = 5 and pK a,HB = 6. The dashed lines indicate the location of the two equivalence points. 314 Modern Analytical Chemistry 0.00 140.0020.00 40.00 60.00 80.00 100.00 pH Volume of titrant 0.0 4.0 6.0 8.0 10.0 12.0 2.0 120.00 14.0 0.00 140.0020.00 40.00 60.00 80.00 100.00 pH Volume of titrant 0.0 4.0 6.0 8.0 10.0 12.0 2.0 120.00 14.0 (a) (b) complexation titration A titration in which the reaction between the analyte and titrant is a complexation reaction. performing automated titrations, however, concerns about analysis time are less of a problem. When performing a titration manually the equipment needs are few (a buret and possibly a pH meter), inexpensive, routinely available in most laborato- ries, and easy to maintain. Instrumentation for automatic titrations can be pur- chased for around $3000. 9 C Titrations Based on Complexation Reactions The earliest titrimetric applications involving metal–ligand complexation were the determinations of cyanide and chloride using, respectively, Ag + and Hg 2+ as titrants. Both methods were developed by Justus Liebig (1803–1873) in the 1850s. The use of a monodentate ligand, such as Cl – and CN – , however, limited the utility of com- plexation titrations to those metals that formed only a single stable complex, such as Ag(CN) 2 – and HgCl 2 . Other potential metal–ligand complexes, such as CdI 4 2– , were not analytically useful because the stepwise formation of a series of metal– ligand complexes (CdI + , CdI 2 , CdI 3 – , and CdI 4 2– ) resulted in a poorly defined end point. The utility of complexation titrations improved following the introduction by Schwarzenbach, in 1945, of aminocarboxylic acids as multidentate ligands capable of forming stable 1:1 complexes with metal ions. The most widely used of these new ligands was ethylenediaminetetraacetic acid, EDTA, which forms strong 1:1 complexes with many metal ions. The first use of EDTA as a titrant occurred in 1400-CH09 9/9/99 2:13 PM Page 314 Figure 9.25 Structures of (a) EDTA, and (b) a six- coordinate metal–EDTA complex. 1946, when Schwarzenbach introduced metallochromic dyes as visual indicators for signaling the end point of a complexa- tion titration. 9 C.1 Chemistry and Properties of EDTA Ethylenediaminetetraacetic acid, or EDTA, is an aminocar- boxylic acid. The structure of EDTA is shown in Figure 9.25a. EDTA, which is a Lewis acid, has six binding sites (the four car- boxylate groups and the two amino groups), providing six pairs of electrons. The resulting metal–ligand complex, in which EDTA forms a cage-like structure around the metal ion (Figure 9.25b), is very stable. The actual number of coordination sites depends on the size of the metal ion; however, all metal–EDTA complexes have a 1:1 stoichiometry. Metal—EDTA Formation Constants To illustrate the formation of a metal–EDTA complex consider the reaction between Cd 2+ and EDTA Cd 2+ (aq)+Y 4– (aq) t CdY 2– (aq) where Y 4– is a shorthand notation for the chemical form of EDTA shown in Figure 9.25. The formation constant for this reaction 9.11 is quite large, suggesting that the reaction’s equilibrium position lies far to the right. Formation constants for other metal–EDTA complexes are found in Appendix 3C. EDTA Is a Weak Acid Besides its properties as a ligand, EDTA is also a weak acid. The fully protonated form of EDTA, H 6 Y 2+ , is a hexaprotic weak acid with succes- sive pK a values of pK a1 = 0.0 pK a2 = 1.5 pK a3 = 2.0 pK a4 = 2.68 pK a5 = 6.11 pK a6 = 10.17 The first four values are for the carboxyl protons, and the remaining two values are for the ammonium protons. A ladder diagram for EDTA is shown in Figure 9.26. The species Y 4– becomes the predominate form of EDTA at pH levels greater than 10.17. It is only for pH levels greater than 12 that Y 4– becomes the only significant form of EDTA. Conditional Metal—Ligand Formation Constants Recognizing EDTA’s acid–base properties is important. The formation constant for CdY 2– in equation 9.11 as- sumes that EDTA is present as Y 4– . If we restrict the pH to levels greater than 12, then equation 9.11 provides an adequate description of the formation of CdY 2– . For pH levels less than 12, however, K f overestimates the stability of the CdY 2– complex. At any pH a mass balance requires that the total concentration of unbound EDTA equal the combined concentrations of each of its forms. C EDTA =[H 6 Y 2+ ]+[H 5 Y + ]+[H 4 Y]+[H 3 Y – ]+[H 2 Y 2– ] + [HY 3– ]+[Y 4– ] To correct the formation constant for EDTA’s acid–base properties, we must ac- count for the fraction, a Y 4– , of EDTA present as Y 4– . 9.12 α Y EDTA Y 4 4 ? = ? [] C K f 2 4 CdY Cd Y ==× ? +? [] [][] . 2 16 29 10 Chapter 9 Titrimetric Methods of Analysis 315 H 2 C H 2 C – OOC – OOC : N N : CH 2 CH 2 COO – COO – O O NO O O N O O M O (a) (b) Y 4– HY 3– H 2 Y 2– H 3 Y – H 5 Y + H 6 Y 2+ H 4 Y 0.0 10.17 6.11 2.68 1.5 2.0 pH Figure 9.26 Ladder diagram for EDTA. 1400-CH09 9/9/99 2:13 PM Page 315 Values of a Y 4– are shown in Table 9.12. Solving equation 9.12 for [Y 4– ] and substi- tuting into the equation for the formation constant gives If we fix the pH using a buffer, then a Y 4– is a constant. Combining a Y 4– with K f gives 9.13 where K f ′ is a conditional formation constant whose value depends on the pH. As shown in Table 9.13 for CdY 2– , the conditional formation constant becomes smaller, and the complex becomes less stable at lower pH levels. EDTA Must Compete with Other Ligands To maintain a constant pH, we must add a buffering agent. If one of the buffer’s components forms a metal–ligand complex with Cd 2+ , then EDTA must compete with the ligand for Cd 2+ . For example, an NH 4 + /NH 3 buffer includes the ligand NH 3 , which forms several stable Cd 2+ –NH 3 complexes. EDTA forms a stronger complex with Cd 2+ and will displace NH 3 . The presence of NH 3 , however, decreases the stability of the Cd 2+ –EDTA complex. We can account for the effect of an auxiliary complexing agent, such as NH 3 , in the same way we accounted for the effect of pH. Before adding EDTA, a mass balance on Cd 2+ requires that the total concentration of Cd 2+ , C Cd , be C Cd = [Cd 2+ ] + [Cd(NH 3 ) 2+ ] + [Cd(NH 3 ) 2 2+ ] + [Cd(NH 3 ) 3 2+ ] + [Cd(NH 3 ) 4 2+ ] ′ =×=? ? + KK C f Y f 2 2 EDTA CdY Cd α 4 [] [] K C f 2 2 Y EDTA CdY Cd 4 = ? + ? [] []α 316 Modern Analytical Chemistry Table 9 .12 Values of a Y 4– for Selected pHs pH a Y 4- pH a Y 4- 2 3.7 · 10 –14 8 5.4 · 10 –3 3 2.5 · 10 –11 9 5.2 · 10 –2 4 3.6 · 10 –9 10 0.35 5 3.5 · 10 –7 11 0.85 6 2.2 · 10 –5 12 0.98 7 4.8 · 10 –4 13 1.00 Table 9 .1 3 Conditional Formation Constants for CdY 2– pH K ’ f pH K ’ f 2 1.1 · 10 3 8 1.6 · 10 14 3 7.3 · 10 5 9 1.5 · 10 15 4 1.0 · 10 8 10 1.0 · 10 16 5 1.0 · 10 10 11 2.5 · 10 16 6 6.4 · 10 11 12 2.8 · 10 16 7 1.4 · 10 13 13 2.9 · 10 16 auxiliary complexing agent A second ligand in a complexation titration that initially binds with the analyte but is displaced by the titrant. conditional formation constant The equilibrium formation constant for a metal–ligand complex for a specific set of solution conditions, such as pH. 1400-CH09 9/9/99 2:13 PM Page 316 The fraction, a Cd 2+ , present as uncomplexed Cd 2+ is 9.14 Solving equation 9.14 for [Cd 2+ ] and substituting into equation 9.13 gives If the concentration of NH 3 is held constant, as it usually is when using a buffer, then we can rewrite this equation as 9.15 where K f ? is a new conditional formation constant accounting for both pH and the presence of an auxiliary complexing agent. Values of a M n+ for several metal ions are provided in Table 9.14. 9 C.2 Complexometric EDTA Titration Curves Now that we know something about EDTA’s chemical properties, we are ready to evaluate its utility as a titrant for the analysis of metal ions. To do so we need to know the shape of a complexometric EDTA titration curve. In Section 9B we saw that an acid–base titration curve shows the change in pH following the addition of titrant. The analogous result for a titration with EDTA shows the change in pM, where M is the metal ion, as a function of the volume of EDTA. In this section we learn how to calculate the titration curve. We then show how to quickly sketch the titration curve using a minimum number of calculations. Calculating the Titration Curve As an example, let’s calculate the titration curve for 50.0 mL of 5.00 · 10 –3 M Cd 2+ with 0.0100 M EDTA at a pH of 10 and in the presence of 0.0100 M NH 3 . The formation constant for Cd 2+ –EDTA is 2.9 · 10 16 . Since the titration is carried out at a pH of 10, some of the EDTA is present in forms other than Y 4– . In addition, the presence of NH 3 means that the EDTA must compete for the Cd 2+ . To evaluate the titration curve, therefore, we must use the appropriate conditional formation constant. From Tables 9.12 and 9.14 we find that a Y 4– is 0.35 at a pH of 10, and that a Cd 2+ is 0.0881 when the ′′ =××=+? ? KK CC f Cd Y f 2 Cd EDTA CdY αα24 [] ′ =×=? + ? KK CC f Y f 2 Cd Cd EDTA CdY α α 4 2 [] α Cd Cd Cd 2 2 + = + [] C Chapter 9 Titrimetric Methods of Analysis 317 Table 9 .1 4 Values of a M n+ for Selected Concentrations of Ammonia [NH 3 ] (M) a Ag + a Ca 2+ a Cd 2+ a Co 2+ a Cu 2+ a Mg 2+ a Ni 2+ a Zn 2+ 1 1.00 · 10 –7 5.50 · 10 –1 6.09 · 10 –8 1.00 · 10 –6 3.79 · 10 –14 1.76 · 10 –1 9.20 · 10 –10 3.95 · 10 –10 0.5 4.00 · 1 0 –7 7.36 · 1 0 –1 1.05 · 1 0 –6 2.22 · 1 0 –5 6.86 · 1 0 –13 4.13 · 1 0 –1 3.44 · 1 0 –8 6.27 · 1 0 –9 0.1 9.98 · 1 0 –6 9.39 · 1 0 –1 3.51 · 1 0 –4 6.64 · 1 0 –3 4.63 · 1 0 –10 8.48 · 1 0 –1 5.12 · 1 0 –5 3.68 · 1 0 –6 0.05 3.99 · 1 0 –5 9.69 · 1 0 –1 2.72 · 1 0 –3 3.54 · 1 0 –2 7.17 · 1 0 –9 9.22 · 1 0 –1 6.37 · 1 0 –4 5.45 · 1 0 –5 0.01 9.83 · 1 0 –4 9.94 · 1 0 –1 8.81 · 1 0 –2 3.55 · 1 0 –1 3.22 · 1 0 –6 9.84 · 1 0 –1 4.32 · 1 0 –2 1.82 · 1 0 –2 0.005 3.86 · 1 0 –3 9.97 · 1 0 –1 2.27 · 1 0 –1 5.68 · 1 0 –1 3.62 · 1 0 –5 9.92 · 1 0 –1 1.36 · 1 0 –1 1.27 · 1 0 –1 0.001 7.95 · 1 0 –2 9.99 · 1 0 –1 6.90 · 1 0 –1 8.84 · 1 0 –1 4.15 · 1 0 –3 9.98 · 1 0 –1 5.76 · 1 0 –1 7.48 · 1 0 –1 1400-CH09 9/9/99 2:13 PM Page 317 concentration of NH 3 is 0.0100 M. Using these values, we calculate that the con- ditional formation constant is K f ? = a Y 4– · a Cd 2+ · K f = (0.35)(0.0881)(2.9 · 10 16 ) = 8.9 · 10 14 Because K f ? is so large, we treat the titration reaction as though it proceeds to completion. The first task in calculating the titration curve is to determine the volume of EDTA needed to reach the equivalence point. At the equivalence point we know that Moles EDTA = moles Cd 2+ or M EDTA V EDTA = M Cd V Cd Solving for the volume of EDTA shows us that 25.0 mL of EDTA is needed to reach the equivalence point. Before the equivalence point, Cd 2+ is in excess, and pCd is determined by the concentration of free Cd 2+ remaining in solution. Not all the untitrated Cd 2+ is free (some is complexed with NH 3 ), so we will have to account for the presence of NH 3 . For example, after adding 5.0 mL of EDTA, the total concentration of Cd 2+ is To calculate the concentration of free Cd 2+ we use equation 9.14. [Cd 2+ ]=a Cd 2+ · C Cd = (0.0881)(3.64 · 10 –3 M) = 3.21 · 10 –4 M Thus, pCd is pCd = –log[Cd 2+ ] = –log(3.21 · 10 –4 ) = 3.49 At the equivalence point, all the Cd 2+ initially present is now present as CdY 2– . The concentration of Cd 2+ , therefore, is determined by the dissociation of the CdY 2– complex. To find pCd we must first calculate the concentration of the complex. Letting the variable x represent the concentration of Cd 2+ due to the dissociation of the CdY 2– complex, we have ′′ == ×? =× ==× ?? ? K CC x xx xC f 2 Cd EDTA Cd CdY M []. ()() . . 333 10 894 10 193 10 3 14 9 [] (. )( . ) .. . CdY initial moles Cd total volume M mL mL mL M 2 2+ Cd Cd Cd EDTA ? ? ? == + = × + =× MV VV 500 10 500 50 0 25 0 333 10 3 3 C MV M V VV Cd Cd Cd EDTA EDTA Cd EDTA moles excess Cd total volume M mL M)(5.0 mL mL + 5.0 mL M == ? + = ×? =× + ? ? 2 3 3 5 00 10 50 0 0 0100 50 0 364 10 (. )( . ) (. ) . . V MV M EDTA Cd Cd EDTA M mL M mL== × = ? (. )( . ) . . 500 10 500 0 0100 25 0 3 318 Modern Analytical Chemistry 1400-CH09 9/9/99 2:13 PM Page 318 Figure 9.27 Complexometric titration curve for 50.0 mL of 5.00 · 10 –3 M Cd 2+ with 0.0100 M EDTA at a pH of 10.0 in the presence of 0.0100 M NH 3 . Once again, to find the [Cd 2+ ] we must account for the presence of NH 3 ; thus [Cd 2+ ]=a Cd 2+ · C Cd = (0.0881)(1.93 · 10 –9 M) = 1.70 · 10 –10 M giving pCd as 9.77. After the equivalence point, EDTA is in excess, and the concentration of Cd 2+ is determined by the dissociation of the CdY 2– complex. Examining the equation for the complex’s conditional formation constant (equation 9.15), we see that to calcu- late C Cd we must first calculate [CdY 2– ] and C EDTA . After adding 30.0 mL of EDTA, these concentrations are Substituting these concentrations into equation 9.15 and solving for C Cd gives Thus, [Cd 2+ ]=a Cd 2+ · C Cd = (0.0881)(5.60 · 10 –15 M) = 4.93 · 10 –16 M and pCd is 15.31. Figure 9.27 and Table 9.15 show additional results for this titration. [] . (. ) . . CdY M M Cd EDTA Cd Cd 23 4 14 15 313 10 625 10 894 10 560 10 ?? ? ? = × × =× =× CC C C C MV MV VV EDTA EDTA EDTA Cd Cd Cd EDTA moles excess EDTA total volume M mL M) mL mL + 30.0 mL M == ? + = ?× =× ? ? (. )( . ) (. ( . ) . . 0 0100 30 0 5 00 10 50 0 50 0 625 10 3 4 [] ( . )( . . CdY initial moles Cd total volume M mL) 50.0 mL + 30.0 mL M Cd Cd Cd EDTA 2 2 3 3 500 10 500 313 10 ? + ? ? == + = × =× MV VV Chapter 9 Titrimetric Methods of Analysis 319 pCd Volume of EDTA (mL) 0 0.00 4.00 6.00 8.00 10.00 16.00 18.00 10 20 30 50 2.00 40 12.00 14.00 1400-CH09 9/9/99 2:13 PM Page 319 320 Modern Analytical Chemistry Table 9 .1 5 Data for Titration of 5.00 · 10 –3 M Cd 2+ with 0.0100 M EDTA at a pH of 10.0 and in the Presence of 0.0100 M NH 3 Volume of EDTA (mL) pCd 0.00 3.36 5.00 3.49 10.00 3.66 15.00 3.87 20.00 4.20 23.00 4.62 25.00 9.77 27.00 14.91 30.00 15.31 35.00 15.61 40.00 15.78 45.00 15.91 50.00 16.01 Sketching an EDTA Titration Curve Our strategy for sketching an EDTA titration curve is similar to that for sketching an acid–base titration curve. We begin by drawing axes, placing pM on the y-axis and volume of EDTA on the x-axis. After calculating the volume of EDTA needed to reach the equivalence point, we add a vertical line intersecting the x-axis at this volume. Next we calculate and plot two values of pM for volumes of EDTA before the equivalence point and two values of pM for volumes after the equivalence point. Straight lines are drawn through each pair of points. Finally, a smooth curve is drawn connecting the three straight-line segments. EXAMPLE 9 . 7 Sketch the titration curve for 50.0 mL of 5.00 · 10 –3 M Cd 2+ with 0.010 M EDTA at a pH of 10, and in the presence of an ammonia concentration that is held constant throughout the titration at 0.010 M. This is the same titration for which we previously calculated the titration curve (Table 9.15 and Figure 9.27). SOLUTION We begin by drawing axes for the titration curve (Figure 9.28a). We have already shown that the equivalence point is at 25.0 mL, so we draw a vertical line intersecting the x-axis at this volume (Figure 9.28b). Before the equivalence point, pCd is determined by the excess concentration of free Cd 2+ . Using values from Table 9.15, we plot pCd for 5.0 mL and 10.0 mL of EDTA (Figure 9.28c). After the equivalence point, pCd is determined by the dissociation of the Cd 2+ –EDTA complex. Using values from Table 9.15, we plot pCd for 30.0 mL and 40.0 mL of EDTA (Figure 9.28d). 1400-CH09 9/9/99 2:13 PM Page 320 Chapter 9 Titrimetric Methods of Analysis 321 pCd Volume of titrant (mL) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 14.0 20.00 500 100 200150 50.00 2.0 10.00 30.00 40.00 16.0 12.0 18.0 (a) pCd Volume of titrant (mL) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 14.0 20.00 500 100 200150 50.00 2.0 10.00 30.00 40.00 16.0 12.0 18.0 (b) pCd Volume of titrant (mL) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 14.0 20.00 500 100 200150 50.00 2.0 10.00 30.00 40.00 16.0 12.0 18.0 (c) pCd Volume of titrant (mL) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 14.0 20.00 500 100 200150 50.00 2.0 10.00 30.00 40.00 16.0 12.0 18.0 (d) pCd Volume of titrant (mL) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 14.0 20.00 500 100 200150 50.00 2.0 10.00 30.00 40.00 16.0 12.0 18.0 (e) pCd Volume of titrant (mL) Percent titrated 0.00 0.0 4.0 6.0 8.0 10.0 14.0 20.00 500 100 200150 50.00 2.0 10.00 30.00 40.00 16.0 12.0 18.0 (f) Figure 9.28 How to sketch an EDTA complexometric titration curve; see text for explanation. 1400-CH09 9/9/99 2:13 PM Page 321 An approximate sketch of the titration curve is completed by drawing separate straight lines through the two points before and after the equivalence point (Figure 9.28e). Finally, a smooth curve is drawn to connect the three straight-line segments (Figure 9.28f). Additional examples of Cd 2+ –EDTA titration curves are shown in Figure 9.29. Note in particular, that the change in pCd near the equivalence point is not as great when the titration is carried out at a lower pH or in the presence of a higher concentration of NH 3 . 322 Modern Analytical Chemistry pCd Volume titrant (mL) 0.00 0.0 4.0 6.0 8.0 10.0 16.0 18.0 10.00 20.00 (c) (c) (a) (a) (b) (b) 30.00 50.00 2.0 40.00 12.0 14.0 Figure 9.29 Effect of pH and [NH 3 ] on the titration curve for 50.0 mL of 5.00 · 10 –3 M Cd 2+ with 0.0100 M EDTA. (a) pH = 10, [NH 3 ] = 0; (b) pH = 7, [NH 3 ]=0; (c) pH = 10, [NH 3 ] = 0.5 M. 9 C. 3 Selecting and Evaluating the End Point The equivalence point of a complexation titration occurs when stoichiometri- cally equivalent amounts of analyte and titrant have reacted. For titrations in- volving metal ions and EDTA, the equivalence point occurs when C M and C EDTA are equal and may be located visually by looking for the titration curve’s inflec- tion point. As with acid–base titrations, the equivalence point of a complexation titration is estimated by an experimental end point. A variety of methods have been used to find the end point, including visual indicators and sensors that respond to a change in the solution conditions. Typical examples of sensors include recording a poten- tiometric titration curve using an ion-selective electrode (analogous to measuring pH with a pH electrode),* monitoring the temperature of the titration mixture, and monitoring the absorbance of electromagnetic radiation by the titration mixture. *See Chapter 11 for a further discussion of ion-selective electrodes. 1400-CH09 9/9/99 2:13 PM Page 322 The first two sensors were discussed in Section 9B.3 for acid–base titrations and are not considered further in this section. Finding the End Point with a Visual Indicator Most indicators for complexation titrations are organic dyes that form stable complexes with metal ions. These dyes are known as metallochromic indicators. To function as an indicator for an EDTA titration, the metal–indicator complex must possess a color different from that of the uncomplexed indicator. Furthermore, the formation constant for the metal–indicator complex must be less favorable than that for the metal–EDTA complex. The indicator, In m– , is added to the solution of analyte, forming a colored metal–indicator complex, MIn n-m . As EDTA is added, it reacts first with the free an- alyte, and then displaces the analyte from the metal–indicator complex, affecting a change in the solution’s color. The accuracy of the end point depends on the strength of the metal–indicator complex relative to that of the metal–EDTA com- plex. If the metal–indicator complex is too strong, the color change occurs after the equivalence point. If the metal–indicator complex is too weak, however, the end point is signaled before reaching the equivalence point. Most metallochromic indicators also are weak acids or bases. The condi- tional formation constant for the metal–indicator complex, therefore, depends on the solution’s pH. This provides some control over the indicator’s titration error. The apparent strength of a metal–indicator complex can be adjusted by controlling the pH at which the titration is carried out. Unfortunately, because they also are acid–base indicators, the color of the uncomplexed indicator changes with pH. For example, calmagite, which we may represent as H 3 In, undergoes a change in color from the red of H 2 In – to the blue of HIn 2– at a pH of approximately 8.1, and from the blue of HIn 2– to the red-orange of In 3– at a pH of approximately 12.4. Since the color of calmagite’s metal–indicator complexes are red, it is only useful as a metal- lochromic indicator in the pH range of 9–11, at which almost all the indicator is present as HIn 2– . A partial list of metallochromic indicators, and the metal ions and pH condi- tions for which they are useful, is given in Table 9.16. Even when a suitable indica- tor does not exist, it is often possible to conduct an EDTA titration by introducing a small amount of a secondary metal–EDTA complex, provided that the secondary metal ion forms a stronger complex with the indicator and a weaker complex with EDTA than the analyte. For example, calmagite can be used in the determination of Chapter 9 Titrimetric Methods of Analysis 323 metallochromic indicator A visual indicator used to signal the end point in a complexation titration. Table 9.16 Selected Metallochromic Indicators Indicator Useful pH Range Useful for calmagite 9–11 Ba, Ca, Mg, Zn Eriochrome Black T 7.5–10.5 Ba, Ca, Mg, Zn Eriochrome Blue Black R 8–12 Ca, Mg, Zn, Cu murexide 6–13 Ca, Ni, Cu PAN 2–11 Cd, Cu, Zn salicylic acid 2–3 Fe 1400-CH09 9/9/99 2:13 PM Page 323 Ca 2+ if a small amount of Mg 2+ –EDTA is added to the solution containing the ana- lyte. The Mg 2+ is displaced from the EDTA by Ca 2+ , freeing the Mg 2+ to form the red Mg 2+ –indicator complex. After all the Ca 2+ has been titrated, Mg 2+ is displaced from the Mg 2+ –indicator complex by EDTA, signaling the end point by the pres- ence of the uncomplexed indicator’s blue form. Finding the End Point by Monitoring Absorbance. An important limitation when using a visual indicator is the need to observe the change in color signal- ing the end point. This may be difficult when the solution is already colored. For example, ammonia is used to adjust the pH of solutions containing Cu 2+ before its titration with EDTA. The presence of the intensely colored Cu(NH 3 ) 4 2+ complex obscures the indicator’s color, making an accurate deter- mination of the end point difficult. Other absorbing species present within the sample matrix may also interfere in a similar fashion. This is often a problem when analyzing clinical samples such as blood or environmental samples such as natural waters. As long as at least one species in a complexation titration absorbs electro- magnetic radiation, the equivalence point can be located by monitoring the ab- sorbance of the analytical solution at a carefully selected wavelength.* For ex- ample, the equivalence point for the titration of Cu 2+ with EDTA, in the presence of NH 3 , can be located by monitoring the absorbance at a wavelength of 745 nm, where the Cu(NH 3 ) 4 2+ complex absorbs strongly. At the beginning of the titration the absorbance is at a maximum. As EDTA is added, however, the reaction Cu(NH 3 ) 4 2+ (aq)+Y 4– (aq) fi CuY 2– (aq) + 4NH 3 (aq) occurs, decreasing both the concentration of Cu(NH 3 ) 4 2+ and the absorbance. The absorbance reaches a minimum at the equivalence point and remains essen- tially unchanged as EDTA is added in excess. The resulting spectrophotometric titration curve is shown in Figure 9.30a. In order to keep the individual segments of the titration curve linear, the measured absorbance, A meas , is corrected for dilution where A corr is the corrected absorbance, and V EDTA and V Cu are, respectively, the volumes of EDTA and Cu. The equivalence point is given by the intersection of the linear segments, which are extrapolated if necessary to correct for any curvature in the titration curve. Other common spectrophotometric titration curves are shown in Figures 9.30b–f. 9 C. 4 Representative Method Although every complexation titrimetric method has its own unique considera- tions, the following description for determining the hardness of water provides an instructive example of a typical procedure. AA VV V corr meas EDTA Cu Cu =× + 324 Modern Analytical Chemistry *See Chapter 10 for a further discussion of absorbance and spectroscopy. 1400-CH09 9/9/99 2:13 PM Page 324 Figure 9.30 Spectrophotometric titration curves for the titration of an analyte, A, with a titrant, T, to form a product, P, in the presence of a visual indicator. Titration curves are shown for cases where (a) only A absorbs; (b) only T absorbs; (c) only P absorbs; (d) A and T absorb; (e) P and T absorb; and (f) only the visual indicator absorbs. Chapter 9 Titrimetric Methods of Analysis 325 Corrected absorbance Corrected absorbance Corrected absorbance Corrected absorbance Corrected absorbance Corrected absorbance Volume of titrant Volume of titrant Volume of titrant Volume of titrant Volume of titrant Volume of titrant (a) (c) (e) or or (b) (d) (f) 1400-CH09 9/9/99 2:13 PM Page 325 326 Modern Analytical Chemistry Representative Methods Method 9 .2 Determination of Hardness of Water and Wastewater 12 Description of the Method. The operational definition of water hardness is the total concentration of cations in a sample capable of forming insoluble complexes with soap. Although most divalent and trivalent metal ions contribute to hardness, the most important are Ca 2+ and Mg 2+ . Hardness is determined by titrating with EDTA at a buffered pH of 10. Eriochrome Black T or calmagite is used as a visual indicator. Hardness is reported in parts per million CaCO 3 . Procedure. Select a volume of sample requiring less than 15 mL of titrant to keep the analysis time under 5 min and, if necessary, dilute the sample to 50 mL with distilled water. Adjust the pH by adding 1–2 mL of a pH 10 buffer containing a small amount of Mg 2+ –EDTA. Add 1–2 drops of indicator, and titrate with a standard solution of EDTA until the red-to-blue end point is reached. Questions 1. Why is the sample buffered to a pH of 10? What problems might be expected at higher or lower pHs? Of the cations contributing to hardness, Mg 2+ forms the weakest complex with EDTA and is the last cation to be titrated. Calmagite was selected as the indicator because it gives a distinct end point with Mg 2+ . Because of calmagite’s acid–base properties the indicator is only useful in the pH range of 9–11 (see Table 9.16). Figure 9.31 shows the titration curve for a solution of 10 –3 M Mg 2+ with 10 –2 M EDTA at pHs of 9, 10, and 11. Superimposed on each titration curve is the range of conditions in which the average analyst will find the end point. At a pH of 9 an early end point is possible, leading to a negative determinate error, and at a pH of 11 there is a chance of a late end point and a positive determinate error. pMg Volume titrant (mL) 0.00 0.0 2.0 3.0 4.0 5.0 8.0 10.0 2.00 4.00 Early end point 6.00 10.00 1.0 8.00 6.0 7.0 9.0 pMg Volume titrant (mL) 0.00 0.0 2.0 3.0 4.0 5.0 8.0 10.0 2.00 4.00 6.00 10.00 1.0 8.00 6.0 7.0 9.0 (a) (b) (c) Figure 9.31 Titration curves for 10 –3 M Mg 2+ with 10 –2 M EDTA using calmagite as an indicator at (a) pH = 9, (b) pH = 10, and (c) pH = 11. The range of pMg and volume of titrant over which the indicator is expected to change color is shown for each titration curve. —Continued Late end point pMg Volume titrant (mL) 0.00 0.0 2.0 3.0 4.0 5.0 8.0 10.0 2.00 4.00 6.00 10.00 1.0 8.00 6.0 7.0 9.0 The photo in Colorplate 8b shows the indicator s color change for this titration. 1400-CH09 9/9/99 2:13 PM Page 326 Chapter 9 Titrimetric Methods of Analysis 327 2. Why is a small amount of Mg 2+ –EDTA complex added to the buffer? The titration’s end point is signaled by the indicator calmagite, which gives a good end point with magnesium, but a poor end point with other cations such as calcium. If the sample does not contain any Mg 2+ as a source of hardness, then the titration will have a poorly defined end point and inaccurate results will be obtained. By adding a small amount of Mg 2+ –EDTA to the buffer, a source of Mg 2+ is ensured. When the buffer is added to the sample, the Mg 2+ is displaced by Ca 2+ , because Ca 2+ forms a stronger complex with EDTA. Since the displacement is stoichiometric, the total concentration of hardness cations remains unchanged, and there is no change in the amount of EDTA needed to reach the equivalence point. 3. Why does the procedure specify that the titration take no longer than 5 min? The presence of a time limitation suggests that there must be a kinetically controlled interference, possibly arising from a competing chemical reaction. In this case the interference is the possible precipitation of CaCO 3 . 9 C. 5 Quantitative Applications With a few exceptions, most quantitative applications of complexation titrimetry have been replaced by other analytical methods. In this section we review the gen- eral application of complexation titrimetry with an emphasis on selected applica- tions from the analysis of water and wastewater. We begin, however, with a discus- sion of the selection and standardization of complexation titrants. Selection and Standardization of Titrants EDTA is a versatile titrant that can be used for the analysis of virtually all metal ions. Although EDTA is the most com- monly employed titrant for complexation titrations involving metal ions, it cannot be used for the direct analysis of anions or neutral ligands. In the latter case, stan- dard solutions of Ag + or Hg 2+ are used as the titrant. Solutions of EDTA are prepared from the soluble disodium salt, Na 2 H 2 Y?2H 2 O. Concentrations can be determined directly from the known mass of EDTA; however, for more accurate work, standardization is accomplished by titrating against a solution made from the primary standard CaCO 3 . Solutions of Ag + and Hg 2+ are prepared from AgNO 3 and Hg(NO 3 ) 2 , both of which are sec- ondary standards. Standardization is accomplished by titrating against a solution prepared from primary standard grade NaCl. Inorganic Analysis Complexation titrimetry continues to be listed as a standard method for the determination of hardness, Ca 2+ , CN – , and Cl – in water and waste- water analysis. The evaluation of hardness was described earlier in Method 9.2. The determination of Ca 2+ is complicated by the presence of Mg 2+ , which also reacts with EDTA. To prevent an interference from Mg 2+ , the pH is adjusted to 12–13, precipitating any Mg 2+ as Mg(OH) 2 . Titrating with EDTA using murexide or Eri- ochrome Blue Black R as a visual indicator gives the concentration of Ca 2+ . Cyanide is determined at concentrations greater than 1 ppm by making the sample alkaline with NaOH and titrating with a standard solution of AgNO 3 , forming the soluble Ag(CN) 2 – complex. The end point is determined using p-dimethylaminobenzalrhodamine as a visual indicator, with the solution turn- ing from yellow to a salmon color in the presence of excess Ag + . 1400-CH09 9/9/99 2:13 PM Page 327 328 Modern Analytical Chemistry Chloride is determined by titrating with Hg(NO 3 ) 2 , forming soluble HgCl 2 . The sample is acidified to within the pH range of 2.3–3.8 where diphenylcarbazone, which forms a colored complex with excess Hg 2+ , serves as the visual indicator. Xy- lene cyanol FF is added as a pH indicator to ensure that the pH is within the desired range. The initial solution is a greenish blue, and the titration is carried out to a purple end point. Quantitative Calculations The stoichiometry of complexation reactions is given by the conservation of electron pairs between the ligand, which is an electron-pair donor, and the metal, which is an electron-pair acceptor (see Section 2C); thus This is simplified for titrations involving EDTA where the stoichiometry is always 1:1 regardless of how many electron pairs are involved in the formation of the metal–ligand complex. EXAMPLE 9 .8 The concentration of a solution of EDTA was determined by standardizing against a solution of Ca 2+ prepared from the primary standard CaCO 3 . A 0.4071-g sample of CaCO 3 was transferred to a 500-mL volumetric flask, dissolved using a minimum of 6 M HCl, and diluted to volume. A 50.00-mL portion of this solution was transferred into a 250-mL Erlenmeyer flask and the pH adjusted by adding 5 mL of a pH 10 NH 3 –NH 4 Cl buffer containing a small amount of Mg 2+ –EDTA. After adding calmagite as a visual indicator, the solution was titrated with the EDTA, requiring 42.63 mL to reach the end point. Report the molar concentration of the titrant. SOLUTION Conservation of electron pairs for the titration reaction requires that Moles EDTA = moles Ca 2+ Making appropriate substitution for the moles of EDTA and Ca 2+ gives M EDTA · V EDTA = M Ca · V Ca The molarity of the Ca 2+ solution is Substituting known values and solving for M EDTA gives MV V Ca Ca EDTA M mL mL M EDTA= × =× ? ? (. )( . ) . . 8 135 10 50 00 42 63 9 541 10 3 3 moles CaCO g CaCO FW CaCO g 100.09 g/mol L M Ca 3 flask 3 3 flask 2+ VV = × = × =× ? 0 4071 0 5000 8 135 10 3 . . . moles of electron pairs donated mole ligand moles ligand = moles of electron pairs accepted mole metal moles metal × × 1400-CH09 9/9/99 2:13 PM Page 328 Chapter 9 Titrimetric Methods of Analysis 329 The principle of the conservation of electron pairs is easily extended to other com- plexation reactions, as shown in the following example. EXAMPLE 9 . 9 The concentration of Cl – in a 100.0-mL sample of water drawn from a fresh water acquifer suffering from encroachment of sea water, was determined by titrating with 0.0516 M Hg(NO 3 ) 2 . The sample was acidified and titrated to the diphenylcarbazone end point, requiring 6.18 mL of the titrant. Report the concentration of Cl – in parts per million. SOLUTION Conservation of electron pairs requires that Moles Cl – =2· moles Hg 2+ Making appropriate substitutions for the moles of Cl – and Hg 2+ and rearranging leaves us with g Cl – =2· M Hg · V Hg · AW Cl Substituting known values and solving gives 2 · 0.0516 M · 0.00618 L · 35.453 g/mol = 0.0226 g Cl – The concentration of Cl – in parts per million, therefore, is Finally, quantitative problems involving multiple analytes and back titrations also can be solved by applying the principle of conservation of electron pairs. EXAMPLE 9 .10 An alloy of chromel containing Ni, Fe, and Cr was analyzed by a complexation titration using EDTA as the titrant. A 0.7176-g sample of the alloy was dissolved in HNO 3 and diluted to 250 mL in a volumetric flask. A 50.00-mL aliquot of the sample, treated with pyrophosphate to mask the Fe and Cr, required 26.14 mL of 0.05831 M EDTA to reach the murexide end point. A second 50.00-mL aliquot was treated with hexamethylenetetramine to mask the Cr. Titrating with 0.05831 M EDTA required 35.43 mL to reach the murexide end point. Finally, a third 50.00-mL aliquot was treated with 50.00 mL of 0.05831 M EDTA, and back titrated to the murexide end point with 6.21 mL of 0.06316 M Cu 2+ . Report the weight percents of Ni, Fe, and Cr in the alloy. mg Cl liter mg 0.1000 L ppm ? == 22 6 226 . g Cl AW Cl Hg Hg ? =× ×2 MV 1400-CH09 9/9/99 2:13 PM Page 329 SOLUTION Conservation of electron pairs for the three titrations requires that for Titration 1: moles Ni = moles EDTA1 (Fe, Cr masked) Titration 2: moles Ni + moles Fe = moles EDTA2 (Cr masked) Titration 3: moles Ni + moles Fe + moles Cr + moles Cu = moles EDTA3 Note that the third titration is a back titration. Titration 1 can be used to determine the amount of Ni in the alloy. Once the amount of Ni is known, the amount of Fe can be determined from the results for titration 2. Finally, titration 3 can be solved for the amount of Cr. Titration 1 Titration 2 Titration 3 Each of these titrations was conducted on a 50.00-mL aliquot of the original 250.0-mL sample. The mass of each analyte, therefore, must be corrected by multiplying by a factor of 5. Thus, the grams of Ni, Fe, and Cr in the original sample are Moles EDTA2 + moles Cr moles Cu moles EDTA3 g Cr AW Cr moles EDTA3 moles EDTA2 moles Cu g Cr AW Cr M 0.05000 L M L M L) 51.996 g/mol = 0.02378 g Cr EDTA3 EDTA3 EDTA2 EDTA2 Cu Cu += =?? =×?×?×× ×? × ×× () (. . . –. . MVMVMV 0 05831 0 05831 0 03543 0 06316 0 00621 Moles EDTA1 + moles Fe moles EDTA2 g Fe AW Fe moles EDTA2 moles EDTA1 g Fe AW Fe M 0.03543 L M L) g/mol g Fe EDTA2 EDTA2 EDTA1 EDTA1 = =? =×?×× ×? × ×= () (. . . .. MVMV 0 05831 0 05831 0 02614 55 847 0 03025 g Ni AW Ni g Ni AW Ni 0.05831 M L g / mol g Ni EDTA1 EDTA1 EDTA1 EDTA1 =× =×× ×× = MV MV 0 02614 58 69 0 08946.. . 330 Modern Analytical Chemistry 1400-CH09 9/9/99 2:13 PM Page 330 Figure 9.32 Titration curve for 10 –2 M Ca 2+ with 10 –2 M EDTA at (a) pH = 9 and (b) pH = 3. Figure 9.33 Spectrophotometric titration curve for the complexation titration of a mixture. Chapter 9 Titrimetric Methods of Analysis 331 (a) (b) pCa Volume of EDTA (mL) 0.0 0 2 3 4 5 8 10 20.0 40.0 60.0 100.0 1 80.0 6 7 9 Corrected absorbance Volume of titrant Second equivalence point First equivalence point 0.08946 g · 5 = 0.4473 g Ni 0.03025 g · 5 = 0.1513 g Fe 0.02378 g · 5 = 0.1189 g Cr and the %w/w for each metal is 9 C.6 Evaluation of Complexation Titrimetry The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving complexation titrations are similar to those described earlier for acid–base titrimetric methods. Compared with acid–base titrations, however, complexation titrations are more selective. Despite the ability of EDTA to form strong complexes with virtually all metal ions, carefully controlling the pH at which the titration is carried out makes it possible to analyze samples containing two or more analytes (see Example 9.10). The reason that pH can be used to provide selectivity is easily appreciated by examining Figure 9.32. A titration of Ca 2+ at a pH of 9 gives a dis- tinct break in the titration curve because the conditional formation constant (K ′ f = 2.6 · 10 9 ) is large enough to ensure that the reaction of Ca 2+ and EDTA goes to completion. At a pH of 3, however, the conditional formation constant (K ′ f = 1.23) is so small that very little Ca 2+ reacts with the EDTA. Spectrophotometric titrations are particularly useful for the analysis of mixtures if a suitable difference in absorbance exists between the analytes and products, or titrant. For example, the analysis of a two-component mixture can be accomplished if there is a difference between the absorbance of the two metal–ligand complexes (Figure 9.33). 9 D Titrations Based on Redox Reactions Redox titrations were introduced shortly after the development of acid–base titrimetry. The earliest methods took advantage of the oxidizing power of chlorine. In 1787, Claude Berthollet introduced a method for the quantitative analysis of chlo- rine water (a mixture of Cl 2 , HCl, and HOCl) based on its ability to oxidize solutions of the dye indigo (indigo is colorless in its oxidized state). In 1814, Joseph Louis Gay- Lussac (1778–1850), developed a similar method for chlorine in bleaching powder. In both methods the end point was signaled visually. Before the equivalence point, the solution remains clear due to the oxidation of indigo. After the equivalence point, however, unreacted indigo imparts a permanent color to the solution. 0 1189 0 7176 100 16 57 . . .% g g w/w Cr×= 0 1513 100 21 08 . .% g 0.7176 g w/w Fe×= 0 4473 100 62 33 . .% g 0.7176 g w/w Ni×= redox titration A titration in which the reaction between the analyte and titrant is an oxidation/reduction reaction. 1400-CH09 9/9/99 2:13 PM Page 331 The number of redox titrimetric methods increased in the mid-1800s with the introduction of MnO 4 – , Cr 2 O 7 2– and I 2 as oxidizing titrants, and Fe 2+ and S 2 O 3 2– as reducing titrants. Even with the availability of these new titrants, however, the rou- tine application of redox titrimetry to a wide range of samples was limited by the lack of suitable indicators. Titrants whose oxidized and reduced forms differ signifi- cantly in color could be used as their own indicator. For example, the intensely pur- ple MnO 4 – ion serves as its own indicator since its reduced form, Mn 2+ , is almost colorless. The utility of other titrants, however, required a visual indicator that could be added to the solution. The first such indicator was diphenylamine, which was introduced in the 1920s. Other redox indicators soon followed, increasing the applicability of redox titrimetry. 9 D.1 Redox Titration Curves To evaluate a redox titration we must know the shape of its titration curve. In an acid–base titration or a complexation titration, a titration curve shows the change in concentration of H 3 O + (as pH) or M n+ (as pM) as a function of the volume of titrant. For a redox titration, it is convenient to monitor electrochemical potential. You will recall from Chapter 6 that the Nernst equation relates the electro- chemical potential to the concentrations of reactants and products participating in a redox reaction. Consider, for example, a titration in which the analyte in a reduced state, A red , is titrated with a titrant in an oxidized state, T ox . The titration reaction is A red + T ox t T red + A ox The electrochemical potential for the reaction is the difference between the reduc- tion potentials for the reduction and oxidation half-reactions; thus, E rxn = E Tox/Tred – E Aox/Ared After each addition of titrant, the reaction between the analyte and titrant reaches a state of equilibrium. The reaction’s electrochemical potential, E rxn , therefore, is zero, and E Tox/Tred = E Aox /Ared Consequently, the potential for either half-reaction may be used to monitor the titration’s progress. Before the equivalence point the titration mixture consists of appreciable quan- tities of both the oxidized and reduced forms of the analyte, but very little unreacted titrant. The potential, therefore, is best calculated using the Nernst equation for the analyte’s half-reaction Although E? Aox /Ared is the standard-state potential for the analyte’s half-reaction, a matrix-dependent formal potential is used in its place. After the equivalence point, the potential is easiest to calculate using the Nernst equation for the titrant’s half-reaction, since significant quantities of its oxidized and reduced forms are present. EE RT nF T T TT TT ox red ox red red ox / ––ln [] [] = ° EE RT nF A A AA AA ox red ox red red ox / ln [] [] =? ° 332 Modern Analytical Chemistry formal potential The potential of a redox reaction for a specific set of solution conditions, such as pH and ionic composition. 1400-CH09 9/9/99 2:13 PM Page 332 Calculating the Titration Curve As an example, let’s calculate the titration curve for the titration of 50.0 mL of 0.100 M Fe 2+ with 0.100 M Ce 4+ in a matrix of 1 M HClO 4 . The reaction in this case is Fe 2+ (aq)+Ce 4+ (aq) t Ce 3+ (aq)+Fe 3+ (aq) 9.16 The equilibrium constant for this reaction is quite large (it is approximately 6 · 10 15 ), so we may assume that the analyte and titrant react completely. The first task is to calculate the volume of Ce 4+ needed to reach the equivalence point. From the stoichiometry of the reaction we know Moles Fe 2+ = moles Ce 4+ or M Fe V Fe = M Ce V Ce Solving for the volume of Ce 4+ gives the equivalence point volume as 50.0 mL. Before the equivalence point the concentration of unreacted Fe 2+ and the con- centration of Fe 3+ produced by reaction 9.16 are easy to calculate. For this reason we find the potential using the Nernst equation for the analyte’s half-reaction 9.17 The concentrations of Fe 2+ and Fe 3+ after adding 5.0 mL of titrant are Substituting these concentrations into equation 9.17 along with the formal potential for the Fe 3+ /Fe 2+ half-reaction from Appendix 3D, we find that the potential is At the equivalence point, the moles of Fe 2+ initially present and the moles of Ce 4+ added are equal. Because the equilibrium constant for reaction 9.16 is large, the concentrations of Fe 2+ and Ce 4+ are exceedingly small and difficult to calculate without resorting to a complex equilibrium problem. Consequently, we cannot cal- culate the potential at the equivalence point, E eq , using just the Nernst equation for the analyte’s half-reaction or the titrant’s half-reaction. We can, however, calculate E =+ ? × × ? ? ? ? ? ? =+ ? ? 0 767 0 05916 818 10 909 10 0 711 2 3 . . log . . . V V [] (. . Fe moles Ce added total volume M)(5.0 mL) 50.0 ml + 5.0 mL M Ce Ce Fe Ce 3 4 3 0 100 909 10 + + ? == + ==× MV VV [] (. ) (. . . Fe moles unreacted Fe total volume M)(50.0 mL M)(5.0 mL) 50.0 mL mL M Fe Fe Ce Ce Fe Ce 2 2 2 0 100 0 100 50 818 10 + + ? == ? + = ? + =× MV MV VV EE=? ° + + ++ Fe Fe Fe Fe 32 0 05916 2 3 / .ln [] [] V MV M Ce Fe Fe Ce M)(50.0 mL) (0.100 M) mL== = (. . 0 100 50 0 Chapter 9 Titrimetric Methods of Analysis 333 1400-CH09 9/9/99 2:13 PM Page 333 E eq by combining the two Nernst equations. To do so we recognize that the poten- tials for the two half-reactions are the same; thus, Adding together these two Nernst equations leaves us with 9.18 At the equivalence point, the titration reaction’s stoichiometry requires that [Fe 2+ ] = [Ce 4+ ] [Fe 3+ ] = [Ce 3+ ] The ratio in the log term of equation 9.18, therefore, equals one and the log term is zero. Equation 9.18 simplifies to After the equivalence point, the concentrations of Ce 3+ and excess Ce 4+ are easy to calculate. The potential, therefore, is best calculated using the Nernst equation for the titrant’s half-reaction. 9.19 For example, after adding 60.0 mL of titrant, the concentrations of Ce 3+ and Ce 4+ are Substituting these concentrations into equation 9.19 gives the potential as Additional results for this titration curve are shown in Table 9.17 and Figure 9.34. E =+ ? × × = ? ? 1 70 0 05916 455 10 909 10 166 2 3 . . log . . . V V [] (. (. . Ce moles excess Ce total volume M)(60.0 mL) M)(50.0 mL) 50.0 mL + 60.0 mL M 4+ 4+ Ce Ce Fe Fe Fe Ce == ? + = ? =× ? MV MV VV 0 100 0 100 909 10 3 [] (. . Ce intial moles Fe total volume M)(50.0 mL) 50.0 mL + 60.0 mL M 3+ 2+ Fe Fe Fe Ce == + ==× ? MV VV 0 100 455 10 2 EE=? ° Ce Ce 3+ 4+ 4+ 3+ Ce Ce / . log [] [] 0 05916 E EE eq Fe Fe Ce Ce 3+ 2+ 4+ 3+ V V 2 V= + = + = °° // .. . 2 0 767 1 70 123 2 0 05916EE E eq Fe Fe Ce Ce 2+ 3+ 3+ 4+ 3+ 2+ 4+ 3+ Fe Ce Fe Ce =+ ? °° // . log [][ ] [][ ] EE eq Ce Ce 3+ 4+ 4+ 3+ Ce Ce =? ° / . log [] [] 0 05916 EE eq Fe Fe 2+ 3+ 2+ Fe Fe =?+ ° 3 0 05916 / . log [] [] 334 Modern Analytical Chemistry 1400-CH09 9/9/99 2:13 PM Page 334 Figure 9.34 Redox titration curve for 50.0 mL of 0.100 M Fe 2+ with 0.100 M Ce 4+ in 1 M HClO 4 . Chapter 9 Titrimetric Methods of Analysis 335 E Volume of Ce 4+ 0 0.6 1.8 10080604020 1.6 1.4 1.2 1 0.8 Table 9 .1 7 Data for Titration of 50.0 mL of 0.100 M Fe 2+ with 0.100 M Ce 4+ Volume Ce 4+ Volume Ce 4+ (mL) E (V) (mL) E (V) 5.00 0.711 55.00 1.64 10.00 0.731 60.00 1.66 15.00 0.745 65.00 1.67 20.00 0.757 70.00 1.68 25.00 0.767 75.00 1.68 30.00 0.777 80.00 1.69 35.00 0.789 85.00 1.69 40.00 0.803 90.00 1.69 45.00 0.823 95.00 1.70 50.00 1.23 100.00 1.70 Sketching a Redox Titration Curve As we have done for acid–base and complexo- metric titrations, we now show how to quickly sketch a redox titration curve using a minimum number of calculations. EXAMPLE 9 .11 Sketch a titration curve for the titration of 50.0 mL of 0.100 M Fe 2+ with 0.100 M Ce 4+ in a matrix of 1 M HClO 4 . This is the same titration for which we previously calculated the titration curve (Table 9.17 and Figure 9.34). SOLUTION We begin by drawing axes for the titration curve (Figure 9.35a). Having shown that the equivalence point volume is 50.0 mL, we draw a vertical line intersecting the x-axis at this volume (Figure 9.35b). Before the equivalence point, the solution’s electrochemical potential is determined by the concentration of excess Fe 2+ and the concentration of Fe 3+ produced by the titration reaction. Using values from Table 9.17, we plot E for 5.0 mL and 45.0 mL of titrant (Figure 9.35c). After the equivalence point, the solution’s electrochemical potential is determined by the concentration of excess Ce 4+ and the concentration of Ce 3+ . Using values from Table 9.17, we plot points for 60.0 mL and 80.0 mL of titrant (Figure 9.35d). To complete an approximate sketch of the titration curve, we draw separate straight lines through the two points before and after the equivalence point (Figure 9.35e). Finally, a smooth curve is drawn to connect the three straight-line segments (Figure 9.35f). 1400-CH09 9/9/99 2:13 PM Page 335 336 Modern Analytical Chemistry P otential (V) Volume of titrant (mL) Percent titrated 0 0.000 0.400 0.600 0.800 1.000 1.400 40 500 100 200150 100 0.200 20 60 80 1.600 1.200 1.800 (a) P otential (V) Volume of titrant (mL) Percent titrated 0 0.000 0.400 0.600 0.800 1.000 1.400 40 500 100 200150 100 0.200 20 60 80 1.600 1.200 1.800 (c) P otential (V) Volume of titrant (mL) Percent titrated 0 0.000 0.400 0.600 0.800 1.000 1.400 40 500 100 200150 100 0.200 20 60 80 1.600 1.200 1.800 (e) P otential (V) Volume of titrant (mL) Percent titrated 0 0.000 0.400 0.600 0.800 1.000 1.400 40 500 100 200150 100 0.200 20 60 80 1.600 1.200 1.800 (b) P otential (V) Volume of titrant (mL) Percent titrated 0 0.000 0.400 0.600 0.800 1.000 1.400 40 500 100 200150 100 0.200 20 60 80 1.600 1.200 1.800 (d) P otential (V) Volume of titrant (mL) Percent titrated 0 0.000 0.400 0.600 0.800 1.000 1.400 40 500 100 200150 100 0.200 20 60 80 1.600 1.200 1.800 (f) Figure 9.35 How to sketch a redox titration curve; see text for explanation. 1400-CH09 9/9/99 2:13 PM Page 336 Chapter 9 Titrimetric Methods of Analysis 337 9 D.2 Selecting and Evaluating the End Point The equivalence point of a redox titration occurs when stoichiometrically equiva- lent amounts of analyte and titrant react. As with other titrations, any difference be- tween the equivalence point and the end point is a determinate source of error. Where Is the Equivalence Point? In discussing acid–base titrations and com- plexometric titrations, we noted that the equivalence point is almost identical with the inflection point located in the sharply rising part of the titration curve. If you look back at Figures 9.8 and 9.28, you will see that for acid–base and com- plexometric titrations the inflection point is also in the middle of the titration curve’s sharp rise (we call this a symmetrical equivalence point). This makes it relatively easy to find the equivalence point when you sketch these titration curves. When the stoichiometry of a redox titration is symmetrical (one mole analyte per mole of titrant), then the equivalence point also is symmetrical. If the stoichiometry is not symmetrical, then the equivalence point will lie closer to the top or bottom of the titration curve’s sharp rise. In this case the equivalence point is said to be asymmetrical. Example 9.12 shows how to calculate the equiv- alence point potential in this situation. EXAMPLE 9 .12 Derive a general equation for the electrochemical potential at the equivalence point for the titration of Fe 2+ with MnO 4 – ; the reaction is 5Fe 2+ (aq) + MnO 4 – (aq)+8H 3 O + (aq) t 5Fe 3+ (aq)+Mn 2+ (aq) + 12H 2 O(l) SOLUTION The redox half-reactions for the analyte and the titrant are Fe 2+ (aq) t Fe 3+ (aq)+e – MnO 4 – (aq)+8H 3 O + (aq)+5e – t Mn 2+ (aq) + 12H 2 O(l) for which the Nernst equations are Before adding together these two equations, the second equation must be multiplied by 5 so that the log terms can be combined; thus At the equivalence point, we know that [Fe 2+ ]=5· [MnO 4 – ] [Fe 3+ ]=5· [Mn 2+ ] 6 5 0 05916 8 EE E eq Fe Fe MnO Mn 2+ 2+ 3+ 43 + 3+ 2+ 4 2+ Fe Mn Fe MnO H O =+ ? °° ? ?/ / . log [][ ] [][ ][ ] EE eq MnO Mn 43 + 4 2+ Mn [MnO H O =? ? ° + ? / . log [] ][ ] 0 05916 5 2 8 EE eq Fe Fe 2+ 3+ 3+ 2+ Fe Fe =? ° / . log [] [] 0 05916 1400-CH09 9/9/99 2:13 PM Page 337 Substituting these equalities into the equation for E eq and rearranging gives For this titration the electrochemical potential at the equivalence point consists of two terms. The first term is a weighted average of the standard state or formal potentials for the analyte and titrant, in which the weighting factors are the number of electrons in their respective redox half-reactions. The second term shows that E eq is pH-dependent. Figure 9.36 shows a typical titration curve for the analysis of Fe 2+ by titration with MnO 4 – , showing the asymmetrical equivalence point. Note that the change in potential near the equivalence point is sharp enough that selecting an end point near the middle of the titration curve’s sharply rising portion does not introduce a significant titration error. E EE EE EE eq Fe Fe MnO Mn 4 2+ 2+ 43 + Fe Fe MnO Mn 3 + Fe Fe MnO Mn 3+ 2+ 4 2+ 3+ 2+ 4 2+ 3+ 2+ 4 2+ MnO Mn Mn MnO H O HO = + ? = + ? = + + °° ? ? °° °° ? ? ? / / / / / / . log [][] [][ ][] . log [] 5 6 0 05916 6 5 5 5 6 0 05916 6 1 5 6 8 8 ( . )( ) log[ ] . / / 0 05916 8 6 5 6 0 07888 HO pH 3 + Fe Fe MnO Mn 3+ 2+ 4 2+ = + ? °° ? EE 338 Modern Analytical Chemistry E Volume of MnO 4 – 0 0.6 1.6 10080604020 1.4 1.2 1 0.8 Figure 9.36 Titration curve for Fe 2+ with MnO 4 – in 1 M H 2 SO 4 ; equivalence point is shown by the symbol . Finding the End Point with a Visual Indicator Three types of visual indicators are used to signal the end point in a redox titration. A few titrants, such as MnO 4 – , have oxidized and reduced forms whose colors in solution are signifi- cantly different. Solutions of MnO 4 – are intensely purple. In acidic solutions, however, permanganate’s reduced form, Mn 2+ , is nearly colorless. When MnO 4 – is used as an oxidizing titrant, the solution remains colorless until the first drop of excess MnO 4 – is added. The first permanent tinge of purple signals the end point. A few substances indicate the presence of a specific oxidized or reduced species. Starch, for example, forms a dark blue complex with I 3 – and can be used to signal the presence of excess I 3 – (color change: colorless to blue), or the completion of a reaction in which I 3 – is consumed (color change: blue to colorless). Another exam- ple of a specific indicator is thiocyanate, which forms a soluble red-colored com- plex, Fe(SCN) 2+ , with Fe 3+ . 1400-CH09 9/9/99 2:13 PM Page 338 The most important class of redox indicators, however, are substances that do not participate in the redox titration, but whose oxidized and reduced forms differ in color. When added to a solution containing the analyte, the indicator im- parts a color that depends on the solution’s electrochemical potential. Since the indicator changes color in response to the electrochemical potential, and not to the presence or absence of a specific species, these compounds are called general redox indicators. The relationship between a redox indicator’s change in color and the solution’s electrochemical potential is easily derived by considering the half-reaction for the indicator In ox + ne – t In red where In ox and In red are, respectively, the indicator’s oxidized and reduced forms. The Nernst equation for this reaction is If we assume that the indicator’s color in solution changes from that of In ox to that of In red when the ratio [In red ]/[In ox ] changes from 0.1 to 10, then the end point occurs when the solution’s electrochemical potential is within the range A partial list of general redox indicators is shown in Table 9.18. Examples of appropriate and inappropriate indicators for the titration of Fe 2+ with Ce 4+ are shown in Figure 9.37. Finding the End Point Potentiometrically Another method for locating the end point of a redox titration is to use an appropriate electrode to monitor the change in electrochemical potential as titrant is added to a solution of analyte. The end point can then be found from a visual inspection of the titration curve. The simplest experimental design (Figure 9.38) consists of a Pt indica- tor electrode whose potential is governed by the analyte’s or titrant’s redox half-reaction, and a reference electrode that has a fixed potential. A further discussion of potentiometry is found in Chapter 11. EE n =± ° In In ox red / .0 05916 EE n =? ° In In red ox ox red In [In / . log [] ] 0 05916 Chapter 9 Titrimetric Methods of Analysis 339 redox indicator A visual indicator used to signal the end point in a redox titration. Table 9 .18 Selected General Redox Indicators Indicator Oxidized Color Reduced Color E° (V) indigo tetrasulfonate blue colorless 0.36 methylene blue blue colorless 0.53 diphenylamine violet colorless 0.75 diphenylamine sulfonic acid red-violet colorless 0.85 tris(2,2’-bipyridine)iron pale blue red 1.120 ferroin pale blue red 1.147 tris(5-nitro-1,10-phenanthroline)iron pale blue red-violet 1.25 P otential (V) Volume of titrant (mL) 0 0.000 0.400 0.800 1.600 1.400 1.000 0.600 0.200 1.800 20 40 Diphenylamine sulfonic acid Ferroin 60 10080 1.200 Potentiometer Pt indicator electrode Reference electrode Buret Figure 9.38 Experimental arrangement for recording a potentiometric redox titration curve. Figure 9.37 Titration curve for 50.00 mL of 0.0500 M Fe 2+ with 0.0500 M Ce 4+ showing the range of E and volume of titrant over which the indicators ferroin and diphenylamine sulfonic acid are expected to change color. 1400-CH09 9/9/99 2:13 PM Page 339 340 Modern Analytical Chemistry Representative Methods 9 D. 3 Representative Method Although every redox titrimetric method has its own unique considerations, the fol- lowing description of the determination of total residual chlorine in water provides an instructive example of a typical procedure. Method 9 . 3 Determination of Total Chlorine Residual 13 Description of the Method. The chlorination of public water supplies results in the formation of several chlorine-containing species, the combined concentration of which is called the total chlorine residual. Chlorine may be present in a variety of states including free residual chlorine, consisting of Cl 2 , HOCl, and OCl – , and combined chlorine residual, consisting of NH 2 Cl, NHCl 2 , and NCl 3 . The total chlorine residual is determined by using the oxidizing power of chlorine to convert I – to I 3 – . The amount of I 3 – formed is then determined by a redox titration using S 2 O 3 2– as a titrant and starch as an indicator. Regardless of its form, the total chlorine residual is calculated as if all the chlorine were available as Cl 2 , and is reported as parts per million of Cl. Procedure. Select a volume of sample requiring less than 20 mL of S 2 O 3 2– to reach the end point. Using glacial acetic acid, acidify the sample to a pH in the range of 3 to 4, and add about 1 g of KI. Titrate with Na 2 S 2 O 3 until the yellow color due to I 3 – begins to disappear. Add 1 mL of a starch indicator solution, and continue titrating until the blue color of the starch–I 3 – complex disappears. The volume of titrant needed to reach the end point should be corrected for reagent impurities by conducting a blank titration. Questions 1. Is this an example of a direct or an indirect analysis? This is an indirect method of analysis because the chlorine-containing species do not react with the titrant. Instead the total chlorine residual oxidizes I – to I 3 – , and the amount of I 3 – is determined by the redox titration with Na 2 S 2 O 3 . 2. Why is the procedure not carried out directly using KI as a titrant? The redox half-reaction when I – is used as a titrant is 3I – t I 3 – +2e – Because the product of the reaction, I 3 – , is itself colored, the color of the solution containing the analyte changes with each addition of titrant. For this reason it is difficult to find a suitable visual indicator for the titration’s end point. 3. Both oxidizing and reducing agents can interfere with this analysis. Explain what effect each of these interferents will have on the result of an analysis. An interferent that is an oxidizing agent will convert additional I – to I 3 – . This extra I 3 – requires an additional volume of S 2 O 3 2– to reach the end point, overestimating the total chlorine residual. If the interferent is a reducing agent, it reduces some of the I 3 – , produced by the reaction between the total chlorine residual and iodide, back to I – . The result is an underestimation of the total chlorine residual. The photo in Colorplate 8c shows the indicator s color change for this titration. 1400-CH09 9/9/99 2:13 PM Page 340 Figure 9.39 Schematic diagram of a reductor column. 9 D. 4 Quantitative Applications As with acid–base and complexation titrations, redox titrations are not frequently used in modern analytical laboratories. Nevertheless, several important applications continue to find favor in environmental, pharmaceutical, and industrial laborato- ries. In this section we review the general application of redox titrimetry. We begin, however, with a brief discussion of selecting and characterizing redox titrants, and methods for controlling the analyte’s oxidation state. Adjusting the Analyte s Oxidation State If a redox titration is to be used in a quantitative analysis, the analyte must initially be present in a single oxidation state. For example, the iron content of a sample can be determined by a redox titration in which Ce 4+ oxidizes Fe 2+ to Fe 3+ . The process of preparing the sample for analysis must ensure that all iron is present as Fe 2+ . Depending on the sample and the method of sample preparation, however, the iron may initially be present in both the +2 and +3 oxidation states. Before titrating, any Fe 3+ that is present must be re- duced to Fe 2+ . This type of pretreatment can be accomplished with an auxiliary re- ducing or oxidizing agent. Metals that are easily oxidized, such as Zn, Al, and Ag, can serve as auxiliary re- ducing agents. The metal, as a coiled wire or powder, is placed directly in the solu- tion where it reduces the analyte. Of course any unreacted auxiliary reducing agent will interfere with the analysis by reacting with the titrant. The residual auxiliary re- ducing agent, therefore, must be removed once the analyte is completely reduced. This can be accomplished by simply removing the coiled wire or by filtering. An alternative approach to using an auxiliary reducing agent is to immobilize it in a column. To prepare a reduction column, an aqueous slurry of the finely divided metal is packed in a glass tube equipped with a porous plug at the bottom (Figure 9.39). The sample is placed at the top of the column and moves through the column under the influence of gravity or vacuum suction. The length of the reduction col- umn and the flow rate are selected to ensure the analyte’s complete reduction. Two common reduction columns are used. In the Jones reductor the column is filled with amalgamated Zn prepared by briefly placing Zn granules in a solution of HgCl 2 to form Zn(Hg). Oxidation of the amalgamated Zn Zn(Hg)(s) t Zn 2+ (aq) + Hg(l)+2e – provides the electrons for reducing the analyte. In the Walden reductor the column is filled with granular Ag metal. The solution containing the analyte is acidified with HCl and passed through the column where the oxidation of Ag Ag(s)+Cl – (aq) t AgCl(s)+e – provides the necessary electrons for reducing the analyte. Examples of both reduc- tion columns are shown in Table 9.19. Several reagents are commonly used as auxiliary oxidizing agents, including ammonium peroxydisulfate, (NH 4 ) 2 S 2 O 8 , and hydrogen peroxide, H 2 O 2 . Ammo- nium peroxydisulfate is a powerful oxidizing agent S 2 O 8 2– (aq)+2e – t 2SO 4 2– (aq) capable of oxidizing Mn 2+ to MnO 4 – , Cr 3+ to Cr 2 O 7 2– , and Ce 3+ to Ce 4+ . Excess per- oxydisulfate is easily destroyed by briefly boiling the solution. The reduction of hy- drogen peroxide in acidic solution H 2 O 2 (aq)+2H 3 O + (aq)+2e – t 4H 2 O(l) Chapter 9 Titrimetric Methods of Analysis 341 Aqueous sample Solid reductant Porous plug auxiliary reducing agent A reagent used to reduce the analyte before its analysis by a redox titration. Jones reductor A reduction column using a Zn amalgam as a reducing agent. Walden reductor A reduction column using granular Ag as a reducing agent. auxiliary oxidizing agent A reagent used to oxidize the analyte before its analysis by a redox titration. 1400-CH09 9/9/99 2:13 PM Page 341 provides another method for oxidizing an analyte. Excess H 2 O 2 also can be de- stroyed by briefly boiling the solution. Selecting and Standardizing a Titrant In quantitative work the titrant’s concen- tration must remain stable during the analysis. Since titrants in a reduced state are susceptible to air oxidation, most redox titrations are carried out using an oxidizing agent as the titrant. The choice of which of several common oxidizing titrants is best for a particular analysis depends on the ease with which the analyte can be oxidized. Analytes that are strong reducing agents can be successfully titrated with a relatively weak oxidizing titrant, whereas a strong oxidizing titrant is required for the analysis of analytes that are weak reducing agents. The two strongest oxidizing titrants are MnO 4 – and Ce 4+ , for which the reduc- tion half-reactions are MnO 4 – (aq)+8H 3 O + (aq)+5e – t Mn 2+ (aq) + 12H 2 O(l) Ce 4+ (aq)+e – t Ce 3+ (aq) Solutions of Ce 4+ are prepared from the primary standard cerium ammonium ni- trate, Ce(NO 3 ) 4 2NH 4 NO 3 , in 1 M H 2 SO 4 . When prepared from reagent grade materials, such as Ce(OH) 4 , the solution must be standardized against a primary standard reducing agent such as Na 2 C 2 O 4 or Fe 2+ (prepared using Fe wire). Ferroin is a suitable indicator when standardizing against Fe 2+ (Table 9.20). Despite its availability as a primary standard and its ease of preparation, Ce 4+ is not as fre- quently used as MnO 4 – because of its greater expense. Solutions of MnO 4 – are prepared from KMnO 4 , which is not available as a pri- mary standard. Aqueous solutions of permanganate are thermodynamically unsta- ble due to its ability to oxidize water. 4MnO 4 – (aq)+2H 2 O(l) t 4MnO 2 (s)+3O 2 (g) + 4OH – (aq) This reaction is catalyzed by the presence of MnO 2 , Mn 2+ , heat, light, and the pres- ence of acids and bases. Moderately stable solutions of permanganate can be pre- pared by boiling for an hour and filtering through a sintered glass filter to remove any solid MnO 2 that precipitates. Solutions prepared in this fashion are stable for 1–2 weeks, although the standardization should be rechecked periodically. Stan- dardization may be accomplished using the same primary standard reducing agents that are used with Ce 4+ , using the pink color of MnO 4 – to signal the end point (Table 9.20). 342 Modern Analytical Chemistry Table 9.19 Selected Reductions Using Metal Reductor Columns Oxidized Species Walden Reductor Jones Reductor Cr 3+ –– a Cr 3+ + e – fi Cr 2+ Cu 2+ Cu 2+ + e – fi Cu + Cu 2+ +2e – fi Cu Fe 3+ Fe 3+ + e – fi Fe 2+ Fe 3+ + e – fi Fe 2+ TiO 2+ — a TiO 2+ +2H 3 O + + e – fi Ti 3+ +3H 2 O MoO 2 2+ MoO 2 2+ + e – fi MoO 2 + MoO 2 2+ +4H 3 O + +3e – fi Mo 3+ +6H 2 O VO 2 + VO 2 + +2H 3 O + + e – fi VO 2+ +3H 2 OVO 2 + +4H 3 O + +3e – fi V 2+ +6H 2 O a No reaction. 1400-CH09 9/9/99 2:13 PM Page 342 Potassium dichromate is a relatively strong oxidizing agent whose principal ad- vantages are its availability as a primary standard and the long-term stability of its solutions. It is not, however, as strong an oxidizing agent as MnO 4 – or Ce 4+ , which prevents its application to the analysis of analytes that are weak reducing agents. Its reduction half-reaction is Cr 2 O 7 2– (aq) + 14H 3 O + (aq)+6e – t 2Cr 3+ (aq) + 21H 2 O(l) Although solutions of Cr 2 O 7 2– are orange and those of Cr 3+ are green, neither color is intense enough to serve as a useful indicator. Diphenylamine sulfonic acid, whose oxidized form is purple and reduced form is colorless, gives a very distinct end point signal with Cr 2 O 7 2– . Iodine is another commonly encountered oxidizing titrant. In comparison with MnO 4 – , Ce 4+ , and Cr 2 O 7 2– , it is a weak oxidizing agent and is useful only for the analysis of analytes that are strong reducing agents. This apparent limitation, how- ever, makes I 2 a more selective titrant for the analysis of a strong reducing agent in the presence of weaker reducing agents. The reduction half-reaction for I 2 is I 2 (aq)+2e – t 2I – (aq) Because of iodine’s poor solubility, solutions are prepared by adding an excess of I – . The complexation reaction I 2 (aq)+I – (aq) t I 3 – (aq) increases the solubility of I 2 by forming the more soluble triiodide ion, I 3 – . Even though iodine is present as I 3 – instead of I 2 , the number of electrons in the reduc- tion half-reaction is unaffected. I 3 – (aq)+2e – t 3I – (aq) Solutions of I 3 – are normally standardized against Na 2 S 2 O 3 (see Table 9.20) using starch as a specific indicator for I 3 – . Oxidizing titrants such as MnO 4 – , Ce 4+ , Cr 2 O 7 2– and I 3 – , are used to titrate ana- lytes that are in a reduced state. When the analyte is in an oxidized state, it can be reduced with an auxiliary reducing agent and titrated with an oxidizing titrant. Al- ternatively, the analyte can be titrated with a suitable reducing titrant. Iodide is a relatively strong reducing agent that potentially could be used for the analysis of an- alytes in higher oxidation states. Unfortunately, solutions of I – cannot be used as a direct titrant because they are subject to the air oxidation of I – to I 3 – . 3I – (aq) t I 3 – (aq)+2e – Chapter 9 Titrimetric Methods of Analysis 343 Table 9 .20 Standardization Reactions for Selected Redox Titrants Titration Reaction Ce 4+ +Fe 2+ fi Ce 3+ +Fe 3+ 2Ce 4+ +H 2 C 2 O 4 +2H 2 O fi 2Ce 3+ + 2CO 2 +2H 3 O + MnO 4 – + 5Fe 2+ +8H 3 O + fi Mn 2+ + 5Fe 3+ + 12H 2 O 2MnO 4 – +5H 2 C 2 O 4 +6H 3 O + fi 2Mn 2+ + 10CO 2 + 14H 2 O I 3 – +2S 2 O 3 2– fi 3I – +S 4 O 6 2– I 2 +2S 2 O 3 2– fi 2I – +S 4 O 6 2– 1400-CH09 9/9/99 2:13 PM Page 343 Instead, an excess of KI is added, reducing the analyte and liberating a stoichiomet- ric amount of I 3 – . The amount of I 3 – produced is then determined by a back titra- tion using Na 2 S 2 O 3 as a reducing titrant. 2S 2 O 3 2– (aq) t S 4 O 6 2– (aq)+2e – Solutions of Na 2 S 2 O 3 are prepared from the pentahydrate and must be stan- dardized before use. Standardization is accomplished by dissolving a carefully weighed portion of the primary standard KIO 3 in an acidic solution containing an excess of KI. When acidified, the reaction between IO 3 – and I – IO 3 – (aq)+8I – (aq)+6H 3 O + (aq) t 3I 3 – (aq)+9H 2 O(l) liberates a stoichiometric amount of I 3 – . Titrating I 3 – using starch as a visual indica- tor allows the determination of the titrant’s concentration. Although thiosulfate is one of the few reducing titrants not readily oxidized by contact with air, it is subject to a slow decomposition to bisulfite and ele- mental sulfur. When used over a period of several weeks, a solution of thiosul- fate should be restandardized periodically. Several forms of bacteria are able to metabolize thiosulfate, which also can lead to a change in its concentration. This problem can be minimized by adding a preservative such as HgI 2 to the solution. Another reducing titrant is ferrous ammonium sulfate, Fe(NH 4 ) 2 (SO 4 ) 2 6H 2 O, in which iron is present in the +2 oxidation state. Solutions of Fe 2+ are normally very susceptible to air oxidation, but when prepared in 0.5 M H 2 SO 4 the solution may remain stable for as long as a month. Periodic restandardization with K 2 Cr 2 O 7 is advisable. The titrant can be used in either a direct titration in which the Fe 2+ is oxidized to Fe 3+ , or an excess of the solution can be added and the quantity of Fe 3+ produced determined by a back titration using a standard solution of Ce 4+ or Cr 2 O 7 2– . Inorganic Analysis Redox titrimetry has been used for the analysis of a wide range of inorganic analytes. Although many of these methods have been replaced by newer methods, a few continue to be listed as standard methods of analysis. In this section we consider the application of redox titrimetry to several important envi- ronmental, public health, and industrial analyses. Other examples can be found in the suggested readings listed at the end of this chapter. One of the most important applications of redox titrimetry is in evaluating the chlorination of public water supplies. In Method 9.3 an approach for determining the total chlorine residual was described in which the oxidizing power of chlorine is used to oxidize I – to I 3 – . The amount of I 3 – formed is determined by a back titration with S 2 O 3 2– . The efficiency of chlorination depends on the form of the chlorinating species. For this reason it is important to distinguish between the free chlorine residual, due to Cl 2 , HOCl, and OCl – , and the combined chlorine residual. The latter form of chlorine results from the reaction of ammonia with the free chlo- rine residual, forming NH 2 Cl, NHCl 2 , and NCl 3 . When a sample of iodide-free chlorinated water is mixed with an excess of the indicator N,N-diethyl-p- phenylenediamine (DPD), the free chlorine oxidizes a stoichiometric portion of DPD to its red-colored form. The oxidized DPD is then titrated back to its color- less form with ferrous ammonium sulfate, with the volume of titrant being pro- portional to the amount of free residual chlorine. Adding a small amount of KI reduces monochloramine, NH 2 Cl, forming I 3 – . The I 3 – then oxidizes a portion of 344 Modern Analytical Chemistry 1400-CH09 9/9/99 2:13 PM Page 344 the DPD to its red-colored form. Titrating the oxidized DPD with ferrous am- monium sulfate yields the amount of NH 2 Cl in the sample. The amount of dichloramine and trichloramine are determined in a similar fashion. The methods described earlier for determining the total, free, or combined chlorine residual also are used in establishing the chlorine demand of a water sup- ply. The chlorine demand is defined as the quantity of chlorine that must be added to a water supply to completely react with any substance that can be oxi- dized by chlorine while also maintaining the desired chlorine residual. It is deter- mined by adding progressively greater amounts of chlorine to a set of samples drawn from the water supply and determining the total, free, or combined chlo- rine residual. Another important example of redox titrimetry that finds applications in both public health and environmental analyses is the determination of dissolved oxygen. In natural waters the level of dissolved O 2 is important for two reasons: it is the most readily available oxidant for the biological oxidation of inorganic and organic pollutants; and it is necessary for the support of aquatic life. In wastewater treat- ment plants, the control of dissolved O 2 is essential for the aerobic oxidation of waste materials. If the level of dissolved O 2 falls below a critical value, aerobic bacte- ria are replaced by anaerobic bacteria, and the oxidation of organic waste produces undesirable gases such as CH 4 and H 2 S. One standard method for determining the dissolved O 2 content of natural wa- ters and wastewaters is the Winkler method. A sample of water is collected in a fash- ion that prevents its exposure to the atmosphere (which might change the level of dissolved O 2 ). The sample is then treated with a solution of MnSO 4 , and then with a solution of NaOH and KI. Under these alkaline conditions Mn 2+ is oxidized to MnO 2 by the dissolved oxygen. 2Mn 2+ (aq) + 4OH – (aq)+O 2 (aq) fi 2MnO 2 (s)+2H 2 O(l) After the reaction is complete, the solution is acidified with H 2 SO 4 . Under the now acidic conditions I – is oxidized to I 3 – by MnO 2 . MnO 2 (s)+3I – (aq)+4H 3 O + (aq) fi Mn 2+ (aq)+I 3 – (aq)+6H 2 O(l) The amount of I 3 – formed is determined by titrating with S 2 O 3 2– using starch as an indicator. The Winkler method is subject to a variety of interferences, and several modifications to the original procedure have been proposed. For example, NO 2 – in- terferes because it can reduce I 3 – to I – under acidic conditions. This interference is eliminated by adding sodium azide, NaN 3 , reducing NO 2 – to N 2 . Other reducing agents, such as Fe 2+ , are eliminated by pretreating the sample with KMnO 4 , and de- stroying the excess permanganate with K 2 C 2 O 4 . Another important example of a redox titration for inorganic analytes, which is important in industrial labs, is the determination of water in nonaqueous solvents. The titrant for this analysis is known as the Karl Fischer reagent and consists of a mixture of iodine, sulfur dioxide, pyridine, and methanol. The concentration of pyridine is sufficiently large so that I 2 and SO 2 are complexed with the pyridine (py) as py I 2 and py SO 2 . When added to a sample containing water, I 2 is reduced to I – , and SO 2 is oxidized to SO 3 . py I 2 +py SO 2 +py+H 2 O fi 2py HI+py SO 3 Methanol is included to prevent the further reaction of py SO 3 with water. The titration’s end point is signaled when the solution changes from the yellow color of the products to the brown color of the Karl Fischer reagent. Chapter 9 Titrimetric Methods of Analysis 345 1400-CH09 9/9/99 2:13 PM Page 345 Organic Analysis Redox titrimetric methods also are used for the analysis of or- ganic analytes. One important example is the determination of the chemical oxygen demand (COD) in natural waters and wastewaters. The COD provides a measure of the quantity of oxygen necessary to completely oxidize all the organic matter in a sample to CO 2 and H 2 O. No attempt is made to correct for organic matter that can- not be decomposed biologically or for which the decomposition kinetics are very slow. Thus, the COD always overestimates a sample’s true oxygen demand. The de- termination of COD is particularly important in managing industrial wastewater treatment facilities where it is used to monitor the release of organic-rich wastes into municipal sewer systems or the environment. The COD is determined by refluxing the sample in the presence of excess K 2 Cr 2 O 7 , which serves as the oxidizing agent. The solution is acidified with H 2 SO 4 , and Ag 2 SO 4 is added as a catalyst to speed the oxidation of low-molecular-weight fatty acids. Mercuric sulfate, HgSO 4 , is added to complex any chloride that is pres- ent, thus preventing the precipitation of the Ag + catalyst as AgCl. Under these con- ditions, the efficiency for oxidizing organic matter is 95–100%. After refluxing for 2h, the solution is cooled to room temperature, and the excess Cr 2 O 7 2– is deter- mined by a back titration, using ferrous ammonium sulfate as the titrant and fer- roin as the indicator. Since it is difficult to completely remove all traces of organic matter from the reagents, a blank titration must be performed. The difference in the amount of ferrous ammonium sulfate needed to titrate the blank and the sample is proportional to the COD. Iodine has been used as an oxidizing titrant for a number of compounds of pharmaceutical interest. Earlier we noted that the reaction of S 2 O 3 2– with I 3 – pro- duces the tetrathionate ion, S 4 O 6 2– . The tetrathionate ion is actually a dimer consist- ing of two thiosulfate ions connected through a disulfide (-S-S-) linkage. In the same fashion, I 3 – can be used to titrate mercaptans of the general formula RSH, forming the dimer RSSR as a product. The amino acid cysteine also can be titrated with I 3 – . The product of this titration is cystine, which is a dimer of cysteine. Triio- dide also can be used for the analysis of ascorbic acid (vitamin C) by oxidizing the enediol functional group to an alpha diketone 346 Modern Analytical Chemistry HC O HC OH HO CH CH 2 OH HC OH + 3OH – HC OH HC OH HO CH CO 2 – CH 2 OH HC OH + 2H 2 O + 2e – HC OH OH OH HO OH O O OH HO + 2H + + 2e – OO O O and for the analysis of reducing sugars, such as glucose, by oxidizing the aldehyde functional group to a carboxylate ion in a basic solution. 1400-CH09 9/9/99 2:13 PM Page 346 Chapter 9 Titrimetric Methods of Analysis 347 Organic compounds containing a hydroxyl, carbonyl, or amine functional group adjacent to a hydoxyl or carbonyl group can be oxidized using metaperio- date, IO 4 – , as an oxidizing titrant. IO 4 – (aq)+H 2 O(l)+2e – fi IO 3 – (aq) + 2OH – (aq) A two-electron oxidation cleaves the C—C bond between the two functional groups, with hydroxyl groups being oxidized to aldehydes or ketones, carbonyl functional groups being oxidized to carboxylic acids, and amines being oxidized to an aldehyde and an amine (ammonia if the original amine was primary). For exam- ple, treatment of serine with IO 4 – results in the following oxidation reaction The analysis is conducted by adding a known excess of IO 4 – to the solution contain- ing the analyte and allowing the oxidation to take place for approximately 1 h at room temperature. When the oxidation is complete, an excess of KI is added, which reacts with the unreacted IO 4 – to form IO 3 – and I 3 – . IO 4 – (aq)+3I – (aq)+H 2 O(l) fi IO 3 – (aq)+I 3 – (aq) + 2OH – (aq) The I 3 – is then determined by titrating with S 2 O 3 2– using starch as an indicator. Quantitative Calculations The stoichiometry of a redox reaction is given by the con- servation of electrons between the oxidizing and reducing agents (see Section 2C); thus Example 9.13 shows how this equation is applied to an analysis based on a direct titration. EXAMPLE 9 .1 3 The amount of Fe in a 0.4891-g sample of an ore was determined by a redox titration with K 2 Cr 2 O 7 . The sample was dissolved in HCl and the iron brought into the +2 oxidation state using a Jones reductor. Titration to the diphenylamine sulfonic acid end point required 36.92 mL of 0.02153 M K 2 Cr 2 O 7 . Report the iron content of the ore as %w/w Fe 2 O 3 . SOLUTION In this titration the analyte is oxidized from Fe 2+ to Fe 3+ , and the titrant is reduced from Cr 2 O 7 2– to Cr 3+ . Oxidation of Fe 2+ requires only a single electron. Reducing Cr 2 O 7 2– , in which chromium is in the +6 oxidation state, requires a total of six electrons. Conservation of electrons for the redox reaction, therefore, requires that Moles Fe 2+ =6· moles Cr 2 O 7 2– A conservation of mass relates the moles of Fe 2+ to the moles of Fe 2 O 3 Moles Fe 2+ =2· moles Fe 2 O 3 Moles of electrons lost Mole reducing agent moles reducing agent = moles of electrons gained mole oxidizing agent moles oxidizing agent × × OH H 2 CCHCO 2 – CO 2 – NH 3 + O CH 2 O HC + 2OH – + NH 4 + + H 2 O + 2e – + 1400-CH09 9/9/99 2:13 PM Page 347 348 Modern Analytical Chemistry Combining the two conservation equations gives 2 · moles Fe 2 O 3 =6· moles Cr 2 O 7 2– Making appropriate substitutions for the moles of Fe 2 O 3 and Cr 2 O 7 2– gives the following equation which we solve for the grams of Fe 2 O 3 . Thus, the %w/w Fe 2 O 3 in the sample of ore is As shown in the following two examples, this approach is easily extended to situa- tions that require an indirect analysis or a back titration. EXAMPLE 9 .1 4 A 25.00-mL sample of a liquid bleach was diluted to 1000 mL in a volumetric flask. A 25-mL portion of the diluted sample was transferred by pipet into an Erlenmeyer flask and treated with excess KI, oxidizing the OCl – to Cl – , and producing I 3 – . The liberated I 3 – was determined by titrating with 0.09892 M Na 2 S 2 O 3 , requiring 8.96 mL to reach the starch indicator end point. Report the %w/v NaOCl in the sample of bleach. SOLUTION Reducing OCl – to Cl – requires two electrons, and oxidizing a single I – to I 3 – requires 2/3 of an electron; thus, conservation of electrons requires that 2 · moles NaOCl = 2/3 · moles I – In the titration, two electrons are needed when I 3 – is reduced, but oxidizing S 2 O 3 2– releases one electron; thus 2 · moles I 3 – = moles S 2 O 3 2– These two equations can be combined by recognizing that a conservation of mass for iodine requires that Moles I – =3· moles I 3 – Thus 2 · moles NaOCl = moles S 2 O 3 2– Substituting for moles of NaOCl and S 2 O 3 2– leaves us with an equation 2 22 × =× ?? grams NaOCl FW NaOCl SO SO 23 23 MV grams Fe O grams sample g 0.4891 g w/w Fe O 23 23 ×= ×=100 0 3808 100 77 86 . .% 6 2 6 0 02153 0 3808 22 ××× = = ?? MV Cr O Cr O 23 23 27 27 FW Fe O M)(0.03692 L)(159.69 g/mol) 2 g Fe O ()(. . 2 6 2 2 × =× × ? grams Fe O FW Fe O 23 23 Cr O Cr O 27 27 MV – 1400-CH09 9/9/99 2:13 PM Page 348 Chapter 9 Titrimetric Methods of Analysis 349 that is solved for the grams of NaOCl. Thus, the %w/v NaOCl in the diluted sample is Since the bleach was diluted by a factor of 40 (25 mL to 1000 mL), the concentration of NaOCl in the bleach is 5.28% (w/v). EXAMPLE 9 .1 5 The amount of ascorbic acid, C 6 H 8 O 6 , in orange juice was determined by oxidizing the ascorbic acid to dehydroascorbic acid, C 6 H 6 O 6 , with a known excess of I 3 – , and back titrating the excess I 3 – with Na 2 S 2 O 3 . A 5.00-mL sample of filtered orange juice was treated with 50.00 mL of excess 0.01023 M I 3 – . After the oxidation was complete, 13.82 mL of 0.07203 M Na 2 S 2 O 3 was needed to reach the starch indicator end point. Report the concentration of ascorbic acid in milligrams per 100 mL. SOLUTION Oxidizing ascorbic acid requires two electrons, and reducing I 3 – to I – also requires two electrons. Thus (Moles I 3 – ) ascorbic acid = moles C 6 H 8 O 6 For the back titration, the stoichiometric relationship between I 3 – and S 2 O 3 2– is (see Example 9.14) (Moles I 3 – ) back titration = 0.5 · moles S 2 O 3 2– The total moles of I 3 – used in the analysis is the sum of that reacting with ascorbic acid and S 2 O 3 2– (Moles I 3 – ) tot = (moles I 3 – ) ascorbic acid + (moles I 3 – ) back titration or Moles I 3 – = moles C 6 H 8 O 6 + 0.5 · moles S 2 O 3 2– Making appropriate substitutions for the moles of I 3 – , C 6 H 8 O 6 , and S 2 O 3 2– and solving for the grams of C 6 H 8 O 6 gives (–. ) [( . )( . ) – ( . )( . )( . )]( . ) . MV M V I I SO SO 686 686 3 3 23 23 FW C H O g/mol CHO ?? ? ? ×××× = = 05 0 01023 0 0500 0 5 0 07203 0 01382 176 13 0 00243 22 ML M L g MV M V II 686 686 SO SO 3 3 23 23 g C H O FW C H O ?? ? ? ×= +× ×05 22 . Grams NaOCl mL sample g 25.00 mL w/v NaOCl×= ×=100 0 03299 100 0 132 . .% MV SO SO 23 23 FW NaOCl 2 M)(0.00896 L)(74.44 g /mol) 2 g NaOCl 22 0 09892 0 03299 ?? ×× = = (. . 1400-CH09 9/9/99 2:13 PM Page 349 Figure 9.40 Titration curve for 50.00 mL of 0.0250 M Sn 2+ and 0.0250 M Fe 2+ with 0.055 M Ce 4+ . 350 Modern Analytical Chemistry Thus, there is 2.43 mg of ascorbic acid in the 5.00-mL sample, or 48.6 mg/100 mL of orange juice. 9 D. 5 Evaluation of Redox Titrimetry The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving redox titrations are similar to those described earlier in the chapter for acid–base and complexometric titrimetric methods. As with acid–base titrations, redox titrations can be extended to the analysis of mix- tures if there is a significant difference in the ease with which the analytes can be oxidized or reduced. Figure 9.40 shows an example of the titration curve for a mixture of Fe 2+ and Sn 2+ , using Ce 4+ as the titrant. The titration of a mixture of analytes whose standard-state potentials or formal potentials differ by at least 200 mV will result in a separate equivalence point for each analyte. 9 E Precipitation Titrations Thus far we have examined titrimetric methods based on acid–base, complexation, and redox reactions. A reaction in which the analyte and titrant form an insoluble precipitate also can form the basis for a titration. We call this type of titration a pre- cipitation titration. One of the earliest precipitation titrations, developed at the end of the eigh- teenth century, was for the analysis of K 2 CO 3 and K 2 SO 4 in potash. Calcium nitrate, Ca(NO 3 ) 2 , was used as a titrant, forming a precipitate of CaCO 3 and CaSO 4 . The end point was signaled by noting when the addition of titrant ceased to generate ad- ditional precipitate. The importance of precipitation titrimetry as an analytical method reached its zenith in the nineteenth century when several methods were de- veloped for determining Ag + and halide ions. 9 E.1 Titration Curves The titration curve for a precipitation titration follows the change in either the ana- lyte’s or titrant’s concentration as a function of the volume of titrant. For example, in an analysis for I – using Ag + as a titrant Ag + (aq)+I – (aq) t AgI(s) the titration curve may be a plot of pAg or pI as a function of the titrant’s volume. As we have done with previous titrations, we first show how to calculate the titra- tion curve and then demonstrate how to quickly sketch the titration curve. Calculating the Titration Curve As an example, let’s calculate the titration curve for the titration of 50.0 mL of 0.0500 M Cl – with 0.100 M Ag + . The reaction in this case is Ag + (aq)+Cl – (aq) t AgCl(s) The equilibrium constant for the reaction is K =(K sp ) –1 = (1.8 · 10 –10 ) –1 = 5.6 · 10 9 Since the equilibrium constant is large, we may assume that Ag + and Cl – react completely. precipitation titration A titration in which the reaction between the analyte and titrant involves a precipitation. P otential (V) Volume of titrant (mL) 0 0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 20 40 60 10080 1400-CH09 9/9/99 2:13 PM Page 350 By now you are familiar with our approach to calculating titration curves. The first task is to calculate the volume of Ag + needed to reach the equivalence point. The stoichiometry of the reaction requires that Moles Ag + = moles Cl – or M Ag V Ag = M Cl V Cl Solving for the volume of Ag + shows that we need 25.0 mL of Ag + to reach the equivalence point. Before the equivalence point Cl – is in excess. The concentration of unreacted Cl – after adding 10.0 mL of Ag + , for example, is If the titration curve follows the change in concentration for Cl – , then we calculate pCl as pCl = –log[Cl – ] = –log(2.50 · 10 –2 ) = 1.60 However, if we wish to follow the change in concentration for Ag + then we must first calculate its concentration. To do so we use the K sp expression for AgCl K sp = [Ag + ][Cl – ] = 1.8 · 10 –10 Solving for the concentration of Ag + gives a pAg of 8.14. At the equivalence point, we know that the concentrations of Ag + and Cl – are equal. Using the solubility product expression K sp = [Ag + ][Cl – ] = [Ag + ] 2 = 1.8 · 10 –10 gives [Ag + ] = [Cl – ] = 1.3 · 10 –5 M At the equivalence point, therefore, pAg and pCl are both 4.89. After the equivalence point, the titration mixture contains excess Ag + . The con- centration of Ag + after adding 35.0 mL of titrant is [] (. (. . Ag moles excess Ag total volume M)(35.0 mL) M)(50.0 mL) 50.0 mL + 35.0 mL M + + Ag Ag Cl Cl Cl Ag == ? + = ? =× ? MV MV VV 0 100 0 0500 118 10 2 [] [] . . .Ag Cl M + sp == × × =× ? ? ? ? K 18 10 250 10 72 10 10 2 9 [] (. (. . Cl moles excess Cl total volume M)(50.0 mL) M)(10.0 mL) 50.0 mL + 10.0 mL M Cl Cl Ag Ag Cl Ag ? ? ? == ? + = ? =× MV M V VV 0 0500 0 100 250 10 2 V MV M Ag Cl Cl Ag M)(50.0 mL) (0.100 M) mL== = (. . 0 0500 25 0 Chapter 9 Titrimetric Methods of Analysis 351 1400-CH09 9/9/99 2:13 PM Page 351 352 Modern Analytical Chemistry pCl or pAg Volume AgNO 3 0 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10 20 30 5040 (b) (a) Table 9 .21 Data for Titration of 50.0 mL of 0.0500 M Cl – with 0.100 M Ag + Volume AgNO 3 (mL) pCl pAg 0.00 1.30 — 5.00 1.44 8.31 10.00 1.60 8.14 15.00 1.81 7.93 20.00 2.15 7.60 25.00 4.89 4.89 30.00 7.54 2.20 35.00 7.82 1.93 40.00 7.97 1.78 45.00 8.07 1.68 50.00 8.14 1.60 Figure 9.41 Precipitation titration curve for 50.0 mL of 0.0500 M Cl – with 0.100 M Ag + . (a) pCl versus volume of titrant; (b) pAg versus volume of titrant. or a pAg of 1.93. The concentration of Cl – is or a pCl of 7.82. Additional results for the titration curve are shown in Table 9.21 and Figure 9.41. Sketching the Titration Curve As we have done for acid–base, complexometric titrations, and redox titrations, we now show how to quickly sketch a precipitation titration curve using a minimum number of calculations. EXAMPLE 9 .16 Sketch a titration curve for the titration of 50.0 mL of 0.0500 M Cl – with 0.100 M Ag + . This is the same titration for which we previously calculated the titration curve (Table 9.21 and Figure 9.41). SOLUTION We begin by drawing axes for the titration curve (Figure 9.42a). Having shown that the equivalence point volume is 25.0 mL, we draw a vertical line intersecting the x-axis at this volume (Figure 9.42b). Before the equivalence point, pCl and pAg are determined by the concentration of excess Cl – . Using values from Table 9.21, we plot either pAg or pCl for 10.0 mL and 20.0 mL of titrant (Figure 9.42c). After the equivalence point, pCl and pAg are determined by the concentration of excess Ag + . Using values from Table 9.21, we plot points for 30.0 mL and 40.0 mL of titrant (Figure 9.42d). To complete an approximate sketch of the titration curve, we draw separate straight lines through the two points before and after the equivalence point (Figure 9.42e). Finally, a smooth curve is drawn to connect the three straight-line segments (Figure 9.42f). [] [] . . .Cl Ag M sp + ? ? ? ? == × × =× K 18 10 118 10 15 10 10 2 8 1400-CH09 9/9/99 2:13 PM Page 352 Chapter 9 Titrimetric Methods of Analysis 353 pCl or pAg Volume of titrant (mL) Percent titrated 0 0.0 2.0 4.0 20.00 500 100 200150 50.0010.00 30.00 40.00 8.0 6.0 pCl or pAg Volume of titrant (mL) Percent titrated 0 0.0 2.0 4.0 20.00 500 100 200150 50.0010.00 30.00 40.00 8.0 6.0 pCl or pAg Volume of titrant (mL) Percent titrated 0 0.0 2.0 4.0 20.00 500 100 200150 50.0010.00 30.00 40.00 8.0 6.0 pCl or pAg Volume of titrant (mL) Percent titrated 0 0.0 2.0 4.0 20.00 500 100 200150 50.0010.00 30.00 40.00 8.0 6.0 pCl or pAg Volume of titrant (mL) Percent titrated 0 0.0 2.0 4.0 20.00 500 100 200150 50.0010.00 30.00 40.00 8.0 6.0 () pCl or pAg Volume of titrant (mL) Percent titrated 0 0.0 2.0 4.0 20.00 500 100 200150 50.0010.00 30.00 40.00 8.0 6.0 (a) (b) (c) (d) (e) (f) Figure 9.42 How to sketch a precipitation titration curve; see text for explanation. 1400-CH09 9/9/99 2:13 PM Page 353 9 E.2 Selecting and Evaluating the End Point Initial attempts at developing precipitation titration methods were limited by a poor end point signal. Finding the end point by looking for the first addition of titrant that does not yield additional precipitate is cumbersome at best. The feasibil- ity of precipitation titrimetry improved with the development of visual indicators and potentiometric ion-selective electrodes. Finding the End Point with a Visual Indicator The first important visual indicator to be developed was the Mohr method for Cl – using Ag + as a titrant. By adding a small amount of K 2 CrO 4 to the solution containing the analyte, the formation of a precipitate of reddish-brown Ag 2 CrO 4 signals the end point. Because K 2 CrO 4 im- parts a yellow color to the solution, obscuring the end point, the amount of CrO 4 2– added is small enough that the end point is always later than the equivalence point. To compensate for this positive determinate error an analyte-free reagent blank is analyzed to determine the volume of titrant needed to effect a change in the indica- tor’s color. The volume for the reagent blank is subsequently subtracted from the experimental end point to give the true end point. Because CrO 4 2– is a weak base, the solution usually is maintained at a slightly alkaline pH. If the pH is too acidic, chromate is present as HCrO 4 – , and the Ag 2 CrO 4 end point will be in significant error. The pH also must be kept below a level of 10 to avoid precipitating silver hydroxide. A second end point is the Volhard method in which Ag + is titrated with SCN – in the presence of Fe 3+ . The end point for the titration reaction Ag + (aq) + SCN – (aq) t AgSCN(s) is the formation of the reddish colored Fe(SCN) 2+ complex. SCN – (aq)+Fe 3+ (aq) t Fe(SCN) 2+ (aq) The titration must be carried out in a strongly acidic solution to achieve the desired end point. A third end point is evaluated with Fajans’ method, which uses an adsorption indicator whose color when adsorbed to the precipitate is different from that when it is in solution. For example, when titrating Cl – with Ag + the anionic dye dichloro- fluoroscein is used as the indicator. Before the end point, the precipitate of AgCl has a negative surface charge due to the adsorption of excess Cl – . The anionic indicator is repelled by the precipitate and remains in solution where it has a greenish yellow color. After the end point, the precipitate has a positive surface charge due to the adsorption of excess Ag + . The anionic indicator now adsorbs to the precipitate’s surface where its color is pink. This change in color signals the end point. Finding the End Point Potentiometrically Another method for locating the end point of a precipitation titration is to monitor the change in concentration for the analyte or titrant using an ion-selective electrode. The end point can then be found from a visual inspection of the titration curve. A further discussion of potentiome- try is found in Chapter 11. 9 E. 3 Quantitative Applications Precipitation titrimetry is rarely listed as a standard method of analysis, but may still be useful as a secondary analytical method for verifying results obtained by other methods. Most precipitation titrations involve Ag + as either an analyte or 354 Modern Analytical Chemistry 1400-CH09 9/9/99 2:13 PM Page 354 Chapter 9 Titrimetric Methods of Analysis 355 Table 9 .22 Representative Examples of Precipitation Titrations Analyte Titrant a End Point b AsO 4 3- AgNO 3 , KSCN Volhard Br – AgNO 3 Mohr or Fajans AgNO 3 , KSCN Volhard Cl – AgNO 3 Mohr or Fajans AgNO 3 , KSCN Volhard * CO 3 2- AgNO 3 , KSCN Volhard * C 2 O 4 2- AgNO 3 , KSCN Volhard * CrO 4 2- AgNO 3 , KSCN Volhard * I – AgNO 3 Fajans AgNO 3 , KSCN Volhard PO 4 3- AgNO 3 , KSCN Volhard * S 2- AgNO 3 , KSCN Volhard * SCN – AgNO 3 , KSCN Volhard a When two reagents are listed, the analysis is by a back titration. The first reagent is added in excess, and the second reagent is used to back titrate the excess. b For Volhard methods identified by an asterisk (*) the precipitated silver salt must be removed before carrying out the back titration. titrant. Those titrations in which Ag + is the titrant are called argentometric titra- tions. Table 9.22 provides a list of several typical precipitation titrations. Quantitative Calculations The stoichiometry of a precipitation reaction is given by the conservation of charge between the titrant and analyte (see Section 2C); thus Example 9.17 shows how this equation is applied to an analysis based on a direct titration. EXAMPLE 9 .1 7 A mixture containing only KCl and NaBr is analyzed by the Mohr method. A 0.3172-g sample is dissolved in 50 mL of water and titrated to the Ag 2 CrO 4 end point, requiring 36.85 mL of 0.1120 M AgNO 3 . A blank titration requires 0.71 mL of titrant to reach the same end point. Report the %w/w KCl and NaBr in the sample. SOLUTION The volume of titrant reacting with the analytes is V Ag = 36.85 mL – 0.71 mL = 36.14 mL Conservation of charge for the titration requires that Moles Ag + = moles KCl + moles NaBr Moles of charge mole titrant moles titrant moles of charge mole analyte moles analyte ×= × argentometric titration A precipitation titration in which Ag + is the titrant. 1400-CH09 9/9/99 2:14 PM Page 355 356 Modern Analytical Chemistry Making appropriate substitutions for the moles of Ag + , KCl, and NaBr gives the following equation. Since the sample contains just KCl and NaBr, we know that g NaBr = 0.3172 g – g KCl and Solving, we find that there is 0.2614 g of KCl and g NaBr = 0.3172 g – 0.2614 g = 0.0558 g 0.0558 g of NaBr. The weight percents for the two analytes, therefore, are The analysis for I – using the Volhard method requires a back titration. A typical calculation is shown in the following example. EXAMPLE 9 .18 The %w/w I – in a 0.6712-g sample was determined by a Volhard titration. After adding 50.00 mL of 0.05619 M AgNO 3 and allowing the precipitate to form, the remaining silver was back titrated with 0.05322 M KSCN, requiring 35.14 mL to reach the end point. Report the %w/w I – in the sample. SOLUTION Conservation of charge for this back titration requires that Moles Ag + = moles I – + moles SCN – Making appropriate substitutions for moles of Ag + , I – , and SCN – leaves us with MV M V Ag Ag – – SCN SCN g I AW I =+ 0 0558 100 17 59 . .% g NaBr 0.3172 g w/w NaBr×= 0 2614 100 82 41 . .% g KCl 0.3172 g w/w KCl×= (. . .. . . . 0 1120 0 3172 4 048 10 1 341 10 9 719 10 9 650 10 0 2614 32 33 34 M)(0.03614 L) g KCl 74.551 g/mol g g KCl 102.89 g/mol (g KCl) + 3.083 10 (g KCl) 3.691 10 (g KCl) g KCl g =+ ? ×= × ×? × ×=× = ?? ? ?? MV Ag Ag g KCl FW KCl g – g KCl FW NaBr =+ 0 3172. MV Ag Ag g KCl FW KCl g NaBr FW NaBr =+ 1400-CH09 9/9/99 2:14 PM Page 356 Figure 9.43 Titration curve for a mixture of I – and Cl – using AgNO 3 as a titrant. Solving for the grams of I – , we find g I – = (AW I – )(M Ag V Ag – M SCN V SCN ) = (126.9 g/mol)[(0.05619 M)(0.05000 L) – (0.05322 M)(0.03514 L)] = 0.1192 g that there is 0.1192 g of iodide. The weight percent iodide, therefore, is 9 E. 4 Evaluation of Precipitation Titrimetry The scale of operations, accuracy, precision, sensitivity, time, and cost of methods involving precipitation titrations are similar to those described earlier in the chapter for other titrimetric methods. Precipitation titrations also can be extended to the analysis of mixtures, provided that there is a significant difference in the solubilities of the precipitates. Figure 9.43 shows an example of the titration curve for a mixture of I – and Cl – using Ag + as a titrant. 0 1192 100 17 76 . .% g 0.6712 g w/w I×= ? Chapter 9 Titrimetric Methods of Analysis 357 pAg Volume AgNO 3 (mL) 0 0.00 16.00 10080604020 14.00 12.00 8.00 6.00 10.00 4.00 2.00 9 F KEY TERMS acid–base titration (p. 278) acidity (p. 301) alkalinity (p. 300) argentometric titration (p. 355) auxiliary complexing agent (p. 316) auxiliary oxidizing agent (p. 341) auxiliary reducing agent (p. 341) back titration (p. 275) buret (p. 277) complexation titration (p. 314) conditional formation constant (p. 316) displacement titration (p. 275) end point (p. 274) equivalence point (p. 274) formal potential (p. 332) Gran plot (p. 293) indicator (p. 274) Jones reductor (p. 341) Kjeldahl analysis (p. 302) leveling (p. 296) metallochromic indicator (p. 323) precipitation titration (p. 350) redox indicator (p. 339) redox titration (p. 331) titrant (p. 274) titration curve (p. 276) titration error (p. 274) titrimetry (p. 274) Walden reductor (p. 341) In a titrimetric method of analysis the volume of titrant reacting stoichiometrically with the analyte provides quantitative infor- mation about the amount of analyte in a sample. The volume of titrant required to achieve this stoichiometric reaction is called the equivalence point. Experimentally we determine the titra- tion’s end point using a visual indicator that changes color near the equivalence point. Alternatively, we can locate the end point by recording a titration curve showing the titration reaction’s progress as a function of the titrant’s volume. In either case, the end point must closely match the equivalence point if a titration is to be accurate. Knowing the shape of a titration curve is critical to evaluating the feasibility of a proposed titri- metric method. Many titrations are direct, in which the titrant reacts with the analyte. Other titration strategies may be used when a direct reac- tion between the analyte and titrant is not feasible. In a back titra- tion a reagent is added in excess to a solution containing the ana- lyte. When the reaction between the reagent and the analyte is complete, the amount of excess reagent is determined titrimetri- cally. In a displacement titration the analyte displaces a reagent, usually from a complex, and the amount of the displaced reagent is determined by an appropriate titration. 9 G SUMMARY 1400-CH09 9/9/99 2:14 PM Page 357 358 Modern Analytical Chemistry The first block of experiments cover acid–base titrimetry. Castillo, C. A.; Jaramillo, A. “An Alternative Procedure for Titration Curves of a Mixture of Acids of Different Strengths,” J. Chem. Educ. 1989, 66, 341. This short paper describes a modification to the traditional Gran plot for determining the concentration of a weak acid in the presence of a strong acid. Clay, J. T.; Walters, E. A.; Brabson, G. D. “A Dibasic Acid Titration for the Physical Chemistry Laboratory,” J. Chem. Educ. 1995, 72, 665–667. Values for K a1 and K a2 for acids of the form H 2 A are determined from a least-squares analysis of data from a potentiometric titration. Crossno, S. K; Kalbus, L. H.; Kalbus, G. E. “Determinations of Carbon Dioxide by Titration,” J. Chem. Educ. 1996, 73, 175–176. Experiments are described for determining CO 2 in carbonated beverages, NaHCO 3 in Alka-Seltzer tablets, and the molecular weight of CO 2 . Carbon dioxide is collected in NaOH and the concentrations of CO 3 2– are determined by titrating with a standard solution of HCl to the phenolphthalein and methyl orange end points. Flowers, P. A. “Potentiometric Measurement of Transition Ranges and Titration Errors for Acid–Base Indicators,” J. Chem. Educ. 1997, 74, 846–847. Dilute solutions of nominally 0.001 M NaOH and HCl are used to demonstrate the effect of an indicator’s color transition range on titration error. Potentiometric titration curves are measured, and the indicator’s color transition range is noted. Titration errors are calculated using the volume of titrant needed to effect the first color change and for a complete color change. Graham. R. C.; DePew, S. “Determination of Ammonia in Household Cleaners,” J. Chem. Educ. 1983, 60, 765–766. A quantitative analysis for NH 3 in several household cleaning products is carried out by titrating with a standard solution of HCl. The titration’s progress is followed thermometrically by monitoring the temperature of the titration mixture as a function of the volume of added titrant. Household cleaning products may contain other basic components, such as sodium citrate or sodium carbonate, that will also be titrated by HCl. By comparing titration curves for prepared samples of NH 3 to titration curves for the samples, it is possible to determine that portion of the thermometric titration curve due to the neutralization of NH 3 . Kalbus, L. H.; Petrucci, R. H.; Forman, J. E.; et al. “Titration of Chromate–Dichromate Mixtures,” J. Chem. Educ. 1991, 68, 677–678. A known amount of HCl in excess is added to a mixture of CrO 4 – and Cr 2 O 7 2– , with the HCl converting CrO 4 2– to HCrO 4 – . The resulting solution is then back titrated with a standard solution of NaOH. The titration is followed potentiometrically, showing two end points. Titration to the first end point provides a quantitative measure of the excess HCl, providing an indirect measure of the amount of CrO 4 2– in the sample. Titration to the second end point gives the total amount of Cr 2 O 7 2– and CrO 4 2– in the sample. Ophardt, C. E. “Acid Rain Analysis by Standard Addition Titration,” J. Chem. Educ. 1985, 62, 257–258. This experiment describes a method for determining the acidity, reported as an equivalent molarity of H 2 SO 4 , of rain water. Because the volume of standard base needed to titrate a sample of rain water is small, the analysis is done by a standard addition. A 10.00-mL sample of nominally 0.005 M H 2 SO 4 is diluted with 100.0 mL of distilled water and standardized by titrating with 0.0100 M NaOH. A second 10.00-mL sample of the sulfuric acid is mixed with 100.0 mL of rain water and titrated with the same solution of NaOH. The difference between the two equivalence point volumes 9 H Suggested EXPERIMENTS The following experiments may be used to illustrate the application of titrimetry to quantitative, qualitative, or characterization problems. Experiments are grouped into four categories based on the type of reaction (acid–base, complexation, redox, and precipitation). A brief description is included with each experiment providing details such as the type of sample analyzed, the method for locating end points, or the analysis of data. Additional experiments emphasizing potentiometric electrodes are found in Chapter 11. Experiments Titrimetric methods have been developed using acid–base, complexation, redox, and precipitation reactions. Acid–base titra- tions use a strong acid or strong base as a titrant. The most com- mon titrant for a complexation titration is EDTA. Because of their stability against air oxidation, most redox titrations use an oxidiz- ing agent as a titrant. Titrations with reducing agents also are pos- sible. Precipitation titrations usually involve Ag + as either the ana- lyte or titrant. — Continued 1400-CH09 9/9/99 2:14 PM Page 358 Chapter 9 Titrimetric Methods of Analysis 359 Experiments gives the volume of NaOH needed to neutralize the acidity in the rain water. Titration curves are measured potentiometrically and the equivalence point determined from a Gran plot. Partanen, J. I.; K?rki, M. H. “Determination of the Thermodynamic Dissociation Constant of a Weak Acid by Potentiometric Acid–Base Titration,” J. Chem. Educ. 1994, 71, A120–A122. Directions are provided in this experiment for determining the dissociation constant for a weak acid. Potentiometric titration data are analyzed by a modified Gran plot. The experiment is carried out at a variety of ionic strengths and the thermodynamic dissociation constant determined by extrapolating to zero ionic strength. Thompson, R. Q. “Identification of Weak Acids and Bases by Titration with Primary Standards,” J. Chem. Educ. 1988, 65, 179–180. Potentiometric titration curves are used to determine the molecular weight and K a or K b for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. Tucker, S. A.; Acree, Jr., W. E. “A Student-Designed Analytical Laboratory Method,” J. Chem. Educ. 1994, 71, 71–74. In this experiment the effect of a mixed aqueous–organic solvent on the color transition range of common indicators is investigated. One goal of the experiment is to design an appropriate titrimetric method for analyzing sparingly soluble acids and bases. Tucker, S. A.; Amszi, V. L.; Acree, Jr., W. E. “Studying Acid- Base Equilibria in Two-Phase Solvent Media,” J. Chem. Educ. 1993, 70, 80–82. This experiment shows how modifying the matrix of the solution containing the analyte can dramatically improve the shape of the titration curve. Trialkylammonium salts, such as lidocaine hydrochloride, are titrated in an aqueous solution containing a surfactant. The presence of the surfactant increases the trialkylammonium salt’s K a , giving a titration curve with a more pronounced break. The effect of adding an immiscible organic solvent, such as methylene chloride or toluene, also is demonstrated. Werner, J. A.; Werner, T. C. “Multifunctional Base Unknowns in the Introductory Analytical Chemistry Lab,” J. Chem. Educ. 1991, 68, 600–601. A potentiometric titration is used to determine if an unknown sample is pure Na 2 CO 3 , a mixture of Na 2 CO 3 and NaHCO 3 , pure Na 3 PO 4 , or a mixture of Na 3 PO 4 and Na 2 HPO 4 . Three experiments involving complexometric titrations are described in this second block of experiments. Fulton, R.; Ross, M.; Schroeder, K. “Spectrophotometric Titration of a Mixture of Calcium and Magnesium,” J. Chem. Educ. 1986, 63, 721–723. In this experiment the concentrations of Ca 2+ and Mg 2+ in aqueous solutions are determined by titrating with EDTA. The titration is followed spectrophotometrically by measuring the absorbance of a visual indicator. The effect of changing the indicator, the pH at which the titration is carried out, and the relative concentrations of Ca 2+ and Mg 2+ are also investigated. Novick, S. G. “Complexometric Titration of Zinc,” J. Chem. Educ. 1997, 74, 1463. Most experiments involving EDTA as a titrant use Ca 2+ and Mg 2+ as an analyte. This experiment describes a quantitative analysis for Zn 2+ in cold lozenges using EDTA as a titrant and xylenol orange as a visual indicator. Smith, R. L.; Popham, R. E. “The Quantitative Resolution of a Mixture of Group II Metal Ions by Thermometric Titration with EDTA,” J. Chem. Educ. 1983, 60, 1076–1077. Binary mixtures of Ca 2+ and Mg 2+ , and ternary mixtures of Ca 2+ , Mg 2+ , and Ba 2+ are determined by titrating with EDTA. The progress of the titration is followed thermometrically. Complexation of Ca 2+ and Ba 2+ with EDTA is exothermic, whereas complexation of Mg 2+ with EDTA is endothermic. As EDTA is added, the temperature initially rises due to the complexation of Ca 2+ . The temperature then falls as Mg 2+ is titrated, rising again as Ba 2+ is titrated. —Continued The following five experiments provide examples of redox titrimetric methods. Guenther, W. B. “Supertitrations: High-Precision Methods,” J. Chem. Educ. 1988, 65, 1097–1098. The purity of ferrous ammonium sulfate is determined by a redox titration with K 2 Cr 2 O 7 , using the weight of the reagents as the signal in place of volume. 1400-CH09 9/9/99 2:14 PM Page 359 360 Modern Analytical Chemistry Continued from page 359 Experiments Haddad, P. “Vitamin C Content of Commercial Orange Juices,” J. Chem. Educ. 1977, 54, 192–193. The content of ascorbic acid, in milligrams per 100 mL, in orange juice is determined by a redox titration using either 2,6-dichlorophenolindephenol or N-bromosuccinimide as the titrant. Harris, D. C.; Hills, M. E.; Hewston, T. A. “Preparation, Iodometric Analysis and Classroom Demonstration of Superconductivity in YBa 2 Cu 3 O 8-x ,” J. Chem. Educ. 1987, 64, 847–850. The superconductor YBa 2 Cu 3 O 8-x contains copper in both the +2 and +3 oxidation states. Procedures are described for synthesizing the superconductor, demonstrating the superconducting effect, and for determining the amount of Cu 2+ and Cu 3+ in the prepared material. Phinyocheep, P.; Tang, I. M. “Determination of the Hole Concentration (Copper Valency) in the High T c Superconductors,” J. Chem. Educ. 1994, 71, A115–A118. This experiment outlines a potentiometric titration for determining the valency of copper in superconductors in place of the visual end point used in the preceding experiment of Harris, Hill, and Hewston. The analysis of several different superconducting materials is described. Powell, J. R.; Tucker, S. A.; Acree, Jr., et al. “A Student–Designed Potentiometric Titration: Quantitative Determination of Iron(II) by Caro’s Acid Titration,” J. Chem. Educ. 1996, 73, 984–986. Caro’s acid, H 2 SO 5 is used as a titrant for determining Fe 2+ . Directions are given for exploring the method of end point detection, the titrant’s shelf-life, the method’s accuracy and precision, and the susceptibility of the method to interference from other species. Finally, the last experiment describes an unusual precipitation titration. Ueno, K.; Kina, K. “Colloid Titration—A Rapid Method for the Determination of Charged Colloid,” J. Chem . Educ. 1985, 62, 627–629. The reaction of a positively charged polyelectrolyte with a negatively charged polyelectrolyte produces a precipitate, forming the basis for a precipitation titration. This paper provides an overview of colloid titrations, discussing compounds that can be used as positively charged titrants or negatively charged titrants. Several methods for detecting the end point, including the use of visual indicators or potentiometric measurements, are discussed. Procedures for titrating positive polyelectrolytes and negative polyelectrolytes also are given. 1. Calculate or sketch (or both) qualitatively correct titration curves for the following acid–base titrations. (a) 25.0 mL of 0.100 M NaOH with 0.0500 M HCl (b) 50.0 mL of 0.0500 M HCOOH with 0.100 M NaOH (c) 50.0 mL of 0.100 M NH 3 with 0.100 M HCl (d) 50.0 mL of 0.0500 M ethylenediamine with 0.100 M HCl (e) 50.0 mL of 0.0400 M citric acid with 0.120 M NaOH (f) 50.0 mL of 0.0400 M H 3 PO 4 with 0.120 M NaOH 2. Locate the equivalence point for each of the titration curves in problem 1. What is the stoichiometric relationship between the moles of acid and moles of base at each of these equivalence points? 3. Suggest an appropriate visual indicator for each of the titrations in problem 1. 4. In sketching the titration curve for a weak acid, we approximated the pH at 10% of the equivalence point volume as pK a – 1, and the pH at 90% of the equivalence point volume as pK a + 1. Show that these assumptions are reasonable. 5. Tartaric acid, H 2 C 4 H 4 O 6 , is a diprotic weak acid with a pK a1 of 3.0 and a pK a2 of 4.4. Suppose you have a sample of impure tartaric acid (%purity > 80) and that you plan to determine its purity by titrating with a solution of 0.1 M NaOH using a visual indicator to signal the end point. Describe how you would carry out the analysis, paying particular attention to how much sample you would use, the desired pH range over which you would like the visual indicator to operate, and how you would calculate the %w/w tartaric acid. 6. The following data were collected with an automatic titrator during the titration of a monoprotic weak acid with a strong base. Prepare normal, first-derivative, second-derivative, and Gran plot titration curves for this data, and locate the equivalence point for each. 9 I PROBLEMS 1400-CH09 9/9/99 2:14 PM Page 360 Volume NaOH Volume NaOH (mL) pH (mL) pH 0.25 3.0 49.97 8.0 0.86 3.2 49.98 8.2 1.63 3.4 49.99 8.4 2.72 3.6 50.00 8.7 4.29 3.8 50.01 9.1 6.54 4.0 50.02 9.4 9.67 4.2 50.04 9.6 13.79 4.4 50.06 9.8 18.83 4.6 50.10 10.0 24.47 4.8 50.16 10.2 30.15 5.0 50.25 10.4 35.33 5.2 50.40 10.6 39.62 5.4 50.63 10.8 42.91 5.6 51.01 11.0 45.28 5.8 51.61 11.2 46.91 6.0 52.58 11.4 48.01 6.2 54.15 11.6 48.72 6.4 56.73 11.8 49.19 6.6 61.11 12.0 49.48 6.8 68.83 12.2 49.67 7.0 83.54 12.4 49.79 7.2 116.14 12.6 49.87 7.4 49.92 7.6 49.95 7.8 7. Schwartz has published some hypothetical data for the titration of a 1.02 · 10 –4 M solution of a monoprotic weak acid (pK a = 8.16) with 1.004 · 10 –3 M NaOH. 14 A 50-mL pipet is used to transfer a portion of the weak acid solution to the titration vessel. Calibration of the pipet, however, shows that it delivers a volume of only 49.94 mL. Prepare normal, first- derivative, second-derivative, and Gran plot titration curves for these data, and determine the equivalence point for each. How do these equivalence points compare with the expected equivalence point? Comment on the utility of each titration curve for the analysis of very dilute solutions of very weak acids. Volume Strong Volume Strong Base Base (mL) pH (mL) pH 0.03 6.212 4.79 8.858 0.09 6.504 4.99 8.926 0.29 6.936 5.21 8.994 0.72 7.367 5.41 9.056 1.06 7.567 5.61 9.118 1.32 7.685 5.85 9.180 1.53 7.776 6.05 9.231 1.76 7.863 6.28 9.283 1.97 7.938 6.47 9.327 2.18 8.009 6.71 9.374 2.38 8.077 6.92 9.414 2.60 8.146 7.15 9.451 2.79 8.208 7.36 9.484 3.01 8.273 7.56 9.514 3.19 8.332 7.79 9.545 3.41 8.398 7.99 9.572 3.60 8.458 8.21 9.599 3.80 8.521 8.44 9.624 3.99 8.584 8.64 9.645 4.18 8.650 8.84 9.666 4.40 8.720 9.07 9.688 4.57 8.784 9.27 9.706 8. Calculate or sketch (or both) the titration curves for 50.0 mL of a 0.100 M solution of a monoprotic weak acid (pK a =8) with 0.1 M strong base in (a) water; and (b) a non-aqueous solvent with K s =10 –20 . You may assume that the change in solvent does not affect the weak acid’s pK a . 9. The titration of a mixture of p-nitrophenol (pK a = 7.0) and m-nitrophenol (pK a = 8.3) can be followed spectrophotometrically. Neither acid absorbs at a wavelength of 545 nm, but their respective conjugate bases do absorb at this wavelength. The m-nitrophenolate ion has a greater absorbance than an equimolar solution of the p-nitrophenolate ion. Sketch the spectrophotometric titration curve for a 50.00-mL mixture consisting of 0.0500 M p-nitrophenol and 0.0500 M m-nitrophenol with 0.100 M NaOH, and compare the curve with the expected potentiometric titration curves. 10. The quantitative analysis for aniline (C 6 H 5 NH 2 ; K b = 3.94 · 10 –10 ) can be carried out by an acid–base titration, using glacial acetic acid as the solvent and HClO 4 as the titrant. A known volume of sample containing 3–4 mmol of aniline is transferred to a 250-mL Erlenmeyer flask and diluted to approximately 75 mL with glacial acetic acid. Two drops of a methyl violet visual indicator are added, and the solution is titrated with previously standardized 0.1000 M HClO 4 (prepared in glacial acetic acid using anhydrous HClO 4 ) until the visual end point is reached. Results are reported as parts per million of aniline. (a) Explain why this titration is conducted using glacial acetic acid as the solvent instead of water. (b) One problem with using glacial acetic acid as solvent is its relatively high coefficient of thermal expansion of 0.11%/°C. Thus, for example, 100.00 mL of glacial acetic acid at 25 °C will occupy 100.22 mL at 27 °C. What is the effect on the reported parts per million of aniline if the standardization of HClO 4 was conducted at a lower temperature than the analysis of the unknown? (c) The procedure calls for a sample containing only 3–4 mmol of aniline. Why was this requirement made? 11. Using a ladder diagram, explain why the presence of dissolved CO 2 leads to a determinate error in the standardization of Chapter 9 Titrimetric Methods of Analysis 361 1400-CH09 9/9/99 2:14 PM Page 361 NaOH when the end point’s pH falls between 6 and 10, but no determinate error is observed when the end point’s pH is less than 6. 12. The acidity of a water sample is determined by titrating to fixed end points of 3.7 and 8.3, with the former providing a measure of the concentration of strong acid, and the latter a measure of the combined concentrations of strong acid and weak acid. Sketch a titration curve for a mixture of 0.10 M HCl and 0.10 M H 2 CO 3 with 0.20 M strong base, and use it to justify the choice of these end points. 13. Ethylenediaminetetraacetic acid, H 4 Y, is a tetraprotic weak acid with successive acid dissociation constants of 0.010, 2.1 · 10 –3 , 7.8 · 10 –7 , and 6.8 · 10 –11 . The titration curve shown here is for H 4 Y with NaOH. What is the stoichiometric relationship between H 4 Y and NaOH at the equivalence point marked with the arrow? 14 3.36 37 6.14 60 12.42 15 3.54 38 10.30 61 12.43 16 3.69 39 11.31 62 12.44 17 3.81 40 11.58 63 12.45 18 3.93 41 11.74 64 12.47 19 4.02 42 11.85 65 12.48 20 4.14 43 11.93 66 12.49 21 4.22 44 12.00 67 12.50 22 4.30 45 12.05 68 12.51 23 4.38 46 12.10 69 12.52 15. Explain why it is not possible for a sample of water to simultaneously have OH – and HCO 3 – as sources of alkalinity. 16. For each of the following, determine the forms of alkalinity (OH – , HCO 3 – , CO 3 2– ) that are present, and their respective concentrations in parts per million. In each case, a 25.00-mL sample is titrated with 0.1198 M HCl to the bromocresol green and phenolphthalein end points. Volume of HCl (mL) to Volume of HCl (mL) the Phenolphthalein to the Bromocresol End Point Green End Point (a) 21.36 21.38 (b) 5.67 21.13 (c) 0.00 14.28 (d) 17.12 34.26 (e) 21.36 25.69 17. A sample may contain any of the following: HCl, NaOH, H 3 PO 4 , H 2 PO 4 – , HPO 4 2– , or PO 4 3– . The composition of a sample is determined by titrating a 25.00-mL portion with 0.1198 M HCl or 0.1198 M NaOH to the phenolphthalein and methyl orange end points. For each of the following, determine which species are present in the sample, and their respective molar concentrations. Volume (mL) to the Volume (mL) to the Phenolphthalein Methyl Orange Titrant End Point End Point (a) HCl 11.54 35.29 (b) NaOH 19.79 9.89 (c) HCl 22.76 22.78 (d) NaOH 39.42 17.48 18. The protein in a 1.2846-g sample of an oat cereal is determined by the Kjeldahl procedure for organic nitrogen. The sample is digested with H 2 SO 4 , the resulting solution made basic with NaOH, and the NH 3 distilled into 50.00 mL of 0.09552 M HCl. The excess HCl is then back titrated using 37.84 mL of 0.05992 M NaOH. Given that the protein in grains averages 17.54% w/w N, report the %w/w protein in the sample of cereal. 19. The concentration of SO 2 in atmospheric samples can be determined by bubbling a sample of air through a trap containing H 2 O 2 . Oxidation of SO 2 by H 2 O 2 results in the 362 Modern Analytical Chemistry p Volume of titrant (mL) 0.00 0.0 14.0 80.0060.0040.0020.00 12.0 8.0 6.0 10.0 4.0 2.0 14. A Gran plot method has been described for determining the quantitative analysis of a mixture consisting of a strong acid and a monoprotic weak acid. 15 A 50.00-mL sample that is a mixture of HCl and CH 3 COOH is transferred to an Erlenmeyer flask and titrated by using a pipet to add 1.00-mL aliquots of 0.09186 M NaOH. The progress of the titration is monitored by recording the pH after the addition of each aliquot of titrant. Using the two papers listed in reference 15 (p. 367), prepare a Gran plot for the following data, and determine the concentrations of HCl and CH 3 COOH. Volume of Volume of Volume of NaOH NaOH NaOH (mL) pH (mL) pH (mL) pH 1 1.83 24 4.45 47 12.14 2 1.86 25 4.53 48 12.17 3 1.89 26 4.61 49 12.20 4 1.92 27 4.69 50 12.23 5 1.95 28 4.76 51 12.26 6 1.99 29 4.84 52 12.28 7 2.03 30 4.93 53 12.30 8 2.10 31 5.02 54 12.32 9 2.18 32 5.13 55 12.34 10 2.31 33 5.23 56 12.36 11 2.51 34 5.37 57 12.38 12 2.81 35 5.52 58 12.39 13 3.16 36 5.75 59 12.40 1400-CH09 9/9/99 2:14 PM Page 362 formation of H 2 SO 4 , the amount of which can be determined by titrating with NaOH. In a typical analysis, a sample of air was passed through the peroxide trap at a rate of 1.25 L/min for 60 min and required 10.08 mL of 0.0244 M NaOH to reach the phenolphthalein end point. Calculate the parts per million of SO 2 (m L/L) in the sample of air. The density of SO 2 at the temperature of the air sample is 2.86 mg/mL. 20. The concentration of CO 2 in air can be determined by an indirect acid–base titration. A sample of the air is bubbled through a solution containing an excess of Ba(OH) 2 , precipitating BaCO 3 . The excess Ba(OH) 2 is back titrated with HCl. In a typical analysis, a 3.5-L sample of air was bubbled through 50.00 mL of 0.0200 M Ba(OH) 2 . Back titrating with 0.0316 M HCl requires 38.58 mL to reach the end point. Determine the parts per million of CO 2 in the sample of air, given that the density of CO 2 at the temperature of the sample is 1.98 g/L. 21. The purity of a synthetic preparation of methylethyl ketone (C 4 H 8 O) can be determined by reacting the ketone with hydroxylamine hydrochloride, liberating HCl (see Table 9.10). In a typical analysis, a 3.00-mL sample was diluted to 50.00 mL and treated with an excess of hydroxylamine hydrochloride. The liberated HCl was titrated with 0.9989 M NaOH, requiring 32.68 mL to reach the end point. Report the percent purity of the sample, given that the density of methylethyl ketone is 0.805 g/mL. 22. Animal fats and vegetable oils are triacylglycerols, or triesters, formed from the reaction of glycerol (1, 2, 3-propanetriol) with three long-chain fatty acids. One of the methods used to characterize a fat or an oil is a determination of its saponification number. When treated with boiling aqueous KOH, an ester is saponified into the parent alcohol and fatty acids (as carboxylate ions). The saponification number is the number of milligrams of KOH required to saponify 1.000 g of the fat or oil. In a typical analysis, a 2.085-g sample of butter is added to 25.00 mL of 0.5131 M KOH. After saponification is complete, the excess KOH is back titrated with 10.26 mL of 0.5000 M HCl. What is the saponification number for this sample of butter? 23. A 250.0-mg sample of an organic weak acid was dissolved in an appropriate solvent and titrated with 0.0556 M NaOH, requiring 32.58 mL to reach the end point. Determine the compound’s equivalent weight. 24. The potentiometric titration curve shown here was recorded on a 0.4300-g sample of a purified amino acid that was dissolved in 50.00 mL of water and titrated with 0.1036 M NaOH. Identify the amino acid from the possibilities listed in the following table. Amino Acid FW K a Amino Acid FW K a alanine 89.1 1.36 · 10 –10 asparagine 150 1.9 · 10 –9 glycine 75.1 1.67 · 10 –10 leucine 131.2 1.79 · 10 –10 methionine 149.2 8.9 · 10 –10 phenylalanine 166.2 4.9 · 10 –10 taurine 125.2 1.8 · 10 –9 valine 117.2 1.91 · 10 –10 Abbreviation: FW=formula weight. 25. Using its titration curve, determine the acid dissociation constant for the weak acid in problem 6. 26. Where in the scale of operations do the microtitration techniques discussed in Section 9B.8 belong? 27. An acid–base titration can be used to determine an analyte’s equivalent weight, but cannot be used to determine its formula weight. Explain why. 28. Commercial washing soda contains approximately 30–40% w/w Na 2 CO 3 . One procedure for the quantitative analysis of washing soda contains the following instructions: Transfer an approximately 4-g sample of the washing soda to a 250-mL volumetric flask. Dissolve the sample in about 100 mL of H 2 O, and then dilute to the mark. Using a pipet, transfer a 25-mL aliquot of this solution to a 125-mL Erlenmeyer flask, and add 25-mL of H 2 O and 2 drops of bromocresol green indicator. Titrate the sample with 0.1 M HCl to the indicator’s end point. What modifications, if any, would need to be made to this procedure if it were to be adapted for the evaluation of the purity of commercial Na 2 CO 3 (>98+% pure)? 29. When standardizing a solution of NaOH against potassium hydrogen phthalate (KHP), a variety of systematic and random errors are possible. Identify, with justification, whether the following are systematic or random sources of error, or if they have no effect. If the error is systematic, then indicate whether the experimentally determined molarity for NaOH will be too high or too low. The standardization reaction is C 8 H 5 O 4 – +OH – fi C 8 H 4 O 4 2– +H 2 O (a) The balance used to weigh the KHP is not properly calibrated and always reads 0.15 g too low. (b) The visual indicator selected for the standardization changes color between a pH of 3 and 4. (c) An air bubble is lodged in the tip of the buret at the beginning of the analysis, but is dislodged during the titration. (d) Samples of KHP are weighed into separate Erlenmeyer flasks, but the balance is only tared with the first flask. (e) The KHP was not dried before it was used. (f) The NaOH was not dried before it was used. (g) The procedure calls for the sample of KHP to be dissolved in 25 mL of water, but it is accidentally dissolved in 35 mL. 30. The concentration of o-phthalic acid in an organic solvent, such as n-butanol, may be determined by an acid–base titration using aqueous NaOH as the titrant. As the titrant is added, the o-phthalic acid is extracted into the aqueous Chapter 9 Titrimetric Methods of Analysis 363 pH Volume of titrant (mL) 0.00 0.0 14.0 50.0030.00 40.0020.0010.00 12.0 8.0 6.0 10.0 4.0 2.0 1400-CH09 9/9/99 2:14 PM Page 363 solution, where it reacts with the titrant. The titrant must be added slowly to allow sufficient time for the extraction to occur. (a) What type of error would you expect if the titration were to be carried out too quickly? (b) Propose an alternative acid–base titration method that would allow for the more rapid determination of the concentration of o-phthalic acid in n-butanol. 31. Calculate or sketch (or both) the titration curves for 50.00 mL of 0.0500 Mg 2+ with 0.0500 M EDTA at a pH of 7 and 10. Locate the equivalence point for each titration curve. 32. Calculate or sketch (or both) the titration curves for 25.0 mL of 0.0500 M Cu 2+ with 0.025 M EDTA at a pH of 10 and in the presence of 10 –3 M and 10 –1 M NH 3 . Locate the equivalence point for each titration curve. 33. Sketch the spectrophotometric titration curve for the titration of a mixture of 5.00 · 10 –3 M Bi 3+ and 5.00 · 10 –3 M Cu 2+ with 0.0100 M EDTA. Assume that only the Cu 2+ –EDTA complex absorbs at the selected wavelength. 34. The EDTA titration of mixtures of Ca 2+ and Mg 2+ can be followed thermometrically because the formation of the Ca 2+ –EDTA complex is exothermic, but the formation of the Mg 2+ –EDTA complex is endothermic. Sketch the thermometric titration curve for a mixture of 5.00 · 10 –3 M Ca 2+ with 5.00 · 10 –3 M Mg + with 0.0100 M EDTA. The heats of formation for CaY 2– and MgY 2– are, respectively, –23.9 kJ/mol and 23.0 kJ/mol. 35. EDTA is one member of a class of aminocarboxylate ligands that form very stable 1:1 complexes with metal ions. The following table shows log K f values for several ligands with Ca 2+ and Mg 2+ . Which ligand is the best choice for the direct titration of Ca 2+ in the presence of Mg 2+ ? Mg 2+ Ca 2+ EDTA ethylenediaminetetraacetic acid 8.7 10.7 HEDTA N-hydroxyethylenediaminetriacetic acid 7.0 8.0 EEDTA ethyletherdiaminetetraacetic acid 8.3 10.0 EGTA ethyleneglycol-bis(b -aminoethylether) -N,N￠ -tetraacetic acid 5.4 10.9 DTPA diethylenetriaminepentaacetic acid 9.0 10.7 CyDTA cyclohexanediaminetetraacetic acid 10.3 12.3 36. The amount of calcium in physiologic fluids can be determined by a complexometric titration with EDTA. In one such analysis, a 0.100-mL sample of blood serum was made basic by adding 2 drops of NaOH and titrated with 0.00119 M EDTA, requiring 0.268 mL to reach the end point. Report the concentration of calcium in the sample as miligrams of Ca per 100 mL. 37. After removing the membranes from an eggshell, the shell is dried and its mass recorded as 5.613 g. The eggshell is transferred to a 250-mL beaker and dissolved in 25 mL of 6 M HCl. After filtering, the solution containing the dissolved eggshell is diluted to 250 mL in a volumetric flask. A 10.00- mL aliquot is placed in a 125-mL Erlenmeyer flask and buffered to a pH of 10. Titrating with 0.04988 M EDTA requires 44.11 mL to reach the end point. Determine the amount of calcium in the eggshell as %w/w CaCO 3 . 38. The concentration of cyanide, CN – , in a copper electroplating bath can be determined by a complexometric titration with Ag + , forming the soluble Ag(CN) 2 – complex. In a typical analysis a 5.00-mL sample from an electroplating bath is transferred to a 250-mL Erlenmeyer flask, and treated with 100 mL of H 2 O, 5 mL of 20% w/v NaOH, and 5 mL of 10% w/v KI. The sample is titrated with 0.1012 M AgNO 3 , requiring 27.36 mL to reach the end point as signaled by the formation of a yellow precipitate of AgI. Report the concentration of cyanide as parts per million of NaCN. 39. Before the introduction of EDTA most complexation titrations used Ag + or CN – as the titrant. The analysis for Cd 2+ , for example, was accomplished indirectly by adding an excess of KCN to form Cd(CN) 4 2– , and back titrating the excess CN – with Ag + , forming Ag(CN) 2 – . In one such analysis, a 0.3000-g sample of an ore was dissolved and treated with 20.00 mL of 0.5000 M KCN. The excess CN – required 13.98 mL of 0.1518 M AgNO 3 to reach the end point. Determine the %w/w Cd in the ore. 40. Solutions containing both Fe 3+ and Al 3+ can be selectively analyzed for Fe 3+ by buffering to a pH of 2 and titrating with EDTA. The pH of the solution is then raised to 5 and an excess of EDTA added, resulting in the formation of the Al 3+ –EDTA complex. The excess EDTA is back titrated using a standard solution of Fe 3+ , providing an indirect analysis for Al 3+ . (a) Show that at a pH of 2 the formation of the Fe 3+ –EDTA complex is favorable, but the formation of the Al 3+ –EDTA complex is not favorable. (b) A 50.00-mL aliquot of a sample containing Fe 3+ and Al 3+ is transferred to a 250- mL Erlenmeyer flask and buffered to a pH of 2. A small amount of salicylic acid is added, forming the soluble red- colored Fe 3+ –salicylic acid complex. The solution is titrated with 0.05002 M EDTA, requiring 24.82 mL to reach the end point as signaled by the disappearance of the Fe 3+ –salicylic acid complex’s red color. The solution is buffered to a pH of 5, and 50.00 mL of 0.05002 M EDTA is added. After ensuring that the formation of the Al 3+ –EDTA complex is complete, the excess EDTA was back titrated with 0.04109 M Fe 3+ , requiring 17.84 mL to reach the end point as signaled by the reappearance of the red-colored Fe 3+ –salicylic acid complex. Report the molar concentrations of Fe 3+ and Al 3+ in the sample. 41. Prada and colleagues recently described a new indirect method for determining sulfate in natural samples, such as sea water and industrial effluents. 16 The method is based on (1) precipitating the sulfate as PbSO 4; (2) dissolving the PbSO 4 in an ammonical solution of excess EDTA to form the soluble PbY 2– complex; and (3) titrating the excess EDTA 364 Modern Analytical Chemistry 1400-CH09 9/9/99 2:14 PM Page 364 with a standard solution of Mg 2+ . The following reactions and equilibrium constants are known PbSO 4 (s) t Pb 2+ +SO 4 2– K sp = 1.6 · 10 –8 Pb 2+ +Y 4– t PbY 2– K f = 1.1 · 10 18 Mg 2+ +Y 4– t MgY 2– K f = 4.9 · 10 8 Zn 2+ +Y 4– t ZnY 2– K f = 3.2 · 10 16 (a) Show that a precipitate of PbSO 4 will dissolve in a solution of Y 4– . (b) Sporek proposed a similar method using Zn 2+ as a titrant and found that the accuracy was frequently poor. 17 One explanation is that Zn 2+ might react with the PbY 2– complex, forming ZnY 2– . Using the preceding equilibrium constants, show that this might be a problem when using Zn 2+ as a titrant, but will not be a problem when using Mg 2+ as a titrant. Would such a displacement of Pb 2+ by Zn 2+ lead to the reporting of too much or too little sulfate? (c) In a typical analysis, a 25.00-mL sample of an industrial effluent was carried through the procedure using 50.00 mL of 0.05000 M EDTA. Titrating the excess EDTA required 12.42 mL of 0.1000 M Mg 2+ . Report the molar concentration of SO 4 2– in the effluent sample. 42. Calculate or sketch (or both) titration curves for the following (unbalanced) redox titration reactions at 25 °C. Assume that the analyte is initially present at a concentration of 0.0100 M and that a 25.0-mL sample is taken for analysis. The titrant, which is the underlined species in each reaction, is 0.0100 M. (a) V 2+ +Ce 4+ fi V 3+ +Ce 3+ (b) Ti 2+ +Fe 3+ fi Ti 3+ +Fe 2+ (c) Fe 2+ + MnO 4 – fi Fe 3+ +Mn 2+ (pH=1) 43. What are the equivalence points for the titrations listed in problem 42? 44. Suggest an appropriate indicator for each of the titrations in problem 42. 45. The iron content of an ore can be determined by a redox titration using K 2 Cr 2 O 7 as the titrant. The ore sample is dissolved in concentrated HCl using Sn 2+ to speed the ore’s dissolution by reducing the Fe 3+ to Fe 2+ . After the sample is dissolved, the Fe 2+ and any excess Sn 2+ are oxidized to Fe 3+ and Sn 4+ , using MnO 4 – . The iron is then carefully reduced to Fe 2+ by adding a 2–3-drop excess of Sn 2+ . A solution of HgCl 2 is added and, if a white precipitate of Hg 2 Cl 2 forms, the analysis is continued by titrating with K 2 Cr 2 O 7 . The sample is discarded without completing the analysis if no precipitate forms, or if a gray precipitate (due to Hg) forms. (a) Explain the role of the HgCl 2 in the analysis. (b) Will a determinate error be introduced into the analysis if the analyst forgets to add Sn 2+ in the step where the iron ore is dissolved? (c) Will a determinate error be introduced if the iron is not quantitatively oxidized back to Fe 3+ by the MnO 4 – ? 46. The amount of Cr 3+ in inorganic salts can be determined by a redox titration. A portion of sample containing approximately 0.25 g of Cr 3+ is accurately weighed and dissolved in 50 mL of H 2 O. The Cr 3+ is oxidized to Cr 2 O 7 2– by adding 20 mL of 0.1 M AgNO 3 , which serves as a catalyst, and 50 mL of 10% w/v (NH 4 ) 2 S 2 O 8 , which serves as the oxidizing agent. After the reaction is complete, the resulting solution is boiled for 20 min to destroy the excess S 2 O 8 2– , cooled to room temperature, and diluted to 250 mL in a volumetric flask. A 50-mL portion of the resulting solution is transferred to an Erlenmeyer flask, treated with 50 mL of a standard solution of Fe 2+ , and acidified with 200 mL of 1 M H 2 SO 4 , reducing the Cr 2 O 7 2– to Cr 3+ . The excess Fe 2+ is then determined by a back titration with a standard solution of K 2 Cr 2 O 7 , using an appropriate visual indicator. The results are reported as %w/w Cr 3+ . (a) Several volume measurements are noted in this procedure by underlining. Which of these measurements must be made using a volumetric pipet? (b) The excess peroxydisulfate, S 2 O 8 2– , which was used to oxidize the Cr 3+ to Cr 2 O 7 2– , is destroyed by boiling the solution. What would be the effect on the reported %w/w Cr 3+ if some of the S 2 O 8 2– was not destroyed during this step? (c) Solutions of Fe 2+ undergo slow air oxidation to Fe 3+ . What would be the effect on the reported %w/w Cr 3+ if the standard solution of Fe 2+ had been inadvertently allowed to partially oxidize? 47. The exact concentration of H 2 O 2 in a solution that is nominally 6% w/v H 2 O 2 can be determined by a redox titration with MnO 4 – . A 25-mL aliquot of the sample is transferred to a 250-mL volumetric flask and diluted to volume with distilled water. A 25 mL aliquot of the diluted sample is added to an Erlenmeyer flask, diluted with 200 mL of distilled water, and acidified with 20 mL of 25% v/v H 2 SO 4 . The resulting solution is then titrated with a standard solution of KMnO 4 until a faint pink color persists for 30 s. The results are reported as %w/v H 2 O 2 . (a) Many commercially available solutions of H 2 O 2 contain an inorganic or organic stabilizer to prevent the autodecomposition of the peroxide to H 2 O and O 2 . What effect would the presence of this stabilizer have on the reported %w/v H 2 O 2 if it also reacts with MnO 4 – ? (b) Laboratory distilled water often contains traces of dissolved organic material that also can react with MnO 4 – . Describe a simple method to correct for this potential interference. (c) What modifications, if any, would need to be made to this procedure if the sample to be analyzed has a nominal concentration of 30% w/v H 2 O 2 . 48. The amount of iron in a meteorite was determined by a redox titration using KMnO 4 as the titrant. A 0.4185-g sample was dissolved in acid and the liberated Fe 3+ quantitatively reduced to Fe 2+ , using a reductor column. Titrating with 0.02500 M KMnO 4 requires 41.27 mL to reach the end point. Determine the %w/w Fe 2 O 3 in the sample of meteorite. 49. Under basic conditions, MnO 4 – can be used as a titrant for the analysis of Mn 2+ , with both the analyte and the titrant ending up as MnO 2 . In the analysis of a mineral sample for Chapter 9 Titrimetric Methods of Analysis 365 1400-CH09 9/9/99 2:14 PM Page 365 manganese, a 0.5165-g sample is dissolved, and the manganese reduced to Mn 2+ . The solution is made basic and titrated with 0.03358 M KMnO 4 , requiring 34.88 mL to reach the end point. Calculate the %w/w Mn in the mineral sample. 50. The amount of uranium in an ore sample can be determined by an indirect redox titration. The analysis is accomplished by dissolving the ore in sulfuric acid and reducing the resulting UO 2 2+ to U 4+ with a Walden reductor. The resulting solution is treated with an excess of Fe 3+ , forming Fe 2+ and U 6+ . The Fe 2+ is titrated with a standard solution of K 2 Cr 2 O 7 to a visual end point. In a typical analysis, a 0.315-g sample of ore is passed through the Walden reductor and treated with an excess of Fe 3+ . Titrating with 0.00987 M K 2 Cr 2 O 7 requires 10.52 mL. What is the %w/w U in the sample? 51. The thickness of the chromium plate on an auto fender was determined by dissolving a 30.0-cm 2 section of the fender in acid and oxidizing the liberated Cr 3+ to Cr 2 O 7 2– with peroxydisulfate. After removing the excess peroxydisulfate by boiling, 500.0 mg of Fe(NH 4 ) 2 (SO 4 ) 2 6H 2 O was added, reducing the Cr 2 O 7 2– to Cr 3+ . The excess Fe 2+ was back- titrated, requiring 18.29 mL of 0.00389 M K 2 Cr 2 O 7 to reach the end point. Determine the average thickness of the chromium plate given that the density of Cr is 7.20 g/cm 3 . 52. The concentration of CO in air can be determined by passing a known volume of air through a tube containing I 2 O 5 , resulting in the formation of CO 2 and I 2 . The I 2 is removed from the tube by distillation and is collected in a solution containing an excess of KI, producing I 3 – . The I 3 – is titrated with a standard solution of Na 2 S 2 O 3 . In a typical analysis, a 4.79-L sample of air was sampled as described here, requiring 7.17 mL of 0.00329 M Na 2 S 2 O 3 to reach the end point. If the air has a density of 1.23 · 10 –3 g/mL, determine the parts per million of CO in the air. 53. The level of dissolved oxygen in a water sample can be determined by the Winkler method. In a typical analysis, a 100.0-mL sample is made basic, and treated with a solution of MnSO 4 , resulting in the formation of MnO 2 . An excess of KI is added, and the solution is acidified, resulting in the formation of Mn 2+ and I 2 . The liberated I 2 is titrated with a solution of 0.00870 M Na 2 S 2 O 3 , requiring 8.90 mL to reach the starch indicator end point. Calculate the concentration of dissolved oxygen as parts per million of O 2 . 54. The analysis for Cl – using the Volhard method requires a back titration. A known amount of AgNO 3 in excess is added, precipitating AgCl. The unreacted Ag + is determined by back titrating with KSCN. There is a complication, however, because AgCl is more soluble than AgSCN. (a) Why do the relative solubilities of AgCl and AgSCN lead to a titration error? (b) Is the resulting titration error a positive or negative determinate error? (c) How might you modify the procedure described here to prevent this source of determinate error? (d) Will this source of determinate error be of concern when using the Volhard method to determine Br – ? 55. Voncina and co-workers suggest that precipitation titrations can be monitored by measuring pH as a function of the volume of titrant if the titrant is a weak base. 18 For example, in titrating Pb 2+ with CrO 4 2– the solution containing the analyte is initially acidified to a pH of 3.50 using HNO 3 . Before the equivalence point, the concentration of CrO 4 2– is controlled by the solubility product of PbCrO 4 . After the equivalence point, the concentration of CrO 4 2– is determined by the amount of excess titrant added. Considering the reactions controlling the concentration of CrO 4 2– , sketch the expected titration curve of pH versus volume of titrant. 56. Calculate or sketch (or both) the titration curve for the titration of 50.0 mL of 0.0250 M KI with 0.0500 M AgNO 3 . Plot both pAg and pI. 57. Calculate or sketch (or both) the titration curve for the titration of 25.0 mL of 0.0500 M KI and 0.0500 M KSCN with 0.0500 M AgNO 3 . 58. A 0.5131-g sample containing KBr is dissolved in 50 mL of distilled water. Titrating with 0.04614 M AgNO 3 requires 25.13 mL to reach the Mohr end point. A blank titration requires 0.65 mL to reach the same end point. Report the %w/w KBr in the sample. 59. A 0.1036-g sample containing only BaCl 2 and NaCl is dissolved in 50 mL of distilled water. Titrating with 0.07916 M AgNO 3 requires 19.46 mL to reach the Fajans end point. Report the %w/w BaCl 2 in the sample. 60. A 0.1093-g sample of impure Na 2 CO 3 was analyzed by the Volhard method. After adding 50.00 mL of 0.06911 M AgNO 3 , the sample was back titrated with 0.05781 M KSCN, requiring 27.36 mL to reach the end point. Report the purity of the Na 2 CO 3 sample. 366 Modern Analytical Chemistry For a general history of titrimetry, see the following sources. Kolthoff, I. M. “Analytical Chemistry in the USA in the First Quarter of This Century,” Anal. Chem. 1994, 66, 241A–249A. Laitinen, H. A.; Ewing, G. W., eds. A History of Analytical Chemistry. The Division of Analytical Chemistry of the American Chemical Society: Washington, DC, 1977, pp. 52–93. The use of weight instead of volume as a signal for titrimetry is reviewed in the following paper. Kratochvil, B.; Maitra, C. “Weight Titrations: Past and Present,” Am. Lab. 1983, January, 22–29. A more thorough discussion of nonaqueous titrations, with numerous practical examples, is provided in the following text. 9 J SUGGESTED READINGS 1400-CH09 9/9/99 2:14 PM Page 366 Fritz, J. S. Acid–Base Titrations in Nonaqueous Solvents. Allyn and Bacon: Boston, 1973. The text listed below provides more details on how the potentiometric titration data may be used to calculate equilibrium constants. This text provides a number of examples and includes a discussion of several computer programs that have been developed to model equilibrium reactions. Meloun, M.; Havel, J.; H?gfeldt, E. Computation of Solution Equilibria. Ellis Horwood Limited: Chichester, England, 1988. The following paper provides additional information about the use of Gran plots. Schwartz, L. M. “Advances in Acid–Base Gran Plot Methodology,” J. Chem. Educ. 1987, 64, 947–950. For a complete discussion of the application of complexation titrimetry, see the following texts. Pribil, R. Applied Complexometry. Pergamon Press: Oxford, 1982. Ringbom, A. Complexation in Analytical Chemistry. John Wiley and Sons, Inc.: New York, 1963. Schwarzenbach, G. Complexometric Titrations. Methuen & Co. Ltd: London, 1957. A good source for additional examples of the application of all forms of titrimetry is Vogel’s Textbook of Quantitative Inorganic Analysis, 4th ed. Longman: London, 1981. Chapter 9 Titrimetric Methods of Analysis 367 1. (a) Willis, C. J. J. Chem. Educ. 1981, 58, 659–663; (b) Nakagawa, K. J. Chem. Educ. 1990, 67, 673–676; (c) Gordus, A. A. J. Chem. Educ. 1991, 68, 759–761; (d) de Levie, R. J. Electroanal. Chem. 1992, 323, 347–355; (e) de Levie, R. J. Chem. Educ. 1993, 70, 209–217; (f) Chaston, S. J. Chem. Educ. 1993, 70, 878–880; (g) de Levie, R. Anal. Chem. 1996, 68, 585–590. 2. (a) Currie, J. O.; Whiteley, R. V. J. Chem. Educ. 1991, 68, 923–926; (b) Breneman, G. L.; Parker, O. J. J. Chem. Educ. 1992, 69, 46–47; (c) Carter, D. R.; Frye, M. S.; Mattson, W. A. J. Chem. Educ. 1993, 70, 67–71; (d) Freiser, H. Concepts and Calculations in Analytical Chemistry, CRC Press: Boca Raton, FL, 1992. 3. Meites, L.; Goldman, J. A. Anal. Chim. Acta 1963, 29, 472–479. 4. Method 13.86 as published in Official Methods of Analysis, 8th Ed., Association of Official Agricultural Chemists: Washington, DC, 1955, p. 226. 5. Ferek, R. J.; Lazrus, A. L.; Haagenson, P. L.; et al. Environ. Sci. Technol. 1983, 17, 315–324. 6. (a) Molnár-Perl, I.; Pintée-Szakás, M. Anal. Chim. Acta 1987, 202, 159; (b) Barbosa, J.; Bosch, E.; Cortina, J. L., et al. Anal. Chim. Acta 1992, 256, 177–181. 7. Gonzalez, A. G.; Asuero, A. G. Anal. Chim. Acta 1992, 257, 29–33. 8. Papanastasiou, G.; Ziogas, I.; Kokkinidis, G. Anal. Chim. Acta 1993, 277, 119–135. 9. Alexio, L. M.; Godinho, O. E. S.; da Costa, W. F. Anal. Chim. Acta 1992, 257, 35–39. 10. Steele, A. W.; Hieftje, G. M. Anal. Chem. 1984, 56, 2884–2888. 11. (a) Gratzl, M.; Yi, C. Anal. Chem. 1993, 65, 2085–2088; (b) Yi, C.; Gratzl, M. Anal. Chem. 1994, 66, 1976–1982; (c) Hui, K. Y.; Gratzl. M. Anal. Chem. 1997, 69, 695–698; (d) Yi, C.; Huang, D.; Gratzl, M. Anal. Chem. 1996, 68, 1580–1584; (e) Xie, H.; Gratzl, M. Anal. Chem. 1996, 68, 3665–3669. 12. Method 2340 C as published in Standard Methods for the Examination of Water and Wastewater, 20th ed. American Public Health Association: Washington, DC, 1998, pp. 2–27 to 2–29. 13. Method 4500-ClA as published in Standard Methods for the Examination of Water and Wastewater, 20th ed. American Public Health Association: Washington, DC, 1998, pp. 4–53 to 4–55. 14. Schwartz, L. M. J. Chem. Educ. 1992, 69, 879–883. 15. (a) Boiani, J. A. J. Chem. Educ. 1986, 63, 724–726; (b) Castillo, C. A.; Jaramillo, A. J. Chem. Educ. 1989, 66, 341. 16. Prada, S.; Guekezian, M.; Suarez-Iha, M. E. V. Anal. Chim. Acta 1996, 329, 197–202. 17. Sporek, K. F. Anal. Chem. 1958, 30, 1032. 18. Voncina, D. B.; Dobcnik, D.; Gomiscek, S. Anal. Chim. Acta 1992, 263, 147–153. 9 K REFERENCES 1400-CH09 9/9/99 2:14 PM Page 367 368 Chapter 10 Spectroscopic Methods of Analysis Before the beginning of the twentieth century most quantitative chemical analyses used gravimetry or titrimetry as the analytical method. With these methods, analysts achieved highly accurate results, but were usually limited to the analysis of major and minor analytes. Other methods developed during this period extended quantitative analysis to include trace level analytes. One such method was colorimetry. One example of an early colorimetric analysis is Nessler s method for ammonia, which was first proposed in 1856. Nessler found that adding an alkaline solution of HgI 2 and KI to a dilute solution of ammonia produced a yellow to reddish brown colloid with the color determined by the concentration of ammonia. A comparison of the sample s color to that for a series of standards was used to determine the concentration of ammonia. Equal volumes of the sample and standards were transferred to a set of tubes with flat bottoms. The tubes were placed in a rack equipped at the bottom with a reflecting surface, allowing light to pass through the solution. The colors of the samples and standards were compared by looking down through the solutions. Until recently, a modified form of this method was listed as a standard method for the analysis of ammonia in water and wastewater. 1 Colorimetry, in which a sample absorbs visible light, is one example of a spectroscopic method of analysis. At the end of the nineteenth century, spectroscopy was limited to the absorption, emission, and scattering of visible, ultraviolet, and infrared electromagnetic radiation. During the twentieth century, spectroscopy has been extended to include other forms of electromagnetic radiation (photon spectroscopy), such as X-rays, microwaves, and radio waves, as well as energetic particles (particle spectroscopy), such as electrons and ions. 2 1400-CH10 9/8/99 4:17 PM Page 368 Figure 10.1 Plane-polarized electromagnetic radiation showing the electric field, the magnetic field, and the direction of propagation. Chapter 10 Spectroscopic Methods of Analysis 369 10A Overview of Spectroscopy The focus of this chapter is photon spectroscopy, using ultraviolet, visible, and in- frared radiation. Because these techniques use a common set of optical devices for dispersing and focusing the radiation, they often are identified as optical spectros- copies. For convenience we will usually use the simpler term “spectroscopy” in place of photon spectroscopy or optical spectroscopy; however, it should be under- stood that we are considering only a limited part of a much broader area of analyti- cal methods. Before we examine specific spectroscopic methods, however, we first review the properties of electromagnetic radiation. 10A.1 What Is Electromagnetic Radiation Electromagnetic radiation, or light, is a form of energy whose behavior is described by the properties of both waves and particles. The optical properties of electromag- netic radiation, such as diffraction, are explained best by describing light as a wave. Many of the interactions between electromagnetic radiation and matter, such as ab- sorption and emission, however, are better described by treating light as a particle, or photon. The exact nature of electromagnetic radiation remains unclear, as it has since the development of quantum mechanics in the first quarter of the twentieth century. 3 Nevertheless, the dual models of wave and particle behavior provide a use- ful description for electromagnetic radiation. Wave Properties of Electromagnetic Radiation Electromagnetic radiation consists of oscillating electric and magnetic fields that propagate through space along a lin- ear path and with a constant velocity (Figure 10.1). In a vacuum, electromagnetic radiation travels at the speed of light, c, which is 2.99792 · 10 8 m/s. Electromagnetic radiation moves through a medium other than a vacuum with a velocity, v, less than that of the speed of light in a vacuum. The difference between v and c is small enough (< 0.1%) that the speed of light to three significant figures, 3.00 · 10 8 m/s, is sufficiently accurate for most purposes. Oscillations in the electric and magnetic fields are perpendicular to each other, and to the direction of the wave’s propagation. Figure 10.1 shows an example of plane-polarized electromagnetic radiation consisting of an oscillating electric field and an oscillating magnetic field, each of which is constrained to a single plane. Normally, electromagnetic radiation is unpolarized, with oscillating electric and Electric field Magnetic field Direction of propagation 1400-CH10 9/8/99 4:17 PM Page 369 Figure 10.2 Electric field component of plane-polarized electromagnetic radiation. magnetic fields in all possible planes oriented perpendicular to the direction of propagation. The interaction of electromagnetic radiation with matter can be explained using either the electric field or the magnetic field. For this reason, only the electric field component is shown in Figure 10.2. The oscillating electric field is described by a sine wave of the form E = A e sin(2pn t + F ) where E is the magnitude of the electric field at time t, A e is the electric field’s maxi- mum amplitude, n is the frequency, or the number of oscillations in the electric field per unit time, and F is a phase angle accounting for the fact that the electric field’s magnitude need not be zero at t = 0. An identical equation can be written for the magnetic field, M M = A m sin(2pn t + F ) where A m is the magnetic field’s maximum amplitude. An electromagnetic wave, therefore, is characterized by several fundamental properties, including its velocity, amplitude, frequency, phase angle, polarization, and direction of propagation. 4 Other properties, which are based on these funda- mental properties, also are useful for characterizing the wave behavior of electro- magnetic radiation. The wavelength of an electromagnetic wave, l , is defined as the distance between successive maxima, or successive minima (see Figure 10.2). For ultraviolet and visible electromagnetic radiation the wavelength is usually expressed in nanometers (nm, 10 –9 m), and the wavelength for infrared radiation is given in microns (m m, 10 –6 m). Unlike frequency, wavelength depends on the electromag- netic wave’s velocity, where Thus, for electromagnetic radiation of frequency, n , the wavelength in vacuum is longer than in other media. Another unit used to describe the wave properties of electromagnetic radiation is the wavenumber, – n , which is the reciprocal of wave- length ν λ = 1 λ νν == vc (in vacuum) 370 Modern Analytical Chemistry Electric field strength A e Time or distance – + l wavelength The distance between any two consecutive maxima or minima of an electromagnetic wave (l ). frequency The number of oscillations of an electromagnetic wave per second (n ). wavenumber The reciprocal of wavelength ( – n ). 1400-CH10 9/8/99 4:17 PM Page 370 Chapter 10 Spectroscopic Methods of Analysis 371 Wavenumbers are frequently used to characterize infrared radiation, with the units given in reciprocal centimeter (cm –1 ). EXAMPLE 10.1 In 1817, Josef Fraunhofer (1787–1826) studied the spectrum of solar radiation, observing a continuous spectrum with numerous dark lines. Fraunhofer labeled the most prominent of the dark lines with letters. In 1859, Gustav Kirchhoff (1824–1887) showed that the “D” line in the solar spectrum was due to the absorption of solar radiation by sodium atoms. The wavelength of the sodium D line is 589 nm. What are the frequency and the wavenumber for this line? SOLUTION The frequency and wavenumber of the sodium D line are Two additional wave properties are power, P, and intensity, I, which give the flux of energy from a source of electromagnetic radiation. Particle Properties of Electromagnetic Radiation When a sample absorbs electro- magnetic radiation it undergoes a change in energy. The interaction between the sample and the electromagnetic radiation is easiest to understand if we assume that electromagnetic radiation consists of a beam of energetic particles called photons. When a photon is absorbed by a sample, it is “destroyed,” and its energy acquired by the sample. 5 The energy of a photon, in joules, is related to its frequency, wave- length, or wavenumber by the following equations where h is Planck’s constant, which has a value of 6.626 · 10 –34 J ? s. EXAMPLE 10.2 What is the energy per photon of the sodium D line (l = 589 nm)? SOLUTION The energy of the sodium D line is E hc == ×× × =× ? ? ? ? λ (. . . 6 626 10 3 00 10 589 10 337 10 34 8 9 19 J s) ( m /s) m J Eh hc hc = = = ν λ ν ν λ == × ×=× ? ? 11 589 10 170 10 9 41 m 1 m 100 cm cm. ν λ == × × =× ? ? c 300 10 589 10 509 10 8 9 14 1 . . m/s m s photon A particle of light carrying an amount of energy equal to hn . intensity The flux of energy per unit time per area (I). power The flux of energy per unit time (P). 1400-CH10 9/8/99 4:17 PM Page 371 Figure 10.3 The electromagnetic spectrum showing the colors of the visible spectrum. The energy of a photon provides an additional characteristic property of electro- magnetic radiation. The Electromagnetic Spectrum The frequency and wavelength of electromagnetic radiation vary over many orders of magnitude. For convenience, electromagnetic radiation is divided into different regions based on the type of atomic or molecular transition that gives rise to the absorption or emission of photons (Figure 10.3). The boundaries describing the electromagnetic spectrum are not rigid, and an overlap between spectral regions is possible. 10A.2 Measuring Photons as a Signal In the previous section we defined several characteristic properties of electromag- netic radiation, including its energy, velocity, amplitude, frequency, phase angle, polarization, and direction of propagation. Spectroscopy is possible only if the pho- ton’s interaction with the sample leads to a change in one or more of these charac- teristic properties. Spectroscopy is conveniently divided into two broad classes. In one class of techniques, energy is transferred between a photon of electromagnetic radiation and the analyte (Table 10.1). In absorption spectroscopy the energy carried by a photon is absorbed by the analyte, promoting the analyte from a lower-energy state to a higher-energy, or excited, state (Figure 10.4). The source of the energetic state de- pends on the photon’s energy. The electromagnetic spectrum in Figure 10.3, for ex- ample, shows that absorbing a photon of visible light causes a valence electron in the analyte to move to a higher-energy level. When an analyte absorbs infrared radi- ation, on the other hand, one of its chemical bonds experiences a change in vibra- tional energy. 372 Modern Analytical Chemistry Wavelength (m) Type of transition Nuclear g -ray Core-level electrons X-ray Valence electrons UV Molecular vibrations IR Visible Nuclear spin Radio wave Molecular rotations; electron spin Microwave Spectral region Frequency (s –1 ) 10 –14 10 22 10 20 10 18 10 16 10 14 10 12 10 10 10 8 10 –12 10 –10 10 –8 10 –6 10 –4 10 –2 10 0 10 2 380 Violet Blue Green Yellow Orange Red Wavelength (nm) 480 580 680 780 E 2 E 1 E 0 Figure 10.4 Simplified energy level diagram showing absorption of a photon. electromagnetic spectrum The division of electromagnetic radiation on the basis of a photon’s energy. Colorplate 9 shows the spectrum of visible light. 1400-CH10 9/8/99 4:17 PM Page 372 The intensity of photons passing through a sample containing the analyte is at- tenuated because of absorption. The measurement of this attenuation, which we call absorbance, serves as our signal. Note that the energy levels in Figure 10.4 have well-defined values (i.e., they are quantized). Absorption only occurs when the pho- ton’s energy matches the difference in energy, D E, between two energy levels. A plot of absorbance as a function of the photon’s energy is called an absorbance spec- trum (Figure 10.5). Emission of a photon occurs when an analyte in a higher-energy state returns to a lower-energy state (Figure 10.6). The higher-energy state can be achieved in several ways, including thermal energy, radiant energy from a photon, or by a Chapter 10 Spectroscopic Methods of Analysis 373 350.0 0.00 430.0 750.0510.0 Wavelength (nm) Absorbance 590.0 670.0 0.300 0.400 0.600 0.800 1.000 Table 10.1 Representative Spectroscopies Involving an Exchange of Energy Type of Energy Transfer Region of the Electromagnetic Spectrum Spectroscopic Technique absorption g -ray Mossbauer spectroscopy X-ray X-ray absorption spectroscopy UV/Vis a UV/Vis spectroscopy b atomic absorption spectroscopy b infrared infrared spectroscopy b raman spectroscopy microwave microwave spectroscopy electron spin resonance spectroscopy radio waves nuclear magnetic resonance spectroscopy emission (thermal excitation) UV/Vis atomic emission spectroscopy b photoluminescence X-ray X-ray fluorescence UV/Vis fluorescence spectroscopy b phosphorescence spectroscopy b atomic fluorescence spectroscopy a UV/Vis: ultraviolet and visible ranges. b Techniques discussed in this text. absorbance The attenuation of photons as they pass through a sample (A). absorbance spectrum A graph of a sample’s absorbance of electromagnetic radiation versus wavelength (or frequency or wavenumber). Figure 10.5 Ultraviolet/visible absorption spectrum for bromothymol blue. emission The release of a photon when an analyte returns to a lower-energy state from a higher-energy state. 1400-CH10 9/8/99 4:17 PM Page 373 Figure 10.7 Photoluminescent spectra for methyltetrahydrofolate and the enzyme methyltransferase. When methyltetrahydrofolate and methyltransferase are mixed, the enzyme is no longer photoluminescent, but the photoluminescence of methyltetrahydrofolate is enhanced. (Spectra courtesy of Dave Roberts, DePauw University.) 374 Modern Analytical Chemistry Table 10.2 Representative Spectroscopies That Do Not Involve an Exchange of Energy Region of the Electromagnetic Spectrum Type of Interaction Spectroscopic Technique X-ray diffraction X-ray diffraction UV/Vis a refraction refractometry scattering nephelometry b turbidimetry b dispersion optical rotary dispersion a UV/Vis: Ultraviolet and visible ranges. b Techniques covered in this text. Emission intensity 300 nm Methyltransferase + Methyltetrahydrofolate Methyltransferase Methyltetrahydrofolate Wavelength 500 nm photoluminescence Emission following absorption of a photon. chemiluminescence Emission induced by a chemical reaction. emission spectrum A graph of emission intensity versus wavelength (or frequency or wavenumber). chemical reaction. Emission following the absorption of a photon is also called photoluminescence, and that following a chemical reaction is called chemilumi- nescence. A typical emission spectrum is shown in Figure 10.7. In the second broad class of spectroscopy, the electromagnetic radiation under- goes a change in amplitude, phase angle, polarization, or direction of propagation as a result of its refraction, reflection, scattering, diffraction, or dispersion by the sample. Several representative spectroscopic techniques are listed in Table 10.2. 10B Basic Components of Spectroscopic Instrumentation The instruments used in spectroscopy consist of several common components, including a source of energy that can be input to the sample, a means for isolat- ing a narrow range of wavelengths, a detector for measuring the signal, and a sig- nal processor to display the signal in a form convenient for the analyst. In this section we introduce the basic components used to construct spectroscopic in- Figure 10.6 Simplified energy level diagram showing emission of a photon. E 2 E 1 E 0 1400-CH10 9/8/99 4:18 PM Page 374 Figure 10.8 Emission spectrum from a typical continuum source. Figure 10.9 Emission spectrum from a typical line source. struments. A more detailed discussion of these components can be found in the suggested end-of-chapter readings. Specific instrument designs are considered in later sections. 10B.1 Sources of Energy All forms of spectroscopy require a source of energy. In absorption and scattering spectroscopy this energy is supplied by photons. Emission and luminescence spec- troscopy use thermal, radiant (photon), or chemical energy to promote the analyte to a less stable, higher energy state. Sources of Electromagnetic Radiation A source of electromagnetic radiation must provide an output that is both intense and stable in the desired region of the elec- tromagnetic spectrum. Sources of electromagnetic radiation are classified as either continuum or line sources. A continuum source emits radiation over a wide range of wavelengths, with a relatively smooth variation in intensity as a function of wave- length (Figure 10.8). Line sources, on the other hand, emit radiation at a few se- lected, narrow wavelength ranges (Figure 10.9). Table 10.3 provides a list of the most common sources of electromagnetic radiation. Sources of Thermal Energy The most common sources of thermal energy are flames and plasmas. Flame sources use the combustion of a fuel and an oxidant such as acetylene and air, to achieve temperatures of 2000–3400 K. Plasmas, which are hot, ionized gases, provide temperatures of 6000–10,000 K. Chemical Sources of Energy Exothermic reactions also may serve as a source of energy. In chemiluminescence the analyte is raised to a higher-energy state by means of a chemical reaction, emitting characteristic radiation when it returns to a lower-energy state. When the chemical reaction results from a biological or enzy- matic reaction, the emission of radiation is called bioluminescence. Commercially available “light sticks” and the flash of light from a firefly are examples of chemilu- minescence and bioluminescence, respectively. Chapter 10 Spectroscopic Methods of Analysis 375 log(Relative intensity) Wavelength Intensity Wavelength continuum source A source that emits radiation over a wide range of wavelengths. line source A source that emits radiation at only select wavelengths. Table 10. 3 Common Sources of Electromagnetic Radiation for Spectroscopy Source Wavelength Region Useful for H 2 and D 2 lamp continuum source from 160–380 nm UV molecular absorption tungsten lamp continuum source from 320–2400 nm Vis molecular absorption Xe arc lamp continuum source from 200–1000 nm molecular fluorescence Nernst glower continuum source from 0.4–20 m m IR molecular absorption globar continuum source from 1–40 m m IR molecular absorption nichrome wire continuum source from 0.75–20 m m IR molecular absorption hollow cathode lamp line source in UV/Vis atomic absorption Hg vapor lamp line source in UV/Vis molecular fluorescence laser line source in UV/Vis atomic and molecular absorption, fluorescence and scattering Abbreviations: UV: ultraviolet; Vis: visible; IR: infrared. 1400-CH10 9/8/99 4:18 PM Page 375 Figure 10.10 Band of radiation exiting wavelength selector showing the nominal wavelength and effective bandpass. 10B.2 Wavelength Selection In Nessler’s original colorimetric method for ammonia, described at the beginning of the chapter, no attempt was made to narrow the wavelength range of visible light passing through the sample. If more than one component in the sample contributes to the absorption of radiation, however, then a quantitative analysis using Nessler’s original method becomes impossible. For this reason we usually try to select a single wavelength where the analyte is the only absorbing species. Unfortunately, we can- not isolate a single wavelength of radiation from a continuum source. Instead, a wavelength selector passes a narrow band of radiation (Figure 10.10) characterized by a nominal wavelength, an effective bandwidth, and a maximum throughput of radiation. The effective bandwidth is defined as the width of the radiation at half the maximum throughput. The ideal wavelength selector has a high throughput of radiation and a nar- row effective bandwidth. A high throughput is desirable because more photons pass through the wavelength selector, giving a stronger signal with less back- ground noise. A narrow effective bandwidth provides a higher resolution, with spectral features separated by more than twice the effective bandwidth being resolved. Generally these two features of a wavelength selector are in opposition (Figure 10.11). Conditions favoring a higher throughput of radiation usually pro- vide less resolution. Decreasing the effective bandwidth improves resolution, but at the cost of a noisier signal. For a qualitative analysis, resolution is generally more important than the throughput of radiation; thus, smaller effective band- widths are desirable. In a quantitative analysis a higher throughput of radiation is usually desirable. 6 Wavelength Selection Using Filters The simplest method for isolating a narrow band of radiation is to use an absorption or interference filter. Absorption filters work by selectively absorbing radiation from a narrow region of the electromagnetic spectrum. Interference filters use constructive and destructive interference to isolate a narrow range of wavelengths. A simple example of an absorption filter is a piece of colored glass. A purple filter, for example, removes the complementary color green from 500–560 nm. Commercially available absorption filters provide effective band- widths from 30–250 nm. The maximum throughput for the smallest effective band- passes, however, may be only 10% of the source’s emission intensity over that range of wavelengths. Interference filters are more expensive than absorption filters, but have narrower effective bandwidths, typically 10–20 nm, with maximum through- puts of at least 40%. Wavelength Selection Using Monochromators One limitation of an absorption or interference filter is that they do not allow for a continuous selection of wavelength. If measurements need to be made at two wavelengths, then the filter must be changed in between measurements. A further limitation is that filters are available for only selected nominal ranges of wavelengths. An alternative approach to wave- length selection, which provides for a continuous variation of wavelength, is the monochromator. The construction of a typical monochromator is shown in Figure 10.12. Radia- tion from the source enters the monochromator through an entrance slit. The radi- ation is collected by a collimating mirror, which reflects a parallel beam of radiation to a diffraction grating. The diffraction grating is an optically reflecting surface with 376 Modern Analytical Chemistry nominal wavelength The wavelength which a wavelength selector is set to pass. effective bandwidth The width of the band of radiation passing through a wavelength selector measured at half the band’s height. resolution In spectroscopy, the separation between two spectral features, such as absorption or emission lines. Radiant po wer Wavelength Nominal wavelength Effective bandwidth filter A wavelength selector that uses either absorption, or constructive and destructive interference to control the range of selected wavelengths. monochromator A wavelength selector that uses a diffraction grating or prism, and that allows for a continuous variation of the nominal wavelength. 1400-CH10 9/8/99 4:18 PM Page 376 Figure 10.11 Effect of the monochromator’s slit width on noise and resolution for the ultraviolet absorption spectrum of benzene. The slit width increases from spectrum (a) to spectrum (d) with effective bandpasses of 0.25 nm, 1.0 nm, 2.0 nm, and 4.0 nm. a large number of parallel grooves (see inset to Figure 10.12). Diffraction by the grating disperses the radiation in space, where a second mirror focuses the radiation onto a planar surface containing an exit slit. In some monochromators a prism is used in place of the diffraction grating. Radiation exits the monochromator and passes to the detector. As shown in Figure 10.12, a polychromatic source of radiation at the entrance slit is converted at the exit slit to a monochromatic source of finite effective bandwidth. The choice of Chapter 10 Spectroscopic Methods of Analysis 377 230.0 –0.200 A 0.488 A 290.0280.0260.0240.0 230.0 –0.050 A 0.571 A 290.0280.0260.0240.0 230.0 –0.038 A 0.532 A 290.0280.0260.0240.0 230.0 –0.037 A 0.401 A 290.0280.0260.0240.0 (a) (b) (c) (d) polychromatic Electromagnetic radiation of more than one wavelength. monochromatic Electromagnetic radiation of a single wavelength. 1400-CH10 9/8/99 4:18 PM Page 377 which wavelength exits the monochromator is determined by rotating the diffraction grating. A narrower exit slit provides a smaller effective bandwidth and better reso- lution, but allows a smaller throughput of radiation. Monochromators are classified as ei- ther fixed-wavelength or scanning. In a fixed-wavelength monochromator, the wavelength is selected by manually rotating the grating. Normally, a fixed-wavelength monochromator is only used for quantita- tive analyses where measurements are made at one or two wavelengths. A scan- ning monochromator includes a drive mechanism that continuously rotates the grating, allowing successive wavelengths to exit from the monochromator. Scan- ning monochromators are used to acquire spectra and, when operated in a fixed- wavelength mode, for quantitative analysis. Interferometers An interferometer pro- vides an alternative approach for wave- length selection. Instead of filtering or dispersing the electromagnetic radiation, an interferometer simultaneously allows source radiation of all wavelengths to reach the detector (Figure 10.13). Radia- tion from the source is focused on a beam splitter that transmits half of the radiation to a fixed mirror, while reflecting the other half to a movable mirror. The radia- tion recombines at the beam splitter, where constructive and destructive inter- ference determines, for each wavelength, the intensity of light reaching the de- tector. As the moving mirror changes position, the wavelengths of light experi- encing maximum constructive interference and maximum destructive interference also changes. The signal at the detector shows intensity as a func- tion of the moving mirror’s position, expressed in units of distance or time. The result is called an interferogram, or a time domain spectrum. The time domain spectrum is converted mathematically, by a process called a Fourier transform, to the normal spectrum (also called a frequency domain spectrum) of intensity as a function of the radiation’s energy. Further details about interferometers and the mathematics of the Fourier transform can be found in the suggested readings listed at the end of the chapter. In comparison with a monochromator, interferometers provide two signifi- cant advantages. The first advantage, which is termed Jacquinot’s advantage, re- sults from the higher throughput of source radiation. Since an interferometer does not use slits and has fewer optical components from which radiation can be scattered and lost, the throughput of radiation reaching the detector is 80–200 times greater than that achieved with a monochromator. The result is an im- 378 Modern Analytical Chemistry Collimating mirror Entrance slit Diffraction grating Exit slit l 1 l 1 l 2 l 2 l 3 l 3 Focusing mirror interferometer A device that allows all wavelengths of light to be measured simultaneously, eliminating the need for a wavelength selector. Figure 10.12 Typical grating monochromator with inset showing the dispersion of the radiation by the diffraction grating. 1400-CH10 9/8/99 4:18 PM Page 378 Figure 10.13 Block diagram of an interferometer. proved signal-to-noise ratio. The second advantage, which is called Fellgett’s ad- vantage, reflects a savings in the time needed to obtain a spectrum. Since all fre- quencies are monitored simultaneously, an entire spectrum can be recorded in approximately 1 s, as compared to 10–15 min with a scanning monochromator. 10B. 3 Detectors The first detector for optical spectroscopy was the human eye, which, of course, is limited both by its accuracy and its limited sensitivity to electromagnetic radiation. Modern detectors use a sensitive transducer to convert a signal consisting of pho- tons into an easily measured electrical signal. Ideally the detector’s signal, S, should be a linear function of the electromagnetic radiation’s power, P, S = kP + D where k is the detector’s sensitivity, and D is the detector’s dark current, or the background electric current when all radiation from the source is blocked from the detector. Photon Transducers Two general classes of transducers are used for optical spectroscopy, several examples of which are listed in Table 10.4. Phototubes and photomultipliers contain a photosensitive surface that absorbs radiation in the ultraviolet, visible, and near infrared (IR), producing an electric current propor- tional to the number of photons reaching the transducer. Other photon detec- tors use a semiconductor as the photosensitive surface. When the semiconductor absorbs photons, valence electrons move to the semiconductor’s conduction band, producing a measurable current. One advantage of the Si photodiode is that it is easily miniaturized. Groups of photodiodes may be gathered together in a linear array containing from 64 to 4096 individual photodiodes. With a width of 25 m m per diode, for example, a linear array of 2048 photodiodes requires only 51.2 mm of linear space. By placing a photodiode array along the mono- chromator’s focal plane, it is possible to monitor simultaneously an entire range of wavelengths. Thermal Transducers Infrared radiation generally does not have sufficient en- ergy to produce a measurable current when using a photon transducer. A thermal transducer, therefore, is used for infrared spectroscopy. The absorption of in- frared photons by a thermal transducer increases its temperature, changing one or more of its characteristic properties. The pneumatic transducer, for example, Chapter 10 Spectroscopic Methods of Analysis 379 signal-to-noise ratio The ratio of the signal’s intensity to the average intensity of the surrounding noise. Beam splitter Sample Detector Source Fixed mirror Mo ving mirror dark current The background current present in a photon detector in the absence of radiation from the source. transducer A device that converts a chemical or physical property, such as pH or photon intensity, to an easily measured electrical signal, such as a voltage or current. photodiode array A linear array of photodiodes providing the ability to detect simultaneously radiation at several wavelengths. 1400-CH10 9/8/99 4:18 PM Page 379 380 Modern Analytical Chemistry Table 10.4 Characteristics of Transducers for Optical Spectroscopy Detector Class Wavelength Range Output Signal phototube photon 200–1000 nm current photomultiplier photon 110–1000 nm current Si photodiode photon 250–1100 nm current photoconductor photon 750–6000 nm change in resistance photovoltaic cell photon 400–5000 nm current or voltage thermocouple thermal 0.8–40 m m voltage thermistor thermal 0.8–40 m m change in resistance pneumatic thermal 0.8–1000 m m membrane displacement pyroelectric thermal 0.3–1000 m m current signal processor A device, such as a meter or computer, that displays the signal from the transducer in a form that is easily interpreted by the analyst. consists of a small tube filled with xenon gas equipped with an IR-transparent window at one end, and a flexible membrane at the other end. A blackened sur- face in the tube absorbs photons, increasing the temperature and, therefore, the pressure of the gas. The greater pressure in the tube causes the flexible mem- brane to move in and out, and this displacement is monitored to produce an electrical signal. 10B. 4 Signal Processors The electrical signal generated by the transducer is sent to a signal processor where it is displayed in a more convenient form for the analyst. Examples of signal proces- sors include analog or digital meters, recorders, and computers equipped with digi- tal acquisition boards. The signal processor also may be used to calibrate the detec- tor’s response, to amplify the signal from the detector, to remove noise by filtering, or to mathematically transform the signal. 10C Spectroscopy Based on Absorption Historically, the first spectroscopic studies involved characterizing the emission of visible light from the sun, from flames, and from salts added to flames. Our survey of spectroscopy, however, begins with absorption because it is the more important technique in modern analytical spectroscopy. 10C.1 Absorbance of Electromagnetic Radiation In absorption spectroscopy a beam of electromagnetic radiation passes through a sam- ple. Much of the radiation is transmitted without a loss in intensity. At selected fre- quencies, however, the radiation’s intensity is attenuated. This process of attenuation is called absorption. Two general requirements must be met if an analyte is to absorb electromagnetic radiation. The first requirement is that there must be a mechanism by which the radiation’s electric field or magnetic field interacts with the analyte. For ul- traviolet and visible radiation, this interaction involves the electronic energy of valence electrons. A chemical bond’s vibrational energy is altered by the absorbance of infrared radiation. A more detailed treatment of this interaction, and its importance in deter- 1400-CH10 9/8/99 4:18 PM Page 380 Chapter 10 Spectroscopic Methods of Analysis 381 mining the intensity of absorption, is found in the suggested readings listed at the end of the chapter. The second requirement is that the energy of the electromagnetic radia- tion must exactly equal the difference in energy, D E, between two of the ana- lytes quantized energy states. Figure 10.4 shows a simplified view of the ab- sorption of a photon. The figure is useful because it emphasizes that the photon’s energy must match the difference in energy between a lower-energy state and a higher-energy state. What is missing, however, is information about the types of energetic states involved, which transitions between states are likely to occur, and the appearance of the resulting spectrum. We can use the energy level diagram in Figure 10.14 to explain an ab- sorbance spectrum. The thick lines labeled E 0 and E 1 represent the analyte’s ground (lowest) electronic state and its first electronic excited state. Superim- posed on each electronic energy level is a series of lines representing vibra- tional energy levels. Infrared Spectra for Molecules and Polyatomic Ions The energy of infrared radiation is sufficient to produce a change in the vibrational energy of a mole- cule or polyatomic ion (see Table 10.1). As shown in Figure 10.14, vibrational energy levels are quantized; that is, a molecule may have only certain, discrete vibrational energies. The energy for allowed vibrational modes, E v , is where v is the vibrational quantum number, which may take values of 0, 1, 2, . . ., and n 0 is the bond’s fundamental vibrational frequency. Values for n 0 are determined by the bond’s strength and the mass at each end of the bond and are characteristic of the type of bond. For example, a carbon–carbon sin- gle bond (C—C) absorbs infrared radiation at a lower energy than a carbon–carbon double bond (C=C) because a C—C bond is weaker than a C=C bond. At room temperature most molecules are in their ground vibrational state (v = 0). A transi- tion from the ground vibrational state to the first vibrational excited state (v = 1) requires the absorption of a photon with an energy of hn 0 . Transitions in which D v is – 1 give rise to the fundamental absorption lines. Weaker absorp- tion lines, called overtones, are due to transi- tions in which D v is – 2 or – 3. The number of possible normal vibrational modes for a linear molecule is 3N – 5, and for a nonlinear mole- cule is 3N – 6, where N is the number of atoms in the molecule. Not surprisingly, infrared spec- tra often show a considerable number of ab- sorption bands. Even a relatively simple mole- cule, such as benzene (C 6 H 6 ), for example, has 30 possible normal modes of vibration, al- though not all of these vibrational modes give rise to an absorption. A typical IR spectrum is shown in Figure 10.15. Ev h ν ν=+ ? ? ? ? ? ? 1 2 0 v 4 v 3 v 2 v 1 v 0 v 4 v 3 v 2 v 1 v 0 E 1 E 0 Figure 10.14 Energy level diagram showing difference between the absorption of infrared radiation (left) and ultraviolet–visible radiation (right). 4000 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 50010001500 Wavenumbers (cm –1 ) P e r cent transmittance 3500 3000 2500 2000 Figure 10.15 Fourier transform infrared (FT–IR) spectrum of polyvinylchloride. 1400-CH10 9/8/99 4:18 PM Page 381 Figure 10.16 Splitting of d-orbitals in an octahedral field. UV/Vis Spectra for Molecules and Ions When a molecule or ion absorbs ultravio- let or visible radiation it undergoes a change in its valence electron configuration. The valence electrons in organic molecules, and inorganic anions such as CO 3 2– , oc- cupy quantized sigma bonding, s , pi bonding, p , and nonbonding, n, molecular or- bitals. Unoccupied sigma antibonding, s *, and pi antibonding, p *, molecular or- bitals often lie close enough in energy that the transition of an electron from an occupied to an unoccupied orbital is possible. Four types of transitions between quantized energy levels account for molecu- lar UV/Vis spectra. The approximate wavelength ranges for these absorptions, as well as a partial list of bonds, functional groups, or molecules that give rise to these transitions is shown in Table 10.5. Of these transitions, the most important are the n fip * and pfip *, because they involve functional groups that are characteristic of the analyte and wavelengths that are easily accessible. The bonds and functional groups that give rise to the absorption of ultraviolet and visible radiation are called chromophores. Many transition metal ions, such as Cu 2+ and Co 2+ , form solutions that are colored because the metal ion absorbs visible light. The transitions giving rise to this absorption are due to valence electrons in the metal ion’s d-orbitals. For a free metal ion, the five d-orbitals are of equal energy. In the presence of a com- plexing ligand or solvent molecule, however, the d-orbitals split into two or more groups that differ in energy. For example, in the octahedral complex Cu(H 2 O) 6 2+ the six water molecules perturb the d-orbitals into two groups as shown in Figure 10.16. The resulting d–d transitions for transition metal ions are relatively weak. A more important source of UV/Vis absorption for inorganic metal–ligand complexes is charge transfer, in which absorbing a photon produces an excited state species that can be described in terms of the transfer of an electron from the metal, M, to the ligand, L. M—L + hnfi M + —L – Charge-transfer absorption is important because it produces very large absorbances, providing for a much more sensitive analytical method. One important example of a charge-transfer complex is that of o-phenanthroline with Fe 2+ , the UV/Vis spec- trum for which is shown in Figure 10.17. Charge-transfer absorption in which the electron moves from the ligand to the metal also is possible. Comparing the IR spectrum in Figure 10.15 to the UV/Vis spectrum in Fig- ure 10.17, we note that UV/Vis absorption bands are often significantly broader than those for IR absorption. Figure 10.14 shows why this is true. When a species 382 Modern Analytical Chemistry Table 10. 5 Electronic Transitions Involving n, s , and p Molecular Orbitals Wavelength Range Transition (nm) Examples sfis * < 200 C––C, C––H n fis * 160–260 H 2 O, CH 3 OH, CH 3 Cl pfip * 200–500 C=C, C=O, C=N, C;C n fi p * 250–600 C=O, C=N, N=N, N=O chromophore The specific bonds or functional groups in a molecule responsible for the absorption of a particular wavelength of light. d xy d xz d yz d z 2 d x 2 –y 2 hn d z 2 d x 2 –y 2 d xz d yz d xy 1400-CH10 9/8/99 4:18 PM Page 382 absorbs UV/Vis radiation, the transition between electronic energy levels may also include a transition between vibrational energy levels. The result is a num- ber of closely spaced absorption bands that merge together to form a single broad absorption band. UV/Vis Spectra for Atoms As noted in Table 10.1, the energy of ultra- violet and visible electromagnetic radiation is sufficient to cause a change in an atom’s valence electron configuration. Sodium, for exam- ple, with a valence shell electron configuration of [Ne] 3s 1 , has a single valence electron in its 3s atomic orbital. Unoccupied, higher energy atomic orbitals also exist. Figure 10.18 shows a partial energy level dia- gram for sodium’s occupied and unoccupied valence shell atomic or- bitals. This configuration of atomic orbitals, which shows a splitting of the p orbitals into two levels with slightly different energies, may differ from that encountered in earlier courses. The reasons for this splitting, however, are beyond the level of this text, and unimportant in this context. Absorption of a photon is accompanied by the excitation of an electron from a lower-energy atomic orbital to an orbital of higher energy. Not all possible transitions between atomic orbitals are al- lowed. For sodium the only allowed transitions are those in which there is a change of – 1 in the orbital quantum number (l); thus transitions from sfi p orbitals are allowed, but transitions from sfi d orbitals are forbidden. The wavelengths of electromagnetic ra- diation that must be absorbed to cause several allowed transitions are shown in Figure 10.18. The atomic absorption spectrum for Na is shown in Figure 10.19 and is typical of that found for most atoms. The most obvi- ous feature of this spectrum is that it consists of a few, discrete ab- sorption lines corresponding to transitions between the ground state (the 3s atomic orbital) and the 3p and 4p atomic orbitals. Ab- sorption from excited states, such as that from the 3p atomic or- bital to the 4s or 3d atomic orbital, which are included in the en- ergy level diagram in Figure 10.18, are too weak to detect. Since the Chapter 10 Spectroscopic Methods of Analysis 383 Wavelength (nm) Absorbance 400.0 0.000 0.200 0.400 0.600 0.800 1.000 640.0 700.0460.0 520.0 580.0 Figure 10.17 UV/Vis spectrum for Fe(o-phenanthroline) 3 2+ . 5s 4s Energy 3s 5p 5p 4p 3p3p 4p 330.3 330.2 818.3 819.5 589.0 589.6 1138.3 1140.4 4d 3d Figure 10.18 Valence shell energy diagram for sodium. 1400-CH10 9/8/99 4:18 PM Page 383 384 Modern Analytical Chemistry Figure 10.20 (a) Schematic diagram showing the attenuation of radiation passing through a sample; P 0 is the radiant power from the source and P T is the radiant power transmitted by the sample. (b) Schematic diagram showing that P 0 is redefined as the radiant power transmitted by the blank, correcting the transmittance in (a) for any loss of radiation due to scattering, reflection or absorption by the cuvette, and absorption by the sample’s matrix. lifetime of an excited state is short, typically 10 –7 –10 –8 s, an atom in the ex- cited state is likely to return to the ground state before it has an opportu- nity to absorb a photon. Another feature of the spectrum shown in Figure 10.19 is the narrow width of the absorption lines, which is a consequence of the fixed difference in energy between the ground and excited states. Natural line widths for atomic absorption, which are governed by the uncertainty principle, are ap- proximately 10 –5 nm. Other contributions to broadening increase this line width to approximately 10 –3 nm. 10C.2 Transmittance and Absorbance The attenuation of electromagnetic radiation as it passes through a sample is described quantitatively by two separate, but related terms: transmittance and absorbance. Transmittance is defined as the ratio of the electromagnetic radi- ation’s power exiting the sample, P T , to that incident on the sample from the source, P 0 , (Figure 10.20a). 10.1 Multiplying the transmittance by 100 gives the percent transmittance (%T), which varies between 100% (no absorption) and 0% (complete absorption). All methods of detection, whether the human eye or a modern photoelectric transducer, measure the transmittance of electromagnetic radiation. Attenuation of radiation as it passes through the sample leads to a trans- mittance of less than 1. As described, equation 10.1 does not distinguish be- tween the different ways in which the attenuation of radiation occurs. Besides absorption by the analyte, several additional phenomena contribute to the net attenuation of radiation, including reflection and absorption by the sample container, absorption by components of the sample matrix other than the an- alyte, and the scattering of radiation. To compensate for this loss of the elec- tromagnetic radiation’s power, we use a method blank (Figure 10.20b). The radiation’s power exiting from the method blank is taken to be P 0 . An alternative method for expressing the attenuation of electromagnetic radiation is absorbance, A, which is defined as 10.2 Absorbance is the more common unit for expressing the attenuation of radiation because, as shown in the next section, it is a linear function of the analyte’s concentration. EXAMPLE 10. 3 A sample has a percent transmittance of 50.0%. What is its absorbance? SOLUTION With a percent transmittance of 50.0%, the transmittance of the sample is 0.500. Substituting into equation 10.2 gives A = –log T = –log(0.500) = 0.301 AT P P P P =? =? =log log log T T0 0 T P P = T 0 Absorbance Wavelength (nm) 329.5 330 330.5 588.5 589 589.5 590 Figure 10.19 Atomic absorption spectrum for sodium. P 0 P T Sample Sample container (a) P 0 Blank (b) transmittance The ratio of the radiant power passing through a sample to that from the radiation’s source (T). 1400-CH10 9/8/99 4:18 PM Page 384 Figure 10.21 Factors used in deriving the Beer–Lambert law. Chapter 10 Spectroscopic Methods of Analysis 385 P 0 P T P dx x = 0 x = b P – dP Equation 10.1 has an important consequence for atomic absorption. Because of the narrow line width for atomic absorption, a continuum source of radiation can- not be used. Even with a high-quality monochromator, the effective bandwidth for a continuum source is 100–1000 times greater than that for an atomic absorption line. As a result, little of the radiation from a continuum source is absorbed (P o ? P T ), and the measured absorbance is effectively zero. For this reason, atomic ab- sorption requires a line source. 10C. 3 Absorbance and Concentration: Beer s Law When monochromatic electromagnetic radiation passes through an infinitesimally thin layer of sample, of thickness dx, it experiences a decrease in power of dP (Fig- ure 10.21). The fractional decrease in power is proportional to the sample’s thick- ness and the analyte’s concentration, C; thus 10.3 where P is the power incident on the thin layer of sample, and a is a proportionality constant. Integrating the left side of equation 10.3 from P = P 0 to P = P T , and the right side from x = 0 to x = b, where b is the sample’s overall thickness gives Converting from ln to log, and substituting equation 10.2, gives A = abC 10.4 where a is the analyte’s absorptivity with units of cm –1 conc –1 . When concentration is expressed using molarity, the absorptivity is replaced by the molar absorptivity, e (with units of cm –1 M –1 ) A = e bC 10.5 The absorptivity and molar absorptivity give, in effect, the probability that the ana- lyte will absorb a photon of given energy. As a result, values for both a and e depend on the wavelength of electromagnetic radiation. EXAMPLE 10. 4 A 5.00 · 10 –4 M solution of an analyte is placed in a sample cell that has a pathlength of 1.00 cm. When measured at a wavelength of 490 nm, the absorbance of the solution is found to be 0.338. What is the analyte’s molar absorptivity at this wavelength? SOLUTION Solving equation 10.5 for e and making appropriate substitutions gives ε= = × = ? ?? A bC 0 338 1 00 5 00 10 676 4 11 . (. . cm)( M) cm M ln P P bC 0 T ? ? ? ? ? ? =α ?= = = = = ∫∫ dP P Cdx PP PP x xb 0 0 T α ? = dP P Cdxα 1400-CH10 9/8/99 4:18 PM Page 385 Figure 10.22 Calibration curves showing positive and negative deviations from Beer’s law. Equations 10.4 and 10.5, which establish the linear relationship between absorbance and concentration, are known as the Beer–Lambert law, or more commonly, as Beer’s law. Calibration curves based on Beer’s law are used routinely in quantitative analysis. 10C. 4 Beer s Law and Multicomponent Samples Beer’s law can be extended to samples containing several absorbing components provided that there are no interactions between the components. Individual ab- sorbances, A i , are additive. For a two-component mixture of X and Y, the total ab- sorbance, A tot , is A tot = A X + A Y = e X bC X + e Y bC Y Generalizing, the absorbance for a mixture of n components, A m , is given as 10.6 10C.5 Limitations to Beer s Law According to Beer’s law, a calibration curve of absorbance versus the concentration of analyte in a series of standard solutions should be a straight line with an intercept of 0 and a slope of ab or e b. In many cases, however, calibration curves are found to be nonlinear (Figure 10.22). Deviations from linearity are divided into three cate- gories: fundamental, chemical, and instrumental. Fundamental Limitations to Beers Law Beer’s law is a limiting law that is valid only for low concentrations of analyte. There are two contributions to this funda- mental limitation to Beer’s law. At higher concentrations the individual particles of analyte no longer behave independently of one another. The resulting interaction between particles of analyte may change the value of e . A second contribution is that the absorptivity, a, and molar absorptivity, e , depend on the sample’s refractive index. Since the refractive index varies with the analyte’s concentration, the values of a and e will change. For sufficiently low concentrations of analyte, the refractive index remains essentially constant, and the calibration curve is linear. Chemical Limitations to Beer s Law Chemical deviations from Beer’s law can occur when the absorbing species is involved in an equilibrium reaction. Consider, as an example, an analysis for the weak acid, HA. To construct a Beer’s law calibra- tion curve, several standards containing known total concentrations of HA, C tot , are prepared and the absorbance of each is measured at the same wavelength. Since HA is a weak acid, it exists in equilibrium with its conjugate weak base, A – HA+H 2 O t H 3 O + +A – If both HA and A – absorb at the selected wavelength, then Beers law is written as A = e HA bC HA + e A bC A 10.7 where C HA and C A are the equilibrium concentrations of HA and A – . Since the weak acid’s total concentration, C tot , is C tot = C HA + C A AA bC i i n ii i n m == == ∑∑ 11 ε 386 Modern Analytical Chemistry Beer’s law The relationship between a sample’s absorbance and the concentration of the absorbing species (A = e bC). Absorbance Concentration Ideal Positive deviation Negative deviation 1400-CH10 9/8/99 4:18 PM Page 386 Figure 10.23 Effect of wavelength on the linearity of a Beer’s law calibration curve. the concentrations of HA and A – can be written as C HA = a HA C tot 10.8 C A =(1–a HA )C tot 10.9 where a HA is the fraction of weak acid present as HA. Substituting equations 10.8 and 10.9 into equation 10.7, and rearranging, gives A =(e HA a HA + e A – e A a HA )bC tot 10.10 Because values of a HA may depend on the concentration of HA, equation 10.10 may not be linear. A Beer’s law calibration curve of A versus C tot will be linear if one of two conditions is met. If the wavelength is chosen such that e HA and e A are equal, then equation 10.10 simplifies to A = e A bC tot and a linear Beer’s law calibration curve is realized. Alternatively, if a HA is held con- stant for all standards, then equation 10.10 will be a straight line at all wavelengths. Because HA is a weak acid, values of a HA change with pH. To maintain a constant value for a HA , therefore, we need to buffer each standard solution to the same pH. Depending on the relative values of e HA and e A , the calibration curve will show a positive or negative deviation from Beer’s law if the standards are not buffered to the same pH. Instrumental Limitations to Beer s Law There are two principal instrumental limi- tations to Beer’s law. The first limitation is that Beer’s law is strictly valid for purely monochromatic radiation; that is, for radiation consisting of only one wavelength. As we learned in Section 10B.2, however, even the best wavelength selector passes radiation with a small, but finite effective bandwidth. Using polychromatic radia- tion always gives a negative deviation from Beer’s law, but is minimized if the value of e is essentially constant over the wavelength range passed by the wavelength se- lector. For this reason, as shown in Figure 10.23, it is preferable to make absorbance measurements at a broad absorption peak. In addition, deviations from Beer’s law are less serious if the effective bandwidth from the source is less than one tenth of the natural bandwidth of the absorbing species. 7,8 When measurements must be made on a slope, linearity is improved by using a narrower effective bandwidth. Stray radiation is the second contribution to instrumental deviations from Beer’s law. Stray radiation arises from imperfections within the wavelength selector Chapter 10 Spectroscopic Methods of Analysis 387 Absorbance Absorbance Wavelength Dl passed by wavelength selector Concentration stray radiation Any radiation reaching the detector that does not follow the optical path from the source to the detector. 1400-CH10 9/8/99 4:18 PM Page 387 388 Modern Analytical Chemistry that allows extraneous light to “leak” into the instrument. Stray radiation adds an additional contribution, P stray, to the radiant power reaching the detector; thus For small concentrations of analyte, P stray is significantly smaller than P 0 and P T , and the absorbance is unaffected by the stray radiation. At higher concentrations of ana- lyte, however, P stray is no longer significantly smaller than P T and the absorbance is smaller than expected. The result is a negative deviation from Beer’s law. 10D Ultraviolet-Visible and Infrared Spectrophotometry The earliest routine application of molecular absorption spectroscopy, which dates to the 1830s, was colorimetry, in which visible light was absorbed by a sample. The concentration of analyte was determined visually by comparing the sample’s color to that of a set of standards using Nessler tubes (as described at the beginning of this chapter), or by using an instrument called a colorimeter. The development of visi- ble absorption spectroscopy as a routine analytical technique was limited by the te- dious nature of making visual color comparisons. Furthermore, although infrared radiation was discovered in 1800 and ultraviolet radiation in 1801, their use in opti- cal molecular absorption spectroscopy was limited by the lack of a convenient means for detecting the radiation. During the 1930s and 1940s, advances in elec- tronics resulted in the introduction of photoelectric transducers for ultraviolet and visible radiation, and thermocouples for infrared radiation. As a result, “modern” instrumentation for absorption spectroscopy routinely became available in the 1940s. Progress in these fields has been rapid ever since. 10D.1 Instrumentation Frequently an analyst must select, from several instruments of different design, the one instrument best suited for a particular analysis. In this section we examine some of the different types of instruments used for molecular absorption spec- troscopy, emphasizing their advantages and limitations. Methods of sample intro- duction are also covered in this section. Instrument Designs for Molecular UV/Vis Absorption The simplest instrument cur- rently used for molecular UV/Vis absorption is the filter photometer shown in Fig- ure 10.24, which uses an absorption or interference filter to isolate a band of radia- tion. The filter is placed between the source and sample to prevent the sample from decomposing when exposed to high-energy radiation. A filter photometer has a sin- gle optical path between the source and detector and is called a single-beam instru- ment. The instrument is calibrated to 0% T while using a shutter to block the source radiation from the detector. After removing the shutter, the instrument is calibrated to 100% T using an appropriate blank. The blank is then replaced with the sample, and its transmittance is measured. Since the source’s incident power and the sensitiv- ity of the detector vary with wavelength, the photometer must be recalibrated when- ever the filter is changed. In comparison with other spectroscopic instruments, pho- tometers have the advantage of being relatively inexpensive, rugged, and easy to maintain. Another advantage of a photometer is its portability, making it a useful in- strument for conducting spectroscopic analyses in the field. A disadvantage of a pho- tometer is that it cannot be used to obtain an absorption spectrum. A PP PP = + + log 0 stray T stray filter photometer A simple instrument for measuring absorbance that uses absorption or interference filters to select the wavelength. 1400-CH10 9/8/99 4:18 PM Page 388 Figure 10.24 Block diagram for a filter photometer with photo showing a typical hand-held instrument suitable for field work. Colorimeter? is manufactured by Hach Company/photo courtesy of Hach Company. Instruments using monochromators for wavelength selection are called spectrometers. In absorbance spectroscopy, where the transmittance is a ratio of two radiant powers, the instrument is called a spectrophotometer. The simplest spectrophotometer is a single-beam instrument equipped with a fixed- wavelength monochromator, the block diagram for which is shown in Fig- ure 10.25. Single-beam spectrophotometers are calibrated and used in the same manner as a photometer. One common example of a single-beam spectropho- tometer is the Spectronic-20 manufactured by Milton-Roy. The Spectronic-20 can be used from 340 to 625 nm (950 nm with a red-sensitive detector), and has a fixed effective bandwidth of 20 nm. Because its effective bandwidth is fairly large, this instrument is more appropriate for a quantitative analysis than for a qualitative analysis. Battery-powered, hand-held single-beam spectrophotome- ters are available, which are easily transported and can be used for on-site analy- ses. Other single-beam spectrophotometers are available with effective band- widths of 2–8 nm. Fixed-wavelength single-beam spectrophotometers are not practical for recording spectra since manually adjusting the wavelength and re- calibrating the spectrophotometer is awkward and time-consuming. In addition, the accuracy of a single-beam spectrophotometer is limited by the stability of its source and detector over time. Chapter 10 Spectroscopic Methods of Analysis 389 Source Filter Sample Shutter Blank Detector Signal processor spectrophotometer An instrument for measuring absorbance that uses a monochromator to select the wavelength. Source Monochromator Sample Shutter Blank Detector Signal processor Figure 10.25 Block diagram for a single-beam fixed- wavelength spectrophotometer with photo of a typical instrument. Photo courtesy of Fisher Scientific. 1400-CH10 9/8/99 4:18 PM Page 389 Figure 10.26 Block diagram for a double-beam in-time scanning spectrophotometer with photo of a typical instrument. Photo courtesy of Varian, Inc. The limitations of fixed-wavelength, single-beam spectrophotometers are mini- mized by using the double-beam in-time spectrophotometer as shown in Figure 10.26. A chopper, similar to that shown in the insert, controls the radiation’s path, alternat- ing it between the sample, the blank, and a shutter. The signal processor uses the chop- per’s known speed of rotation to resolve the signal reaching the detector into that due to the transmission of the blank (P 0 ) and the sample (P T ). By including an opaque sur- face as a shutter it is possible to continuously adjust the 0% T response of the detector. The effective bandwidth of a double-beam spectrophotometer is controlled by means of adjustable slits at the entrance and exit of the monochromator. Effective band- widths of between 0.2 nm and 3.0 nm are common. A scanning monochromator al- lows for the automated recording of spectra. Double-beam instruments are more ver- satile than single-beam instruments, being useful for both quantitative and qualitative analyses; they are, however, more expensive. The instrument designs considered thus far use a single detector and can only monitor one wavelength at a time. A linear photodiode array consists of multiple de- tectors, or channels, allowing an entire spectrum to be recorded in as little as 0.1 s. A block diagram for a typical multichannel spectrophotometer is shown in Figure 10.27. Source radiation passing through the sample is dispersed by a grating. The linear pho- todiode array is situated at the grating’s focal plane, with each diode recording the ra- diant power over a narrow range of wavelengths. 390 Modern Analytical Chemistry Source Monochromator Sample Mirror Mirror Grid mirror Sample Chopper Blank Blank Detector Signal processor Source Shutter Detector Signal processor Blank Sample Grating Figure 10.27 Block diagram for a diode array spectrophotometer. 1400-CH10 9/8/99 4:18 PM Page 390 One advantage of a linear photodiode array is the speed of data acquisition, which makes it possible to collect several spectra for a single sample. Individual spec- tra are added and averaged to obtain the final spectrum. This process of signal aver- aging improves a spectrum’s signal-to-noise ratio. When a series of spectra is added, the sum of the signal at any point increases as (nS x ), where n is the number of spec- tra, and S x is the signal for the spectrum’s x-th point. The propagation of noise, which is a random event, increases as , where N x is the noise level for the spectrum’s x-th point. The signal-to-noise ratio (S/N) at the x-th data point, there- fore, increases by a factor of where (S x /N x ) is the signal-to-noise ratio for a single scan. The effect of signal averaging is shown in Figure 10.28. The spectrum in Figure 10.28a shows the total signal for a single scan. Although there is an apparent peak near the cen- ter of the spectrum, the level of background noise makes it difficult to mea- sure the peak’s signal. Figures 10.28b and Figure 10.28c demonstrate the im- provement in signal-to-noise ratio achieved by signal averaging. One disadvantage of a linear photodiode array is that the effective bandwidth per diode is roughly an order of magnitude larger than that obtainable with a high-quality monochromator. The sample compartment for the instruments in Figures 10.24–10.27 provides a light-tight environment that prevents the loss of radiation, as well as the addition of stray radiation. Samples are normally in the liquid or solu- tion state and are placed in cells constructed with UV/Vis-transparent materi- als, such as quartz, glass, and plastic (Figure 10.29). Quartz or fused-silica cells are required when working at wavelengths of less than 300 nm where other materials show a significant absorption. The most common cell has a pathlength of 1 cm, although cells with shorter (? 1 mm) and longer path- lengths (≤ 10 cm) are available. Cells with a longer pathlength are useful for the analysis of very dilute solutions or for gaseous samples. The highest qual- ity cells are constructed in a rectangular shape, allowing the radiation to strike the cell at a 90° angle, where losses to reflection are minimal. These cells, which are usually available in matched pairs having identical optical proper- ties, are the cells of choice for double-beam instruments. Cylindrical test tubes are often used as a sample cell for simple, single-beam instruments, al- though differences in the cell’s pathlength and optical properties add an addi- tional source of error to the analysis. In some circumstances it is desirable to monitor a system without physi- cally removing a sample for analysis. This is often the case, for example, with the on-line monitoring of industrial production lines or waste lines, for physi- ological monitoring, and for monitoring environmental systems. With the use of a fiber-optic probe it is possible to analyze samples in situ. A simple exam- ple of a remote-sensing, fiber-optic probe is shown in Figure 10.30a and con- sists of two bundles of fiber-optic cable. One bundle transmits radiation from the source to the sample cell, which is designed to allow for the easy flow of sample through the cell. Radiation from the source passes through the solu- tion, where it is reflected back by a mirror. The second bundle of fiber-optic cable transmits the nonabsorbed radiation to the wavelength selector. In an alternative design (Figure 10.30b), the sample cell is a membrane containing a reagent phase capable of reacting with the analyte. When the analyte diffuses S N nS nN n S N x x x x == n ()nN x Chapter 10 Spectroscopic Methods of Analysis 391 signal averaging The adding together of successive spectra to improve the signal-to-noise ratio. Signal (arbitrar y units) 0.16 Wavelength 0.12 0.08 0.04 Signal (arbitrar y units) 0.60 Wavelength 0.45 0.30 0.15 Signal (arbitrar y units) 1.25 Wavelength 1.00 0.75 0.50 Figure 10.28 Effect of signal averaging on a spectrum’s signal- to-noise ratio: (a) spectrum for a single scan; (b) spectrum after co-adding five spectra; (c) spectrum after co-adding ten spectra. (a) (b) (c) 1400-CH10 9/8/99 4:18 PM Page 391 Figure 10.29 Typical cells used in UV/Vis spectroscopy. Courtesy of Fisher Scientific. 392 Modern Analytical Chemistry Source Wavelength selector Mirror Source Wavelength selector Membrane Reagent (a) (b) Figure 10.30 Example of fiber-optic probes. 1400-CH10 9/8/99 4:18 PM Page 392 across the membrane, it reacts with the reagent phase, producing a product that ab- sorbs UV or visible radiation. Nonabsorbed radiation from the source is reflected or scattered back to the detector. Fiber-optic probes that show chemical selectivity are called optrodes. 9,10 Instrument Designs for Infrared Absorption The simplest instrument for IR ab- sorption spectroscopy is a filter photometer similar to that shown in Figure 10.24 for UV/Vis absorption. These instruments have the advantage of portability and typically are used as dedicated analyzers for gases such as HCN and CO. Infrared instruments using a monochromator for wavelength selection are con- structed using double-beam optics similar to that shown in Figure 10.26. Double- beam optics are preferred over single-beam optics because the sources and detectors for infrared radiation are less stable than that for UV/Vis radiation. In addition, it is easier to correct for the absorption of infrared radiation by atmospheric CO 2 and H 2 O vapor when using double-beam optics. Resolutions of 1–3 cm –1 are typical for most instruments. In a Fourier transform, infrared spectrometer, or FT–IR, the monochromator is replaced with an interferometer (see Figure 10.13). Because an FT–IR includes only a single optical path, it is necessary to collect a separate spectrum to compen- sate for the absorbance of atmospheric CO 2 and H 2 O vapor. This is done by collect- ing a background spectrum without the sample and storing the result in the instru- ment’s computer memory. The background spectrum is removed from the sample’s spectrum by ratioing the two signals. In comparison to other IR instruments, an FT–IR provides for rapid data acquisition, allowing an enhancement in signal-to- noise ratio through signal averaging. Infrared spectroscopy is routinely used for the analysis of samples in the gas, liquid, and solid states. Sample cells are made from materials, such as NaCl and KBr, that are transparent to infrared radiation. Gases are analyzed using a cell with a pathlength of approximately 10 cm. Longer pathlengths are obtained by using mir- rors to pass the beam of radiation through the sample several times. Liquid samples are analyzed in one of two ways. For nonvolatile liquids a suit- able sample can be prepared by placing a drop of the liquid between two NaCl plates, forming a thin film that typically is less than 0.01 mm thick. Volatile liquids must be placed in a sealed cell to prevent their evaporation. The analysis of solution samples is limited by the solvent’s IR-absorbing prop- erties, with CCl 4 , CS 2 , and CHCl 3 being the most common solvents. Solutions are placed in cells containing two NaCl windows separated by a Teflon spacer. By changing the Teflon spacer, pathlengths from 0.015 to 1.0 mm can be obtained. Sealed cells with fixed or variable pathlengths also are available. The analysis of aqueous solutions is complicated by the solubility of the NaCl cell window in water. One approach to obtaining infrared spectra on aqueous solu- tions is to use attenuated total reflectance (ATR) instead of transmission. Figure 10.31 shows a diagram of a typical ATR sampler, consisting of an IR-transparent crystal of high- refractive index, such as ZnSe, surrounded by a sample of lower-refractive index. Radiation from the source enters the ATR crystal, where it undergoes a series of total internal reflec- tions before exiting the crystal. During each reflection, the ra- diation penetrates into the sample to a depth of a few microns. The result is a selec- tive attenuation of the radiation at those wavelengths at which the sample absorbs. Chapter 10 Spectroscopic Methods of Analysis 393 From source To detector ATR crystal Sample Sample Figure 10.31 Attenuated total reflectance (ATR) cell for use in infrared spectroscopy. 1400-CH10 9/8/99 4:18 PM Page 393 Figure 10.32 Method for determining absorbance from an IR transmission spectrum. ATR spectra are similar, but not identical, to those obtained by measuring the transmission of radiation. Transparent solid samples can be analyzed directly by placing them in the IR beam. Most solid samples, however, are opaque and must be dispersed in a more transparent medium before recording a traditional transmission spectrum. If a suit- able solvent is available, then the solid can be analyzed by preparing a solution and analyzing as described earlier. When a suitable solvent is not available, solid samples may be analyzed by preparing a mull of the finely powdered sample with a suitable oil. Alternatively, the powdered sample can be mixed with KBr and pressed into an optically transparent pellet. Solid samples also can be analyzed by means of reflectance. The ATR sampler (see Figure 10.31) described for the analysis of aqueous solutions can be used for the analysis of solid samples, provided that the solid can be brought into contact with the ATR crystal. Examples of solids that have been analyzed by ATR include polymers, fibers, fabrics, powders, and biological tissue samples. Another re- flectance method is diffuse reflectance, in which radiation is reflected from a rough surface, such as a powder. Powdered samples are mixed with a nonabsorbing mate- rial, such as powdered KBr, and the reflected light is collected and analyzed. As with ATR, the resulting spectrum is similar to that obtained by conventional transmis- sion methods. Further details about these and other methods for preparing solids for infrared analysis can be found in the suggested readings listed at the end of the chapter. 10D.2 Quantitative Applications The determination of an analyte’s concentration based on its absorption of ultravi- olet or visible radiation is one of the most frequently encountered quantitative ana- lytical methods. One reason for its popularity is that many organic and inorganic compounds have strong absorption bands in the UV/Vis region of the electromag- netic spectrum. In addition, analytes that do not absorb UV/Vis radiation, or that absorb such radiation only weakly, frequently can be chemically coupled to a species that does. For example, nonabsorbing solutions of Pb 2+ can be reacted with dithizone to form the red Pb–dithizonate complex. An additional advantage to UV/Vis absorption is that in most cases it is relatively easy to adjust experimental and instrumental conditions so that Beer’s law is obeyed. Quantitative analyses based on the absorption of infrared radiation, although important, are less frequently encountered than those for UV/Vis absorption. One reason is the greater tendency for instrumental deviations from Beer’s law when using infrared radiation. Since infrared absorption bands are relatively narrow, de- viations due to the lack of monochromatic radiation are more pronounced. In addi- tion, infrared sources are less intense than sources of UV/Vis radiation, making stray radiation more of a problem. Differences in pathlength for samples and stan- dards when using thin liquid films or KBr pellets are a problem, although an inter- nal standard can be used to correct for any difference in pathlength. Finally, estab- lishing a 100% T (A = 0) baseline is often difficult since the optical properties of NaCl sample cells may change significantly with wavelength due to contamination and degradation. This problem can be minimized by determining absorbance rela- tive to a baseline established for the absorption band. Figure 10.32 shows how this is accomplished. The applications of Beer’s law for the quantitative analysis of samples in envi- ronmental chemistry, clinical chemistry, industrial chemistry and forensic chem- istry are numerous. Examples from each of these fields follow. 394 Modern Analytical Chemistry P e r cent transmittance Wavenumber P 0 P T P 0 A = log P T 1400-CH10 9/8/99 4:18 PM Page 394 Environmental Applications Methods for the analysis of waters and wastewaters relying on the absorption of UV/Vis radiation are among some of the most fre- quently employed analytical methods. Many of these methods are outlined in Table 10.6, and a few are described later in more detail. Although the quantitative analysis of metals in water and wastewater is ac- complished primarily by atomic absorption or atomic emission spectroscopy, many metals also can be analyzed following the formation of a colored metal– ligand complex. One advantage to these spectroscopic methods is that they are easily adapted to the field analysis of samples using a filter photometer. One lig- and used in the analysis of several metals is diphenylthiocarbazone, also known as dithizone. Dithizone is insoluble in water, but when a solution of dithizone in CHCl 3 is shaken with an aqueous solution containing an appropriate metal ion, a colored metal–dithizonate complex forms that is soluble in CHCl 3 . The selectivity Chapter 10 Spectroscopic Methods of Analysis 395 Table 10.6 Selected Examples of the Application of UV/Vis Molecular Absorption to the Analysis of Waters and Wastewaters l Analyte Method (nm) Trace Metals aluminum reaction with Eriochrome cyanide R dye at pH 6 produces red to pink complex 535 arsenic reduce to AsH 3 using Zn and react with silver diethyldithiocarbamate to form red complex 535 cadmium extraction into CHCl 3 containing dithizone from sample made basic with NaOH; pink to red complex 518 chromium oxidize to Cr(VI) and react with diphenylcarbazide in acidic solution to give red-violet product 540 copper react with neocuprine in neutral to slightly acid solution; extract into CHCl 3 /CH 3 OH to give yellow solution 457 iron react with o-phenanthroline in acidic solution to form orange-red complex 510 lead extraction into CHCl 3 containing dithizone from sample made basic with ammoniacal buffer; cherry red complex 510 manganese oxidize to MnO 4 – with persulfate 525 mercury extraction into CHCl 3 containing dithizone from acidic sample; orange complex 492 zinc reaction with zincon at pH 9 to form blue complex 620 Inorganic Nonmetals ammonia reaction with ammonia, hypochlorite, and phenol produces blue indophenol; catalyzed by manganous salt 630 cyanide convert to CNCl by reaction with chloramine-T, followed by reaction with a pyridine-barbituric acid to form red-blue dye 578 fluoride reaction with red Zr-SPADNS lake results in formation of ZrF 6 2– and decrease in concentration of the lake 570 chlorine (residual) oxidation of leuco crystal violet to form product with a bluish color 592 nitrate reduction to NO 2 – by Cd, colored azo dye formed by reaction with sulfanilamide and N-(1-naphthyl)-ethelyenediamine 543 phosphate reaction with ammonium molybdate followed by reduction with stannous chloride to form molybdenum blue 690 Organics phenol reaction with 4-aminoantipyrine and K 3 Fe(CN) 6 to form antipyrine dye 460 surfactants formation of blue ion pair between anionic surfactant and the cationic dye methylene blue, which is extracted into CHCl 3 652 1400-CH10 9/8/99 4:18 PM Page 395 of dithizone is controlled by adjusting the pH of the aqueous sample. For exam- ple, Cd 2+ is extracted from solutions that are made strongly basic with NaOH, Pb 2+ from solutions that are made basic with an ammoniacal buffer, and Hg 2+ from solutions that are slightly acidic. When chlorine is added to water that portion available for disinfection is called the chlorine residual. Two forms of the chlorine residual are recognized. The free chlorine residual includes Cl 2 , HOCl, and OCl – . The combined chlorine residual, which forms from the reaction of NH 3 with HOCl, consists of monochloroamine, NH 2 Cl, dichlororamine, NHCl 2 , and trichloroamine, NCl 3 . Since the free chlorine residual is more efficient at disinfection, analytical methods have been developed to determine the concentration of both forms of residual chlorine. One such method is the leuco crystal violet method. Free residual chlorine is determined by adding leuco crystal violet to the sample, which instantaneously oxidizes giving a bluish color that is monitored at 592 nm. Completing the analysis in less than 5 min pre- vents a possible interference from the combined chlorine residual. The total chlo- rine residual (free + combined) is determined by reacting a separate sample with io- dide, which reacts with both chlorine residuals to form HOI. When the reaction is complete, leuco crystal violet is added and oxidized by HOI, giving the same bluish colored product. The combined chlorine residual is determined by difference. The concentration of fluoride in drinking water may be determined indirectly by its ability to form a complex with zirconium. In the presence of the dye SPADNS,* solutions of zirconium form a reddish colored compound, called a “lake,” that absorbs at 570 nm. When fluoride is added, the formation of the stable ZrF 6 2– complex causes a portion of the lake to dissociate, decreasing the absorbance. A plot of absorbance versus the concentration of fluoride, therefore, has a negative slope. Spectroscopic methods also are used in determining organic constituents in water. For example, the combined concentrations of phenol, and ortho- and meta- substituted phenols are determined by using steam distillation to separate the phe- nols from nonvolatile impurities. The distillate is reacted with 4-aminoantipyrine at pH 7.9 – 0.1 in the presence of K 3 Fe(CN) 6 , forming a colored antipyrine dye. The dye is extracted into CHCl 3 , and the absorbance is monitored at 460 nm. A calibration curve is prepared using only the unsubstituted phenol, C 6 H 5 OH. Be- cause the molar absorptivities of substituted phenols are generally less than that for phenol, the reported concentration represents the minimum concentration of phe- nolic compounds. Molecular absorption also can be used for the analysis of environmentally sig- nificant airborne pollutants. In many cases the analysis is carried out by collecting the sample in water, converting the analyte to an aqueous form that can be analyzed by methods such as those described in Table 10.6. For example, the concentration of NO 2 can be determined by oxidizing NO 2 to NO 3 – . The concentration of NO 3 – is then determined by reducing to NO 2 – with Cd and reacting the NO 2 – with sulfanil- amide and N-(1-naphthyl)-ethylenediamine to form a brightly colored azo dye. An- other important application is the determination of SO 2 , which is determined by collecting the sample in an aqueous solution of HgCl 4 2– where it reacts to form Hg(SO 3 ) 2 2– . Addition of p-rosaniline and formaldehyde results in the formation of a bright purple complex that is monitored at 569 nm. Infrared absorption has proved useful for the analysis of organic vapors, including HCN, SO 2 , nitrobenzene, methyl mercaptan, and vinyl chloride. Frequently, these analyses are accomplished using portable, dedicated infrared photometers. 396 Modern Analytical Chemistry *SPADNS is the acronym for the sodium salt of 2-(4-sulfophenylazo)-1,8-dihydroxy-3,6-naphthalenedisulfonic acid. 1400-CH10 9/8/99 4:18 PM Page 396 Clinical Applications UV/Vis molecular absorption is one of the most commonly employed techniques for the analysis of clinical samples, several examples of which are listed in Table 10.7. The analysis of clinical samples is often complicated by the complexity of the sample matrix, which may contribute a significant background absorption at the desired wavelength. The determination of serum barbiturates provides one example of how this problem is overcome. The barbiturates are extracted from a sample of serum with CHCl 3 , and extracted from the CHCl 3 into 0.45 M NaOH (pH ? 13). The absorbance of the aqueous extract is measured at 260 nm and includes contri- butions from the barbiturates as well as other components extracted from the serum sample. The pH of the sample is then lowered to approximately 10 by adding NH 4 Cl, and the absorbance remeasured. Since the barbiturates do not absorb at this pH, the absorbance at pH 10 is used to correct the absorbance at pH 13; thus Industrial Analysis UV/Vis molecular absorption is used for the analysis of a di- verse array of industrial samples, including pharmaceuticals, food, paint, glass, and metals. In many cases the methods are similar to those described in Tables 10.6 and 10.7. For example, the iron content of food can be determined by bringing the iron into solution and analyzing using the o-phenanthroline method listed in Table 10.6. Many pharmaceutical compounds contain chromophores that make them suit- able for analysis by UV/Vis absorption. Products that have been analyzed in this fashion include antibiotics, hormones, vitamins, and analgesics. One example of the use of UV absorption is in determining the purity of aspirin tablets, for which the active ingredient is acetylsalicylic acid. Salicylic acid, which is produced by the hy- drolysis of acetylsalicylic acid, is an undesirable impurity in aspirin tablets, and should not be present at more than 0.01% w/w. Samples can be screened for unac- ceptable levels of salicylic acid by monitoring the absorbance at a wavelength of AA VV V A barb pH 13 samp NH Cl samp pH 10 4 =? +? ? ? ? ? ? Chapter 10 Spectroscopic Methods of Analysis 397 Table 10. 7 Selected Examples of the Application of UV/Vis Molecular Absorption to the Analysis of Clinical Samples l Analyte Method (nm) total serum protein reaction with protein, NaOH, and Cu 2+ produces blue-violet complex 540 serum cholesterol reaction with Fe 3+ in presence of isopropanol, acetic acid, and H 2 SO 4 produces blue-violet complex 540 uric acid reaction with phosphotungstic acid produces tungsten blue 710 serum barbiturates barbiturates are extracted into CHCl 3 , and then into 0.45 M NaOH 260 glucose reaction with o-toludine at 100 °C produces blue-green complex 630 protein-bound iodine decompose protein to release iodide; I – determined by its catalytic effect on redox reaction between Ce 4+ and As 3+ 420 1400-CH10 9/8/99 4:18 PM Page 397 398 Modern Analytical Chemistry Representative Methods —Continued 312 nm. Acetylsalicylic acid absorbs at 280 nm, but absorbs poorly at 312 nm. Con- ditions for preparing the sample are chosen such that an absorbance of greater than 0.02 signifies an unacceptable level of salicylic acid. Forensic Applications UV/Vis molecular absorption is routinely used in the analy- sis of narcotics and for drug testing. One interesting forensic application is the de- termination of blood alcohol using the Breathalyzer test. In this test a 52.5-mL breath sample is bubbled through an acidified solution of K 2 Cr 2 O 7 . Any ethanol present in the breath sample is oxidized by the dichromate, producing acetic acid and Cr 3+ as products. The concentration of ethanol in the breath sample is deter- mined from the decrease in absorbance at 440 nm where the dichromate ion ab- sorbs. A blood alcohol content of 0.10%, which is the legal limit in most states, cor- responds to 0.025 mg of ethanol in the breath sample. Developing a Quantitative Method for a Single Component In developing a quan- titative analytical procedure, the conditions under which Beer’s law is obeyed must be established. First, the most appropriate wavelength for the analysis is determined from an absorption spectrum. In most cases the best wavelength corresponds to an absorption maximum because it provides greater sensitivity and is less susceptible to instrumental limitations to Beer’s law due to the lack of monochromatic radia- tion. Second, if an instrument with adjustable slits is being used, then an appropri- ate slit width needs to be chosen. The absorption spectrum also aids in selecting a slit width. Generally the slit width should be as wide as possible to increase the throughput of radiation from the source, while being narrow enough to avoid in- strumental limitations to Beer’s law. Finally, a calibration curve is constructed to determine the range of concentrations for which Beer’s law is valid. Additional con- siderations that are important in any quantitative method are the effect of potential interferents and establishing an appropriate blank. Method 10.1 Determination of Iron in Water and Wastewater 11 Description of Method. Iron in the +2 oxidation state reacts with o-phenanthroline to form the orange-red Fe(o-phen) 3 2+ complex (see Figure 10.17). The intensity of the complex’s color is independent of solution acidity between a pH of 3 and 9. Due to a faster rate of complex formation in more acidic solutions, the reaction is usually carried out within the pH range of 3.0–3.5. Any iron present in the +3 oxidation state is reduced with hydroxylamine before adding o-phenanthroline. The most important interferents are strong oxidizing agents; polyphosphates; and metal ions, such as Cu 2+ , Zn 2+ , Ni 2+ , and Cd 2+ . Interferences from oxidizing agents are minimized by adding an excess of hydroxylamine, whereas the interference from polyphosphate is minimized by boiling the sample in the presence of acid. The absorbance of samples and standards is measured at a wavelength of 510 nm using a 1-cm cell (longer-pathlength cells may be used as well). Beer’s law is obeyed for concentrations of iron within the range of 0.2–4.0 ppm. o-Phenanthroline NN 1400-CH10 9/8/99 4:18 PM Page 398 Chapter 10 Spectroscopic Methods of Analysis 399 Procedure. For samples containing less than 2 ppm Fe, directly transfer a 50-mL portion to a 125-mL Erlenmeyer flask. Samples containing more than 2 ppm Fe must be suitably diluted before acquiring the 50-mL portion. Add 2 mL of concentrated HCl and 1 mL of hydroxylamine to the sample in the Erlenmeyer flask. Heat the solution to boiling, and continue boiling until the solution’s volume is reduced to between 15 and 20 mL. After cooling to room temperature, transfer the solution to a 50-mL volumetric flask, add 10 mL of an ammonium acetate buffer, 2 mL of a 1000 ppm solution of o-phenanthroline, and dilute to volume. Allow 10–15 min for color development before measuring the absorbance, using distilled water to set 100% T. Calibration standards, including a blank, are prepared by the same procedure using a stock solution containing a known concentration of Fe 2+ . Questions 1. Explain why strong oxidizing agents interfere with this analysis and why an excess of hydroxylamine prevents such interferences from occurring. To obtain an accurate result it is necessary that only the Fe(o-phen) 3 2+ complex be present. Strong oxidizing agents interfere by oxidizing some of the Fe 2+ to Fe 3+ , producing the weakly absorbing Fe(o-phen) 3 3+ complex. The excess hydroxylamine reacts with the oxidizing agents, removing them from the solution. 2. The color intensity of the complex is stable between pH levels of 3 and 9. What are some possible complications at more acidic or more basic pHs? As with EDTA, which we encountered in Chapter 9, o-phenanthroline is a ligand possessing acid–base properties. The formation of the Fe(o-phen) 3 2+ complex, therefore, is less favorable at lower pH levels, where o-phenanthroline is protonated. The result is a decrease in absorbance. When the pH is greater than 9, competition for Fe 2+ between OH – and o-phenanthroline also leads to a decrease in absorbance. In addition, if the pH is sufficiently basic there is a risk that the iron will precipitate as Fe(OH) 2 . 3. Cadmium is an interferant because it forms a precipitate with o-phenanthroline. What effect would the formation of a Cd-o-phenanthroline precipitate have on the determination of the parts per million of Fe in a sample? Since o-phenanthroline is present in large excess (2000 m g of o-phenanthroline for 100 m g of Fe 2+ ), it is not likely that the interference is due to an insufficient amount of o-phenanthroline being available to react with the Fe 2+ . The presence of a precipitate in the sample cell results in the scattering of radiation and an apparent increase in absorbance. Since the measured absorbance is too high, the reported concentration also is too high. 4. Even high-quality ammonium acetate contains a significant amount of iron. Why is this source of iron not a problem? Since all samples and standards are prepared using the same volume of ammonium acetate buffer, the contribution of this source of iron is accounted for by the calibration curve’s reagent blank. 1400-CH10 9/8/99 4:18 PM Page 399 400 Modern Analytical Chemistry Quantitative Analysis for a Single Analyte The concentration of a single analyte is determined by measuring the absorbance of the sample and applying Beer’s law (equation 10.5) using any of the standardization methods described in Chapter 5. The most common methods are the normal calibration curve and the method of standard additions. Single-point standardizations also can be used, provided that the validity of Beer’s law has been demonstrated. EXAMPLE 10. 5 The determination of Fe in an industrial waste stream was carried out by the o-phenanthroline described in Method 10.1. Using the data shown in the following table, determine the concentration of Fe in the waste stream. ppm Fe Absorbance 0.00 0.000 1.00 0.183 2.00 0.364 3.00 0.546 4.00 0.727 unknown 0.269 SOLUTION Linear regression of absorbance versus the concentration of Fe in the standards gives A = 0.0006 + 0.1817 · (ppm Fe) Substituting the unknown’s absorbance into the calibration expression gives the concentration of Fe in the waste stream as 1.48 ppm. Quantitative Analysis of Mixtures The analysis of two or more components in the same sample is straightforward if there are regions in the sample’s spectrum in which each component is the only absorbing species. In this case each component can be analyzed as if it were the only species in solution. Unfortunately, UV/Vis ab- sorption bands are so broad that it frequently is impossible to find appropriate wavelengths at which each component of a mixture absorbs separately. Earlier we learned that Beer’s law is additive (equation 10.6); thus, for a two-component mix- ture of X and Y, the mixture’s absorbance, A m , is (A m ) l 1 =(e X ) l 1 bC X +(e Y ) l 1 bC Y 10.11 where l 1 is the wavelength at which the absorbance is measured. Since equation 10.11 includes terms for both the concentrations of X and Y, the absorbance at one wave- length does not provide sufficient information to determine either C X or C Y . If we measure the absorbance at a second wavelength, l 2, (A m ) l 2 =(e X ) l 2 bC X +(e Y ) l 2 bC Y 10.12 then C X and C Y can be determined by solving equations 10.11 and 10.12. Of course, it is necessary to determine values for e for each component at both wavelengths. In general, for a mixture of n components, the absorbance must be measured at n dif- ferent wavelengths. 1400-CH10 9/8/99 4:18 PM Page 400 Chapter 10 Spectroscopic Methods of Analysis 401 EXAMPLE 10.6 The concentrations of Fe 3+ and Cu 2+ in a mixture can be determined following their reaction with hexacyanoruthenate (II), Ru(CN) 6 4– , which forms a purple- blue complex with Fe 3+ (l max = 550 nm), and a pale green complex with Cu 2+ (l max = 396 nm). 12 The molar absorptivities (M –1 cm –1 ) for the metal complexes at the two wavelengths are summarized in the following table. e 550 e 396 Fe 3+ 9970 84 Cu 2+ 34 856 When a sample containing Fe 3+ and Cu 2+ is analyzed in a cell with a pathlength of 1.00 cm, the absorbance at 550 nm is 0.183, and the absorbance at 396 nm is 0.109. What are the molar concentrations of Fe 3+ and Cu 2+ in the sample? SOLUTION Substituting known values into equations 10.11 and 10.12 gives A 550 = 0.183 = 9970C Fe +34C Cu A 396 = 0.109 = 84C Fe + 856C Cu To determine the C Fe and C Cu we solve the first equation for C Cu and substitute the result into the second equation. Solving for C Fe gives the concentration of Fe 3+ as 1.80 · 10 –5 M. Substituting this concentration back into the equation for the mixture’s absorbance at a wavelength of 396 nm gives the concentration of Cu 2+ as 1.26 · 10 –4 M. To obtain results with good accuracy and precision the two wavelengths should be selected so that e X > e Y at one wavelength and e Y < e X at the other wavelength. The optimum precision is obtained when the difference in molar absorptivities is as large as possible. One method for locating the optimum wavelengths, therefore, is to plot e X /e Y as a function of wavelength and determine the wavelengths at which e X /e Y reaches maximum and minimum values. 13 Two additional methods for determining the composition of a mixture deserve mention. In multiwavelength linear regression analysis (MLRA) the absorbance of a mixture is compared with that of standard solutions at several wavelengths. 14 If A SX and A SY are the absorbances of standard solutions of components X and Y at any wavelength, then A SX = e X bC SX 10.13 A SY = e Y bC SY 10.14 0 109 84 856 0 183 9970 34 4 607 2 51 10 5 . . .(. )=+ ?? ? ? ? ? ? =?×C C C Fe Fe Fe C C Cu Fe = ?0 183 9970 34 . 1400-CH10 9/8/99 4:18 PM Page 401 where C SX and C SY are the known concentrations of X and Y in the standard solu- tions. Solving equations 10.13 and 10.14 for e X and e Y , substituting into equation 10.11 (the wavelength designation can be dropped), and rearranging gives To determine C X and C Y , the mixture’s absorbance and the absorbances of the stan- dard solutions are measured at several wavelengths. Plotting A m /A SX versus A SY /A SX gives a straight line with a slope of C Y /C SY and a y-intercept of C X /C SX . The generalized standard addition method (GSAM) extends the analysis of mixtures to situations in which matrix effects prevent the determination of e X and e Y using external standards. 15 When adding a known concentration of analyte to a solution containing an unknown concentration of analyte, the concentrations usu- ally are not additive (see question 9 in Chapter 5). Conservation of mass, however, is always obeyed. Equation 10.11 can be written in terms of moles, n, by using the relationship 10.15 where V is the total solution volume. Substituting equation 10.15 into 10.11 and 10.12 gives Q l 1 = V(A m ) l 1 =(e X ) l 1 bn X +(e Y ) l 1 bn Y 10.16 Q l 2 = V(A m ) l 2 =(e X ) l 2 bn X +(e Y ) l 2 bn Y 10.17 where Q is the volume-corrected absorbance. If a standard is added to the sample, the moles of X and Y increase by the amount D n X and D n Y , and the new volume- corrected absorbances are (Q l 1 )￠ =(e X ) l 1 b(n X + D n X )+(e Y ) l 1 b(n Y + D n Y ) 10.18 (Q l 2 )￠ =(e X ) l 2 b(n X + D n X )+(e Y ) l 2 b(n Y + D n Y ) 10.19 Subtracting equation 10.16 from 10.18, and 10.17 from 10.19 gives D Q l 1 =(e X ) l 1 b D n X +(e Y ) l 1 b D n Y D Q l 2 =(e X ) l 2 b D n X +(e Y ) l 2 b D n Y Values for (e X ) l 1 , (e Y ) l 1 , (e X ) l 2 , and (e Y ) l 2 are obtained by plotting D Q l 1 versus D n X , D Q l 1 versus D n Y , D Q l 2 versus D n X , and D Q l 2 versus D n Y and determining the slopes. Equations 10.16 and 10.17 can then be solved to determine n X and n Y . 10D. 3 Qualitative Applications As discussed earlier in Section 10C.1, ultraviolet, visible and infrared absorption bands result from the absorption of electromagnetic radiation by specific valence electrons or bonds. The energy at which the absorption occurs, as well as the inten- sity of the absorption, is determined by the chemical environment of the absorbing moiety. For example, benzene has several ultraviolet absorption bands due to pfip * transitions. The position and intensity of two of these bands, 203.5 nm (e = 7400) and 254 nm (e = 204), are very sensitive to substitution. For benzoic acid, in which a carboxylic acid group replaces one of the aromatic hydrogens, the C n V = A A C C C C A A m SX X SX Y SY SY SX =+ ? ? ? ? ? ? 402 Modern Analytical Chemistry 1400-CH10 9/8/99 4:18 PM Page 402 two bands shift to 230 nm (e = 11,600) and 273 nm (e = 970). Several rules have been developed to aid in correlating UV/Vis absorption bands to chemical struc- ture. Similar correlations have been developed for determining structures using in- frared absorption bands. For example the carbonyl, C=O, stretch is very sensitive to adjacent functional groups, occurring at 1650 cm –1 for acids, 1700 cm –1 for ke- tones, and 1800 cm –1 for acid chlorides. The qualitative manual interpretation of UV/Vis and IR spectra receives adequate coverage elsewhere in the chemistry cur- riculum, notably in organic chemistry and is therefore not considered further in this text. With the availability of computerized data acquisition and storage it is possible to build database libraries of standard reference spectra. When a spectrum of an un- known compound is obtained, its identity can often be determined by searching through a library of reference spectra. This process is known as spectral searching. Comparisons are made by an algorithm that calculates the cumulative difference between the absorbances of the sample and reference spectra. For example, one simple algorithm uses the following equation where D is the cumulative difference, A s is the absorbance of the sample at wave- length or wavenumber i, A r is the absorbance of the reference compound at the same wavelength or wavenumber, and n is the number of points for which the spec- tra were digitized. The cumulative difference is calculated for each reference spec- trum. The reference compound with the smallest value of D provides the closest match to the unknown compound. The accuracy of spectral searching is limited by the number and type of compounds included in the library and by the effect of the sample’s matrix on the spectrum. Another advantage of computerized data acquisition is the ability to subtract one spectrum from another. When coupled with spectral searching it may be pos- sible, by repeatedly searching and subtracting reference spectra, to determine the identity of several components in a sample without the need of a prior separation step. An example is shown in Figure 10.33 in which the composition of a two- component mixture consisting of mannitol and cocaine hydrochloride was identi- fied by successive searching and subtraction. Figure 10.33a shows the spectrum of the mixture. A search of the spectral library selects mannitol (Figure 10.33b) as a likely component of the mixture. Subtracting mannitol’s spectrum from the mix- ture’s spectrum leaves a result (Figure 10.33c) that closely matches the spectrum of cocaine hydrochloride (Figure 10.33d) in the spectral library. Subtracting leaves only a small residual signal (Figure 10.33e). 10D. 4 Characterization Applications Molecular absorption, particularly in the UV/Vis range, has been used for a variety of different characterization studies, including determining the stoichiometry of metal–ligand complexes and determining equilibrium constants. Both of these ex- amples are examined in this section. Stoichiometry of a Metal—Ligand Complex The stoichiometry for a metal–ligand complexation reaction of the following general form M+yL t ML y DAA ii i n =? = ∑ () () sr 1 Chapter 10 Spectroscopic Methods of Analysis 403 spectral searching The matching of a spectrum for an unknown compound to a reference spectrum stored in a computer database. 1400-CH10 9/8/99 4:18 PM Page 403 Figure 10.33 Spectral searching and stripping in the analysis of a mixture of mannitol and cocaine hydrochloride. (a) IR spectrum for the mixture; (b) Library IR spectrum of mannitol; (c) Result of subtracting mannitol’s IR spectrum from that of the mixture; (d) Library IR spectrum for cocaine hydrochloride; (e) Result of subtracting the spectrum of cocaine hydrochloride from the mixture’s IR spectrum. can be determined by one of three methods: the method of continuous variations, the mole-ratio method, and the slope-ratio method. Of the three methods, the method of continuous variations, also called Job’s method, is the most popular. In this method a series of solutions is prepared such that the total moles of metal and ligand, n tot , in each solution is the same. Thus, if (n M ) i and (n L ) i are, respectively, the moles of metal and ligand in the i-th solution, then n tot =(n M ) i +(n L ) i The relative amount of ligand and metal in each solution is expressed as the mole fraction of ligand, (X L ) i , and the mole fraction of metal, (X M ) i , The concentration of the metal–ligand complex is determined by the limiting reagent, with the greatest concentration occurring when the metal and ligand are mixed stoichiometrically. If the reaction is monitored at a wavelength where only the metal–ligand complex absorbs, a plot of absorbance versus the mole fraction of ligand will show two linear branches: one when the ligand is the limiting reagent and a second when the metal is the limiting reagent. The intersection of these two branches occurs when a stoichiometric mixing of metal and ligand is reached. The () () () XX n n ii i ML M tot =? =1 () () X n n i i L L tot = 404 Modern Analytical Chemistry 0 50 E 100 Wavenumbers (cm -1 ) 6008001000120014001600200030003500 2500 1800 0 50 100 0 50 100 0 50 100 P e r cent transmittance P e r cent transmittance P e r cent transmittance P e r cent transmittance P e r cent transmittance 0 50 100 D C B A method of continuous variations A procedure for determining the stoichiometry between two reactants by preparing solutions containing different mole fractions of one reactant; also known as Job’s method. 1400-CH10 9/8/99 4:18 PM Page 404 Colorplate 10 shows the solutions used to generate this data. Chapter 10 Spectroscopic Methods of Analysis 405 mole fraction of ligand at this intersection is used to determine the value of y for the metal–ligand complex, ML y. If there is no wavelength where only the metal–ligand complex absorbs, then the measured absorbances must be corrected for the absorbance that would be exhib- ited if the metal and ligand did not react to form ML y . A corr = A meas – e M bC M – e L bC L In essence, the corrected absorbance gives the change in absorbance due to the for- mation of the metal–ligand complex. An example of the application of the method of continuous variations is shown in Example 10.7. EXAMPLE 10. 7 To determine the formula for the complex between Fe 2+ and o-phenanthroline, a series of solutions was prepared in which the total concentration of metal and ligand was held constant at 3.15 · 10 –4 M. The absorbance of each solution was measured at a wavelength of 510 nm. Using the following data, determine the formula for the complex. X L Absorbance X L Absorbance 0.0 0.000 0.6 0.693 0.1 0.116 0.7 0.809 0.2 0.231 0.8 0.693 0.3 0.347 0.9 0.347 0.4 0.462 1.0 0.000 0.5 0.578 SOLUTION To determine the formula of the complex, we plot absorbance versus the mole fraction of ligand, obtaining the result shown in the graph. The maximum absorbance is determined by extrapolating the two linear portions of the plot. The intersection of the extrapolated lines corresponds to a mole fraction of ligand of 0.75. Solving for the value of y gives and, the formula for the metal–ligand complex is Fe(o-phenanthroline) 3 2+ . y X X = ? = ? = L L 1 075 1075 3 . . y n n X X X X == = ? L M L M L L 1 Absorbance 0 0.0 1.0 X L X L = 0.75 10.2 0.4 0.6 0.8 0.8 0.6 0.4 0.2 1400-CH10 9/8/99 4:18 PM Page 405 Several precautions are necessary when using the method of continuous varia- tions. First, the method of continuous variations requires that a single metal–ligand complex be formed. To determine if this condition is true, plots of absorbance ver- sus X L should be constructed for several different wavelengths and for several differ- ent values of n tot . If the maximum absorbance does not occur at the same value of X L for each set of conditions, then more than one metal–ligand complex must be present. A second precaution is that the metal–ligand complex must obey Beer’s law for the range of concentrations used in constructing the plot of absorbance versus X L . Third, if the metal–ligand complex’s formation constant is relatively small, the plot of absorbance versus X L may show significant curvature. In this case it is often difficult to determine the stoichiometry by extrapolation. Finally, since the stability of the metal–ligand complex may be influenced by solution conditions, the compo- sition of the solutions must be carefully controlled. When the ligand is a weak base, for example, the solutions must be buffered to the same pH. In the mole-ratio method the moles of one reactant, usually the metal, are held constant, while the moles of the other reactant are varied. The absorbance is moni- tored at a wavelength at which the metal–ligand complex absorbs. A plot of ab- sorbance as a function of the ligand-to-metal mole ratio (n L /n M ) has two linear branches that intersect at a mole ratio corresponding to the formula of the complex. Figure 10.34a shows a mole-ratio plot for the formation of a 1:1 complex in which the absorbance is monitored at a wavelength at which only the complex absorbs. Figure 10.34b shows a mole-ratio plot for a 1:2 complex in which the metal, the ligand, and the complex absorb at the selected wavelength. Unlike the method of continuous variations, the mole-ratio method can be used for complexation reac- tions that occur in a stepwise fashion, provided that the molar absorptivities of the metal–ligand complexes differ and the formation constants are sufficiently differ- ent. A typical mole-ratio plot for the stepwise formation of ML and ML 2 is shown in Figure 10.34c. Both the method of continuous variations and the mole-ratio method rely on an extrapolation of absorbance data collected under conditions in which a linear re- lationship exists between absorbance and the relative amounts of metal and ligand. When a metal–ligand complex is very weak, a plot of absorbance versus X L or n L /n M may be curved, making it impossible to determine the stoichiometry by extrapola- tion. In this case the slope ratio may be used. 406 Modern Analytical Chemistry mole-ratio method A procedure for determining the stoichiometry between two reactants by preparing solutions containing different mole ratios of two reactants. Absorbance 0 Moles ligand Moles metal 41 2 ML 3 Absorbance 0 Moles ligand Moles metal 41 2 ML 2 3 ML Absorbance 0 Moles ligand Moles metal 41 2 ML 2 3 (a) (b) (c) Figure 10.34 Mole-ratio plots used to determine the stoichiometry of a metal–ligand complexation reaction. 1400-CH10 9/8/99 4:18 PM Page 406 In the slope-ratio method two sets of solutions are prepared. The first set con- sists of a constant amount of metal and a variable amount of ligand, chosen such that the total concentration of metal, C M , is much greater than the total concentra- tion of ligand, C L . Under these conditions we may assume that essentially all the ligand is complexed. The concentration of a metal–ligand complex of the general form M x L y is If absorbance is monitored at a wavelength where only M x L y absorbs, then and a plot of absorbance versus C L will be linear with a slope, s L , of A second set of solutions is prepared with a fixed concentration of ligand that is much greater than the variable concentration of metal; thus The mole ratio of ligand-to-metal is determined from the ratio of the two slopes. An important assumption in the slope-ratio method is that the complexation reac- tion continues to completion in the presence of a sufficiently large excess of metal or ligand. The slope-ratio method also is limited to systems that obey Beer’s law and in which only a single complex is formed. Determination of Equilibrium Constants Another important application of molec- ular absorption is the determination of equilibrium constants. Let’s consider, as a simple example, an acid–base reaction of the general form HIn+H 2 O t H 3 O + +In – where HIn and In – are the conjugate weak acid and weak base forms of a visual acid–base indicator. The equilibrium constant for this reaction is To determine the equilibrium constant’s value, we prepare a solution in which the reaction exists in a state of equilibrium and determine the equilibrium concentration of H 3 O + , HIn, and In – . The concentration of H 3 O + is easily determined by measuring K a 3 + HO In HIn] = ? [][] [ s s bx by y x M L == ε ε / / s b x M = ε Ab bC x xy ==ε ε []ML M []ML M xy C x = s b y L = ε Ab bC y xy ==ε ε []ML L []ML L xy C y = Chapter 10 Spectroscopic Methods of Analysis 407 slope-ratio method A procedure for determining the stoichiometry between two reactants by measuring the relative change in absorbance under conditions when each reactant is the limiting reagent. 1400-CH10 9/8/99 4:18 PM Page 407 408 Modern Analytical Chemistry the solution’s pH, whereas the concentration of HIn and In – may be determined by measuring the solution’s absorbance. If both HIn and In – absorb at the selected wavelength, then, from equation 10.6, we know that A = e HIn b[HIn] + e In b[In – ] 10.20 where e HIn and e In are the molar absorptivities for HIn and In – . The total concentra- tion of indicator, C, is given by a mass balance equation C = [HIn] + [In – ] 10.21 Solving equation 10.21 for [HIn] and substituting into equation 10.20 gives A = e HIn b(C – [In – ]) + e In b[In – ] which simplifies to A = e HIn bC – e HIn b[In – ]+e In b[In – ] A = A HIn + b[In – ](e In – e HIn ) 10.22 where A HIn , which is equal to e HIn bC, is the absorbance when the pH is acidic enough that essentially all the indicator is present as HIn. Solving equation 10.22 for the concentration of In – gives 10.23 Proceeding in the same fashion, we can derive a similar equation for the concentra- tion of HIn; thus 10.24 where A In , which is equal to e In bC, is the absorbance when the pH is basic enough that only In – contributes to the absorbance. Substituting equations 10.23 and 10.24 into the equilibrium constant expression for HIn gives 10.25 Using equation 10.25, the value of K a can be determined in one of two ways. The simplest approach is to prepare three solutions, each of which contains the same amount, C, of indicator. The pH of one solution is made acidic enough that [HIn] >> [In – ]. The absorbance of this solution gives A HIn . The value of A In is determined by adjusting the pH of the second solution such that [In – ] >> [HIn]. Finally, the pH of the third solution is adjusted to an intermediate value, and the pH and ab- sorbance, A, are recorded. The value of K a can then be calculated by making appro- priate substitutions into equation 10.25. EXAMPLE 10.8 The acidity constant for an acid–base indicator was determined by preparing three solutions, each of which has a total indicator concentration of 5.00 · 10 –5 M. The first solution was made strongly acidic with HCl and has an absorbance of 0.250. The second solution was made strongly basic and has an absorbance of 1.40. The pH of the third solution was measured at 2.91, with an absorbance of 0.662. What is the value of K a for the indicator? K AA AA a3 HIn In HO= ? ? ? ? ? ? ? ? + [] [ () HIn] = A In In HIn ? ? A b εε [] () In HIn In HIn ? = ? ? AA b εε 1400-CH10 9/8/99 4:18 PM Page 408 SOLUTION The value of K a is determined by making appropriate substitutions into 10.25; thus A second approach is to prepare a series of solutions, each of which contains the same amount of indicator. Two solutions are used to determine values for A HIn and A In . Rewriting equation 10.25 in logarithmic form and rearranging shows that a plot of log [(A – A HIn )/(A In – A)] versus pH is linear, with a slope of +1 and a y-intercept of –pK a . In developing this treatment for determining equilibrium constants, we have considered a relatively simple system in which the absorbance of HIn and In – were easily measured, and for which it is easy to determine the concentration of H 3 O + . In addition to acid–base reactions, the same approach can be applied to any reaction of the general form X+Y t Z including metal–ligand complexation and redox reactions, provided that the con- centration of the product, Z, and one of the reactants can be determined spec- trophotometrically and the concentration of the other reactant can be determined by another method. With appropriate modifications, more-complicated systems, in which one or more of these parameters cannot be measured, also can be treated. 16 10D. 5 Evaluation Scale of Operation Molecular UV/Vis absorption is routinely used for the analysis of trace analytes in macro and meso samples. Major and minor analytes can be de- termined by diluting samples before analysis, and concentrating a sample may allow for the analysis of ultratrace analytes. The scale of operations for infrared absorp- tion is generally poorer than that for UV/Vis absorption. Accuracy Under normal conditions relative errors of 1–5% are easily obtained with UV/Vis absorption. Accuracy is usually limited by the quality of the blank. Examples of the type of problems that may be encountered include the presence of particulates in a sample that scatter radiation and interferents that react with analytical reagents. In the latter case the interferant may react to form an absorbing species, giving rise to a positive determinate error. Interferents also may prevent the analyte from reacting, leading to a negative determinate error. With care, it may be possible to improve the accuracy of an analysis by as much as an order of magnitude. Precision In absorption spectroscopy, precision is limited by indeterminate er- rors, or instrumental “noise,” introduced when measuring absorbance. Precision is generally worse with very low absorbances due to the uncertainty of distin- guishing a small difference between P 0 and P T , and for very high absorbances when P T approaches 0. We might expect, therefore, that precision will vary with transmittance. log AA AA K ? ? ? ? ? ? ? ? =? HIn In a pH p K a =× ? ? ? ? ? ? ? ? =× ?? (. ) .. .. .123 10 0 662 0 250 1 40 0 662 687 10 34 Chapter 10 Spectroscopic Methods of Analysis 409 1400-CH10 9/8/99 4:18 PM Page 409 We can derive an expression between precision and transmittance by applying the propagation of uncertainty as described in Chapter 4. To do so we write Beer’s law as 10.26 Using Table 4.9, the absolute uncertainty in the concentration, s C , is given as 10.27 where s T is the absolute uncertainty for the transmittance. Dividing equation 10.27 by equation 10.26 gives the relative uncertainty in concentration, s C /C, as Thus, if s T is known, the relative uncertainty in concentration can be determined for any transmittance. Calculating the relative uncertainty in concentration is complicated by the fact that s T may be a function of the transmittance. Three categories of indeterminate instrumental error have been observed. 17 Table 10.8 provides a summary of these categories. A constant s T is observed for the uncertainty associated with the reading %T from a meter’s analog or digital scale. Typical values are – 0.2–0.3% (k 1 of – 0.002–0.003) for an analog scale, and – 0.001% (k 1 of – 0.00001) for a digital scale. A constant s T also is observed for the thermal transducers used in infrared spec- trophotometers. The effect of a constant s T on the relative uncertainty in concen- tration is shown by curve A in Figure 10.35. Note that the relative uncertainty is very large for both high and low absorbances, reaching a minimum when the ab- sorbance is 0.434. This source of indeterminate error is important for infrared spectrophotometers and for inexpensive UV/Vis spectrophotometers. To obtain a relative uncertainty in concentration of – 1–2%, the absorbance must be kept be- tween 0.1 and 1. Values of s T are a complex function of transmittance when indeterminate er- rors are dominated by the noise associated with photon transducers. Curve B in Figure 10.35 shows that the relative uncertainty in concentration is very large for low absorbances, but is less affected by higher absorbances. Although the relative uncertainty reaches a minimum when the absorbance is 0.96, there is little change in the relative uncertainty for absorbances between 0.5 and 2. This source of inde- s C s TT CT = 0 434. log s b s T C T =? × 0 434. ε C b T=? 1 ε log 410 Modern Analytical Chemistry Table 10.8 Effect of Indeterminate Instrumental Errors on Relative Uncertainty in Concentration Equation for Relative Uncertainty Category Sources of Indeterminate Error in Concentration s T = k 1 %T readout resolution, noise in thermal detectors noise in photon detectors s T = k 3 T positioning of sample cell, fluctuations in source intensity s C k T C = 0 434 3 . log s C k TT C =+ 0 434 1 1 2 . log skT T T =+ 2 2 s C k TT C = 0 434 1 . log 1400-CH10 9/8/99 4:18 PM Page 410 Figure 10.35 Relative uncertainties for absorption spectrophotometry as a function of absorbance for the three categories of indeterminate instrumental errors (see Table 10.8 for equations). terminate error generally limits the precision of high-quality UV/Vis spectropho- tometers for mid-to-high absorbances. Finally, values of s T are directly proportional to transmittance for indetermi- nate errors due to fluctuations in source intensity and for uncertainty in positioning the sample cell within the spectrometer. The latter is of particular importance since the optical properties of any sample cell are not uniform. As a result, repositioning the sample cell may lead to a change in the intensity of transmitted radiation. As shown by curve C in Figure 10.35, the effect of this source of indeterminate error is only important at low absorbances. This source of indeterminate errors is usually the limiting factor for high-quality UV/Vis spectrophotometers when the ab- sorbance is relatively small. When the relative uncertainty in concentration is limited by the %T readout res- olution, the precision of the analysis can be improved by redefining the standards used to define 100% T and 0% T. Normally 100% T is established using a blank, and 0% T is established while using a shutter to prevent source radiation from reaching the detector. When the absorbance is too high, precision can be improved by reset- ting 100% T using a standard solution of analyte whose concentration is less than that of the sample (Figure 10.36a). For a sample whose absorbance is too low, preci- sion can be improved by redefining 0% T, using a standard solution of analyte whose concentration is greater than that of the analyte (Figure 10.36b). In this case a cali- bration curve is required because a linear relationship between absorbance and con- centration no longer exists. Precision can be further increased by combining these two methods (Figure 10.36c). Again, a calibration curve is necessary because the rela- tionship between absorbance and concentration is no longer linear. Sensitivity The sensitivity of a molecular absorption analysis is equivalent to the slope of a Beer’s-law calibration curve and is determined by the product of the an- alyte’s absorptivity and the pathlength of the sample cell. Sensitivity is improved by selecting a wavelength when absorbance is at a maximum or by increasing the pathlength. Chapter 10 Spectroscopic Methods of Analysis 411 Relative uncer tainty in concentration [( s C / C ) · 100] 0 0.0 5.0 Absorbance 30.5 1 1.5 Curve A s T = k 1 Curve B s T = k 2 T 2 + T Curve C s T = k 3 T 2 2.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 %T Sa St 100 %T 0 %T SaSt 100 %T 0 %T SaStSt 100 %T Figure 10.36 Methods for improving the precision of absorption measurements: (a) high-absorbance method; (b) low-absorbance method; and (c) maximum precision method. Abbreviations: Sa = sample; St = standard. (a) (b) (c) 1400-CH10 9/8/99 4:18 PM Page 411 Figure 10.37 Photo of a typical atomic absorption spectrophotometer. Courtesy of Varian, Inc. 412 Modern Analytical Chemistry atomization The process of converting an analyte into a free atom. Selectivity Selectivity is rarely a problem in molecular absorption spectrophotom- etry. In many cases it is possible to find a wavelength at which only the analyte ab- sorbs or to use chemical reactions in a manner such that the analyte is the only species that absorbs at the chosen wavelength. When two or more species con- tribute to the measured absorbance, a multicomponent analysis is still possible, as shown in Example 10.6. Time, Cost, and Equipment The analysis of a sample by molecular absorption spec- troscopy is relatively rapid, although additional time may be required when it is nec- essary to use a chemical reaction to transform a nonabsorbing analyte into an ab- sorbing form. The cost of UV/Vis instrumentation ranges from several hundred dollars for a simple, manually operated, single-beam instrument equipped with an inexpensive grating, to as much as $50,000 for a computer-controlled, high-resolu- tion, double-beam instrument equipped with variable slits and operating over an ex- tended range of wavelengths. Fourier transform infrared spectrometers can be ob- tained for as little as $15,000–$20,000, although more expensive models are available. 10E Atomic Absorption Spectroscopy Atomic absorption, along with atomic emission, was first used by Guystav Kirch- hoff and Robert Bunsen in 1859 and 1860, as a means for the qualitative identifica- tion of atoms. Although atomic emission continued to develop as an analytical technique, progress in atomic absorption languished for almost a century. Modern atomic absorption spectroscopy was introduced in 1955 as a result of the indepen- dent work of A. Walsh and C. T. J. Alkemade. 18 Commercial instruments were in place by the early 1960s, and the importance of atomic absorption as an analytical technique was soon evident. 10E.1 Instrumentation Atomic absorption spectrophotometers (Figure 10.37) are designed using either the single-beam or double-beam optics described earlier for molecular absorption spec- trophotometers (see Figures 10.25 and 10.26). There are, however, several impor- tant differences that are considered in this section. Atomization The most important difference between a spectrophotometer for atomic absorption and one for molecular absorption is the need to convert the ana- lyte into a free atom. The process of converting an analyte in solid, liquid, or solu- tion form to a free gaseous atom is called atomization. In most cases the sample containing the analyte undergoes some form of sample preparation that leaves the analyte in an organic or aqueous solution. For this reason, only the introduction of solution samples is considered in this text. Two general methods of atomization are used: flame atomization and electrothermal atomization. A few elements are atom- ized using other methods. Flame Atomizers In flame atomization the sample is first converted into a fine mist consisting of small droplets of solution. This is accomplished using a nebulizer assembly similar to that shown in the inset to Figure 10.38. The sample is aspirated into a spray chamber by passing a high-pressure stream consisting of one or more combustion gases, past the end of a capillary tube immersed in the sample. The im- pact of the sample with the glass impact bead produces an aerosol mist. The aerosol 1400-CH10 9/8/99 4:18 PM Page 412 mist mixes with the combustion gases in the spray chamber before passing to the burner where the flame’s thermal energy desolvates the aerosol mist to a dry aerosol of small, solid particles. Subsequently, thermal energy volatilizes the particles, pro- ducing a vapor consisting of molecular species, ionic species, and free atoms. Thermal energy in flame atomization is provided by the combustion of a fuel–oxidant mixture. Common fuels and oxidants and their normal temperature ranges are listed in Table 10.9. Of these, the air–acetylene and nitrous oxide-acetylene flames are used most frequently. Normally, the fuel and oxidant are mixed in an ap- proximately stoichiometric ratio; however, a fuel-rich mixture may be desirable for atoms that are easily oxidized. The most common design for the burner is the slot burner shown in Figure 10.38. This burner provides a long path length for monitoring absorbance and a stable flame. The burner is mounted on an adjustable stage that allows the entire burner as- sembly to move horizontally and vertically. Horizontal adjustment is necessary to ensure that the flame is aligned with the instrument’s optical path. Vertical adjust- ments are needed to adjust the height within the flame from which absorbance is Chapter 10 Spectroscopic Methods of Analysis 413 Nebulizer assembly Sample Sample Impact bead Combustion gas To waste Nebulizer assembly Oxidant Fuel Spray chamber Optical path Optical path Burner head Figure 10.38 Flame atomization assembly equipped with spray chamber and slot burner. The inset shows the nebulizer assembly. 1400-CH10 9/8/99 4:18 PM Page 413 monitored. This is important because two competing processes affect the concen- tration of free atoms in the flame. An increased residence time in the flame results in a greater atomization efficiency; thus, the production of free atoms increases with height. On the other hand, longer residence times may lead to the formation of metal oxides that absorb at a wavelength different from that of the atom. For easily oxidized metals, such as Cr, the concentration of free atoms is greatest just above the burner head. For metals, such as Ag, which are difficult to oxidize, the concen- tration of free atoms increases steadily with height (Figure 10.39). Other atoms show concentration profiles that maximize at a characteristic height. The most common means for introducing samples into a flame atomizer is continuous aspiration, in which the sample is continuously passed through the burner while monitoring the absorbance. Continuous aspiration is sample- intensive, typically requiring 2–5 mL of sample. Flame microsampling provides a means for introducing a discrete sample of fixed volume and is useful when the volume of sample is limited or when the sample’s matrix is incompatible with the flame atomizer. For example, the continuous aspiration of a sample contain- ing a high concentration of dissolved solids, such as sea water, may result in the build-up of solid deposits on the burner head. These deposits partially obstruct the flame, lowering the absorbance. Flame microsampling is accomplished using a micropipet to place 50–250 m L of sample in a Teflon funnel connected to the nebulizer, or by dipping the nebulizer tubing into the sample for a short time. Dip sampling is usually accomplished with an automatic sampler. The signal for flame microsampling is a transitory peak whose height or area is proportional to the amount of analyte that is injected. The principal advantage of flame atomization is the reproducibility with which the sample is introduced into the spectrophotometer. A significant disadvantage to flame atomizers is that the efficiency of atomization may be quite poor. This may occur for two reasons. First, the majority of the aerosol mist produced during nebu- lization consists of droplets that are too large to be carried to the flame by the com- bustion gases. Consequently, as much as 95% of the sample never reaches the flame. A second reason for poor atomization efficiency is that the large volume of combus- tion gases significantly dilutes the sample. Together, these contributions to the effi- ciency of atomization reduce sensitivity since the analyte’s concentration in the flame may be only 2.5 · 10 –6 of that in solution. 19 Electrothermal Atomizers A significant improvement in sensitivity is achieved by using resistive heating in place of a flame. A typical electrothermal atomizer, also known as a graphite furnace, consists of a cylindrical graphite tube approximately 414 Modern Analytical Chemistry graphite furnace An electrothermal atomizer that relies on resistive heating to atomize samples. Figure 10.39 Absorbance profile for Ag and Cr in flame atomic absorption spectroscopy. Absorbance Height (cm) Ag Cr Table 10. 9 Fuels and Oxidants Used for Flame Combustion Temperature Range Fuel Oxidant (°C) natural gas air 1700–1900 hydrogen air 2000–2100 acetylene air 2100–2400 acetylene nitrous oxide 2600–2800 acetylene oxygen 3050–3150 1400-CH10 9/8/99 4:18 PM Page 414 Figure 10.40 Diagram of an electrothermal analyzer. 1–3 cm in length, and 3–8 mm in diameter (Figure 10.40). The graphite tube is housed in an assembly that seals the ends of the tube with optically transparent win- dows. The assembly also allows for the passage of a continuous stream of inert gas, protecting the graphite tube from oxidation, and removing the gaseous products produced during atomization. A power supply is used to pass a current through the graphite tube, resulting in resistive heating. Samples between 5 and 50 m L are injected into the graphite tube through a small-diameter hole located at the top of the tube. Atomization is achieved in three stages. In the first stage the sample is dried using a current that raises the tempera- ture of the graphite tube to about 110 °C. Desolvation leaves the sample as a solid residue. In the second stage, which is called ashing, the temperature is increased to 350–1200 °C. At these temperatures, any organic material in the sample is con- verted to CO 2 and H 2 O, and volatile inorganic materials are vaporized. These gases are removed by the inert gas flow. In the final stage the sample is atomized by rapidly increasing the temperature to 2000–3000 °C. The result is a transient ab- sorbance peak whose height or area is proportional to the absolute amount of ana- lyte injected into the graphite tube. The three stages are complete in approximately 45–90 s, with most of this time used for drying and ashing the sample. Electrothermal atomization provides a significant improvement in sensitivity by trapping the gaseous analyte in the small volume of the graphite tube. The ana- lyte’s concentration in the resulting vapor phase may be as much as 1000 times greater than that produced by flame atomization. 20 The improvement in sensitivity, and the resulting improvement in detection limits, is offset by a significant decrease in precision. Atomization efficiency is strongly influenced by the sample’s contact with the graphite tube, which is difficult to control reproducibly. Miscellaneous Atomization Methods A few elements may be atomized by a chemi- cal reaction that produces a volatile product. Elements such as As, Se, Sb, Bi, Ge, Sn, Te, and Pb form volatile hydrides when reacted with NaBH 4 in acid. An inert gas carries the volatile hydrides to either a flame or to a heated quartz observation tube situated in the optical path. Mercury is determined by the cold-vapor method in which it is reduced to elemental mercury with SnCl 2 . The volatile Hg is carried by an inert gas to an unheated observation tube situated in the instrument’s optical path. 10E.2 Quantitative Applications Atomic absorption using either flame or electrothermal atomization is widely used for the analysis of trace metals in a variety of sample matrices. Using the atomic ab- sorption analysis for zinc as an example, procedures have been developed for its de- termination in samples as diverse as water and wastewater, air, blood, urine, muscle Chapter 10 Spectroscopic Methods of Analysis 415 Optical window Optical window Optical path Optical path Sample Inert gas Inert gas 1400-CH10 9/8/99 4:18 PM Page 415 Figure 10.41 Schematic diagram of a hollow cathode lamp showing mechanism by which atomic emission is obtained. tissue, hair, milk, breakfast cereals, shampoos, alloys, industrial plating baths, gaso- line, oil, sediments, and rocks. Developing a quantitative atomic absorption method requires several consider- ations, including choosing a method of atomization, selecting the wavelength and slit width, preparing the sample for analysis, minimizing spectral and chemical in- terferences, and selecting a method of standardization. Each of these topics is con- sidered in this section. Flame Versus Electrothermal Atomization The choice of atomization method is determined primarily by the analyte’s concentration in the samples being analyzed. Because of its greater sensitivity, detection limits for most elements are significantly lower when using electrothermal atomization (Table 10.10). A better precision when using flame atomization makes it the method of choice when the analyte’s concentration is significantly greater than the detection limit for flame atomization. In addition, flame atomization is subject to fewer interferences, allows for a greater throughput of samples, and requires less expertise from the operator. Electrother- mal atomization is the method of choice when the analyte’s concentration is lower than the detection limit for flame atomization. Electrothermal atomization is also useful when the volume of sample is limited. Selecting the Wavelength and Slit Width The source for atomic absorption is a hollow cathode lamp consisting of a cathode and anode enclosed within a glass tube filled with a low pressure of Ne or Ar (Figure 10.41). When a potential is applied across the electrodes, the filler gas is ionized. The positively charged ions collide with the negatively charged cathode, dislodging, or “sputtering,” atoms from the cathode’s surface. Some of the sputtered atoms are in the excited state and emit ra- diation characteristic of the metal from which the cathode was manufactured. By fashioning the cathode from the metallic analyte, a hollow cathode lamp provides emission lines that correspond to the analyte’s absorption spectrum. The sensitivity of an atomic absorption line is often described by its characteris- tic concentration, which is the concentration of analyte giving an absorbance of 0.00436 (corresponding to a percent transmittance of 99%). For example, Table 10.11 shows a list of wavelengths and characteristic concentrations for copper. Usually the wavelength providing the best sensitivity is used, although a less sensitive wavelength may be more appropriate for a high concentration of analyte. A less sensitive wavelength also may be appropriate when significant interferences occur at the most sensitive wavelength. For example, atomizing a sample produces atoms of not only the analyte, but also of other components present in the sam- ple’s matrix. The presence of other atoms in the flame does not result in an inter- ference unless the absorbance lines for the analyte and the potential interferant are within approximately 0.01 nm. When this is a problem, an interference may be 416 Modern Analytical Chemistry characteristic concentration The concentration of analyte giving an absorbance of 0.00436. Ne + MCathode Anode Shield M* M M + hn Optically transparent window 1400-CH10 9/8/99 4:18 PM Page 416 Table 10.10 Atomic Absorption Detection Limits for Selected Elements Detection Limits (ppb) Element Flame Atomization Electrothermal Atomization Ag 0.9 0.001 Al 20 0.01 As a 20 0.08 Au 6 0.01 B 700 15 Ba 8 0.04 Be 1 0.003 Bi a 20 0.1 Ca 0.5 0.01 Cd 0.5 0.0002 Co 2 0.008 Cr 2 0.004 Cs 8 0.04 Fe 3 0.01 Ga 50 0.01 Ge 50 0.1 Hg b 200 0.2 K 1 0.004 Li 0.3 0.01 Mg 0.1 0.0002 Mn 0.8 0.0006 Mo 10 0.02 Na 0.2 0.004 Ni 2 0.05 Pb 10 0.007 Pd 10 0.05 Pt 40 0.2 Sb a 30 0.08 Se a 100 0.05 Si 20 0.005 Sn 10 0.03 Sr 2 0.01 Ti 10 0.3 V 20 0.1 Zn 0.8 0.0006 Source: Compiled from Parson, M. L.; Major, S.; Forster, A. R. Appl. Spectrosc. 1983, 37, 411–418; Weltz, B. Atomic Absorption Spectrometry, VCH: Deerfield Beach, FL, 1985. a Detection limit by hydride vaporization method: As 0.02 ppb Bi 0.02 ppb Sb 0.1 ppb Se 0.02 ppb b Detection limit by cold-vapor method: Hg 0.001 ppb 417 1400-CH10 9/8/99 4:18 PM Page 417 avoided by selecting another wavelength at which the analyte, but not the interfer- ant, absorbs. The emission spectrum from a hollow cathode lamp includes, besides emission lines for the analyte, additional emission lines for impurities present in the metallic cathode and the filler gas. These additional lines serve as a potential source of stray radiation that may lead to an instrumental deviation from Beer’s law. Normally the monochromator’s slit width is set as wide as possible, improving the throughput of radiation, while being narrow enough to eliminate this source of stray radiation. Preparing the Sample Flame and electrothermal atomization require that the sample be in a liquid or solution form. Samples in solid form are prepared for analysis by dissolving in an appropriate solvent. When the sample is not soluble, it may be digested, either on a hot plate or by microwave, using HNO 3 , H 2 SO 4 , or HClO 4 . Alternatively, the analyte may be extracted via a Soxhlet extraction. Liquid samples may be analyzed directly or may be diluted or extracted if the matrix is in- compatible with the method of atomization. Serum samples, for instance, may be difficult to aspirate when using flame atomization and may produce unacceptably high background absorbances when using electrothermal atomization. A liquid–liquid extraction using an organic solvent containing a chelating agent is frequently used to concentrate analytes. Dilute solutions of Cd 2+ , Co 2+ , Cu 2+ , Fe 3+ , Pb 2+ , Ni 2+ , and Zn 2+ , for example, can be concentrated by extracting with a solu- tion of ammonium pyrrolidine dithiocarbamate in methyl isobutyl ketone. Minimizing Spectral Interference A spectral interference occurs when an analyte’s absorption line overlaps with an interferant’s absorption line or band. As noted pre- viously, the overlap of two atomic absorption lines is seldom a problem. On the other hand, a molecule’s broad absorption band or the scattering of source radia- tion is a potentially serious spectral interference. An important question to consider when using a flame as an atomization source, is how to correct for the absorption of radiation by the flame. The prod- ucts of combustion consist of molecular species that may exhibit broad-band ab- sorption, as well as particulate material that may scatter radiation from the source. If this spectral interference is not corrected, then the intensity of the transmitted radiation decreases. The result is an apparent increase in the sam- 418 Modern Analytical Chemistry Table 10.11 Absorption Lines and Characteristic Concentrations for Copper Wavelength Characteristic Concentration (nm) (ppm) 324.8 0.04 327.4 0.1 217.9 0.6 222.6 2.0 249.2 10 244.2 40 1400-CH10 9/8/99 4:18 PM Page 418 ple’s absorbance. Fortunately, absorption and scattering of radiation by the flame are corrected by analyzing a blank. Spectral interferences also occur when components of the sample’s matrix react in the flame to form molecular species, such as oxides and hydroxides. Ab- sorption and scattering due to components in the sample matrix other than the analyte constitute the sample’s background and may present a significant prob- lem, particularly at wavelengths below 300 nm, at which the scattering of radia- tion becomes more important. If the composition of the sample’s matrix is known, then standards can be prepared with an identical matrix. In this case the background absorption is the same for both the samples and standards. Alterna- tively, if the background is due to a known matrix component, then that compo- nent can be added in excess to all samples and standards so that the contribution of the naturally occurring interferant is insignificant. Finally, many interferences due to the sample’s matrix can be eliminated by adjusting the flame’s composi- tion. For example, by switching to a higher temperature flame it may be possible to prevent the formation of interfering oxides and hydroxides. When the identity of the matrix interference is unknown, or when it is impossi- ble to adjust the flame to eliminate the interference, then other means must be used to compensate for the background interference. Several methods have been devel- oped to compensate for matrix interferences, and most atomic absorption spec- trophotometers include one or more of these methods. One of the most common methods for background correction is the use of a continuum source, such as a D 2 lamp. Since the D 2 lamp is a continuum source, the absorbance of its radiation by the analyte’s narrow absorption line is negligible. Any absorbance of radiation from the D 2 lamp, therefore, is due to the background. Absorbance of radiation from the hollow cathode lamp, how- ever, is due to both the analyte and the background. Subtracting the absorbance for the D 2 lamp from that for the hollow cathode lamp gives an absorbance that has been corrected for the background interference. Although this method of background correction may be quite effective, it assumes that the background absorbance is constant over the range of wavelengths passed by the monochro- mator. When this is untrue, subtracting the two absorbances may under- or over-correct for the background. Other methods of background correction have been developed, including Zee- man effect background correction and Smith–Hieftje background correction, both of which are included in some commercially available atomic absorption spec- trophotometers. Further details about these methods can be found in several of the suggested readings listed at the end of the chapter. Minimizing Chemical Interferences The quantitative analysis of some elements is complicated by chemical interferences occurring during atomization. The two most common chemical interferences are the formation of nonvolatile com- pounds containing the analyte and ionization of the analyte. One example of a chemical interference due to the formation of nonvolatile compounds is ob- served when PO 4 3– or Al 3+ is added to solutions of Ca 2+ . In one study, for exam- ple, adding 100 ppm Al 3+ to a solution of 5 ppm Ca 2+ decreased the calcium ion’s absorbance from 0.50 to 0.14, whereas adding 500 ppm PO 4 3– to a similar solution of Ca 2+ decreased the absorbance from 0.50 to 0.38. 21 These interfer- ences were attributed to the formation of refractory particles of Ca 3 (PO 4 ) 2 and an Al–Ca–O oxide. Chapter 10 Spectroscopic Methods of Analysis 419 background correction In atomic absorption spectroscopy, the correction of the net absorbance from that due to the sample matrix. 1400-CH10 9/8/99 4:18 PM Page 419 The formation of nonvolatile compounds often can be minimized by increas- ing the temperature of the flame, either by changing the fuel-to-oxidant ratio or by switching to a different combination of fuel and oxidant. Another approach is to add a releasing agent or protecting agent to solutions containing the analyte. A releasing agent is a species whose reaction with the interferent is more favorable than that of the analyte. Adding Sr 2+ or La 3+ to solutions of Ca 2+ , for example, minimizes the effect of PO 4 3– and Al 3+ by reacting in place of the analyte. Thus, adding 2000 ppm SrCl 2 to the Ca 2+ /PO 4 3– and Ca 2+ /Al 3+ mixtures discussed in the preceding paragraph gave absorbances for each of 0.48, whereas a solution of 2000 ppm SrCl 2 and Ca 2+ alone gave an absorbance of 0.49. Protecting agents react with the analyte to form a stable volatile complex. Adding 1% w/w EDTA to the Ca 2+ /PO 4 3– solution discussed in the preceding paragraph gave an absorbance of 0.52, compared with an absorbance of 0.55 for just the Ca 2+ and EDTA. On the other hand, EDTA does not serve as a protecting agent for solutions of Ca 2+ and Al 3+ . Ionization interferences occur when thermal energy from the flame or elec- trothermal atomizer is sufficient to ionize the analyte M t M + + e – 10.28 where M is the analyte in atomic form, and M + is the cation of the analyte formed by ionization. Since the absorption spectra for M and M + are different, the position of the equilibrium in reaction 10.28 affects absorbance at wavelengths where M ab- sorbs. If another species is present that ionizes more easily than M, then the equilib- rium in reaction 10.28 shifts to the left. Variations in the concentration of easily ionized species, therefore, may have a significant effect on a sample’s absorbance, resulting in a determinate error. The effect of ionization can be minimized by adding a high concentration of an ionization suppressor, which is simply another species that ionizes more easily than the analyte. If the concentration of the ioniza- tion suppressor is sufficient, then the increased concentration of electrons in the flame pushes reaction 10.28 to the left, preventing the analyte’s ionization. Potas- sium and cesium are frequently used as ionization suppressors because of their low ionization energy. Standardizing the Method Because Beer’s law also applies to atomic absorp- tion, we might expect atomic absorption calibration curves to be linear. In prac- tice, however, most atomic absorption calibration curves are nonlinear, or linear for only a limited range of concentrations. Nonlinearity in atomic absorption is a consequence of instrumental limitations, including stray radiation from the hol- low cathode lamp and a nonconstant molar absorptivity due to the narrow width of the absorption line. Accurate quantitative work, therefore, often requires a suitable means for computing the calibration curve from a set of standards. Non- linear calibration curves may be fit using quadratic and cubic equations, al- though neither works well over a broad range of concentrations. More accurate results may be obtained using some of the methods mentioned in Section 5C.5 in Chapter 5. When possible, a quantitative analysis is best conducted using external stan- dards. Unfortunately, matrix interferences are a frequent problem, particularly when using electrothermal atomization. For this reason the method of standard additions is often used. One limitation to this method of standardization, however, is the re- quirement that there be a linear relationship between absorbance and concentration. 420 Modern Analytical Chemistry protecting agent A reagent that reacts with the analyte, preventing it from transforming into a nonanalyzable form. ionization suppressor A reagent that is more easily ionizable than the analyte. releasing agent A reagent whose reaction with an interferant is more favorable than the interferant’s reaction with the analyte. 1400-CH10 9/8/99 4:18 PM Page 420 Chapter 10 Spectroscopic Methods of Analysis 421 Representative Methods Method 10.2 Determination of Cu and Zn in Tissue Samples 22 Description of Method. Copper and zinc are isolated by digesting tissue samples after extracting any fatty tissue. The concentration of copper and zinc in the supernatant are determined by atomic absorption using an air–acetylene flame. Procedure. Tissue samples are obtained by a muscle needle biopsy and are dried for 24–30 hours at 105 °C to remove all traces of moisture. The fatty tissue in the dried samples is removed by extracting overnight with anhydrous ether. After removing the ether, the sample is dried to obtain the fat-free dry tissue weight (FFDT). The sample is digested at 68 °C for 20–24 h using 3 mL of 0.75 M HNO 3 . After centrifuging at 2500 rpm for 10 min, the supernatant is transferred to a 5-mL volumetric flask. The digestion is repeated two more times, for 2–4 h each, using 0.9-mL aliquots of 0.75 M HNO 3 . These supernatants are added to the 5-mL volumetric flask, which is diluted to volume with 0.75 M HNO 3 . The concentration of Cu and Zn in the diluted supernatant is determined by atomic absorption spectroscopy using an air–acetylene flame and external standards. Copper is analyzed at a wavelength of 324.8 nm with a slit width of 0.5 nm, and zinc is analyzed at 213.9 nm with a slit width of 1.0 nm. Background correction is used for zinc. Results are reported as micrograms of Cu or Zn per gram of FFDT. Questions 1. What is the proper matrix for the external standards and the blank? The matrix for the standards and the blank should match that of the samples; thus, an appropriate matrix is 0.75 M HNO 3 . Any interferences from other components of the sample matrix are minimized by background correction. 2. Why is background correction necessary for the analysis of Zn, but not for the analysis of Cu? Background correction is used to compensate for background absorption and scattering due to interferents in the sample. Such interferences are most severe for analytes, such as Zn, that absorb at wavelengths of less than 300 nm. 3. The following absorbances were obtained for a set of Cu calibration standards ppm Cu Absorbance 0.000 0.000 0.100 0.006 0.200 0.013 0.300 0.020 0.400 0.026 0.500 0.033 0.600 0.039 0.700 0.046 1.000 0.066 What is the concentration of copper, in micrograms per gram FFDT, for a 11.23- mg FFDT tissue sample that yields an absorbance of 0.023? Linear regression of the calibration standards gives the relationship between absorbance and concentration as A = –0.0002 + 0.0661(ppm Cu) Substituting the sample’s absorbance into the preceding equation gives the concentration of copper in solution as 0.351 ppm. The concentration in the tissue sample, therefore, is (.0 351 156 g/mL)(5 mL) 0.01123 g g Cu/g FFDT μ μ= 1400-CH10 9/8/99 4:18 PM Page 421 10E. 3 Evaluation Scale of Operation Atomic absorption spectroscopy is ideally suited for the analy- sis of trace and ultratrace analytes, particularly when using electrothermal atomiza- tion. By diluting samples, atomic absorption also can be applied to minor and major analytes. Most analyses use macro or meso samples. The small volume re- quirement for electrothermal atomization or flame microsampling, however, allows the use of micro, or even ultramicro samples. Accuracy When spectral and chemical interferences are minimized, accuracies of 0.5–5% are routinely possible. With nonlinear calibration curves, higher accuracy is obtained by using a pair of standards whose absorbances closely bracket the sam- ple’s absorbance and assuming that the change in absorbance is linear over the lim- ited concentration range. Determinate errors for electrothermal atomization are frequently greater than that obtained with flame atomization due to more serious matrix interferences. Precision For absorbances greater than 0.1–0.2, the relative standard deviation for atomic absorption is 0.3–1% for flame atomization, and 1–5% for electrothermal atomization. The principal limitation is the variation in the concentration of free- analyte atoms resulting from a nonuniform rate of aspiration, nebulization, and at- omization in flame atomizers, and the consistency with which the sample is heated during electrothermal atomization. Sensitivity The sensitivity of an atomic absorption analysis with flame atom- ization is influenced strongly by the flame’s composition and the position in the flame from which absorption is monitored. Normally the sensitivity for an analysis is optimized by aspirating a standard and adjusting operating condi- tions, such as the fuel-to-oxidant ratio, the nebulizer flow rate, and the height of the burner, to give the greatest absorbance. With electrothermal atomization, sensitivity is influenced by the drying and ashing stages that precede atomiza- tion. The temperature and time used for each stage must be worked out for each type of sample. Sensitivity is also influenced by the sample’s matrix. We have already noted, for example, that sensitivity can be decreased by chemical interferences. An increase in sensitivity can often be realized by adding a low-molecular-weight alcohol, ester, or ketone to the solution or by using an organic solvent. Selectivity Due to the narrow width of absorption lines, atomic absorption pro- vides excellent selectivity. Atomic absorption can be used for the analysis of over 60 elements at concentrations at or below the level of parts per million. Time, Cost, and Equipment The analysis time when using flame atomization is rapid, with sample throughputs of 250–350 determinations per hour when using a fully automated system. Electrothermal atomization requires substantially more time per analysis, with maximum sample throughputs of 20–30 determinations per hour. The cost of a new instrument ranges from $10,000 to $50,000 for flame atomization and $18,000 to $70,000 for electrothermal atomization. The more ex- pensive instruments in each price range include double-beam optics and auto- matic samplers, are computer controlled, and can be programmed for multiele- mental analysis by allowing the wavelength and hollow cathode lamp to be changed automatically. 422 Modern Analytical Chemistry 1400-CH10 9/8/99 4:18 PM Page 422 Chapter 10 Spectroscopic Methods of Analysis 423 10F Spectroscopy Based on Emission An analyte in an excited state possesses an energy, E 2 , that is greater than that when it is in a lower energy state, E 1 . When the analyte returns, or relaxes to a lower en- ergy state the excess energy, D E, D E = E 2 – E 1 must be released. Figure 10.5 shows a simplified picture of this process. The lifetime of an analyte in the excited state, A * , is short; typically 10 –5 –10 –9 s for electronic excited states and 10 –15 s for vibrational excited states. Relaxation oc- curs through collisions between A * and other species in the sample, by photochemi- cal reactions, and by the emission of photons. In the first process, which is called vi- brational deactivation, or nonradiative relaxation, the excess energy is released as heat; thus A * fi A + heat Relaxation by a photochemical reaction may involve a decomposition reaction in which A * splits apart A * fi X+Y or a reaction between A * and another species A * +Zfi X+Y In either case the excess energy is used up in the chemical reaction or released as heat. In the third mechanism excess energy is released as a photon of electromagnetic radiation. A * fi A + hn The release of a photon following thermal excitation is called emission, and that fol- lowing the absorption of a photon is called photoluminescence. In chemilumines- cence and bioluminescence, excitation results from a chemical or biochemical reac- tion, respectively. Spectroscopic methods based on photoluminescence are the subject of Section 10G, and atomic emission is covered in Section 10H. 10G Molecular Photoluminescence Spectroscopy Photoluminescence is divided into two categories: fluorescence and phosphores- cence. Absorption of an ultraviolet or visible photon promotes a valence electron from its ground state to an excited state with conservation of the electron’s spin. For example, a pair of electrons occupying the same electronic ground state have oppo- site spins (Figure 10.42a) and are said to be in a singlet spin state. Absorbing a pho- ton promotes one of the electrons to a singlet excited state (Figure 10.42b). Emis- sion of a photon from a singlet excited state to a singlet ground state, or between any two energy levels with the same spin, is called fluorescence. The probability of a fluorescent transition is very high, and the average lifetime of the electron in the ex- cited state is only 10 –5 –10 –8 s. Fluorescence, therefore, decays rapidly after the exci- tation source is removed. In some cases an electron in a singlet excited state is trans- formed to a triplet excited state (Figure 10.42c) in which its spin is no longer paired with that of the ground state. Emission between a triplet excited state and a singlet ground state, or between any two energy levels that differ in their respective lifetime The length of time that an analyte stays in an excited state before returning to a lower-energy state. relaxation Any process by which an analyte returns to a lower-energy state from a higher- energy state. singlet excited state An excited state in which all electron spins are paired. fluorescence Emission of a photon when the analyte returns to a lower-energy state with the same spin as the higher-energy state. triplet excited state An excited state in which unpaired electron spins occur. 1400-CH10 9/8/99 4:18 PM Page 423 Figure 10.42 Difference between singlet and triplet states. spin states, is called phosphorescence. Because the average lifetime for phosphores- cence ranges from 10 –4 to 10 4 s, phosphorescence may continue for some time after removing the excitation source. The use of molecular fluorescence for qualitative analysis and semiquantitative analysis can be traced to the early to mid-1800s, with more accurate quantitative methods appearing in the 1920s. Instrumentation for fluorescence spectroscopy using filters and monochromators for wavelength selection appeared in, respec- tively, the 1930s and 1950s. Although the discovery of phosphorescence preceded that of fluorescence by almost 200 years, qualitative and quantitative applications of molecular phosphorescence did not receive much attention until after the develop- ment of fluorescence instrumentation. 10G.1 Molecular Fluorescence and Phosphorescence Spectra To appreciate the origin of molecular fluorescence and phosphorescence, we must consider what happens to a molecule following the absorption of a photon. Let’s as- sume that the molecule initially occupies the lowest vibrational energy level of its electronic ground state. The ground state, which is shown in Figure 10.43, is a sin- glet state labeled S 0 . Absorption of a photon of correct energy excites the molecule to one of several vibrational energy levels in the first excited electronic state, S 1 , or the second electronic excited state, S 2 , both of which are singlet states. Relaxation to the ground state from these excited states occurs by a number of mechanisms that are either radiationless, in that no photons are emitted, or involve the emission of a photon. These relaxation mechanisms are shown in Figure 10.43. The most likely pathway by which a molecule relaxes back to its ground state is that which gives the shortest lifetime for the excited state. Radiationless Deactivation One form of radiationless deactivation is vibra- tional relaxation, in which a molecule in an excited vibrational energy level loses energy as it moves to a lower vibrational energy level in the same electronic state. Vibrational relaxation is very rapid, with the molecule’s average lifetime in an excited vibrational energy level being 10 –12 s or less. As a consequence, molecules that are excited to different vibrational energy levels of the same excited elec- tronic state quickly return to the lowest vibrational energy level of this excited state. 424 Modern Analytical Chemistry Singlet ground state Singlet excited state Triplet excited state (a) (b) (c) phosphorescence Emission of a photon when the analyte returns to a lower-energy state with the opposite spin as the higher-energy state. vibrational relaxation A form of radiationless relaxation in which an analyte moves from a higher vibrational energy level to a lower vibrational energy level in the same electronic level. 1400-CH10 9/8/99 4:19 PM Page 424 Figure 10.43 Energy level diagram for a molecule showing pathways for deactivation of an excited state: vr is vibrational relaxation; ic is internal conversion; ec is external conversion, and isc is intersystem crossing. The lowest vibrational energy level for each electronic state is indicated by the thicker line. Another form of radiationless relaxation is internal conversion, in which a molecule in the ground vibrational level of an excited electronic state passes directly into a high vibrational energy level of a lower energy electronic state of the same spin state. By a combination of internal conversions and vibrational relaxations, a molecule in an excited electronic state may return to the ground electronic state without emitting a photon. A related form of radiationless relaxation is external conversion in which excess energy is transferred to the solvent or another compo- nent in the sample matrix. A final form of radiationless relaxation is an intersystem crossing in which a molecule in the ground vibrational energy level of an excited electronic state passes into a high vibrational energy level of a lower energy electronic energy state with a different spin state. For example, an intersystem crossing is shown in Figure 10.43 between a singlet excited state, S 1 , and a triplet excited state, T 1 . Fluorescence Fluorescence occurs when a molecule in the lowest vibrational en- ergy level of an excited electronic state returns to a lower energy electronic state by emitting a photon. Since molecules return to their ground state by the fastest mech- anism, fluorescence is only observed if it is a more efficient means of relaxation than the combination of internal conversion and vibrational relaxation. A quantitative expression of the efficiency of fluorescence is the fluorescent quantum yield, F f , which is the fraction of excited molecules returning to the ground state by fluores- cence. Quantum yields range from 1, when every molecule in an excited state un- dergoes fluorescence, to 0 when fluorescence does not occur. The intensity of fluorescence, I f , is proportional to the amount of the radiation from the excitation source that is absorbed and the quantum yield for fluorescence I f= kF f (P 0 – P T ) 10.29 Chapter 10 Spectroscopic Methods of Analysis 425 S 2 S 0 S 1 T 1 vr vr vr vr vr vr vr vr vr vr vr vr ic and ec ic ec isc PhosphorescenceFluorescence vr vrvr vrvr vrvr internal conversion A form of radiationless relaxation in which the analyte moves from a higher electronic energy level to a lower electronic energy level. external conversion A form of radiationless relaxation in which energy is transferred to the solvent or sample matrix. intersystem crossing A form of radiationless relaxation in which the analyte moves from a higher electronic energy level to a lower electronic energy level with a different spin state. quantum yield The fraction of absorbed photons that produce a desired event, such as fluorescence or phosphorescence (F) . 1400-CH10 9/8/99 4:19 PM Page 425 where k is a constant accounting for the efficiency of collecting and detecting the fluorescent emission. From Beer’s law we know that 10.30 where C is the concentration of the fluorescing species. Solving equation 10.30 for P T and substituting into equation 10.29 gives, after simplifying I f = kF f P 0 (1–10 –e bC ) 10.31 For low concentrations of the fluorescing species, where e bC is less than 0.01, this equation simplifies to I f = 2.303kF f P 0 e bC 10.32 The intensity of fluorescence therefore, increases with an increase in quantum effi- ciency, incident power of the excitation source, and the molar absorptivity and con- centration of the fluorescing species. Fluorescence is generally observed with molecules where the lowest energy ab- sorption is a pfip * transition, although some n fip * transitions show weak fluo- rescence. Most unsubstituted, nonheterocyclic aromatic compounds show favorable fluorescence quantum yields, although substitution to the aromatic ring can have a significant effect on F f . For example, the presence of an electron-withdrawing group, such as —NO 2 , decreases F f , whereas adding an electron-donating group, such as —OH, increases F f . Fluorescence also increases for aromatic ring systems and for aromatic molecules with rigid planar structures. A molecule’s fluorescence quantum yield is also influenced by external vari- ables such as temperature and solvent. Increasing temperature generally decreases F f because more frequent collisions between the molecule and the solvent increases external conversion. Decreasing the solvent’s viscosity decreases F f for similar rea- sons. For an analyte with acidic or basic functional groups, a change in pH may change the analyte’s structure and, therefore, its fluorescent properties. Changes in both the wavelength and intensity of fluorescence may be affected. As shown in Figure 10.43, fluorescence may return the molecule to any of sev- eral vibrational energy levels in the ground electronic state. Fluorescence, therefore, occurs over a range of wavelengths. Because the change in energy for fluorescent emission is generally less than that for absorption, a molecule’s fluorescence spec- trum is shifted to higher wavelengths than its absorption spectrum. Phosphorescence A molecule in the lowest vibrational energy level of an excited triplet electronic state normally relaxes to the ground state by an intersystem cross- ing to a singlet state or by external conversion. Phosphorescence is observed when relaxation occurs by the emission of a photon. As shown in Figure 10.43, phospho- rescence occurs over a range of wavelengths, all of which are at a lower energy than the molecule’s absorption band. The intensity of phosphorescence, I p , is given by an equation similar to equation 10.32 for fluorescence I p = 2.303kF p P 0 e bC 10.33 where F p is the quantum yield for phosphorescence. Phosphorescence is most favorable for molecules that have n fip * transitions, which have a higher probability for an intersystem crossing than do pfip * transi- tions. For example, phosphorescence is observed with aromatic molecules contain- ing carbonyl groups or heteroatoms. Aromatic compounds containing halide atoms P P bC T 0 10= ?ε 426 Modern Analytical Chemistry 1400-CH10 9/8/99 4:19 PM Page 426 also have a higher efficiency for phosphorescence. In general, an increase in phos- phorescence corresponds to a decrease in fluorescence. Since the average lifetime for phosphorescence is very long, ranging from 10 –4 to 10 4 s, the quantum yield for phosphorescence is usually quite small. An improve- ment in F p is realized by decreasing the efficiency of external conversion. This may be accomplished in several ways, including lowering the temperature, using a more viscous solvent, depositing the sample on a solid substrate, or trapping the molecule in solution. Excitation Versus Emission Spectra Photoluminescence spectra are recorded by measuring the intensity of emitted radiation as a function of either the excitation wavelength or the emission wavelength. An excitation spectrum is obtained by monitoring emission at a fixed wavelength while varying the excitation wavelength. Figure 10.44 shows the excitation spectrum for the hypothetical system described by the energy level diagram in Figure 10.43. When corrected for variations in source intensity and detector response, a sample’s excitation spectrum is nearly identical to its absorbance spectrum. The excitation spectrum provides a convenient means for selecting the best excitation wavelength for a quantitative or qualitative analysis. In an emission spectrum a fixed wavelength is used to excite the mol- ecules, and the intensity of emitted radiation is monitored as a function of wavelength. Although a molecule has only a single excitation spectrum, it has two emission spectra, one for fluorescence and one for phosphores- cence. The corresponding emission spectra for the hypothetical system in Figure 10.43 are shown in Figure 10.44. 10G.2 Instrumentation The basic design of instrumentation for monitoring molecular fluores- cence and molecular phosphorescence is similar to that found for other spectroscopies. The most significant differences are discussed in the fol- lowing sections. Molecular Fluorescence A typical instrumental block diagram for molec- ular fluorescence is shown in Figure 10.45. In contrast to instruments for absorption spectroscopy, the optical paths for the source and detector are usually positioned at an angle of 90°. Chapter 10 Spectroscopic Methods of Analysis 427 S 0 S 1 Fluorescent emission spectrum Phosphorescent emission spectrum Excitation spectrum S 0 S 2 Emission intensity Wavelength Figure 10.44 Example of molecular excitation and emission spectra. excitation spectrum A fluorescence or phosphorescence spectrum in which the emission intensity at a fixed wavelength is measured as a function of the wavelength used for excitation. Source Sample Detector Excitation wavelength selector Emission wavelength selector Signal processor Chopper Figure 10.45 Block diagram for molecular fluorescence spectrometer. 1400-CH10 9/8/99 4:19 PM Page 427 Figure 10.46 Block diagram for molecular phosphorescence spectrometer with inset showing how choppers are used to isolate excitation and emission. Two basic instrumental designs are used for measuring molecular fluorescence. In a fluorometer the excitation and emission wavelengths are selected with absorp- tion or interference filters. The excitation source for a fluorometer is usually a low- pressure mercury vapor lamp that provides intense emission lines distributed throughout the ultraviolet and visible region (254, 312, 365, 405, 436, 546, 577, 691, and 773 nm). When a monochromator is used to select the excitation and emission wavelengths, the instrument is called a spectrofluorometer. With a monochroma- tor, the excitation source is usually a high-pressure Xe arc lamp, which has a contin- uum emission spectrum. Either instrumental design is appropriate for quantitative work, although only a spectrofluorometer can be used to record an excitation or emission spectrum. The sample cells for molecular fluorescence are similar to those for optical molecular absorption. Remote sensing with fiber-optic probes (see Figure 10.30) also can be adapted for use with either a fluorometer or spectrofluorometer. An an- alyte that is fluorescent can be monitored directly. For analytes that are not fluores- cent, a suitable fluorescent probe molecule can be incorporated into the tip of the fiber-optic probe. The analyte’s reaction with the probe molecule leads to an in- crease or decrease in fluorescence. Molecular Phosphorescence Instrumentation for molecular phosphorescence must discriminate between phosphorescence and fluorescence. Since the lifetime for fluorescence is much shorter than that for phosphorescence, discrimination is easily achieved by incorporating a delay between exciting and measuring phospho- rescent emission. A typical instrumental design is shown in Figure 10.46. As shown 428 Modern Analytical Chemistry fluorometer An instrument for measuring fluorescence that uses filters to select the excitation and emission wavelengths. spectrofluorometer An instrument for measuring fluorescence that uses a monochromator to select the excitation and emission wavelengths. Source Detector Excitation wavelength selector Emission wavelength selector Signal processor Sample ChopperChopper Sample Chopper Chopper From source (a) Excitation To detector Sample Chopper Chopper From source (b) Emission To detector 1400-CH10 9/8/99 4:19 PM Page 428 in the inset, the two choppers are rotated out of phase, such that fluorescent emis- sion is blocked from the detector when the excitation source is focused on the sam- ple, and the excitation source is blocked from the sample when measuring the phos- phorescent emission. Because phosphorescence is such a slow process, provision must be made to prevent deactivation of the excited state by external conversion. Traditionally, this has been accomplished by dissolving the sample in a suitable organic solvent, usu- ally a mixture of ethanol, isopentane, and diethyl ether. The resulting solution is frozen at liquid-N 2 temperatures, forming an optically clear solid. The solid matrix minimizes external conversion due to collisions between the analyte and the sol- vent. External conversion also is minimized by immobilizing the sample on a solid substrate, allowing the measurement of phosphorescence at room temperature. One approach is to place a drop of solution containing the analyte on a small filter paper disk mounted on a sample probe. After drying the sample under a heat lamp, the sample probe is placed in the spectrofluorometer for analysis. Other solid surfaces that have been used include silica gel, alumina, sodium acetate, and sucrose. This approach is particularly useful for the analysis of thin-layer chromatography plates. 10G. 3 Quantitative Applications Using Molecular Luminescence Molecular fluorescence and, to a lesser extent, phosphorescence have been used for the direct or indirect quantitative analysis of analytes in a variety of matrices. A di- rect quantitative analysis is feasible when the analyte’s quantum yield for fluores- cence or phosphorescence is favorable. When the analyte is not fluorescent or phos- phorescent or when the quantum yield for fluorescence or phosphorescence is unfavorable, an indirect analysis may be feasible. One approach to an indirect analysis is to react the analyte with a reagent, forming a product with fluorescent properties. Another approach is to measure a decrease in fluorescence when the an- alyte is added to a solution containing a fluorescent molecule. A decrease in fluores- cence is observed when the reaction between the analyte and the fluorescent species enhances radiationless deactivation, or produces a nonfluorescent product. The ap- plication of fluorescence and phosphorescence to inorganic and organic analytes is considered in this section. Inorganic Analytes Except for a few metal ions, most notably UO 2 + , most inor- ganic ions are not sufficiently fluorescent to allow a direct analysis. Many metal ions can be determined indirectly by reacting with an organic ligand to form a fluorescent, or less commonly, a phosphorescent metal–ligand complex. One ex- ample of a chelating ligand is the sodium salt of 2,4,3￠ -trihydroxyazobenzene-5￠ - sulfonic acid, also known as alizarin garnet R, which forms a fluorescent metal–ligand complex with Al 3+ (Figure 10.47). The analysis is carried out using an excitation wavelength of 470 nm, with fluorescence monitored at 500 nm. Other examples of chelating reagents that form fluorescent metal–ligand com- plexes with metal ions are listed in Table 10.12. A few inorganic nonmetals are determined by their ability to decrease, or quench, the fluorescence of another species. One example is the analysis for F – , which is based on its ability to quench the fluorescence of the Al 3+ –alizarin garnet R complex. Organic Analytes As noted earlier, organic compounds containing aromatic rings generally are fluorescent, but aromatic heterocycles are often phosphores- cent. Many important biochemical, pharmaceutical, and environmental com- pounds are aromatic and, therefore, can be analyzed quantitatively by fluorometry Chapter 10 Spectroscopic Methods of Analysis 429 N NHO OH alizarin garnet R (a) (b) HO SO 3 Na N N HO O fluorescent complex O SO 3 Na Al 3+ Figure 10.47 Structure of (a) alizarin garnet R, and (b) its metal–ligand complex with Al 3+ . 1400-CH10 9/8/99 4:19 PM Page 429 or phosphorometry. Several examples are listed in Table 10.13. When an organic ana- lyte is not naturally fluorescent or phosphorescent, it may be possible to incorporate it into a chemical reaction that produces a fluorescent or phosphorescent product. For example, the enzyme creatine phosphokinase can be determined by using it to catalyze the formation of creatine from phosphocreatine. The creatine that is formed reacts with ninhydrin, producing a fluorescent product of unknown structure. 430 Modern Analytical Chemistry Table 10.12 Selected Chelating Agents for the Fluorometric Analysis of Inorganic Metal Ions Chelating Agent Metal Ions 8-hydroxyquinoline Al 3+ , Be 2+ , Zn 2+ , Li + , Mg 2+ (and others) flavonal Zr 2+ , Sn 4+ benzoin B 4 O 7 2– , Zn 2+ 2￠ ,3,4￠ ,5,7-pentahydroxyflavone Be 2+ 2-(o-hydroxyphenyl) benzoxazole Cd 2+ Table 10.1 3 Selected Examples of Organic Compounds of Biochemical, Pharmaceutical, and Environmental Significance That Show Natural Fluorescence or Phosphorescence Class Compounds a aromatic amino acids phenylalanine (F) tyrosine (F) tryptophan (F, P) vitamins vitamin A (F) vitamin B 2 (F) vitamin B 6 (F) vitamin B 12 (F) vitamin E (F) folic acid (F) catecholamines dopamine (F) norepinephrine (F) pharmaceuticals and drugs quinine (F) salicylic acid (F, P) morphine (F) barbiturates (F) LSD (F) codeine (P) caffeine (P) sulfanilamide (P) environmental pollutants polycyclic aromatic hydrocarbons: pyrene (F) benzo[a]pyrene (F) organothiophosphorous pesticides (F) carbamate insecticides (F) DDT (P) a Abbreviations: F = fluorescence, P = phosphorescence, LSD = lysergic acid diethylamide, DDT = dichlorodiphenyltrichloroethane. 1400-CH10 9/8/99 4:19 PM Page 430 Chapter 10 Spectroscopic Methods of Analysis 431 Representative Methods —Continued Standardizing the Method Equations 10.32 and 10.33 show that the intensity of fluorescent or phosphorescent emission is proportional to the concentration of the photoluminescent species, provided that the absorbance of radiation from the exci- tation source (A = e bC) is less than approximately 0.01. Quantitative methods are usually standardized using a set of external standards. Calibration curves are linear over as much as four to six orders of magnitude for fluorescence and two to four orders of magnitude for phosphorescence. Calibration curves become nonlinear for high concentrations of the photoluminescent species at which the intensity of emis- sion is given by equation 10.31. Nonlinearity also may be observed at low concen- trations due to the presence of fluorescent or phosphorescent contaminants. As dis- cussed earlier, the quantum efficiency for emission is sensitive to temperature and sample matrix, both of which must be controlled if external standards are to be used. In addition, emission intensity depends on the molar absorptivity of the pho- toluminescent species, which is sensitive to the sample matrix. Method 10. 3 Determination of Quinine in Urine 23 Description of Method. Quinine is an alkaloid used in treating malaria (it also is found in tonic water). It is a strongly fluorescent compound in dilute solutions of H 2 SO 4 (F f = 0.55). The excitation spectrum of quinine shows two absorption bands at 250 nm and 350 nm, and the emission spectrum shows a single emission band at 450 nm. Quinine is rapidly excreted from the body in urine and is easily determined by fluorescence following its extraction from the urine sample. Procedure. Transfer a 2.00-mL sample of urine to a 15-mL test tube, and adjust the pH to between 9 and 10 using 3.7 M NaOH. Add 4 mL of a 3:1 (v/v) mixture of chloroform and isopropanol, and shake the contents of the test tube for 1 min. Allow the organic and aqueous (urine) layers to separate, and transfer the organic phase to a clean test tube. Add 2.00 mL of 0.05 M H 2 SO 4 to the organic phase, and shake the contents for 1 min. Allow the organic and aqueous layers to separate, and transfer the aqueous phase to the sample cell. Measure the fluorescent emission at 450 nm, using an excitation wavelength of 350 nm. Determine the concentration of quinine in the urine sample using a calibration curve prepared with a set of external standards in 0.05 M H 2 SO 4 and a distilled water blank. The external standards are prepared from a 100.0-ppm solution of quinine in 0.05 M H 2 SO 4 . Questions 1. Chloride ion is known to quench the intensity of quinine’s fluorescent emission. For example, the presence of 100 ppm NaCl (61 ppm Cl – ) gives an emission intensity that is only 83% of that without chloride, whereas the presence of 1000 ppm NaCl (610 ppm Cl – ) gives a fluorescent emission that is only 29% as intense. The concentration of chloride in urine typically ranges from 4600 to 6700 ppm Cl – . How is an interference from chloride avoided in this procedure? Extracting the aqueous urine sample with a mixture of chloroform and isopropanol separates the quinine and chloride, with the chloride remaining in the urine sample. 2. Samples of urine that do not contain quinine still contain a small amount of fluorescent material after the extraction steps. How can the quantitative procedure described earlier be modified to take this into account? 1400-CH10 9/8/99 4:19 PM Page 431 432 Modern Analytical Chemistry Continued from page 431 One approach is to prepare a sample blank using urine known to be free of quinine. The fluorescent signal for the sample blank is subtracted from the urine sample’s measured fluorescence. 3. The fluorescent emission for quinine at 450 nm can be induced using an excitation frequency of either 250 nm or 350 nm. The fluorescent quantum efficiency is known to be the same for either excitation wavelength, and the UV absorption spectrum shows that e 250 is greater than e 350 . Nevertheless, fluorescent emission intensity is greater when using 350 nm as the excitation wavelength. Speculate on why this is the case. From equation 10.32, I f is a function of k, F f, P 0 , e , b, and C. Since F f , b, and C are the same for either excitation wavelength, and e is larger for a wavelength of 250 nm, the greater I f for an excitation wavelength of 350 nm must be due to larger values for P 0 or k at a wavelength of 350 nm. In fact, P 0 at 350 nm for a high-pressure Xe arc lamp is about 170% of that at 250 nm. In addition, the sensitivity of a typical photomultiplier transducer (which contributes to the value of k) at 350 nm is about 140% of that at 250 nm. 10G. 4 Evaluation Scale of Operation Molecular photoluminescence can be used for the routine analysis of trace and ultratrace analytes in macro and meso samples. Detection limits for fluorescence spectroscopy are strongly influenced by the analyte’s quantum yield. For analytes with F f > 0.5, detection limits in the picomolar range are possible when using a high-quality spectrofluorometer. As an example, the detection limit for qui- nine sulfate, for which F f is 0.55, is generally between 1 ppb and 1 ppTr (part per trillion). Detection limits for phosphorescence are somewhat poorer than those for fluorescence, with typical values in the nanomolar range for low-temperature phos- phorometry and in the micromolar range for room-temperature phosphorometry using a solid substrate. Accuracy The accuracy of a fluorescence method is generally 1–5% when spectral and chemical interferences are insignificant. Accuracy is limited by the same types of problems affecting other spectroscopic methods. In addition, accuracy is affected by interferences influencing the fluorescent quantum yield. The accuracy of phos- phorescence is somewhat greater than that for fluorescence. Precision When the analyte’s concentration is well above the detection limit, the relative standard deviation for fluorescence is usually 0.5–2%. The limiting instru- mental factor affecting precision is the stability of the excitation source. The preci- sion for phosphorescence is often limited by reproducibility in preparing samples for analysis, with relative standard deviations of 5–10% being common. Sensitivity From equations 10.32 and 10.33 we can see that the sensitivity of a flu- orescent or phosphorescent method is influenced by a number of parameters. The importance of quantum yield and the effect of temperature and solution composi- tion on F f and F p already have been considered. Besides quantum yield, the sensi- tivity of an analysis can be improved by using an excitation source that has a greater 1400-CH10 9/8/99 4:19 PM Page 432 Figure 10.48 Use of slit orientation to change the volume from which fluorescence is measured: (a) normal; (b) rotated 90°. emission intensity (P 0 ) at the desired wavelength and by selecting an excitation wavelength that corresponds to an absorption maximum (e ). Another approach that can be used to increase sensitivity is to increase the volume in the sample from which emission is monitored. Figure 10.48 shows how a 90° rotation of the slits used to focus the excitation source on the sample and to collect emission from the sample can produce a 5–30-fold increase in the signal. Selectivity The selectivity of molecular fluorescence and phosphorescence is superior to that of absorption spectrophotometry for two reasons: first, not every compound that absorbs radiation is fluorescent or phosphorescent, and, second, selectivity between an analyte and an interferant is possible if there is a difference in either their excitation or emission spectra. In molecular lumines- cence the total emission intensity is a linear sum of that from each fluorescent or phosphorescent species. The analysis of a sample containing n components, therefore, can be accomplished by measuring the total emission intensity at n wavelengths. Time, Cost, and Equipment As with other optical spectroscopic methods, fluores- cent and phosphorescent methods provide a rapid means of analysis and are capa- ble of automation. Fluorometers are relatively inexpensive, ranging from several Chapter 10 Spectroscopic Methods of Analysis 433 From source From source To detector To detector (a) (b) 1400-CH10 9/8/99 4:19 PM Page 433 434 Modern Analytical Chemistry hundred to several thousand dollars, and often are very satisfactory for quantitative work. Spectrofluorometers are more expensive, with some models costing as much as $50,000. 10H Atomic Emission Spectroscopy In principle, emission spectroscopy can be applied to both atoms and molecules. Molecular infrared emission, or blackbody radiation played an important role in the early development of quantum mechanics and has been used for the analysis of hot gases generated by flames and rocket exhausts. Although the availability of FT–IR instrumentation extended the application of IR emission spectroscopy to a wider array of samples, its applications remain limited. For this reason IR emission is not considered further in this text. Molecular UV/Vis emission spectroscopy is of little importance since the thermal energies needed for excitation generally result in the sample’s decomposition. The focus of this section is the emission of ultraviolet and visible radiation fol- lowing thermal or electrical excitation of atoms. Atomic emission spectroscopy has a long history. Qualitative applications based on the color of flames were used in the smelting of ores as early as 1550 and were more fully developed around 1830 with the observation of atomic spectra generated by flame emission and spark emis- sion. 24 Quantitative applications based on the atomic emission from electrical sparks were developed by Norman Lockyer (1836–1920) in the early 1870s, and quantitative applications based on flame emission were pioneered by H. G. Lunde- gardh in 1930. Atomic emission based on emission from a plasma was introduced in 1964. 10H.1 Atomic Emission Spectra Atomic emission occurs when a valence electron in a higher-energy atomic orbital returns to a lower-energy atomic orbital. Figure 10.23 shows a portion of the energy level diagram for sodium used earlier to discuss atomic absorption spectra. An atomic emission spectrum, therefore, consists of a series of discrete lines at wave- lengths corresponding to the difference in energy between two atomic orbitals. The intensity, I, of an emission line is proportional to the number of atoms, N*, populating the excited state I = kN* 10.34 where k is a constant related to the efficiency of the transition. For a system in ther- mal equilibrium, the population of the excited state is related to the total concentra- tion of atoms, N, by the Boltzmann distribution. For many elements at tempera- tures of less than 5000 K the Boltzmann distribution for the ith excited state is approximated as 10.35 where g i and g 0 are statistical factors accounting for the number of equivalent energy levels for the excited state and ground state, E i is the energy of the ex- cited state relative to that of the ground state (E 0 = 0), k is Boltzmann’s constant NN g g e * i E/kT i = ? ? ? ? ? ? ? 0 1400-CH10 9/8/99 4:19 PM Page 434 (1.3807 · 10 –23 J/K), and T is the temperature in kelvin. From equation 10.35 we can see that excited states with lower energies have larger populations and, therefore, the most intense emission lines. Furthermore, emission intensity in- creases with temperature. 10H.2 Equipment Instrumentation for atomic emission spectroscopy is similar in design to that used for atomic absorption. In fact, most flame atomic absorption spectrometers are eas- ily adapted for use as flame atomic emission spectrometers by turning off the hol- low cathode lamp and monitoring the difference between the intensity of radiation emitted when aspirating the sample and that emitted when aspirating a blank. Many atomic emission spectrometers, however, are dedicated instruments designed to take advantage of features unique to atomic emission, including the use of plasmas, arcs, sparks, and lasers, as atomization and excitation sources and have an enhanced capability for multielemental analysis. Atomization and Excitation Atomic emission requires a means for converting an analyte in solid, liquid, or solution form to a free gaseous atom. The same source of thermal energy usually serves as the excitation source. The most common methods are flames and plasmas, both of which are useful for liquid or solution samples. Solid samples may be analyzed by dissolving in solution and using a flame or plasma atomizer. Flame Sources Atomization and excitation in flame atomic emission is accom- plished using the same nebulization and spray chamber assembly used in atomic absorption (see Figure 10.38). The burner head consists of single or multiple slots or a Meker-style burner. Older atomic emission instruments often used a total con- sumption burner in which the sample is drawn through a capillary tube and in- jected directly into the flame. Plasma Sources A plasma consists of a hot, partially ionized gas, containing an abundant concentration of cations and electrons that make the plasma a conductor. The plasmas used in atomic emission are formed by ionizing a flowing stream of argon, producing argon ions and electrons. The high temperatures in a plasma re- sult from resistive heating that develops due to the movement of the electrons and argon ions. Because plasmas operate at much higher temperatures than flames, they provide better atomization and more highly populated excited states. Besides neu- tral atoms, the higher temperatures of a plasma also produce ions of the analyte. A schematic diagram of the inductively coupled plasma (ICP) torch is shown in Figure 10.49. The ICP torch consists of three concentric quartz tubes, sur- rounded at the top by a radio-frequency induction coil. The sample is mixed with a stream of Ar using a spray chamber nebulizer similar to that used for flame emission and is carried to the plasma through the torch’s central tube. Plasma formation is initiated by a spark from a Tesla coil. An alternating radio- frequency current in the induction coils creates a fluctuating magnetic field that induces the argon ions and electrons to move in a circular path. The resulting collisions with the abundant unionized gas give rise to resistive heating, provid- Chapter 10 Spectroscopic Methods of Analysis 435 Sample + Ar Optional Ar flow Tangential Ar flow RF induction coils Plasma Figure 10.49 Diagram of an inductively coupled plasma torch. plasma A hot, ionized gas containing an abundance of ions and electrons. 1400-CH10 9/8/99 4:19 PM Page 435 Figure 10.50 Schematic diagram of a multichannel atomic emission spectrometer, showing the arrangement of multiple exit slits and detectors for the simultaneous analysis of several elements. ing temperatures as high as 10,000 K at the base of the plasma, and between 6000 and 8000 K at a height of 15–20 mm above the coil, where emission is usually measured. At these high temperatures the outer quartz tube must be thermally isolated from the plasma. This is accomplished by the tangential flow of argon shown in the schematic diagram. Multielemental Analysis Atomic emission spectroscopy is ideally suited for multi- elemental analysis because all analytes in a sample are excited simultaneously. A scanning monochromator can be programmed to move rapidly to an analyte’s de- sired wavelength, pausing to record its emission intensity before moving to the next analyte’s wavelength. Proceeding in this fashion, it is possible to analyze three or four analytes per minute. Another approach to multielemental analysis is to use a multichannel instru- ment that allows for the simultaneous monitoring of many analytes. A simple de- sign for a multichannel spectrometer consists of a standard diffraction grating and 48–60 separate exit slits and detectors positioned in a semicircular array around the diffraction grating at positions corresponding to the desired wave- lengths (Figure 10.50). 436 Modern Analytical Chemistry l 1 l 2 l 3 l 4 1400-CH10 9/8/99 4:19 PM Page 436 10H. 3 Quantitative Applications Atomic emission is used for the analysis of the same types of samples that may be analyzed by atomic absorption. The development of a quantitative atomic emission method requires several considerations, including choosing a source for atomiza- tion and excitation, selecting a wavelength and slit width, preparing the sample for analysis, minimizing spectral and chemical interferences, and selecting a method of standardization. Choice of Atomization and Excitation Source Except for the alkali metals, detection limits when using an ICP are significantly better than those obtained with flame emission (Table 10.14). Plasmas also are subject to fewer spectral and chemical interferences. For these rea- sons a plasma emission source is usually the better choice. Selecting the Wavelength and Slit Width The choice of wavelength is dictated by the need for sensitivity and freedom from interference due to unresolved emis- sion lines from other constituents in the sample. Be- cause an atomic emission spectrum usually has an abun- dance of emission lines, particularly when using a high- temperature plasma source, it is inevitable that some overlap will occur between emission lines. For example, an analysis for Ni using the atomic emission line at 349.30 nm is complicated by the atomic emission line for Fe at 349.06 nm. Narrower slit widths provide for better resolution. The easiest approach to selecting a wavelength is to obtain an emission spectrum for the sample and then to look for an emission line for the an- alyte that provides an intense signal and is resolved from other emission lines. Preparing the Sample Flame and plasma sources are best suited for the analysis of samples in solution and liquid form. Although solids can be analyzed by direct insertion into the flame or plasma, they usually are first brought into solution by digestion or extraction. Minimizing Spectral Interferences The most impor- tant spectral interference is a continuous source of background emission from the flame or plasma and emission bands from molecular species. This back- ground emission is particularly severe for flames in which the temperature is insufficient to break down re- fractory compounds, such as oxides and hydroxides. Background corrections for flame emission are made by scanning over the emission line and drawing a baseline (Figure 10.51). Because the temperature of a plasma is Chapter 10 Spectroscopic Methods of Analysis 437 Table 10.1 4 Atomic Emission Detection Limits for Selected Elements Detection Limits (ppb) Element Flame Emission ICP Ag 2 0.2 Al 3 0.2 As 2000 2 Au 500 0.9 B 50 0.1 Ba 1 0.01 Be 100 0.003 Bi 1000 10 Ca 0.1 0.0001 Cd 300 0.07 Co 5 0.1 Cr 1 0.08 Cs 0.02 — Fe 10 0.09 Ga 5 0.6 Ge 400 0.5 Hg 150 1 K 0.01 30 Li 0.001 0.02 Mg 1 0.003 Mn 1 0.01 Mo 10 0.2 Na 0.01 0.1 Ni 10 0.2 Pb 0.2 1 Pd 40 2 Pt 2000 0.9 Sb 100 10 Se — 1 Si — 2 Sn 100 3 Sr 0.1 0.002 Ti 30 0.03 V 7 0.06 Zn 1000 0.1 Source: Compiled from Parsons, M. L.; Major, S.; Forster, A. R., App. Spectrosc. 1983, 37, 411–418. 1400-CH10 9/8/99 4:19 PM Page 437 Figure 10.52 Atomic emission line at (a) low concentration of analyte, and (b) high concentration of analyte showing the effect of self-absorption. 438 Modern Analytical Chemistry much higher, background interferences due to molecular emission are less prob- lematic. Emission from the plasma’s core is strong but is insignificant at a height of 10–30 mm above the core, where measurements normally are made. Minimizing Chemical Interferences Flame emission is subject to the same types of chemical interferences as atomic absorption. These interferences are minimized by adjusting the flame composition and adding protecting agents, releasing agents, and ionization suppressors. An additional chemical inter- ference results from self-absorption. Since the temperature of a flame is greatest at its center, the concentration of analyte atoms in an excited state is greater at the center than at the outer edges. If an excited state atom in the center of the flame emits radiation while returning to its ground state, then ground state atoms in the cooler, outer regions of the flame may absorb the radiation, thereby decreasing emission intensity. At high analyte concentra- tions a self-reversal may be seen in which the center of the emission band de- creases (Figure 10.52). Chemical interferences with plasma sources generally are insignificant. The higher temperature of the plasma limits the formation of nonvolatile species. For example, the presence of PO 4 3– in solutions being analyzed for Ca 2+ , which is a significant interferant for flame emission, has a negligi- ble effect when using a plasma source. In addition, the high concentration of electrons from the ionization of argon minimizes the effects of ionization interferences. Standardizing the Method Equation 10.34 shows that emission intensity is proportional to the population of the excited state, N*, from which the emis- sion line originates. If the emission source is in thermal equilibrium, then the excited state population is proportional to the total population of analyte atoms, N, through the Boltzmann distribution (equation 10.35). Calibration curves for flame emission are generally linear over two to three orders of magnitude, with chemical interferences due to ionization limiting linearity for lower concentrations of analyte, and self-absorption limiting linear- ity for higher concentrations of analyte. Plasma sources, which suffer from fewer chemical interferences, often yield calibration curves that are linear over four to five orders of magnitude and that are not affected significantly by changes in the matrix of the standards. When possible, quantitative analyses are best conducted using external stan- dards. Emission intensity, however, is affected significantly by many parameters, including the temperature of the excitation source and the efficiency of atomiza- tion. An increase in temperature of 10 K, for example, results in a 4% change in the fraction of Na atoms present in the 3p excited state. The method of internal standards can be used when variations in source parameters are difficult to con- trol. In this case an internal standard is selected that has an emission line close to that of the analyte to compensate for changes in the temperature of the excita- tion source. In addition, the internal standard should be subject to the same chemical interferences to compensate for changes in atomization efficiency. To accurately compensate for these errors, the analyte and internal standard emis- sion lines must be monitored simultaneously. The method of standard additions also can be used. Baseline self-absorption In atomic emission, the decrease in emission intensity when light emitted by excited state atoms in the center of a flame or plasma is absorbed by atoms in the outer portion of the flame. Figure 10.51 Method for background correction in flame atomic emission. (a) (b) 1400-CH10 9/8/99 4:19 PM Page 438 Chapter 10 Spectroscopic Methods of Analysis 439 Representative Methods —Continued Method 10. 4 Determination of Sodium in a Salt Substitute 25 Description of Method. Salt substitutes, which are used in place of table salt for individuals on a low-sodium diet, contain KCl. Depending on the brand, fumaric acid, calcium hydrogen phosphate, or potassium tartrate also may be present. Typically, the concentration of sodium in a salt substitute is about 100 ppm. The concentration of sodium is easily determined by flame atomic emission. Because it is difficult to match the matrix of the standards to that of the sample, the analysis is accomplished by the method of standard additions. Procedure. A sample is prepared by placing an approximately 10-g portion of the salt substitute in 10 mL of 3 M HCl and 100 mL of distilled water. After dissolving the sample, it is transferred to a 250-mL volumetric flask and diluted to volume with distilled water. A series of standard additions is prepared by placing 25-mL portions of the diluted sample into separate 50-mL volumetric flasks, spiking each with a known amount of an approximately 10-ppm standard solution of Na + and diluting to volume. After zeroing the instrument with an appropriate blank, the instrument is optimized at a wavelength of 589.0 nm while aspirating the standard solution of Na + . The emission intensity is measured for each of the standard addition samples, and the concentration of sodium in the salt substitute is reported in parts per million. Questions 1. What type of chemical interference in the sample necessitates the use of the method of standard additions? Potassium is more easily ionized than sodium. The high concentration of potassium in the sample suppresses the ionization of sodium, increasing its emission relative to that of a standard of equal concentration that does not contain potassium. 2. Why is it inadvisable to match the matrix of the standards to the sample by adding KCl to each standard? Sodium is a common contaminant found in many chemicals. Reagent grade KCl, for example, may contain 40–50 ppm sodium. This is a significant source of sodium, given that its concentration in the salt substitute is about 100 ppm. 3. One problem with the analysis of salt samples is their tendency to clog the aspirator and burner assembly. What effect would this have on the analysis? Clogging the aspirator and burner assembly decreases the rate of aspiration, decreasing the analyte’s concentration in the flame. The result is a decrease in the signal and the introduction of a determinate error. 4. The following results were obtained for the analysis of a 10.0077-g sample of salt substitute that was analyzed by the procedure described earlier. Concentration of Added Sodium (ppm) Emission 0.000 1.79 0.420 2.63 1.051 3.54 2.102 4.94 3.153 6.18 1400-CH10 9/8/99 4:19 PM Page 439 440 Modern Analytical Chemistry Continued from page 439 What is the concentration of sodium, in parts per million, in the salt substitute? A standard addition calibration curve of emission versus the concentration of added sodium gives, by linear regression, an equation of I = 1.97 + 1.37 · (ppm Na) The concentration of sodium in the standard addition samples is determined from the absolute value of the x-intercept (see Figure 5.7b); thus, substituting zero for the emission intensity gives the concentration of sodium as 1.44 ppm. The concentration of sodium in the salt substitute, therefore, is 10H. 4 Evaluation Scale of Operation The scale of operations for atomic emission is ideal for the direct analysis of trace and ultratrace analytes in macro and meso samples. With appropriate dilutions, atomic emission also can be applied to major and minor analytes. Accuracy When spectral and chemical interferences are insignificant, atomic emission is capable of producing quantitative results with accuracies of 1–5%. Ac- curacy in flame emission frequently is limited by chemical interferences. Because the higher temperature of a plasma source gives rise to more emission lines, accu- racy when using plasma emission often is limited by stray radiation from overlap- ping emission lines. Precision For samples and standards in which the concentration of analyte ex- ceeds the detection limit by at least a factor of 50, the relative standard deviation for both flame and plasma emission is about 1–5%. Perhaps the most important factor affecting precision is the stability of the flame’s or plasma’s temperature. For exam- ple, in a 2500 K flame a temperature fluctuation of – 2.5 K gives a relative standard deviation of 1% in emission intensity. Significant improvements in precision may be realized when using internal standards. Sensitivity Sensitivity in flame atomic emission is strongly influenced by the tem- perature of the excitation source and the composition of the sample matrix. Nor- mally, sensitivity is optimized by aspirating a standard solution and adjusting the flame’s composition and the height from which emission is monitored until the emission intensity is maximized. Chemical interferences, when present, decrease the sensitivity of the analysis. With plasma emission, sensitivity is less influenced by the sample matrix. In some cases, for example, a plasma calibration curve prepared using standards in a matrix of distilled water can be used for samples with more complex matrices. Selectivity The selectivity of atomic emission is similar to that of atomic absorp- tion. Atomic emission has the further advantage of rapid sequential or simultane- ous analysis. (. / . . 144 10 0077 71 9 g Na mL solution) (50.00 mL/25.00 mL) 250.0 mL g ppm Na + + μ× × = 1400-CH10 9/8/99 4:19 PM Page 440 Figure 10.53 Distribution of radiation for (a) Rayleigh scattering and (b) large-particle scattering. Chapter 10 Spectroscopic Methods of Analysis 441 Time, Cost, and Equipment Sample throughput with atomic emission is very rapid when using automated systems capable of multielemental analysis. For exam- ple, sampling rates of 3000 determinations per hour have been achieved using an ICP with simultaneous analysis, and 300 determinations per hour with a sequential ICP. Flame emission is often accomplished using an atomic absorption spectrome- ter, which typically costs $10,000–50,000. Sequential ICPs range in price from $55,000 to $150,000, whereas an ICP capable of simultaneous multielemental analy- sis costs $80,000–200,000. Combination ICPs that are capable of both sequential and simultaneous analysis range in price from $150,000 to $300,000. The cost of Ar, which is consumed in significant quantities, cannot be overlooked when consider- ing the expense of operating an ICP. 10I Spectroscopy Based on Scattering The blue color of the sky during the day and the red color of the sun at sunset result from the scattering of light by small particles of dust, molecules of water, and other gases in the atmosphere. The efficiency with which light is scattered depends on its wavelength. The sky is blue because violet and blue light are scattered to a greater extent than other, longer wavelengths of light. For the same reason, the sun appears to be red when observed at sunset because red light is less efficiently scattered and, therefore, transmitted to a greater extent than other wavelengths of light. The scat- tering of radiation has been studied since the late 1800s, with applications begin- ning soon thereafter. The earliest quantitative applications of scattering, which date from the early 1900s, used the elastic scattering of light to determine the concentra- tion of colloidal particles in a suspension. 10I.1 Origin of Scattering A focused, monochromatic beam of radiation of wavelength l , passing through a medium containing particles whose largest dimensions are less than l is ob- served to scatter in all directions. For example, visible radiation of 500 nm is scat- tered by particles as large as 750 nm in the longest dimension. With larger parti- cles, radiation also may be reflected or refracted. Two general categories of scattering are recognized. In elastic scattering, radiation is absorbed by the analyte and re-emitted without a change in the radiation’s energy. When the radiation is re-emitted with a change in energy, the scattering is said to be inelastic. Only elas- tic scattering is considered in this text. Elastic scattering is divided into two types: Rayleigh, or small-particle scatter- ing, and large-particle scattering. Rayleigh scattering occurs when the scattering particles largest dimension is less than 5% of the radiation’s wavelength. The inten- sity of the scattered radiation is symmetrically distributed (Figure 10.53a) and is proportional to its frequency to the fourth power (n 4 ), accounting for the greater scattering of blue light compared with red light. For larger particles, the distribution of scattered light increases in the forward direction and decreases in the backward direction as the result of constructive and destructive interferences (Figure 10.53b). 10I.2 Turbidimetry and Nephelometry Turbidimetry and nephelometry are two related techniques in which an incident source of radiation is elastically scattered by a suspension of colloidal particles. In turbidimetry, the detector is placed in line with the radiation source, and the 3 2 Source Source (a) (b) turbidimetry A method in which the decrease in transmitted radiation due to scattering is measured. 1400-CH10 9/8/99 4:19 PM Page 441 Figure 10.54 Block diagrams for (a) a turbidometer and (b) a nephelometer. decrease in the radiation’s transmitted power is measured. In nephelometry, scat- tered radiation is measured at an angle of 90° to the radiation source. The similarity of the measurement of turbidimetry to absorbance, and of nephelometry to fluores- cence, is evident in the block instrumental designs shown in Figure 10.54. In fact, turbidity can be measured using a UV/Vis spectrophotometer, such as a Spectronic- 20, whereas a spectrofluorometer is suitable for nephelometry. Turbidimetry Versus Nephelometry Choosing between turbidimetry and neph- elometry is determined by two principal factors. The most important consideration is the intensity of the transmitted or scattered radiation relative to the intensity of radiation from the source. When the solution contains a small concentration of scattering particles, the intensity of the transmitted radiation, I T , will be very similar to the intensity of the radiation source, I 0 . As we learned earlier in the section on molecular absorption, determining a small difference between two intense signals is subject to a substantial uncertainty. Thus, nephelometry is a more appropriate choice for samples containing few scattering particles. On the other hand, tur- bidimetry is a better choice for samples containing a high concentration of scatter- ing particles. The second consideration in choosing between turbidimetry and nephelometry is the size of the scattering particles. For nephelometry, the intensity of scattered ra- diation at 90° will be greatest if the particles are small enough that Rayleigh scatter- ing is in effect. For larger particles, as shown in Figure 10.37, scattering intensity is diminished at 90°. When using an ultraviolet or visible source of radiation, the opti- mum particle size is 0.1–1 m m. The size of the scattering particles is less important for turbidimetry, in which the signal is the relative decrease in transmitted radia- tion. In fact, turbidimetric measurements are still feasible even when the size of the scattering particles results in an increase in reflection and refraction (although a lin- ear relationship between the signal and the concentration of scattering particles may no longer hold). Determining Concentration by Turbidimetry In turbidimetry the measured trans- mittance, T, is the ratio of the transmitted intensity of the source radiation, I T , to the intensity of source radiation transmitted by a blank, I 0 . T I I = T 0 442 Modern Analytical Chemistry nephelometry A method in which the intensity of scattered radiation is measured at an angle of 90° to the source. Source Sample Detector Wavelength selector Source Sample Wavelength selector Signal processor Detector Signal processor (b) (a) 1400-CH10 9/8/99 4:19 PM Page 442 The relationship between transmittance and the concentration of the scattering par- ticles is similar to that given by Beer’s law –logT = kbC 10.36 where C is the concentration of the scattering particles in mass per unit volume (w/v), b is the pathlength, and k is a constant that depends on several factors, in- cluding the size and shape of the scattering particles and the wavelength of the source radiation. As with Beer’s law, equation 10.36 may show appreciable devia- tions from linearity. The exact relationship is established by a calibration curve pre- pared using a series of standards of known concentration. Determining Concentration by Nephelometry In nephelometry, the relationship between the intensity of scattered radiation, I S , and the concentration (% w/v) of scattering particles is given as I S = k S I 0 C 10.37 where k S is an empirical constant for the system, and I 0 is the intensity of the inci- dent source radiation. The value of k S is determined from a calibration curve pre- pared using a series of standards of known concentration. Selecting a Wavelength for the Incident Radiation Selecting a wavelength for the incident radiation is based primarily on the need to minimize potential interfer- ences. For turbidimetry, where the incident radiation is transmitted through the sample, it is necessary to avoid radiation that is absorbed by the sample. Since ab- sorption is a common problem, the wavelength must be selected with some care, using a filter or monochromator for wavelength selection. For nephelometry, the absorption of incident radiation is not a problem unless it induces fluorescence from the sample. With a nonfluorescent sample there is no need for wavelength se- lection, and a source of white light may be used as the incident radiation. When using a filter or monochromator, other considerations include the dependence of scattering intensity, transducer sensitivity, and source intensity on the wavelength. For example, many common photon transducers are more sensitive to radiation at 400 nm than at 600 nm. Preparing the Sample Although equations 10.36 and 10.37 relate scattering to the concentration of scattering particles, the intensity of scattered radiation is also in- fluenced strongly by the particle’s size and shape. For example, samples containing the same number of scattering particles may show significantly different values for –logT or I s depending on the average diameter of the particles. For a quantitative analysis, therefore, it is necessary to maintain a uniform distribution of particle sizes throughout the sample and between samples and standards. Most turbidimetric and nephelometric methods rely on the formation of the scattering particles by precipitation. As we learned in the discussion of precipitation gravimetry (Chapter 8), the properties of a precipitate are determined by the condi- tions used to effect the precipitation. To maintain a reproducible distribution of particle sizes between samples and standards, it is necessary to control parameters such as the concentration of reagents, order of adding reagents, pH, temperature, agitation or stirring rate, ionic strength, and time between the precipitate’s initial formation and the measurement of transmittance or scattering. In many cases a Chapter 10 Spectroscopic Methods of Analysis 443 1400-CH10 9/8/99 4:19 PM Page 443 Figure 10.55 Synthetic scheme for the preparation of formazin. 444 Modern Analytical Chemistry N N NN hexamethylenetetramine + 6H 2 O + 2H 2 SO 4 6H 2 CO + 2(NH 4 ) 2 SO 4 nH 2 CO + H 2 N — NH 2 n 2 formazin + nH 2 O n 4 N N N N surface-active agent, such as glycerol, gelatin, or dextrin, is added to stabilize the precipitate in a colloidal state and to prevent the coagulation of the particles. Applications Turbidimetry and nephelometry are widely used to determine the clar- ity of water, beverages, and food products. For example, the turbidity of water is de- termined using nephelometry by comparing the sample’s scattering to that of a set of standards. The primary standard for measuring turbidity is formazin (Figure 10.55), which is an easily prepared, stable polymer suspension. 26 Formazin prepared by mix- ing a 1 g/100 mL solution of hydrazine sulfate, N 2 H 4 ?H 2 SO 4 , with a 10-g/100 mL solu- tion of hexamethylenetetramine produces a suspension that is defined as 4000 neph- elometric turbidity units (NTU). A set of standards with NTUs between 0 and 40 is prepared and used to construct a calibration curve. This method is readily adapted to the analysis of the clarity of orange juice, beer, and maple syrup. A number of inorganic cations and anions can be determined by precipitating them under well-defined conditions and measuring the transmittance or scattering of radiation from the precipitated particles. The transmittance or scattering, as given by equation 10.36 or 10.37 is proportional to the concentration of the scatter- ing particles, which, in turn, is related by the stoichiometry of the precipitation re- action to the analyte’s concentration. Examples of analytes that have been deter- mined in this way are listed in Table 10.15. The turbidimetric determination of SO 4 2– in water following its precipitation as BaSO 4 is described in Method 10.5. Table 10.1 5 Selected Precipitates Used in Turbidimetric and Nephelometric Methods Analyte Precipitant Precipitate Ag + NaCl AgCl Ca 2+ Na 2 C 2 O 4 CaC 2 O 4 Cl – AgNO 3 AgCl CN – AgNO 3 AgCN CO 3 2– BaCl 2 BaCO 3 F – CaCl 2 CaF 2 SO 4 2– BaCl 2 BaSO 4 1400-CH10 9/8/99 4:19 PM Page 444 Chapter 10 Spectroscopic Methods of Analysis 445 Representative Methods Method 10.5 Turbidimetric Determination of Sulfate in Water 27 Description of Method. Adding BaCl 2 to an acidified sample precipitates SO 4 2– as BaSO 4 . The concentration of SO 4 2– may be determined either by turbidimetry or nephelometry using an incident source of radiation of 420 nm. External standards containing known concentrations of SO 4 2– are used to standardize the method. Procedure. A 100-mL sample is transferred to a 250-mL Erlenmeyer flask along with 5.00 mL of a conditioning reagent. The composition of the conditioning reagent is 50 mL glycerol, 30 mL concentrated HCl, 300 mL distilled water, 100 mL 95% ethyl or isopropyl alcohol, and 75 g NaCl. The sample and conditioning reagent are placed on a magnetic stirrer that is operated at the same speed for all samples and standards. A portion of crystalline BaCl 2 is added using a measuring spoon whose capacity is 0.2–0.3 mL, precipitating the SO 4 2– as BaSO 4 . Timing begins when the BaCl 2 is added and the solution and precipitate are allowed to stir for exactly 1 min. At the end of the stirring period a portion of the suspension is poured into the turbidimeter or nephelometer cell, and the transmittance or scattering intensity measured at 30-s intervals for 4 min. The minimum transmittance or maximum scattering intensity recorded during this period is the analytical signal. Calibration curves over the range 0–40 ppm SO 4 2– are prepared by diluting a 100-ppm SO 4 2– standard. The standards and a reagent blank must be treated in exactly the same fashion as samples. Questions 1. What is the role of the conditioning reagent? The conditioning reagent is used to stabilize the precipitate of BaSO 4 . The high ionic strength and acidity, due to NaCl and HCl, prevent the formation of microcrystalline particles of BaSO 4 , and glycerol and alcohol help stabilize the precipitate’s suspension. 2. Why is it important to use the same stirring rate and time for all samples and standards? The stirring speed and time influence the size of the precipitate’s particles, as well as their suspension in solution. 3. The following results were obtained for the turbidimetric analysis of a set of standard SO 4 2– solutions using the procedure described earlier Transmittance ppm SO 4 2– (T) 0.00 1.00 10.00 0.646 20.00 0.417 30.00 0.269 40.00 0.174 When a sample of a surface water was carried through this procedure, the transmittance was found to be 0.538. What is the concentration of sulfate in the sample? A calibration curve of –log(T) versus concentration gives a standardization equation of –log(T) = –1.04 · 10 –5 + 0.0190(ppm SO 4 2– ) Substituting the sample’s transmittance into the standardization equation gives the concentration of sulfate in the sample as 14.2 ppm. 1400-CH10 9/8/99 4:19 PM Page 445 446 Modern Analytical Chemistry 10J KEY TERMS absorbance (p. 373) absorbance spectrum (p. 373) atomization (p. 412) background correction (p. 419) Beer’s law (p. 386) characteristic concentration (p. 416) chemiluminescence (p. 374) chromophore (p. 382) continuum source (p. 375) dark current (p. 379) effective bandwidth (p. 376) electromagnetic spectrum (p. 372) emission (p. 373) emission spectrum (p. 374) excitation spectrum (p. 427) external conversion (p. 425) filter (p. 376) filter photometer (p. 388) fluorescence (p. 423) fluorometer (p. 428) frequency (p. 370) graphite furnace (p. 414) intensity (p. 371) interferometer (p. 378) internal conversion (p. 425) intersystem crossing (p. 425) ionization suppressor (p. 420) lifetime (p. 423) line source (p. 375) method of continuous variations (p. 404) mole-ratio method (p. 406) monochromatic (p. 377) monochromator (p. 376) nephelometry (p. 442) nominal wavelength (p. 376) phosphorescence (p. 424) photodiode array (p. 379) photoluminescence (p. 374) photon (p. 371) plasma (p. 435) polychromatic (p. 377) power (p. 371) protecting agent (p. 420) quantum yield (p. 425) relaxation (p. 423) releasing agent (p. 420) resolution (p. 376) self-absorption (p. 438) signal averaging (p. 391) signal processor (p. 380) signal-to-noise ratio (p. 379) singlet excited state (p. 423) slope-ratio method (p. 407) spectral searching (p. 403) spectrofluorometer (p. 428) spectrophotometer (p. 389) stray radiation (p. 387) transducer (p. 379) transmittance (p. 384) triplet excited state (p. 423) turbidimetry (p. 441) vibrational relaxation (p. 424) wavelength (p. 370) wavenumber (p. 370) Spectroscopic methods of analysis covered in this chapter include those based on the absorption, emission, or scattering of electro- magnetic radiation. When a molecule absorbs UV/Vis radiation, it undergoes a change in its valence shell configuration, whereas a change in vibrational energy results for the absorption of IR radia- tion. Experimentally we measure the fraction of radiation trans- mitted (T) through the sample. Instrumentation for molecular ab- sorption requires a source of electromagnetic radiation, a means for selecting the wavelength for which transmittance is measured, and a detector for measuring the transmittance. Beer’s law relates absorbance to both transmittance and the concentration of the ab- sorbing species (A = –logT = e bC). In atomic absorption we measure the absorption of radiation by gaseous atoms. Samples are atomized using thermal energy from either a flame or a graphite furnace. Because the width of an atom’s absorption band is so narrow, the continuum sources com- mon for molecular absorption cannot be used. Instead, a hollow cathode lamp provides the necessary line source of radiation. Atomic absorption suffers from a number of spectral and chemical interferences. The absorption or scattering of radiation from the sample’s matrix is an important spectral interference that may be minimized by background correction. Chemical interferences in- clude the formation of nonvolatile forms of the analyte in the flame and ionization of the analyte. The former interference is minimized by using a releasing agent or a protecting agent, and an ionization suppressor helps minimize the latter interference. When a molecule absorbs radiation, it moves from a lower- energy state to a higher-energy state. In returning to the lower- energy state the molecule may emit radiation. This process is called photoluminescence. One form of photoluminescence is fluorescence, in which the analyte emits a photon without un- dergoing a change in its spin state. In phosphorescence, emission occurs with a change in the analyte’s spin state. For low concen- trations of analyte, both fluorescent and phosphorescent emission intensities are a linear function of the analyte’s concentration. Thermally excited atoms also emit radiation, forming the basis for atomic emission spectroscopy. Thermal excitation is achieved using either a flame or a plasma. Spectroscopic measurements may also involve the scattering of light by a particulate form of the analyte. In turbidimetry, the de- crease in the radiation’s transmittance through the sample is mea- sured and related to the analyte’s concentration through Beer’s law. In nephelometry we measure the intensity of scattered radia- tion, which varies linearly with the analyte’s concentration. 10K SUMMARY 1400-CH10 9/8/99 4:19 PM Page 446 Chapter 10 Spectroscopic Methods of Analysis 447 The first group of experiments are for the analysis of samples using UV/Vis absorption. Allen, H. C.; Brauers, T.; Finlayson-Pitts, B. J. “Illustrating Deviations in the Beer–Lambert Law in an Instrumental Analysis Laboratory: Measuring Atmospheric Pollutants by Differential Optical Absorption Spectrometry,” J. Chem. Educ. 1997, 74, 1459–1462. This experiment demonstrates the chemical limitations to Beer’s law using the NO 2 –N 2 O 4 equilibrium as an example. Blanco, M.; Iturriaga, H.; Maspoch, S.; et al. “A Simple Method for Spectrophotometric Determination of Two- Components with Overlapped Spectra,” J. Chem. Educ. 1989, 66, 178–180. This experiment describes the application of multiwavelength linear regression to the analysis of two-component mixtures. Directions are given for the analysis of permanganate–dichromate mixtures, Ti(IV)–V(V) mixtures and Cu(II)–Zn(II) mixtures. Bruneau, E.; Lavabre, D.; Levy, G.; et al. “Quantitative Analysis of Continuous-Variation Plots with a Comparison of Several Methods,” J. Chem. Educ. 1992, 69, 833–837. In this experiment the method of continuous variations is used to determine the stoichiometry and equilibrium constant for the organic complex of 3-aminopyridine with picric acid in CHCl 3 , and the inorganic complex of Fe 3+ with salicylic acid. Crisp, P. T.; Eckert, J. M.; Gibson, N. A. “The Determination of Anionic Surfactants in Natural and Waste Waters,” J. Chem. Educ. 1983, 60, 236–238. The concentration of anionic surfactants at the sub-ppm level in natural waters and industrial waters are determined spectrophotometrically. The anionic surfactants are extracted into a nonaqueous solvent following the formation of an ion association complex with a suitable cation. Domínguez, A.; Fernández, A.; González, N.; et al. “Determination of Critical Micelle Concentration of Some Surfactants by Three Techniques,” J. Chem. Educ. 1997, 74, 1227–1231. Surfactants are long-chain compounds containing a hydrophobic tail and an ionic head. In polar solvents the surfactants arrange themselves in a “spherical” structure known as a micelle in which the hydrophobic tails form the core. The concentration of surfactant needed to form micelles is called the critical micelle concentration (CMC). This experiment describes both a UV absorption and a fluorescence method for determining the CMC. Directions also are given for determining the CMC by measuring electrical conductivity. Hill, Z. D.; MacCarthy, P. “Novel Approach to Job’s Method,” J. Chem. Educ. 1986, 63, 162–167. Data from the spectrophotometric titrations of Fe 3+ with SCN – , and of Cu 2+ with EDTA are used to determine the stoichiometry of the resulting complexes using the method of continuous variations. Long, J. R.; Drago, R. S. “The Rigorous Evaluation of Spectrophotometric Data to Obtain an Equilibrium Constant,” J. Chem. Educ. 1982, 59, 1037–1039. Although this experiment is written as a “dry-lab,” it can be adapted to the laboratory. Details are given for the determination of the equilibrium constant for the binding of the Lewis base 1-methylimidazole to the Lewis acid cobalt(II)4-trifluoromethyl-o-phenylene-4,6- methoxysalicylideniminate in toluene. The equilibrium constant is found by a linear regression analysis of the absorbance data to a theoretical equilibrium model. McDevitt, V. L.; Rodriquez, A.; Williams, K. R. “Analysis of Soft Drinks: UV Spectrophotometry, Liquid Chromatography, and Capillary Electrophoresis,” 1998, 75, 625–629. Procedures for determining the concentrations of caffeine, benzoic acid and aspartame in soda by these three methods are provided. In the example provided in this paper, the concentrations of caffeine and benzoic acid in Mello Yellow are determined spectrophotometrically. Mehra, M. C.; Rioux, J. “An Analytical Chemistry Experiment in Simultaneous Spectrophotometric Determination of Fe(III) and Cu(II) with Hexacyanoruthenate(II) Reagent,” J. Chem. Educ. 1982, 59, 688–689. This experiment describes a standard multicomponent analysis for two analytes based on measuring the absorbance at two wavelengths. Hexacyanoruthenate(II) is used as a complexing agent, forming a purple-blue complex with Fe(III) and a pale green complex with Cu(II). 10L Suggested EXPERIMENTS The following experiments may be used to illustrate the application of spectroscopy to quantitative or characterization problems. Experiments are divided into four groups: those using UV/Vis absorption, those using IR absorption, those using atomic absorption or atomic emission, and those using fluorescence. Other experiments are described in the highlighted method sections. A brief description is included with each experiment providing details such as the type of sample analyzed and the method of standardization. Experiments —Continued 1400-CH10 9/8/99 4:19 PM Page 447 448 Modern Analytical Chemistry Continued from page 447 Experiments ?rstan, A.; Wojcik, J. F. “Spectroscopic Determination of Protein-Ligand Binding Constants,” J. Chem. Educ. 1987, 64, 814–816. This experiment provides a nice example of the application of spectroscopy to biochemistry. After presenting the basic theory for the spectroscopic treatment of protein–ligand interactions, a procedure for characterizing the binding of methyl orange to bovine serum albumin is described. Pandey, S.; Powell, J. R.; McHale, M. E. R.; et al. “Quantitative Determination of Cr(III) and Co(II) Using a Spectroscopic H-Point Standard Addition,” J. Chem. Educ. 1997, 74, 848–850. Another nice example of a multicomponent analysis based on current research is presented in this experiment. Although the H-point standard addition is not discussed in this text, this paper provides adequate theory and references to the original literature. Raymond, M.; Jochum, C.; Kowalski, B. R. “Optimal Multicomponent Analysis Using the Generalized Standard Addition Method,” J. Chem. Educ. 1983, 60, 1072–1073. This experiment demonstrates the application of the generalized standard additions method for the analysis of mixtures of K 2 Cr 2 O 7 and KMnO 4 . Tucker, S.; Robinson, R.; Keane, C.; et al. “Colorimetric Determination of pH,” J. Chem. Educ. 1989, 66, 769–771. In this experiment mixtures of dyes are used to provide a means for determining spectrophotometrically a sample’s pH. Walmsley, F. “Aggregation in Dyes: A Spectrophotometric Study,” J. Chem. Educ. 1992, 69, 583. A dye used in the coloration of materials, such as fibers, must be present as the monomer if it is to adsorb to the material’s surface. This experiment describes how spectrophotometry can be used to determine the equilibrium constant between a monomer and a dimer for the dye pinacyanol iodide in water. Williams, K. R.; Cole, S. R.; Boyette, S. E.; et al. “The Use of Dristan Nasal Spray as the Unknown for Simultaneous Spectrophotometric Analysis of a Mixture,” J. Chem. Educ. 1990, 67, 535. The analysis of a pharmaceutical preparation for pheniramine maleate and phenylephrine hydrochloride is described in this experiment. Yarnelle, M. K.; West, K. J. “Modification of an Ultraviolet Spectrophotometric Determination of the Active Ingredients in APC Tablets,” J. Chem. Educ. 1989, 66, 601–602. The analysis of APC tablets (a mixture of aspirin, phenacetin, and caffeine) has been a common undergraduate laboratory experiment. This experiment describes modifications to the standard analysis for APC tablets in which paracetamol (also known as acetaminophen) replaces phenacetin. The second block of experiments provide examples of the application of infrared spectroscopy. Frohlich, H. “Using Infrared Spectroscopy Measurements to Study Intermolecular Hydrogen Bonding,” J. Chem. Educ. 1993, 70, A3–A6. This experiment describes a characterization analysis in which the degree of association, equilibrium constant, and hydrogen bond energy are measured for benzyl alcohol and phenol in CCl 4 . Mathias, L. J.; Hankins, M. G.; Bertolucci, C. M.; et al. “Quantitative Analysis by FT-IR: Thin Films of Copolymers of Ethylene and Vinyl Acetate,” J. Chem. Educ. 1992, 69, A217–A219. The % w/w vinyl acetate in a copolymer is determined using a ratio of the peak at 1020 cm –1 due to C—O and the polyethylene C—H peak at 720 cm –1 . Seasholtz, M. B.; Pence, L. E.; Moe Jr., O. A. “Determination of Carbon Monoxide in Automobile Exhaust by FT-IR Spectroscopy,” J. Chem. Educ. 1988, 65, 820–823. Samples of car exhaust are collected using a 4-L glass bottle evacuated to a level of less than 2 torr. A normal calibration curve using external standards of known P CO is used to determine the P CO in the exhaust samples. —Continued The next set of experiments describe suitable applications of atomic absorption spectroscopy. Gilles de Pelichy, L. D.; Adams, C.; Smith, E. T. “Analysis of the Essential Nutrient Strontium in Marine Aquariums by Atomic Absorption Spectroscopy,” J. Chem. Educ. 1997, 74, 1192–1194. The concentration of strontium in a sea water aquarium is determined by atomic absorption using the method of standard additions. The effect of adding La 3+ as a releasing agent also is explored. Hoskins, L. C.; Reichardt, P. B.; Stolzberg, R. J. “Determination of the Extraction Constant for Zinc Pyrrolidinecarbodithioate,” J. Chem. Educ. 1981, 58, 580–581. 1400-CH10 9/8/99 4:19 PM Page 448 Chapter 10 Spectroscopic Methods of Analysis 449 Experiments Aqueous solutions buffered to a pH of 5.2 and containing known total concentrations of Zn 2+ are prepared. A solution containing ammonium pyrrolidinecarbodithioate (APCD) is added along with methyl isobutyl ketone (MIBK). The mixture is shaken briefly and then placed on a rotary shaker table for 30 min. At the end of the extraction period the aqueous and organic phases are separated and the concentration of zinc in the aqueous layer determined by atomic absorption. The concentration of zinc in the organic phase is determined by difference and the equilibrium constant for the extraction calculated. Lehman, T. A.; Everett, W. W. “Solubility of Lead Sulfate in Water and in Sodium Sulfate Solutions,” J. Chem. Educ. 1982, 59, 797. Saturated solutions are prepared by adding PbSO 4 to solutions containing 0 – 2 · 10 –3 M Na 2 SO 4 . After allowing time for equilibration, the aqueous solution is isolated by decantation and analyzed by atomic absorption for the concentration of Pb 2+ . The apparent K sp for PbSO 4 is calculated, showing the effect of ionic strength on solubility products. Markow, P. G. “Determining the Lead Content of Paint Chips,” J. Chem. Educ. 1996, 73, 178–179. The %w/w lead in a lead-based paint Standard Reference Material and in unknown paint chips is determined by atomic absorption using external standards. Masina, M. R.; Nkosi, P. A.; Rasmussen, P. W.; et al. “Determination of Metal Ions in Pineapple Juice and Effluent of a Fruit Canning Industry,” J. Chem. Educ. 1989, 66, 342–343. The concentrations of iron, lead, tin, and aluminum are determined using the method of standard additions. Quigley, M. N. “Determination of Calcium in Analgesic Tablets Using Atomic Absorption Spectrophotometry,” J. Chem. Educ. 1994, 71, 800. Analgesic tablets are ground into a fine powder, dissolved in HCl, and analyzed for calcium by atomic absorption. A releasing agent of La 3+ is used to prevent an interference due to the formation of calcium pyrophosphate. Quigley, M. N.; Vernon, F. “Determination of Trace Metal Ion Concentrations in Seawater,” J. Chem. Educ. 1996, 73, 671–675. Trace metals in sea water are preconcentrated either by coprecipitating with Fe(OH) 3 and recovering by dissolving the precipitate or by ion exchange. The concentrations of several trace metals are determined by standard additions using graphite furnace atomic absorption spectrometry. Rheingold, A. L.; Hues, S.; Cohen, M. N. “Strontium and Zinc Content in Bones as an Indication of Diet,” J. Chem. Educ., 1983, 60, 233–234. Samples of animal bones weighing approximately 3 g are ashed at 600 °C until the entire bone is ash-white. Samples are then crushed in a mortar and pestle. A portion of the sample is digested in HCl and diluted to a known volume. The concentrations of zinc and strontium are determined by atomic absorption. The analysis for strontium illustrates the use of a protecting agent as La(NO 3 ) 3 is added to prevent an interference due to the formation of refractory strontium phosphate. Rocha, F. R. P.; Nóbrega, J. A. “Effects of Solution Physical Properties on Copper and Chromium Signals in Flame Atomic Absorption Spectrometry,” J. Chem. Educ. 1996, 73, 982–984. A nice experiment illustrating the importance of a sample’s matrix. The effect on the absorbance of copper for solutions with different %v/v ethanol, and the effect on the absorbance of chromium for solutions with different concentrations of added surfactants are evaluated. The last set of experiments illustrate the use of fluorescence spectroscopy. Buccigross, J. M.; Bedell, C. M.; Suding-Moster, H. L. “Fluorescent Measurement of TNS Binding to Calmodulin,” J. Chem. Educ. 1996, 73, 275–278. Although intended for the biochemistry lab, this experiment provides analytical students with a practical characterization analysis. Of particular interest is the use of Job’s method to determine the number of TNS (2-p-toludinylnaphthalene-6- sulfonate) binding sites on calmodulin. Fluorescence is measured at 475 nm using an excitation wavelength of 330 nm. Henderleiter, J. A.; Hyslopo, R. M. “The Analysis of Riboflavin in Urine by Fluorescence,” J. Chem. Educ. 1996, 73, 563–564. Samples of urine are analyzed for riboflavin before and after taking a vitamin tablet containing riboflavin. Concentrations are determined using external standards or by the method of standard additions. Fluorescence is monitored at 525 nm using an excitation wavelength of 280 nm. 1400-CH10 9/8/99 4:19 PM Page 449 450 Modern Analytical Chemistry 1. Provide the missing information in the following table Wavelength Frequency Wavenumber Energy (m) (s –1 ) (cm –1 ) (J/molecule) 4.50 · 10 –9 1.33 · 10 15 3215 7.20 · 10 –19 2. Provide the missing information in the following table Molor [Analyte] Absorb- % Trans- absorptivity Pathlength (M) ance mittance (M –1 cm –1 ) (cm) 1.40 · 10 –4 1120 1.00 0.563 750 1.00 2.56 · 10 –4 0.225 440 1.55 · 10 –3 0.167 5.00 33.3 565 1.00 4.35 · 10 –3 21.2 1550 1.20 · 10 –4 81.3 10.00 3. The transmittance of a solution is found to be 35.0%. What is the transmittance if the solution is diluted in half? 4. The transmittance of a solution is found to be 85.0% when measured in a cell whose pathlength is 1.00 cm. What is the percent transmittance if the pathlength is increased to 10.00 cm? 5. The accuracy of a spectrophotometer can be evaluated by preparing a solution of 60.06-ppm K 2 Cr 2 O 7 in 0.0050 M H 2 SO 4 and measuring its absorbance at a wavelength of 350 nm using a cell with a pathlength of 1.00 cm. The absorbance should be 0.640. What is the molar absorptivity of K 2 Cr 2 O 7 at this wavelength? 6. Chemical deviations to Beer’s law may occur when the concentration of an absorbing species is affected by the position of an equilibrium reaction. Consider a weak acid, HA, for which K a is 2 · 10 –5 . Construct Beer’s law calibration curves of absorbance versus the total concentration of weak acid (C tot =[HA]+[A – ]), using values for C tot of 1.0 · 10 –5 , 3.0 · 10 –5 , 5.0 · 10 –5 , 9.0 · 10 –5 , 11 · 10 –5 , and 13 · 10 –5 M for the following sets of conditions: (a) e HA = e A - = 2000, and solution is not buffered; (b) e HA = 2000 and e A - = 500, and solution is not buffered; and, (c) e HA = 2000 and e A - = 500, and solution is buffered to a pH of 4.50. Assume a constant pathlength of 1.00 cm for all samples. All values of e have units of M –1 cm –1 . 7. One instrumental limitation to Beer’s law is the use of polychromatic radiation instead of monochromatic radiation. Consider a radiation source that emits two wavelengths of radiation, l￠ and l? . When treated separately, the absorbances at these wavelengths, A￠ and A? , are When both wavelengths are measured simultaneously, the absorbance is given as (a) Show that when the molar absorptivity at l￠ and l? are the same (e￠ = e? = e ), the absorbance is equivalent to A = e bC (b) Construct Beer’s law calibration curves over the concentration range of zero to 1 · 10 –4 M using e￠ = 1000 and e? = 1000, and e￠ = 1900 and e? = 100. Assume a value of 1.00 cm for the pathlength and that P 0 ￠ = P 0 ? . Explain the difference between the two curves. 8. A second instrumental limitation to Beer’s law is stray radiation. The following data were obtained using a cell with a pathlength of 1.00 cm, when stray light is insignificant (P stray = 0). [Analyte] (M) Absorbance 0.000 0.00 0.002 0.40 0.004 0.80 0.006 1.20 0.008 1.60 0.010 2.00 Calculate the absorbance of each solution when P stray is 5% of P 0 , and plot Beer’s law calibration curves for both sets of data. Explain any differences between the two curves. (Hint: Assume that P 0 is 100). 9. In the process of performing a spectrophotometric determination of Fe, an analyst prepares a calibration curve using a single-beam spectrometer, such as a Spec-20. After preparing the calibration curve, the analyst drops the cuvette used for the method blank and the standards. The analyst acquires a new cuvette, measures the absorbance of the sample, and determines the %w/w Fe in the sample. Will the change in cuvette lead to a determinate error in the analysis? Explain. A PP PP = ′ + ″ ′ + ″ log () () 00 TT ′′ = ″ ″ = ′′A P P bClog 0 T ε ′ = ′ ′ = ′A P P bClog 0 T ε 10M PROBLEMS 1400-CH10 9/8/99 4:19 PM Page 450 10. One method for the analysis of Fe 3+ that can be used with a variety of sample matrices, is to form the highly colored Fe 3+ –thioglycolic acid complex. The complex absorbs strongly at 535 nm. Standardizing the method is accomplished using external standards. A 10.00 ppm Fe 3+ working standard is prepared by transferring a 10-mL aliquot of a 100.0 ppm stock solution of Fe 3+ to a 100-mL volumetric flask and diluting to volume. Calibration standards of 1.00, 2.00, 3.00, 4.00, and 5.00 ppm are then prepared by transferring appropriate amounts of the 100.0-ppm working solution into separate 50-mL volumetric flasks, each containing 5 mL of thioglycolic acid, 2 mL of 20% w/v ammonium citrate, and 5 mL of 0.22 M NH 3 . After diluting to volume and mixing, the absorbances of the external standards are measured against an appropriate blank. Samples are prepared for analysis by taking a portion known to contain approximately 0.1 g of Fe 3+ , dissolving in a minimum amount of HNO 3 , and diluting to volume in a 1-L volumetric flask. A 1.00-mL aliquot of this solution is transferred to a 50-mL volumetric flask, along with 5 mL of thioglycolic acid, 2 mL of 20% w/v ammonium citrate, and 5 mL of 0.22 M NH 3 and diluted to volume. The absorbance of this solution is used to determine the concentration of Fe 3+ in the sample. (a) What is an appropriate blank for this procedure? (b) Ammonium citrate is added to prevent the precipitation of Al 3+ . What effect would the presence of trace amount of Fe 3+ in the ammonium citrate have on the reported concentration of iron in the sample? (c) Why does the procedure call for taking an amount of sample containing approximately 0.1 grams of Fe 3+ ? (d) Unbeknownst to the analyst, the 100-mL volumetric flask used to prepare the 10.00 ppm working standard of Fe 3+ has a volume that is significantly smaller than 100.0mL. What effect will this have on the reported concentration of iron in the sample? 11. A spectrophotometric method for the analysis of iron has a linear calibration curve for standards of 0.00, 5.00, 10.00, 15.00, and 20.00 ppm. An iron ore sample with an expected iron content of 40–60% w/w is to be analyzed by this method. An approximately 0.5-g sample is taken, dissolved in a minimum of concentrated HCl, and diluted to 1 L in a volumetric flask using distilled water. A 5.00-mL aliquot is removed with a pipet. To what volume (10, 25, 50, 100, 250, 500, or 1000 mL) should it be diluted to minimize the uncertainty in the analysis? Explain. 12. In a recent paper, Lozano-Calero and colleagues describe a new method for the quantitative analysis of phosphorus in cola beverages. 28 The method is based on the formation of an intensely blue-colored phosphomolybdate complex, (NH 4 ) 3 [PO 4 (MoO 3 ) 12 ]. The complex is formed by adding (NH 4 ) 6 Mo 7 O 24 to the sample in the presence of a reducing agent, such as ascorbic acid. The concentration of the complex is determined spectrophotometrically at a wavelength of 830 nm, using a normal calibration curve as a method of standardization. In a typical analysis, a set of standard solutions containing known amounts of phosphorus was prepared by placing appropriate volumes of a 4.00-ppm solution of P 2 O 5 in a 5-mL volumetric flask, adding 2 mL of the ascorbic acid reducing solution, and diluting to volume with distilled water. Cola beverages were prepared for analysis by pouring a sample into a beaker and allowing it to stand for 24 h to expel the dissolved gases (CO 2 ). A 2.50-mL sample of the degassed sample was transferred to a 50-mL volumetric flask and diluted to volume. A 250-m L aliquot of the diluted sample was then transferred to a 5-mL volumetric flask, treated with 2 mL of the ascorbic acid reducing solution, and diluted to volume with distilled water. (a) The authors note that this method only can be applied to noncolored cola beverages, such as Crystal Pepsi. Clearly explain why this is the case. (b) How might you modify this method so that it could be applied to any cola beverage? (c) Why is it necessary to ensure that the dissolved gases have been removed? (d) How would you prepare a blank for this method? (e) The calibration curve reported by the authors yields a regression line of Abs = –0.02 + 0.72(ppm P 2 O 5 ) A sample of Crystal Pepsi, analyzed as described here, yields an absorbance of 0.565. What is the concentration of phosphorus, reported as parts per million of P, in the original sample of Crystal Pepsi? 13. EDTA forms colored complexes with a variety of metal ions that may serve as the basis for a quantitative spectrophotometric method of analysis. The molar absorptivities of the EDTA complexes of Cu 2+ , Co 2+ , and Ni 2+ at three wavelengths are summarized in the following table (all values of e are in M –1 cm –1 ) Metal e 462.9 e 732.0 e 378.7 Co 2+ 15.8 2.11 3.11 Cu 2+ 2.32 95.2 7.73 Ni 2+ 1.79 3.03 13.5 Using this information, determine (a) the concentration of Cu 2+ in a solution that has an absorbance of 0.338 at a wavelength of 732.0 nm; (b) the concentrations of Cu 2+ and Co 2+ in a solution that has an absorbance of 0.453 at a wavelength of 732.0 nm and 0.107 at a wavelength of 462.9 nm; and (c) the concentrations of Cu 2+ , Co 2+ , and Ni 2+ in a sample that has an absorbance of 0.423 at a wavelength of 732.0 nm, 0.184 at a wavelength of 462.9 nm, and 0.291 at a wavelength of 378.7 nm. The pathlength, b, for all measurements is 1.00 cm. 14. The concentration of phenol in a water sample is determined by separating the phenol from nonvolatile impurities by steam distillation, followed by reacting with 4- aminoantipyrine and K 3 Fe(CN) 6 at pH 7.9 to form a colored antipyrine dye. A phenol standard with a concentration of Chapter 10 Spectroscopic Methods of Analysis 451 1400-CH10 9/8/99 4:19 PM Page 451 4.00 ppm has an absorbance of 0.424 at a wavelength of 460 nm using a 1.00-cm cell. A water sample is steam-distilled, and a 50.00-mL aliquot of the distillate is placed in a 100-mL volumetric flask and diluted to volume with distilled water. The absorbance of this solution is found to be 0.394. What is the concentration of phenol (in parts per million) in the water sample? 15. Saito described a quantitative spectrophotometric procedure for iron based on a solid-phase extraction using bathophenanthroline in a poly(vinyl chloride) membrane. 29 In the absence of Fe 2+ , the membrane is colorless, but when immersed in a solution of Fe 2+ and I – , the membrane develops a red color as a result of the formation of a Fe 2+ –bathophenanthroline complex. A calibration curve determined using a set of external standards with known molar concentrations of Fe 2+ gave a standardization relationship of A = (8.60 · 10 3 M –1 )[Fe 2+ ] What is the concentration of iron in parts per million for a sample with an absorbance of 0.100? 16. In the DPD colorimetric method for the free chlorine residual, which is reported as parts per million of Cl 2 , the oxidizing power of free chlorine converts the colorless amine N,N-diethyl-p-phenylenediamine to a colored dye that absorbs strongly over the wavelength range of 440–580 nm. Analysis of a set of calibration standards gave the following results ppm Cl 2 absorbance 0 0.000 0.50 0.270 1.00 0.543 1.50 0.813 2.00 1.084 A sample from a public water supply is analyzed to determine the free chlorine residual, giving an absorbance of 0.113. What is the free chlorine residual for the sample in parts per million Cl 2 ? 17. Brown and Lin reported a quantitative method for methanol based on its effect on the visible spectrum of methylene blue. 30 In the absence of methanol, the visible spectrum for methylene blue shows two prominent absorption bands centered at approximately 610 nm and 660 nm, corresponding to the monomer and dimer, respectively. In the presence of methanol, the intensity of the dimer’s absorption band decreases, and that of the monomer increases. For concentrations of methanol between 0 and 30% v/v, the ratio of the absorbance at 663 nm, A 663 , to that at 610 nm, A 610 , is a linear function of the amount of methanol. Using the following standardization data, determine the %v/v methanol in a sample for which A 610 is 0.75 and A 663 is 1.07. % Methanol A 663 (v/v) A 610 0 1.21 5.0 1.29 10.0 1.42 15.0 1.52 20.0 1.62 25.0 1.74 30.0 1.84 18. The concentration of the barbiturate barbital in a blood sample was determined by extracting 3.00 mL of the blood with 15 mL of CHCl 3 . The chloroform, which now contains the barbital, is then extracted with 10.0 mL of 0.045 M NaOH (pH ? 13). A 3.00-mL sample of the aqueous extract is placed in a 1.00-cm cell, and an absorbance of 0.115 is measured. The pH of the sample in the absorption cell is then adjusted to approximately 10 by adding 0.5 mL of 16% w/v NH 4 Cl, giving an absorbance of 0.023. When 3.00 mL of a standard barbital solution with a concentration of 3.0 mg/100 mL is taken through the same procedure, the absorbance at pH 13 is 0.295, whereas that at a pH of 10 is 0.002. Report the concentration of barbital, (milligrams per 100 mL), in the sample. 19. Jones and Thatcher developed a spectrophotometric method for analyzing analgesic tablets containing aspirin, phenacetin, and caffeine. 31 The sample is dissolved in CHCl 3 and extracted with an aqueous solution of NaHCO 3 to remove the aspirin. After the extraction is complete, the chloroform is then transferred to a 250-mL volumetric flask and diluted to volume with CHCl 3 . A 2.00-mL portion of this solution is diluted to volume in a 200-mL volumetric flask with CHCl 3 . The absorbance of the final solution is measured at wavelengths of 250 nm and 275 nm, at which the absorptivities, in ppm –1 cm –1 , for caffeine and phenacetin are caffeine: a 250 = 0.0131 and a 275 = 0.0485 phenacetin: a 250 = 0.0702 and a 275 = 0.0159 Aspirin is determined by neutralizing the NaHCO 3 in the aqueous solution and extracting the aspirin into CHCl 3 . The combined extracts are diluted to 500 mL in a volumetric flask. A 20.00-mL portion of the solution is placed in a 100-mL volumetric flask and diluted to volume with CHCl 3 . The absorbance of this solution is measured at 277 nm, where the absorptivity of aspirin is 0.00682 ppm –1 cm –1 . An analgesic tablet treated by this procedure is found to have absorbances of 0.466 at 250 nm, 0.164 at 275 nm, and 0.600 at 277 nm when using a cell with a 1.00-cm pathlength. Report the milligrams of aspirin, caffeine, and phenacetin in the analgesic tablet. 20. The concentration of SO 2 in a sample of air was determined by the p-rosaniline method. The SO 2 was collected in a 10.00- mL solution of HgCl 4 2– , where it forms Hg(SO 3 ) 2 2– , by pulling the air through the solution for 75 min at a rate of 1.6 L/min. 452 Modern Analytical Chemistry 1400-CH10 9/8/99 4:19 PM Page 452 After adding p-rosaniline and formaldehyde, the colored solution was diluted to 25 mL in a volumetric flask. The absorbance was measured at 569 nm in a 1-cm cell, yielding a value of 0.485. A standard sample was prepared by substituting a 1.00-mL sample of a standard solution containing the equivalent of 15.00 ppm SO 2 for the air sample. The absorbance of the standard was found to be 0.181. Report the concentration of SO 2 in the air in parts per million. The density of air may be taken as 1.18 g/L. 21. Seasholtz and colleagues describe a method for the quantitative analysis of CO in automobile exhaust based on the measurement of infrared radiation at 2170 cm –1 . 32 A calibration curve was prepared by filling a 10-cm infrared gas cell with a known pressure of CO and measuring the absorbance using an FT–IR. The standardization relationship was found to be A = –1.1 · 10 –4 + 9.9 · 10 –4 (P CO in torr) Samples were prepared by using a vacuum manifold to fill the gas cell. After measuring the total pressure, the absorbance of the sample at 2170 cm –1 was measured. Results are reported as %CO (P CO /P tot ). Five exhaust samples were obtained from a 1973 coupe, yielding the following results P tot (torr) Absorbance 595 0.1146 354 0.0642 332 0.0591 233 0.0412 143 0.0254 Determine the %CO for each sample, and report the mean value and the 95% confidence interval. 22. A recent innovation in IR sample preparation is the use of disposable sample cards made from thin sheets of either polyethylene (PE) or polytetrafluoroethylene (PTFE). Samples of analyte are dissolved in a suitable solvent and placed on the IR card. After the solvent evaporates, the sample’s spectrum is obtained. Because the thickness of the PE or PTFE film is not uniform, the primary use for IR cards has been for qualitative analysis. Zhao and Malinowski showed how a quantitative analysis for polystyrene could be performed by adding an internal standard of KSCN to the sample. 33 Polystyrene was monitored at 1494 cm –1 and KSCN at 2064 cm –1 . Standard solutions were prepared by placing weighed portions of polystyrene in a 10-mL volumetric flask and diluting to volume with a solution of 10 g/L KSCN in methyl isobutyl ketone. Results obtained with a PE card are shown below g polystyrene 0.1609 0.3290 0.4842 0.6402 0.8006 A 1494 cm –1 0.0452 0.1138 0.1820 0.3275 0.3195 A 2064 cm –1 0.1948 0.2274 0.2525 0.3580 0.2703 When a 0.8006-g sample of a poly(styrene/maleic anhydride) copolymer was prepared in the same manner, the following results were obtained Replicate A 1494 cm –1 A 2064 cm –1 1 0.2729 0.3582 2 0.2074 0.2820 3 0.2785 0.3642 What is the %w/w polystyrene in the copolymer? Given that the reported %w/w polystyrene is 67%, is there any evidence for a determinate error at a = 0.05? 23. The following table lists the molar absorptivities for the Arsenazo complexes of copper and barium at selected wavelengths. 34 Determine the optimum wavelengths for the analysis of a mixture of copper and barium. Wavelength (nm) e Cu e Ba 595 11900 7100 600 15500 7200 607 18300 7400 611 19300 6900 614 19300 7000 620 17800 7100 626 16300 8400 635 10900 9900 641 7500 10500 645 5300 10000 650 3500 8600 655 2200 6600 658 1900 6500 665 1500 3900 670 1500 2800 680 1800 1500 24. Blanco and co-workers 14 reported several examples of the application of multiwavelength linear regression analysis for the simultaneous determination of mixtures containing two components with overlapping spectra. For each of the following, determine the molar concentration of each analyte in the mixture. (a) A mixture of MnO 4 – and Cr 2 O 7 2– , and standards of 1.0 · 10 –4 M KMnO 4 and 1.0 · 10 –4 M K 2 Cr 2 O 7 gave the following results AbsorbancesWavelength (nm) MnO 4 – standard Cr 2 O 7 2– standard Mixture 266 0.042 0.410 0.766 288 0.082 0.283 0.571 320 0.168 0.158 0.422 350 0.125 0.318 0.672 360 0.056 0.181 0.366 Chapter 10 Spectroscopic Methods of Analysis 453 1400-CH10 9/8/99 4:19 PM Page 453 (b) Titanium and vanadium were determined by forming the H 2 O 2 complexes. Results for a mixture of Ti(IV) and V(V) and standards of 63.1 ppm Ti(IV) and 96.4 ppm V(V) are listed in the following table AbsorbancesWavelength (nm) Ti(IV) standard V(V) standard Mixture 390 0.895 0.326 0.651 430 0.884 0.497 0.743 450 0.694 0.528 0.665 470 0.481 0.512 0.547 510 0.173 0.374 0.314 (c) Copper and zinc were determined by forming colored complexes with 2-pyridyl-azo-resorcinol (PAR). The absorbances for PAR, a mixture of Cu 2+ and Zn 2+ , and standards of 1.0-ppm Cu 2+ and 1.0-ppm Zn 2+ are listed in the following table. Note that you must correct the absorbances for the metal for the absorbance due to the PAR. AbsorbancesWavelength (nm) PAR Cu 2+ standard Zn 2+ standard Mixture 480 0.211 0.698 0.971 0.656 496 0.137 0.732 1.018 0.668 510 0.100 0.732 0.891 0.627 526 0.072 0.602 0.672 0.498 540 0.056 0.387 0.306 0.290 25. Jochum and associates 15b reported the following results for the analysis of Ni 2+ and Cu 2+ by the generalized standard addition method D n Ni D n Cu (mol) (mol) D Q l 1 D Q l 2 1.00 · 10 –5 0 0.174 0.058 2.00 · 10 –5 0 0.345 0.093 3.00 · 10 –5 0 0.532 0.198 4.00 · 10 –5 0 0.732 0.266 4.00 · 10 –5 2.00 · 10 –5 1.11 1.81 4.00 · 10 –5 4.00 · 10 –5 1.49 3.28 4.00 · 10 –5 6.00 · 10 –5 1.89 4.88 4.00 · 10 –5 8.00 · 10 –5 2.31 6.44 Determine the moles of Ni 2+ and Cu 2+ in a sample if the volume corrected absorbances at l 1 and l 2 are, respectively, 0.442 and 0.869 before the standard additions. 26. The stoichiometry for a metal–ligand complex was determined by the method of continuous variations. A series of solutions was prepared in which the combined concentrations of the metal and ligand were held constant at 5.15 · 10 –4 M. The absorbances of these solutions were measured at a wavelength at which only the metal–ligand complex absorbs. Using the following data, determine the formula of the complex. Mole fraction M Mole fraction L Absorbance 1.0 0.0 0.001 0.9 0.1 0.126 0.8 0.2 0.260 0.7 0.3 0.389 0.6 0.4 0.515 0.5 0.5 0.642 0.4 0.6 0.775 0.3 0.7 0.771 0.2 0.8 0.513 0.1 0.9 0.253 0.0 1.0 0.000 27. The stoichiometry of a metal–ligand complex was determined by the mole-ratio method. A series of solutions was prepared in which the concentration of the metal was held constant at 3.65 · 10 –4 M, and the concentration of the ligand varied from 1 · 10 –4 M to 1 · 10 –3 M. Using the following data, determine the stoichiometry of the metal–ligand complex. [Ligand] (M) Absorbance 1.0 · 10 –4 0.122 2.0 · 10 –4 0.251 3.0 · 10 –4 0.376 4.0 · 10 –4 0.496 5.0 · 10 –4 0.625 6.0 · 10 –4 0.752 7.0 · 10 –4 0.873 8.0 · 10 –4 0.937 9.0 · 10 –4 0.962 1.0 · 10 –3 1.001 28. The stoichiometry of a metal–ligand complex was determined by the slope-ratio method. Two sets of solutions were prepared. For the first set of solutions the concentration of the metal was held constant at 0.010 M, and the concentration of the ligand was varied. When the absorbance of these solutions was measured at a wavelength at which only the metal–ligand complex absorbs the following data were obtained. [Ligand] (M) Absorbance 1.0 · 10 –5 0.012 2.0 · 10 –5 0.029 3.0 · 10 –5 0.042 4.0 · 10 –5 0.055 5.0 · 10 –5 0.069 For the second set of solutions the concentration of the ligand was held constant at 0.010 M, and the concentration of the metal was varied, yielding the following absorbances. 454 Modern Analytical Chemistry 1400-CH10 9/8/99 4:19 PM Page 454 [Metal] (M) Absorbance 1.0 · 10 –5 0.040 2.0 · 10 –5 0.085 3.0 · 10 –5 0.125 4.0 · 10 –5 0.162 5.0 · 10 –5 0.206 Using this data, determine the stoichiometry of the metal–ligand complex. 29. Kawakami and Igarashi 35 developed a new spectrophotometric method for nitrite based on its reaction with 5,10,15,20-tetrakis(4-aminophenyl)porphrine (TAPP). As part of their study, they investigated the stoichiometry of the reaction between TAPP and NO 2 – ; the following data are derived from a figure in their paper. [TAPP] [NO 2 – ] (M) (M) Absorbance 8.0 · 10 –7 1.6 · 10 –7 0.227 8.0 · 10 –7 3.2 · 10 –7 0.192 8.0 · 10 –7 4.8 · 10 –7 0.158 8.0 · 10 –7 8.0 · 10 –7 0.126 8.0 · 10 –7 1.6 · 10 –6 0.065 8.0 · 10 –7 2.4 · 10 –6 0.047 8.0 · 10 –7 3.2 · 10 –6 0.042 8.0 · 10 –7 4.0 · 10 –6 0.042 What is the stoichiometry of the reaction? 30. The equilibrium constant for an acid–base indicator is determined by preparing three solutions, each of which has a total indicator concentration of 1.35 · 10 –5 M. The pH of the first solution is adjusted until it is acidic enough to ensure that only the acid form of the indicator is present, yielding an absorbance of 0.673. The absorbance of the second solution, whose pH was adjusted to give only the base form of the indicator, was measured at 0.118. The pH of the third solution was adjusted to 4.17 and had an absorbance of 0.439. What is the acidity constant for the acid–base indicator? 31. The acidity constant for an organic weak acid was determined by measuring its absorbance as a function of pH while maintaining a constant total concentration of the acid. Using the data in the following table, determine the acidity constant for the organic weak acid. pH Absorbance 1.53 0.010 2.20 0.010 3.66 0.035 4.11 0.072 4.35 0.103 4.75 0.169 4.88 0.193 5.09 0.227 5.69 0.288 7.20 0.317 7.78 0.317 32. When using a spectrophotometer for which the precision of absorbance measurements is limited by the uncertainty of reading %T, the analysis of highly absorbing solutions can lead to an unacceptable level of indeterminate errors. Consider the analysis of a sample for which the molar absorptivity is 1.0 · 10 4 M –1 cm –1 , and the pathlength is 1.00 cm. (a) What is the relative uncertainty in concentration for an analyte whose concentration is 2.0 · 10 –4 M if s T is – 0.002? (b) What is the relative uncertainty in the concentration if the spectrophotometer is calibrated using a blank consisting of a 1.0 · 10 –4 M solution of the analyte? 33. Hobbins reported the following calibration data for the flame atomic absorption analysis for phosphorus. 36 ppm P Absorbance 2130 0.048 4260 0.110 6400 0.173 8530 0.230 To determine the purity of a sample of Na 2 HPO 4 , a 2.469-g sample is dissolved and diluted to volume in a 100-mL volumetric flask. Analysis of the resulting solution gives an absorbance of 0.135. What is the purity of the Na 2 HPO 4 ? 34. Bonert and Pohl reported results for the atomic absorption analysis of several metals in caustic suspensions produced during the manufacture of soda by the ammonia-soda process. 37 (a) The concentration of Cu was determined by acidifying a 200-mL sample of the caustic solution with 20 mL of concentrated HNO 3 , adding 1 mL of 27% w/v H 2 O 2 and boiling for 30 min. The resulting solution was diluted to 500 mL, filtered, and analyzed by flame atomic absorption using matrix-matched standards. The results for a typical analysis are shown in the following table. Solution ppm Cu Absorbance blank 0 0.007 standard 1 0.200 0.014 standard 2 0.500 0.036 standard 3 1.000 0.072 standard 4 2.000 0.146 sample 0.027 Determine the concentration of Cu in the caustic suspension. (b) The determination of Cr was accomplished by acidifying a 200-mL sample of the caustic solution with 20 mL of concentrated HNO 3 , adding 0.2 g of Na 2 SO 3 and boiling for 30 min. The Cr was isolated from the sample by adding 20 mL Chapter 10 Spectroscopic Methods of Analysis 455 1400-CH10 9/8/99 4:19 PM Page 455 of NH 3 , producing a precipitate that includes the chromium as well as other oxides. The precipitate was isolated by filtration, washed, and transferred with wash water to a beaker. After acidifying with 10 mL of HNO 3 , the solution was evaporated to dryness. The residue was redissolved in a combination of HNO 3 and HCl and evaporated to dryness. Finally, the residue was dissolved in 5 mL of HCl, filtered, diluted to volume in a 50-mL volumetric flask, and analyzed by atomic absorption using the method of standard additions. The atomic absorption results are summarized in the following table. Sample ppm Cr add Absorbance blank 0.001 sample 0.045 standard addition 1 0.200 0.083 standard addition 2 0.500 0.118 standard addition 3 1.000 0.192 Report the concentration of Cr in the caustic suspension. 35. Quigley and Vernon report results for the determination of trace metals in sea water using a graphite furnace atomic absorption spectrophotometer. 38 Calibration was achieved by the method of standard additions. The trace metals were first separated from their complex, high-salt matrix by coprecipitating with Fe 3+ . In a typical analysis a 5.00-mL portion of 2000-ppm Fe 3+ was added to 1.00 L of sea water. The pH was adjusted to 9 using NH 4 OH, and the precipitate of Fe(OH) 3 allowed to stand overnight. After isolating and rinsing the precipitate, the Fe(OH) 3 and coprecipitated metals were dissolved in 2 mL of concentrated HNO 3 and diluted to volume in a 50-mL volumetric flask. To analyze for Mn 2+ , a 1.00-mL sample of this solution was diluted to 100 mL in a volumetric flask. The following samples were injected into the graphite furnace and analyzed Absorbance 2.5-m L sample + 2.5 m L of 0-ppb Mn 2+ standard 0.223 2.5-m L sample + 2.5 m L of 2.5-ppb Mn 2+ standard 0.294 2.5-m L sample + 2.5 m L of 5.0-ppb Mn 2+ standard 0.361 Report the parts per billion of Mn 2+ in the sample of sea water. 36. The concentration of Na in plant materials may be determined by flame atomic emission. The material to be analyzed is prepared by grinding, homogenizing, and drying at 103 °C. A sample of approximately 4 g is transferred to a quartz crucible and heated on a hot plate to char the organic material. The sample is heated in a muffle furnace at 550 °C for several hours. After cooling to room temperature the residue is dissolved by adding 2 mL of 1:1 HNO 3 and evaporated to dryness. The residue is redissolved in 10 mL of 1:9 HNO 3 , filtered, and diluted to 50 mL in a volumetric flask. The following data were obtained during a typical analysis for the concentration of Na in a 4.0264-g sample of oat bran Emission Sample ppm Na (arbitrary units) blank 0 0 standard 1 2.00 90.3 standard 2 4.00 181 standard 3 6.00 272 standard 4 8.00 363 standard 5 10.00 448 sample 238 Determine the parts per million Na in the sample of oat bran. 37. Gluodenis describes the use of ICP to analyze samples containing Pb and Ni in brass. 39 The analysis for Pb uses external standards prepared from brass samples containing known amounts of lead. Results are shown in the following table. %w/w Pb Emission Intensity 0.000 4.29 · 10 4 0.0100 1.87 · 10 5 0.0200 3.20 · 10 5 0.0650 1.28 · 10 6 0.350 6.22 · 10 6 0.700 1.26 · 10 7 1.04 1.77 · 10 7 2.24 3.88 · 10 7 3.15 5.61 · 10 7 9.25 1.64 · 10 8 What is the %w/w Pb in a sample of brass that gives an emission intensity of 9.25 · 10 4 ? The analysis for Ni uses an internal standard. Results for a typical calibration are shown in the following table. %w/w Ni Ratio of Emission Intensity 0.000 0.00267 0.0140 0.00154 0.0330 0.00312 0.130 0.120 0.280 0.246 0.280 0.247 0.560 0.533 1.30 1.20 4.82 4.44 What is the %w/w Ni in a sample for which the ratio of emission intensity is 1.10 · 10 –3 ? 38. Yan and associates developed a method for the analysis of iron based on its formation of a fluorescent metal–ligand complex with the ligand 5-(4-methylphenylazo)-8- aminoquinoline. 40 In the presence of the surfactant cetyltrimethyl ammonium bromide the analysis is carried out 456 Modern Analytical Chemistry 1400-CH10 9/8/99 4:19 PM Page 456 using an excitation wavelength of 316 nm with emission monitored at 528 nm. Standardization with external standards gives I = –0.03 + 1.594(ppm Fe 3+ ) A 0.5113-g sample of dry dog food was ashed to remove organic materials, and the residue dissolved in a small amount of HCl and diluted to volume in a 50-mL volumetric flask. Analysis of the resulting solution gave a fluorescent emission intensity of 5.72. Determine the parts per million of Fe in the sample of dog food. 39. A solution of 5.00 · 10 –5 M 1,3-dihydroxynaphthelene in 2 M NaOH has a fluorescence intensity of 4.85 at a wavelength of 459 nm. What is the concentration of 1,3- dihydroxynaphthelene in a solution with a fluorescence intensity of 3.74 under identical conditions? 40. The following data was recorded for the phosphorescence intensity for several standard solutions of benzo[a]pyrene. Molar Concentration Intensity of benzo[a]pyrene Phosphorescent Emission 00 1.00 · 10 –5 0.98 3.00 · 10 –5 3.22 6.00 · 10 –5 6.25 1.00 · 10 –4 10.21 What is the concentration of benzo[a]pyrene in a sample yielding a phosphorescent emission intensity of 4.97? 41. The concentration of acetylsalicylic acid, C 9 H 8 O 4 , in aspirin tablets can be determined by hydrolyzing to the salicylate ion, C 7 H 5 O 2 – , and determining the concentration of the salicylate ion spectrofluorometrically. A stock standard solution is prepared by weighing 0.0774 g of salicylic acid, C 7 H 6 O 2 , into a 1-L volumetric flask and diluting to volume with distilled water. A set of calibration standards is prepared by pipeting 0, 2.00, 4.00, 6.00, 8.00, and 10.00 mL of the stock solution into separate 100-mL volumetric flasks containing 2.00 mL of 4 M NaOH and diluting to volume with distilled water. The fluorescence of the calibration standards was measured at an emission wavelength of 400 nm using an excitation wavelength of 310 nm; results are listed in the following table. Milliliters of stock Fluorescence standard solution emission intensity 00 2.00 3.02 4.00 5.98 6.00 9.18 8.00 12.13 10.00 14.96 Several aspirin tablets are ground to a fine powder in a mortar and pestle. A 0.1013-g portion of the powder is placed in a 1-L volumetric flask and diluted to volume with distilled water. A portion of this solution is filtered to remove insoluble binders, and a 10.00-mL aliquot transferred to a 100-mL volumetric flask containing 2.00 mL of 4 M NaOH. After diluting to volume the fluorescence of the resulting solution is found to be 8.69. What is the %w/w acetylsalicylic acid in the aspirin tablets? 42. Selenium (IV) in natural waters can be determined by complexing with ammonium pyrrolidine dithiocarbamate and extracting into CHCl 3 . This step serves to concentrate the Se(IV) and to separate it from Se(VI). The Se(IV) is then extracted back into an aqueous matrix using HNO 3 . After complexing with 2,3-diaminonaphthelene, the complex is extracted into cyclohexane. Fluorescence is measured at 520 nm following its excitation at 380 nm. Calibration is by adding known amounts of Se(IV) to the water sample before beginning the analysis. Given the following results, what is the concentration of Se(IV) in the sample? Concentration of Se(IV) Added Fluorescence (nM) Intensity 0 323 2.00 597 4.00 862 6.00 1123 43. Fibrinogen is a protein found in human plasma that is produced by the liver. Its concentration in plasma is clinically important to blood clotting. Many of the analytical methods used to determine the concentration of fibrinogen in plasma are based on light scattering following its precipitation. da Silva, and colleagues describe such a method in which fibrinogen precipitates in the presence of ammonium sulfate in a guanidine hydrochloride buffer. 41 Light scattering is measured nephelometrically at a wavelength of 340 nm. Analysis of a set of external calibration standards gives the following calibration equation. I = –4.66 + 9907.63C A where I is the intensity of scattered light, and C A is the concentration of fibrinogen in grams per liter. A 9.00-mL sample of plasma was collected from a patient and mixed with 1.00-mL of an anticoagulating agent. A 1.00-mL aliquot was then diluted to 250 mL in a volumetric flask. Analysis of the resulting solution gave a scattering intensity of 44.70. What is the concentration of fibrinogen (in grams per liter) in the plasma sample? Chapter 10 Spectroscopic Methods of Analysis 457 1400-CH10 9/8/99 4:19 PM Page 457 458 Modern Analytical Chemistry The history of spectroscopy is discussed in the following sources. Laitenen, H. A.; Ewing, G. W., eds. A History of Analytical Chemistry, The Division of Analytical Chemistry of the American Chemical Society: Washington, D.C., 1977, pp. 103–243. Thomas, N. C. “The Early History of Spectroscopy,” J. Chem. Educ. 1991, 68, 631–633. An explanation of the many units used to describe the energy of electromagnetic radiation is provided in the following paper. Ball, D. W. “Units! Units! Units!” Spectroscopy 1995, 10(8), 44–47. The following sources present a theoretical treatment of the interaction of electromagnetic radiation with matter. Ingle, J. D.; Crouch, S. R. Spectrochemical Analysis. Prentice Hall, Englewood Cliffs, N.J., 1988. Macomber, R. S. “A Unifying Approach to Absorption Spectroscopy at the Undergraduate Level,” J. Chem. Educ. 1997, 74, 65–67. Orchin, M.; Jaffe, H. H. Symmetry, Orbitals and Spectra. Wiley- Interscience: New York, 1971. Two noncalculus-based approaches to discovering the Beer–Lambert law are found in the following papers. Lykos, P. “The Beer–Lambert Law Revisited: A Development without Calculus,” J. Chem. Educ. 1992, 69, 730–732. Ricci, R. W.; Ditzler, M. A.; Nestor, L. P. “Discovering the Beer–Lambert Law,” J. Chem. Educ. 1994, 71, 983–985. The following sources provide further information on the optical characteristics of gratings and their method of production. Grossman, W. E. L. “The Optical Characteristics and Production of Diffraction Gratings,” J. Chem. Educ. 1993, 70, 741–748. Palmer, C. “Diffraction Gratings,” Spectroscopy 1995, 10(2), 14–15. The evaluation of instrumentation for molecular UV/Vis spectroscopy is reviewed in the following pair of papers. Altermose, I. R. “Evolution of Instrumentation for UV–Visible Spectrophotometry: Part I,” J. Chem. Educ. 1986, 63, A216–A223. Altermose, I. R. “Evolution of Instrumentation for UV–Visible Spectrophotometry: Part II,” J. Chem. Educ. 1986, 63, A262–A266. Listed below is a two-part series on the application of photodiode arrays in UV/Vis spectroscopy. Jones, D. G. “Photodiode Array Detectors in UV–Vis Spectroscopy: Part I,” Anal. Chem. 1985, 57, 1057A–1073A. Jones, D. G. “Photodiode Array Detectors in UV-Vis Spectroscopy: Part II,” Anal. Chem. 1985, 11, 1207A–1214A. There is an abundant literature covering the Fourier transform and its application in spectroscopy, several examples of which are listed here. Bracewell, R. N. “The Fourier Transform,” Sci. American 1989, 260(6), 85–95. Glasser, L. “Fourier Transforms for Chemists: Part I. Introduction to the Fourier Transform,” J. Chem. Educ. 1987, 64, A228–A233. Glasser, L. “Fourier Transforms for Chemists: Part II. Fourier Transforms in Chemistry and Spectroscopy,” J. Chem. Educ. 1987, 64, A260–A266. Glasser, L. “Fourier Transforms for Chemists: Part III. Fourier Transforms in Data Treatment,” J. Chem. Educ. 1987, 64, A306–A313. Graff, D. K. “Fourier and Hadamard: Transforms in Spectroscopy,” J. Chem. Educ. 1995, 72, 304–309. Griffiths, P. R. Chemical Fourier Transform Spectroscopy. Wiley- Interscience: New York, 1975. Griffiths, P. R. ed. Transform Techniques in Chemistry. Plenum Press: New York, 1978. Perkins, W. E. “Fourier Transform Infrared Spectroscopy: Part I. Instrumentation,” J. Chem. Educ. 1986, 63, A5–A10. Perkins, W. E. “Fourier Transform Infrared Spectroscopy: Part II. Advantages of FT–IR,” J. Chem. Educ. 1987, 64, A269–A271. Perkins, W. E. “Fourier Transform Infrared Spectroscopy: Part III. Applications,” J. Chem. Educ. 1987, 64, A296–A305. Strong III, F. C. “How the Fourier Transform Infrared Spectrophotometer Works,” J. Chem. Educ. 1979, 56, 681–684. Consult the following sources for more information about reflectance techniques for IR spectroscopy. Leyden, D. E.; Shreedhara Murthy, R. S. “Surface-Selective Sampling Techniques in Fourier Transform Infrared Spectroscopy,” Spectroscopy 1987, 2(2), 28–36. Optical Spectroscopy: Sampling Techniques Manual. Harrick Scientific Corporation: Ossining, N.Y., 1987. Porro, T. J.; Pattacini, S. C. “Sample Handling for Mid-Infrared Spectroscopy, Part I: Solid and Liquid Sampling,” Spectroscopy 1993, 8(7), 40–47. Porro, T. J.; Pattacini, S. C. “Sample Handling for Mid-Infrared Spectroscopy, Part II: Specialized Techniques,” Spectroscopy 1993, 8(8), 39–44. A thorough treatment of the multicomponent quantitative analysis of samples based on Beer’s law, and the analysis of samples for which the pathlength is indeterminate is found in the following review article. Brown, C. W.; Obremski, R. J. “Multicomponent Quantitative Analysis,” Appl. Spectrosc. Rev. 1984, 20, 373–418. 10N SUGGESTED READINGS 1400-CH10 9/8/99 4:19 PM Page 458 For more information on optical luminescence spectroscopy, see the following sources. Guilbault, G. G. Practical Fluorescence. Decker: New York, 1990. Schenk, G. “Historical Overview of Fluorescence Analysis to 1980,” Spectroscopy 1997, 12, 47–56. Vo-Dinh, T. Room-Temperature Phosphorimetry for Chemical Analysis. Wiley-Interscience: New York, 1984. Winefordner, J. D.; Schulman, S. G.; O’Haver, T. C. Luminescence Spectroscopy in Analytical Chemistry. Wiley-Interscience: New York, 1969. The following sources provide additional information on atomic absorption and atomic emission. Blades, M. W.; Weir, D. G. “Fundamental Studies of the Inductively Coupled Plasma,” Spectroscopy 1994, 9, 14–21. Hieftje, G. M. “Atomic Absorption Spectrometry—Has It Gone or Where Is It Going?” J. Anal. At. Spectrom. 1989, 4, 117–122. Koirtyohann, S. R. “A History of Atomic Absorption Spectrometry from an Academic Perspective,” Anal. Chem. 1991, 63, 1024A–1031A. L’Vov, B. V. “Graphite Furnace Atomic Absorption Spectrometry,” Anal. Chem. 1991, 63, 924A–931A. Slavin, W. “A Comparison of Atomic Spectroscopic Analytical Techniques,” Spectroscopy 1991, 6, 16–21. Van Loon, J. C. Analytical Atomic Absorption Spectroscopy. Academic Press: New York, 1980. Walsh, A. “The Development of Atomic Absorption Methods of Elemental Analysis 1952–1962,” Anal. Chem. 1991, 63, 933A–941A. Welz, B. Atomic Absorption Spectroscopy. VCH: Deerfield Beach, FL, 1985. Several texts may be consulted for further examples of spectroscopic methods of analysis. Christian, G. D.; Callis, J. B. eds. Trace Analysis and Spectroscopic Methods for Molecules. Wiley-Interscience: New York, 1986. Skoog, D. A.; Holler, F. J.; Nieman, T. A. Principles of Instrumental Analysis. Saunders College Publishing: Philadelphia, 1998. Vandecasteele, C.; Block, C. B. Modern Methods for Trace Element Determination. Wiley: Chichester, England, 1994. Van Loon, J. C. Selected Methods of Trace Metal Analysis: Biological and Environmental Samples. Wiley-Interscience: New York, 1985. Chapter 10 Spectroscopic Methods of Analysis 459 1. Method 417B in Standard Methods for the Analysis of Water and Wastewater, American Public Health Association: Washington, D.C.; 15 ed., 1981, pp. 356–360. 2. (a) Sheppard, N. In Andrews, D. L. Ed. Perspectives in Modern Chemical Spectroscopy. Springer-Verlag: Berlin, 1990, pp. 1–41; (b) Tyson, J. Analysis: What Analytical Chemists Do. Royal Society of Chemistry: Letchworth, England, 1988, pp. 49–52. 3. Home, D.; Gribbin, J. New Scientist 1991, 2 November, 30–33. 4. Ball, D. W. Spectroscopy 1994, 9(5), 24–25. 5. Ball, D. W. Spectroscopy 1994, 9(6), 20–21. 6. Jiang, S; Parker, G. A. Am. Lab. 1981, October, 38–43. 7. Strong, F. C., III Anal. Chem. 1976, 48, 2155–2161. 8. Gilbert, D. D. J. Chem. Educ. 1991, 68, A278–A281. 9. Seitz, W. R. Anal. Chem. 1984, 56, 16A–34A. 10. Angel, S. M. Spectroscopy 1987, 2(2), 38–48. 11. Method 3500-Fe-B as published in Standard Methods for the Analysis of Water and Wastewater. American Public Health Association: Washington, D.C., 19th ed., 1998, pp. 3–76 to 3–78. 12. DiTusa, M. R.; Schilt, A. A. J. Chem. Educ. 1985, 62, 541–542. 13. Mehra, M. C.; Rioux, J. J. Chem. Educ. 1982, 59, 688–689. 14. Blanco, M.; Iturriaga, H.; Maspoch, S.; et al. Chem. Educ. 1989, 66, 178–180. 15. (a) Saxberg, B. E. H.; Kowalski, B. R. Anal. Chem. 1979, 51, 1031–1038; (b) Jochum, C.; Jochum, P.; Kowalski, B. R. Anal. Chem. 1981, 53, 85–92; (c) Gerlach, R. W.; Kowalski, B. R. Anal. Chim. Acta. 1982, 134, 119; (d) Kalivas, J. H.; Kowalski, B. R. Anal. Chem. 1981, 53, 2207–2212; Raymond, M.; Jochum, C.; Kowalski, B. R. J. Chem. Educ. 1983, 60, 1072–1073. 16. Ramette, R. W. Chemical Equilibrium and Analysis. Addison-Wesley: Reading, MA, 1981, Chapter 13. 17. Rothman, L. D.; Crouch, S. R.; Ingle, J. D. Jr. Anal. Chem. 1975, 47, 1226–1233. 18. (a) Walsh, A. Anal. Chem. 1991, 63, 933A–941A; (b) Koirtyohann, S. R. Anal. Chem. 1991, 63, 1024A–1031A; (c) Slavin, W. Anal. Chem. 1991, 63, 1033A–1038A. 19. Ingle, J. D.; Crouch, S. R. Spectrochemical Analysis. Prentice-Hall: Englewood Cliffs, N.J., 1988; p. 275. 20. Parsons, M. L.; Major, S.; Forster, A. R. Appl. Spectrosc. 1983, 37, 411–418. 21. Hosking, J. W.; Snell, N. B.; Sturman, B. T. J. Chem. Educ. 1977, 54, 128–130. 22. (a) Bhattacharya, S. K.; Goodwin, T. G.; Crawford, A. J. Anal. Lett. 1984, 17, 1567–1593;(b) Crawford, A. J.; Bhattacharya, S. K. “Microanalysis of Copper and Zinc in Biopsy-Sized Tissue Specimens by Atomic Absorption Spectroscopy Using a Stoichiometric Air–Acetylene Flame”; Varian Instruments at Work, Number AA-46, April 1985. 23. (a) Mule, S. J.; Hushin, P. L. Anal. Chem. 1971, 43, 708–711; (b) O’Reilly, J. E. J. Chem. Educ. 1975, 52, 610–612. 24. Dawson, J. B. J. Anal. At. Spectrosc. 1991, 6, 93–98. 25. Goodney, D. E. J. Chem. Educ. 1982, 59, 875–876. 10O REFERENCES 1400-CH10 9/8/99 4:19 PM Page 459 26. Hach, C. C.; Bryant, M. “Turbidity Standards”; Technical Information Series-Booklet No. 12, Hach Company: Loveland, CO, 1995. 27. Method 4500-SO 4 2– -E in Standard Methods for the Analysis of Water and Wastewater, American Public Health Association: Washington, D.C.; 20th ed., 1998, pp. 4–178 to 4–179. 28. Lozano-Calero, D.; Martín-Palomeque, P.; Madue?o-Loriguillo, S. J. Chem. Educ. 1996, 73, 1173–1174. 29. Saito, T. Anal. Chim. Acta. 1992, 268, 351–355. 30. Lin, J.; Brown, C. W. Spectroscopy 1995, 10(5), 48–51. 31. Jones, M.; Thatcher, R. L. Anal. Chem. 1951, 23, 957–960. 32. Seaholtz, M. B.; Pence, L. E.; Moe, O. A. Jr. J. Chem. Educ. 1988, 65, 820–823. 33. Zhao, Z.; Malinowski, E. R. Spectroscopy 1996, 11(7), 44–49. 34. Grossman, O.; Turanov, A. N. Anal. Chim. Acta. 1992, 257, 195–202. 35. Kawakami, T.; Igarashi, S. Anal. Chim. Acta. 1996, 333, 175–180. 36. Hobbins, W. B. “Direct Determination of Phosphorus in Aqueous Matrices by Atomic Absorption”; Varian Instruments at Work, Number AA-19, February 1982. 37. Bonert, K.; Pohl, B. “The Determination of Cd, Cr, Cu, Ni, and Pb in Concentrated CaCl 2 /NaCl Solutions by AAS”; AA Instruments at Work (Varian), Number 98, November 1990. 38. Quigley, M. N.; Vernon, F. J. Chem. Educ. 1996, 73, 671–673. 39. Gluodenis, Jr., T. J. Am. Lab. November 1998, 245–275. 40. Yan, G.; Shi, G; Liu, Y. Anal. Chim. Acta. 1992, 264, 121–124. 41. da Silva, M. P.; Fernández-Romero, J. M.; Luque de Castro, M. D. Anal. Chim. Acta. 1996, 327, 101–106. 460 Modern Analytical Chemistry 1400-CH10 9/8/99 4:19 PM Page 460 Chapter 11 461 Electrochemical Methods of Analysis In Chapter 10 we examined several analytical methods based on the interaction of electromagnetic radiation with matter. In this chapter we turn our attention to analytical methods in which a measurement of potential, current, or charge in an electrochemical cell serves as the analytical signal. 1400-CH11 9/9/99 2:06 PM Page 461 462 Modern Analytical Chemistry 11A Classification of Electrochemical Methods Although there are only three principal sources for the analytical signal—potential, current, and charge—a wide variety of experimental designs are possible; too many, in fact, to cover adequately in an introductory textbook. The simplest division is be- tween bulk methods, which measure properties of the whole solution, and interfa- cial methods, in which the signal is a function of phenomena occurring at the inter- face between an electrode and the solution in contact with the electrode. The measurement of a solution’s conductivity, which is proportional to the total con- centration of dissolved ions, is one example of a bulk electrochemical method. A determination of pH using a pH electrode is one example of an interfacial electro- chemical method. Only interfacial electrochemical methods receive further consid- eration in this text. 11A.1 Interfacial Electrochemical Methods The diversity of interfacial electrochemical methods is evident from the partial family tree shown in Figure 11.1. At the first level, interfacial electrochemical methods are divided into static methods and dynamic methods. In static methods no current passes between the electrodes, and the concentrations of species in the electrochemical cell remain unchanged, or static. Potentiometry, in which the po- tential of an electrochemical cell is measured under static conditions, is one of the most important quantitative electrochemical methods, and is discussed in detail in Section 11B. The largest division of interfacial electrochemical methods is the group of dy- namic methods, in which current flows and concentrations change as the result of a redox reaction. Dynamic methods are further subdivided by whether we choose to control the current or the potential. In controlled-current coulometry, which is covered in Section 11C, we completely oxidize or reduce the analyte by passing a fixed current through the analytical solution. Controlled-potential methods are subdivided further into controlled-potential coulometry and amperometry, in which a constant potential is applied during the analysis, and voltammetry, in which the potential is systematically varied. Controlled-potential coulometry is dis- cussed in Section 11C, and amperometry and voltammetry are discussed in Section 11D. 11A.2 Controlling and Measuring Current and Potential Electrochemical measurements are made in an electrochemical cell, consisting of two or more electrodes and associated electronics for controlling and measuring the current and potential. In this section the basic components of electrochemical in- strumentation are introduced. Specific experimental designs are considered in greater detail in the sections that follow. The simplest electrochemical cell uses two electrodes. The potential of one of the electrodes is sensitive to the analyte’s concentration and is called the working, or indicator electrode. The second electrode, which is called the counter electrode, serves to complete the electric circuit and provides a reference potential against which the working electrode’s potential is measured. Ideally the counter electrode’s potential remains constant so that any change in the overall cell potential is attrib- uted to the working electrode. In a dynamic method, where the passage of current changes the concentration of species in the electrochemical cell, the potential of the counter electrode may change over time. This problem is eliminated by replacing the counter electrode with two electrodes: a reference electrode, through which no counter electrode The second electrode in a two-electrode cell that completes the circuit. reference electrode An electrode whose potential remains constant and against which other potentials can be measured. indicator electrode The electrode whose potential is a function of the analyte’s concentration (also known as the working electrode). 1400-CH11 9/9/99 2:06 PM Page 462 Figure 11.1 Partial family tree for interfacial electrochemical methods of analysis. current flows and whose potential remains constant; and an auxiliary electrode that completes the electric circuit and through which current is allowed to flow. Although many different electrochemical methods of analysis are possible (Fig- ure 11.1) there are only three basic experimental designs: (1) measuring the potential under static conditions of no current flow; (2) measuring the potential while con- trolling the current; and (3) measuring the current while controlling the potential. Each of these experimental designs, however, is based on Ohm’s law that a current, i, passing through an electric circuit of resistance, R, generates a potential, E; thus E = iR Each of these experimental designs also uses a different type of instrument. To aid in understanding how they control and measure current and potential, these in- struments are described as if they were operated manually. To do so the analyst Chapter 11 Electrochemical Methods of Analysis 463 Static methods (i = 0) Controlled potential Variable potential Stripping voltammetry Fixed potential Cyclic voltammetry Controlled current Controlled current coulometry Polarography and stationary electrode voltammetry Pulse polarography and voltammetry Dynamic methods (i ≠ 0) Potentiometry Stirred solution Quiescent solution Voltammetry Hydrodynamic voltammetry Controlled potential coulometry Amperometry Interfacial electrochemical methods auxiliary electrode The third electrode in a three-electrode cell that completes the circuit. Ohm’s law The statement that the current moving through a circuit is proportional to the applied potential and inversely proportional to the circuit’s resistance (E = iR). 1400-CH11 9/9/99 2:06 PM Page 463 Figure 11.2 Schematic diagram of a manual potentiostat: C = counter electrode; W = working electrode; SW = slide-wire resistor; T = tap key; i = galvanometer. observes a change in current or potential and manually adjusts the instrument’s set- tings to maintain the desired experimental conditions. It is important to understand that modern electrochemical instruments provide an automated, electronic means of controlling and measuring current and potential. They do so by using very differ- ent electronic circuitry than that shown here. Further details about such instru- ments can be found in the suggested readings listed at the end of the chapter. Potentiometers Measuring the potential of an electrochemical cell under condi- tions of zero current is accomplished using a potentiometer. A schematic diagram of a manual potentiometer is shown in Figure 11.2. The current in the upper half of the circuit is where E PS is the power supply’s potential, and R ab is the resistance between points a and b of the slide-wire resistor. In a similar manner, the current in the lower half of the circuit is where E cell is the potential difference between the working electrode and the counter electrode, and R cb is the resistance between the points c and b of the slide-wire resis- tor. When i up = i low =0 no current flows through the galvanometer and the cell potential is given by To make a measurement the tap key is pressed momentarily, and the current is noted at the galvanometer. If a nonzero current is registered, then the slide wire is adjusted and the current remeasured. This process is continued until the gal- vanometer registers a current of zero. Using the tap key minimizes the total amount of current allowed to flow through the cell. Provided that the total cur- rent is negligible, the change in the analyte’s concentration is insignificant. For example, a current of 10 –9 A drawn for 1 s consumes only about 10 –14 mol of analyte. Modern potentiometers use operational amplifiers to create a high- impedance voltmeter capable of measuring potentials while drawing currents of less than 10 –9 A. Galvanostats A galvanostat is used for dynamic methods, such as constant-current coulometry, in which it is necessary to control the current flowing through an elec- trochemical cell. A schematic diagram of a manual constant-current galvanostat is shown in Figure 11.3. If the resistance, R, of the galvanostat is significantly larger than the resistance of the electrochemical cell, and the applied voltage from the power supply is much greater than the cell potential, then the current between the auxiliary and working electrodes is equal to i E R = PS E R R E cb ab cell PS =× i E R low cell cb = i E R ab up PS = 464 Modern Analytical Chemistry potentiometer A device for measuring the potential of an electrochemical cell without drawing a current or altering the cell’s composition. acb ? T SW Electrochemical cell CW i Power supply galvanostat A device used to control the current in an electrochemical cell. 1400-CH11 9/9/99 2:06 PM Page 464 Figure 11.3 Schematic diagram of a galvanostat: R = resistor; i = galvanometer; A = auxiliary electrode; W = working electrode; R = reference electrode; V = voltmeter or potentiometer (optional). Chapter 11 Electrochemical Methods of Analysis 465 i Power supply R A W R V SW AW i Power supply R V Figure 11.4 Schematic diagram of a manual potentiostat: SW = slide-wire resistor; A = auxiliary electrode; R = reference electrode; W = working electrode; V = voltmeter or potentiometer; i = galvanometer. potentiostat A device used to control the potential in an electrochemical cell. The potential of the working electrode, which changes as the composition of the electrochemical cell changes, is monitored by including a reference electrode and a high-impedance potentiometer. Potentiostats A potentiostat is used for dynamic methods when it is necessary to control the potential of the working electrode. Figure 11.4 shows a schematic dia- gram for a manual potentiostat that can be used to maintain a constant cell poten- tial. The potential of the working electrode is monitored by a reference electrode connected to the working electrode through a high-impedance potentiometer. The desired potential is achieved by adjusting the slide-wire resistor connected to the auxiliary electrode. If the working electrode’s potential begins to drift from the de- sired value, then the slide-wire resistor is manually readjusted, returning the poten- tial to its initial value. The current flowing between the auxiliary and working elec- trodes is measured with a galvanostat. Modern potentiostats include waveform generators allowing a time-dependent potential profile, such as a series of potential pulses, to be applied to the working electrode. 11B Potentiometric Methods of Analysis In potentiometry the potential of an electrochemical cell is measured under static conditions. Because no current, or only a negligible current, flows while measuring a solution’s potential, its composition remains unchanged. For this reason, poten- tiometry is a useful quantitative method. The first quantitative potentiometric ap- plications appeared soon after the formulation, in 1889, of the Nernst equation re- lating an electrochemical cell’s potential to the concentration of electroactive species in the cell. 1 When first developed, potentiometry was restricted to redox equilibria at metallic electrodes, limiting its application to a few ions. In 1906, Cremer discov- ered that a potential difference exists between the two sides of a thin glass mem- brane when opposite sides of the membrane are in contact with solutions contain- ing different concentrations of H 3 O + . This discovery led to the development of the glass pH electrode in 1909. Other types of membranes also yield useful potentials. Kolthoff and Sanders, for example, showed in 1937 that pellets made from AgCl could be used to determine the concentration of Ag + . Electrodes based on mem- brane potentials are called ion-selective electrodes, and their continued develop- ment has extended potentiometry to a diverse array of analytes. 1400-CH11 9/9/99 2:06 PM Page 465 Figure 11.5 Electrochemical cell for potentiometry. 466 Modern Analytical Chemistry 11B.1 Potentiometric Measurements Potentiometric measurements are made using a potentiometer to determine the dif- ference in potential between a working or, indicator, electrode and a counter elec- trode (see Figure 11.2). Since no significant current flows in potentiometry, the role of the counter electrode is reduced to that of supplying a reference potential; thus, the counter electrode is usually called the reference electrode. In this section we in- troduce the conventions used in describing potentiometric electrochemical cells and the relationship between the measured potential and concentration. Potentiometric Electrochemical Cells A schematic diagram of a typical potentio- metric electrochemical cell is shown in Figure 11.5. Note that the electrochemical cell is divided into two half-cells, each containing an electrode immersed in a solu- tion containing ions whose concentrations determine the electrode’s potential. This separation of electrodes is necessary to prevent the redox reaction from occurring spontaneously on the surface of one of the electrodes, short-circuiting the electro- chemical cell and making the measurement of cell potential impossible. A salt bridge containing an inert electrolyte, such as KCl, connects the two half-cells. The ends of the salt bridge are fixed with porous frits, allowing ions to move freely be- tween the half-cells and the salt bridge, while preventing the contents of the salt bridge from draining into the half-cells. This movement of ions in the salt bridge completes the electric circuit. By convention, the electrode on the left is considered to be the anode, where oxidation occurs Zn(s) t Zn 2+ (aq)+2e – and the electrode on the right is the cathode, where reduction occurs Ag + (aq)+e – t Ag(s) The electrochemical cell’s potential, therefore, is for the reaction Zn(s) + 2Ag + (aq) t 2Ag(s)+Zn 2+ (aq) 2e – Zn e – Ag Ag + Zn 2+ Cl – Cl – Cl – K + KCl Anode Cathode 0.0167 M ZnCl 2 0.100 M AgNO 3 NO 3 – Salt bridge Porous frit Potentiometer salt bridge A connection between two solutions that allows the movement of current in the form of ionic charge. anode The electrode where oxidation occurs. cathode The electrode where reduction occurs. 1400-CH11 9/9/99 2:06 PM Page 466 Chapter 11 Electrochemical Methods of Analysis 467 Also, by convention, potentiometric electrochemical cells are defined such that the indicator electrode is the cathode (right half-cell) and the reference electrode is the anode (left half-cell). Shorthand Notation for Electrochemical Cells Although Figure 11.5 provides a useful picture of an electrochemical cell, it does not provide a convenient repre- sentation. A more useful representation is a shorthand, or schematic, notation that uses symbols to indicate the different phases present in the electrochemical cell, as well as the composition of each phase. A vertical slash (|) indicates a phase boundary where a potential develops, and a comma (,) separates species in the same phase, or two phases where no potential develops. Shorthand cell nota- tions begin with the anode and continue to the cathode. The electrochemical cell in Figure 11.5, for example, is described in shorthand notation as Zn(s) | ZnCl 2 (aq, 0.0167 M) || AgNO 3 (aq, 0.100 M) | Ag(s) The double vertical slash (||) indicates the salt bridge, the contents of which are nor- mally not indicated. Note that the double vertical slash implies that there is a poten- tial difference between the salt bridge and each half-cell. EXAMPLE 11.1 What are the anodic, cathodic, and overall reactions responsible for the potential in the electrochemical cell shown here? Write the shorthand notation for the electrochemical cell. Ag Pt (0.100 M) HCl KCl AgCl Potentiometer (0.0100 M) FeCl 3 FeCl 2 (0.0500 M) SOLUTION The oxidation of Ag to Ag + occurs at the anode (the left-hand cell). Since the solution contains a source of Cl – , the anodic reaction is Ag(s)+Cl – (aq) t AgCl(s)+e – 1400-CH11 9/9/99 2:06 PM Page 467 The cathodic reaction (the right-hand cell) is the reduction of Fe 3+ to Fe 2+ Fe 3+ (aq)+e – t Fe 2+ (aq) The overall cell reaction, therefore, is Ag(s)+Fe 3+ (aq)+Cl – (aq) t AgCl(s)+Fe 2+ (aq) The electrochemical cell’s shorthand notation is Ag(s) | HCl (aq, 0.100 M), AgCl (sat’d) || FeCl 2 (aq, 0.0100 M), FeCl 3 (aq, 0.0500 M) | Pt Note that the Pt cathode is an inert electrode that carries electrons to the reduction half-reaction. The electrode itself does not undergo oxidation or reduction. Potential and Concentration The Nernst Equation The potential of a potentio- metric electrochemical cell is given as E cell = E c – E a 11.1 where E c and E a are reduction potentials for the reactions occurring at the cathode and anode. These reduction potentials are a function of the concentrations of those species responsible for the electrode potentials, as given by the Nernst equation where E° is the standard-state reduction potential, R is the gas constant, T is the temperature in Kelvins, n is the number of electrons involved in the reduction reaction, F is Faraday’s constant, and Q is the reaction quotient.* Under typical laboratory conditions (temperature of 25 °C or 298 K) the Nernst equation becomes 11.2 where E is given in volts. Using equation 11.2 the potential of the anode and cathode in Figure 11.5 are Note, again, that the Nernst equations for both E c and E a are written for reduction reactions. The cell potential, therefore, is 11.3EE E cell Ag Ag + Zn Zn 2+ +2+ Ag Zn =? ? ? ? ? ? ? ?? ? ? ? ? ? ? °° // . log [] . log [] 0 05916 1 0 05916 2 1 EE c Ag Ag + + Ag =? ° / . log [] 0 05916 1 EE a Zn Zn 2+ 2+ Zn =? ° / . log [] 0 05916 2 1 EE n Q=? ° 0 05916. log EE RT nF Q=? ° ln 468 Modern Analytical Chemistry *See Chapter 6 for a review of the Nernst equation. 1400-CH11 9/9/99 2:06 PM Page 468 Chapter 11 Electrochemical Methods of Analysis 469 Substituting known values for the standard-state reduction potentials (see Appen- dix 3D) and the concentrations of Ag + and Zn 2+ , gives a potential for the electro- chemical cell in Figure 11.5 of EXAMPLE 11.2 What is the potential of the electrochemical cell shown in Example 11.1? SOLUTION The potential for the electrochemical cell is In potentiometry, the concentration of analyte in the cathodic half-cell is gen- erally unknown, and the measured cell potential is used to determine its concentra- tion. Thus, if the potential for the cell in Figure 11.5 is measured at +1.50 V, and the concentration of Zn 2+ remains at 0.0167 M, then the concentration of Ag + is deter- mined by making appropriate substitutions to equation 11.3 Solving for [Ag + ] gives its concentration as 0.0118 M. EXAMPLE 11. 3 What is the concentration of Fe 3+ in an electrochemical cell similar to that shown in Example 11.1 if the concentration of HCl in the left-hand cell is 1.0 M, the concentration of FeCl 2 in the right-hand cell is 0.0151 M and the measured potential is +0.546 V? SOLUTION Making appropriate substitutions into the Nernst equation for the electrochemical cell (see Example 11.2) and solving for [Fe 3+ ] gives its concentration as 0.0136 M. E cell 3+ [Fe =+ = + ? ? ? ? ? ? ? ?+ ?0 546 0 771 0 05916 0 0151 0 2223 0 05916 1 0. . . log . ] [ . . log( . )] E cell + Ag =+ =+ ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? 1 0 7996 0 05916 1 0 7618 0 05916 2 1 0 0167 .50 V . . log [] . . log . EE E cell Fe Fe 2+ 3+ AgCl/Ag 3+ 2+ Fe Fe Cl V =? ? ? ? ? ? ? ?? =+ ? ? ? ? ? ? ? ?+ ? =+ °°? / . log [] [] ( . log[ ]) . . log . . [ . . log ( . )] . 0 05916 0 05916 0 771 0 05916 0 0100 0 0500 0 2223 0 05916 0 100 0 531 E cell V =+ ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? =+ 0 7996 0 05916 1 0 100 0 7618 0 05916 2 1 0 0167 1 555 . . log . . . log . . 1400-CH11 9/9/99 2:06 PM Page 469 Figure 11.6 Origin of liquid junction potential between solutions of 0.1 M HCl and 0.01 M HCl. 470 Modern Analytical Chemistry Despite the apparent ease of determining an analyte’s concentration using the Nernst equation, several problems make this approach impractical. One problem is that standard-state potentials are temperature-dependent, and most values listed in reference tables are for a temperature of 25 °C. This difficulty can be overcome by maintaining the electrochemical cell at a temperature of 25 °C or by measuring the standard-state potential at the desired temperature. Another problem is that the Nernst equation is a function of activities, not con- centrations.* As a result, cell potentials may show significant matrix effects. This problem is compounded when the analyte participates in additional equilibria. For example, the standard-state potential for the Fe 3+ /Fe 2+ redox couple is +0.767 V in 1 M HClO 4 , +0.70 V in 1 M HCl, and +0.53 in 10 M HCl. The shift toward more negative potentials with an increasing concentration of HCl is due to chloride’s ability to form stronger complexes with Fe 3+ than with Fe 2+ . This problem can be minimized by replacing the standard-state potential with a matrix-dependent for- mal potential. Most tables of standard-state potentials also include a list of selected formal potentials (see Appendix 3D). A more serious problem is the presence of additional potentials in the electro- chemical cell, not accounted for by equation 11.1. In writing the shorthand nota- tion for the electrochemical cell in Figure 11.5, for example, we use a double slash (||) for the salt bridge, indicating that a potential difference exists at the interface between each end of the salt bridge and the solution in which it is immersed. The origin of this potential, which is called a liquid junction potential, and its signifi- cance are discussed in the following section. Liquid Junction Potentials A liquid junction potential develops at the interface between any two ionic solutions that differ in composition and for which the mo- bility of the ions differs. Consider, for example, solutions of 0.1 M HCl and 0.01 M HCl separated by a porous membrane (Figure 11.6a). Since the concentration of HCl on the left side of the membrane is greater than that on the right side of the membrane, there is a net diffusion of H + and Cl – in the direction of the arrows. The mobility of H + , however, is greater than that for Cl – , as shown by the difference in the H + Cl – 0.1 M HCl 0.01 M HCl 0.1 M HCl 0.01 M HCl Excess of Cl – Excess of H + – – – – – + + + + + (a) (b) liquid junction potential A potential that develops at the interface between two ionic solutions that differ in composition, because of a difference in the mobilities of the ions (E lj ). * See Chapter 6 for a review of activity. 1400-CH11 9/9/99 2:06 PM Page 470 lengths of their respective arrows. As a result, the solution on the right side of the membrane develops an excess of H + and has a positive charge (Figure 11.6b). Simul- taneously, the solution on the left side of the membrane develops a negative charge due to the greater concentration of Cl – . The difference in potential across the mem- brane is called a liquid junction potential, E lj . The magnitude of the liquid junction potential is determined by the ionic com- position of the solutions on the two sides of the interface and may be as large as 30–40 mV. For example, a liquid junction potential of 33.09 mV has been measured at the interface between solutions of 0.1 M HCl and 0.1 M NaCl. 2 The magnitude of a salt bridge’s liquid junction potential is minimized by using a salt, such as KCl, for which the mobilities of the cation and anion are approximately equal. The magni- tude of the liquid junction potential also is minimized by incorporating a high con- centration of the salt in the salt bridge. For this reason salt bridges are frequently constructed using solutions that are saturated with KCl. Nevertheless, a small liquid junction potential, generally of unknown magnitude, is always present. When the potential of an electrochemical cell is measured, the contribution of the liquid junction potential must be included. Thus, equation 11.1 is rewritten as E cell = E c – E a + E lj Since the junction potential is usually of unknown value, it is normally impossible to directly calculate the analyte’s concentration using the Nernst equation. Quanti- tative analytical work is possible, however, using the standardization methods dis- cussed in Chapter 5. 11B.2 Reference Electrodes Potentiometric electrochemical cells are constructed such that one of the half-cells provides a known reference potential, and the potential of the other half-cell indi- cates the analyte’s concentration. By convention, the reference electrode is taken to be the anode; thus, the shorthand notation for a potentiometric electrochemical cell is Reference || Indicator and the cell potential is E cell = E ind – E ref + E lj The ideal reference electrode must provide a stable potential so that any change in E cell is attributed to the indicator electrode, and, therefore, to a change in the ana- lyte’s concentration. In addition, the ideal reference electrode should be easy to make and to use. Three common reference electrodes are discussed in this section. Standard Hydrogen Electrode The standard hydrogen electrode (SHE) is rarely used for routine analytical work, but is important because it is the reference elec- trode used to establish standard-state potentials for other half-reactions. The SHE consists of a Pt electrode immersed in a solution in which the hydrogen ion activity is 1.00 and in which H 2 gas is bubbled at a pressure of 1 atm (Figure 11.7). A con- ventional salt bridge connects the SHE to the indicator half-cell. The shorthand no- tation for the standard hydrogen electrode is Pt(s), H 2 (g, 1 atm) | H + (aq, a = 1.00) || and the standard-state potential for the reaction 2H + (aq)+e – t H 2 (g) Chapter 11 Electrochemical Methods of Analysis 471 standard hydrogen electrode Reference electrode based on the reduction of H + (aq) to H 2 (g) at a Pt electrode; that is, H + (aq)+e – t H 2 (g). 1 2 1400-CH11 9/9/99 2:06 PM Page 471 Figure 11.7 Schematic diagram of the standard hydrogen electrode (SHE). is, by definition, 0.00 V for all temperatures. Despite its importance as the funda- mental reference electrode against which all other potentials are measured, the SHE is rarely used because it is difficult to prepare and inconvenient to use. Calomel Electrodes Calomel reference electrodes are based on the redox couple between Hg 2 Cl 2 and Hg (calomel is a common name for Hg 2 Cl 2 ). Hg 2 Cl 2 (s) +2e – t 2Hg(l) + 2Cl – (aq) The Nernst equation for the calomel electrode is The potential of a calomel electrode, therefore, is determined by the concentration of Cl – . The saturated calomel electrode (SCE), which is constructed using an aqueous solution saturated with KCl, has a potential at 25 °C of +0.2444 V. A typical SCE is shown in Figure 11.8 and consists of an inner tube, packed with a paste of Hg, Hg 2 Cl 2 , and saturated KCl, situated within a second tube filled with a saturated so- lution of KCl. A small hole connects the two tubes, and an asbestos fiber serves as a salt bridge to the solution in which the SCE is immersed. The stopper in the outer tube may be removed when additional saturated KCl is needed. The shorthand no- tation for this cell is Hg(l)|Hg 2 Cl 2 (sat’d), KCl (aq, sat’d) || The SCE has the advantage that the concentration of Cl – , and, therefore, the poten- tial of the electrode, remains constant even if the KCl solution partially evaporates. On the other hand, a significant disadvantage of the SCE is that the solubility of KCl is sensitive to a change in temperature. At higher temperatures the concentration of Cl – increases, and the electrode’s potential decreases. For example, the potential of EE=? =+? °? ? Hg Cl /Hg 22 Cl Cl 0 05916 2 0 2682 0 05916 2 22 . log[ ] . . log[ ] 472 Modern Analytical Chemistry Pt H 2 To meter To indicator half-cell H 2 (1 atm) Salt bridge H + (activity = 1.00) saturated calomel electrode Reference electrode based on the reduction of Hg 2 Cl 2 to Hg in an aqueous solution saturated with KCl; that is, Hg 2 Cl 2 (s)+2e – t 2Hg(l) + 2Cl – (aq). 1400-CH11 9/9/99 2:06 PM Page 472 Figure 11.8 Schematic diagram of the saturated calomel electrode (SCE). the SCE at 35 °C is +0.2376 V. Electrodes containing unsaturated solutions of KCl have potentials that are less temperature-dependent, but experience a change in po- tential if the concentration of KCl increases due to evaporation. Another disadvan- tage to calomel electrodes is that they cannot be used at temperatures above 80 °C. Silver/Silver Chloride Electrodes Another common reference electrode is the silver/silver chloride electrode, which is based on the redox couple between AgCl and Ag. AgCl(s)+e – t Ag(s)+Cl – (aq) As with the saturated calomel electrode, the potential of the Ag/AgCl electrode is determined by the concentration of Cl – used in its preparation. E + E AgCl/Ag – 0.05916 log [Cl – ] = +0.2223 – 0.05916 log [C1 – ] When prepared using a saturated solution of KCl, the Ag/AgCl electrode has a potential of +0.197 V at 25 °C. Another common Ag/AgCl electrode uses a so- lution of 3.5 M KCl and has a potential of +0.205 at 25 °C. The Ag/AgCl elec- trode prepared with saturated KCl, of course, is more temperature-sensitive than one prepared with an unsaturated solution of KCl. A typical Ag/AgCl electrode is shown in Figure 11.9 and consists of a sil- ver wire, the end of which is coated with a thin film of AgCl. The wire is im- mersed in a solution that contains the desired concentration of KCl and that is saturated with AgCl. A porous plug serves as the salt bridge. The shorthand notation for the cell is Ag(s) | AgCl (sat’d), KCl (x M) || where x is the concentration of KCl. In comparison to the SCE the Ag/AgCl electrode has the advantage of being useful at higher temperatures. On the other hand, the Ag/AgCl electrode is more prone to reacting with solutions to form insoluble silver complexes that may plug the salt bridge between the electrode and the solution. 11B. 3 Metallic Indicator Electrodes The potential of the indicator electrode in a potentiometric electrochemical cell is proportional to the concentration of analyte. Two classes of indicator electrodes are used in potentiometry: metallic electrodes, which are the sub- ject of this section, and ion-selective electrodes, which are covered in the next section. The potential of a metallic electrode is determined by the position of a redox reaction at the electrode–solution interface. Three types of metallic electrodes are commonly used in potentiometry, each of which is considered in the following discussion. Electrodes of the First Kind When a copper electrode is immersed in a solution containing Cu 2+ , the potential of the electrode due to the reaction Cu 2+ (aq)+2e – t Cu(s) is determined by the concentration of copper ion. EE=? =+? ° Cu Cu 2+ 2+ 2+ Cu Cu / . log [] . . log [] 0 05916 2 1 0 3419 0 05916 2 1 Chapter 11 Electrochemical Methods of Analysis 473 To meter Fill hole KCl crystalsHole Asbestos wick Hg Saturated KCl Hg, Hg 2 Cl 2 , KCl silver/silver chloride electrode Reference electrode based on the reduction of AgCl to Ag; that is, AgCl(s)+e – t Ag(s)+Cl – (aq). 1400-CH11 9/9/99 2:06 PM Page 473 If the copper electrode is the indicator electrode in a potentiometric electrochemical cell that also includes a saturated calomel reference electrode SCE || Cu 2+ (unk) | Cu(s) then the cell potential can be used to determine an unknown concentration of Cu 2+ in the indicator half-cell Metallic indicator electrodes in which a metal is in contact with a solution con- taining its ion are called electrodes of the first kind. In general, for a metal M, in a solution of M n+ , the cell potential is given as where K is a constant that includes the standard-state potential for the M n+ /M redox couple, the potential of the reference electrode, and the junction potential. For a variety of reasons, including slow kinetics for electron transfer, the existence of surface oxides and interfering reactions, electrodes of the first kind are limited to Ag, Bi, Cd, Cu, Hg, Pb, Sn, Tl, and Zn. Many of these electrodes, such as Zn, cannot be used in acidic solutions where they are easily oxidized by H + . Electrodes of the Second Kind An electrode of the first kind involving an M n+ /M redox couple will respond to the concentration of another species if that species is in equilibrium with M n+ . For example, the potential of a silver electrode in a solu- tion of Ag + is given by 11.4 If the solution is saturated with AgI, then the solubility reaction AgI(s) t Ag + (aq)+I – (aq) determines the concentration of Ag + ; thus 11.5 where K sp,AgI is the solubility product for AgI. Substituting equation 11.5 into 11.4 shows that the potential of the silver electrode is a function of the concentration of I – . When this electrode is incorporated into a potentiometric electrochemical cell REF || AgI (sat’d), I – (unk) | Ag(s) the cell potential is E cell = K – 0.05916 log [I – ] where K is a constant that includes the standard-state potential for the Ag + /Ag redox couple, the solubility product for AgI, the potential of the reference electrode, and the junction potential. E K =+ ? ? 0 7996 0 05916. . log []I sp,AgI [] [] Ag I + sp,Agl = ? K EE=? =+? ° Ag Ag ++ + Ag Ag / . log [] . . log [] 0 05916 1 0 7996 0 05916 1 EK n K n n n cell M M=? =+ + + 0 05916 1 0 05916. log [] . log[ ] EEEE E cell ind ref lj 2+ lj Cu =?+=+ ? ?+ +0 3419 0 05916 2 1 0 2444. . log [] . 474 Modern Analytical Chemistry To meter Ag wire coated with AgCl KCl solution Porous plug Figure 11.9 Schematic diagram of a Ag/AgCl electrode. electrode of the first kind A metallic electrode whose potential is a function of the concentration of M n+ in an M n+ /M redox half-reaction. 1400-CH11 9/9/99 2:06 PM Page 474 When the potential of an electrode of the first kind responds to the potential of another ion that is in equilibrium with M n+ , it is called an electrode of the second kind. Two common electrodes of the second kind are the calomel and silver/silver chloride reference electrodes. Electrodes of the second kind also can be based on complexation reactions. For example, an electrode for EDTA is constructed by cou- pling a Hg 2+ /Hg electrode of the first kind to EDTA by taking advantage of its for- mation of a stable complex with Hg 2+ . Redox Electrodes Electrodes of the first and second kind develop a potential as the result of a redox reaction in which the metallic electrode undergoes a change in its oxidation state. Metallic electrodes also can serve simply as a source of, or a sink for, electrons in other redox reactions. Such electrodes are called redox electrodes. The Pt cathode in Example 11.1 is an example of a redox electrode because its potential is determined by the concentrations of Fe 2+ and Fe 3+ in the indicator half-cell. Note that the potential of a redox electrode generally responds to the concentration of more than one ion, limiting their usefulness for direct potentiometry. 11B. 4 Membrane Electrodes If metallic electrodes were the only useful class of indicator electrodes, poten- tiometry would be of limited applicability. The discovery, in 1906, that a thin glass membrane develops a potential, called a membrane potential, when oppo- site sides of the membrane are in contact with solutions of different pH led to the eventual development of a whole new class of indicator electrodes called ion- selective electrodes (ISEs). Following the discovery of the glass pH electrode, ion- selective electrodes have been developed for a wide range of ions. Membrane electrodes also have been developed that respond to the concentration of molecu- lar analytes by using a chemical reaction to generate an ion that can be monitored with an ion-selective electrode. The development of new membrane electrodes continues to be an active area of research. Membrane Potentials Ion-selective electrodes, such as the glass pH electrode, function by using a membrane that reacts selectively with a single ion. Figure 11.10 shows a generic diagram for a potentiometric electrochemical cell equipped with an ion-selective electrode. The shorthand notation for this cell is Ref(samp) || [A] samp | [A] int || Ref(int) where the membrane is represented by the vertical slash (|) separating the two solu- tions containing analyte. Two reference electrodes are used; one positioned within the internal solution, and one in the sample solution. The cell potential, therefore, is E cell = E Ref(int) – E Ref(samp) + E mem + E lj 11.6 where E mem is the potential across the membrane. Since the liquid junction poten- tial and reference electrode potentials are constant, any change in the cell’s potential is attributed to the membrane potential. Interaction of the analyte with the membrane results in a membrane potential if there is a difference in the analyte’s concentration on opposite sides of the mem- brane. One side of the membrane is in contact with an internal solution containing a fixed concentration of analyte, while the other side of the membrane is in contact with the sample. Current is carried through the membrane by the movement of ei- ther the analyte or an ion already present in the membrane’s matrix. The membrane potential is given by a Nernst-like equation Chapter 11 Electrochemical Methods of Analysis 475 electrode of the second kind A metallic electrode whose potential is a function of the concentration of X in an MX n /M redox half-reaction. redox electrode An inert electrode that serves as a source or sink for electrons for a redox half- reaction. membrane potential A potential developing across a conductive membrane whose opposite sides are in contact with solutions of different composition. ion-selective electrode An electrode in which the membrane potential is a function of the concentration of a particular ion in solution. 1400-CH11 9/9/99 2:06 PM Page 475 Figure 11.10 Electrochemical cell for potentiometry with an ion-selective membrane electrode. 11.7 where [A] samp and [A] int are the concentrations of analyte in the sample and the internal solution, respectively, and z is the analyte’s charge. Ideally, E mem should be zero when the concentrations of analyte on both sides of the membrane are equal. The term E asym , which is called an asymmetry potential, accounts for the fact that the membrane potential is usually not zero under these conditions. Substituting equation 11.7 into equation 11.6, assuming a temperature of 25 °C and rearranging gives 11.8 where K is a constant accounting for the potentials of the reference electrodes, any liquid junction potentials, the asymmetry potential, and the concentration of ana- lyte in the internal solution. Equation 11.8 is a general equation, and applies to all types of ion-selective electrodes. Selectivity of Membranes Membrane potentials result from a chemical interac- tion between the analyte and active sites on the membrane’s surface. Because the signal depends on a chemical process, most membranes are not selective toward EK z cell samp A]=+ 0 05916. log[ EE RT zF mem asym int samp A] A] =?ln [ [ 476 Modern Analytical Chemistry Ref(sample) Ref(inter nal) Membrane Sample solution Potentiometer [A] samp [A] int ISE Internal solution asymmetry potential The membrane potential when opposite sides of the membrane are in contact with identical solutions yet a nonzero potential is observed. 1400-CH11 9/9/99 2:06 PM Page 476 a single analyte. Instead, the membrane potential is proportional to the concen- tration of all ions in the sample solution capable of interacting at the mem- brane’s active sites. Equation 11.8 can be generalized to include the contribution of an interferent, I, where z A and z I are the charges of the analyte and interferent, and K A,I is a selectivity coefficient accounting for the relative response of the interferent.* The selectivity coefficient is defined as where [A] E and [I] E are the concentrations of analyte and interferent yielding identical cell potentials. When the selectivity coefficient is 1.00, the membrane responds equally to the analyte and interferent. A membrane shows good se- lectivity for the analyte when K A,I is significantly less than 1.00. Selectivity coefficients for most commercially available ion-selective elec- trodes are provided by the manufacturer. If the selectivity coefficient is un- known, it can be determined experimentally. The easiest method for deter- mining K A,I is to prepare a series of solutions, each of which contains the same concentration of interferent, [I] add , but a different concentration of analyte. A plot of cell potential versus the log of the analyte’s concentration has two dis- tinct linear regions (Figure 11.11). When the analyte’s concentration is signif- icantly larger than K A,I [I] add , the potential is a linear function of log [A], as given by equation 11.8. If K A,I [I] add is significantly larger than the analyte’s concentration, however, the cell potential remains constant. The concentra- tion of analyte and interferent at the intersection of these two linear regions is used to calculate K A,I . Glass Ion-Selective Electrodes The first commercial glass electrodes were manu- factured using Corning 015, a glass with a composition of approximately 22% Na 2 O, 6% CaO, and 72% SiO 2 . When immersed in an aqueous solution, the outer approximately 10 nm of the membrane becomes hydrated over the course of sev- eral hours. Hydration of the glass membrane results in the formation of negatively charged sites, G – , that are part of the glass membrane’s silica framework. Sodium ions, which are able to move through the hydrated layer, serve as the counterions. Hydrogen ions from solution diffuse into the membrane and, since they bind more strongly to the glass than does Na + , displace the sodium ions H + (aq)+G – –Na + (s) t G – –H + (s)+Na + (aq) giving rise to the membrane’s selectivity for H + . The transport of charge across the membrane is carried by the Na + ions. The potential of glass electrodes using Corn- ing 015 obeys the equation E cell = K + 0.05916 log [H + ] 11.9 over a pH range of approximately 0.5–9. Above a pH of 9–10, the glass membrane may become more responsive to other cations, such as Na + and K + . K E E zz A,I A] I] AI = [ [ / EK z K zz cell A A,I A] + I] AI =+ 0 05916. log([ [ ) / Chapter 11 Electrochemical Methods of Analysis 477 glass electrode An ion-selective electrode based on a glass membrane in which the potential develops from an ion-exchange reaction on the membrane’s surface. [A] inter log[A] K A,I [I] add >> [A] E cell K A,I = [A] E [I] E z A /z 1 [A] inter [I] add z A /z 1 = [A] >> K A,I [I] add Figure 11.11 Plot of cell potential versus the log of the analyte’s concentration in the presence of a fixed concentration of interferent, showing the determination of the selectivity coefficient. *Note that this treatment of sensitivity is similar to that introduced in Chapter 3. 1400-CH11 9/9/99 2:06 PM Page 477 Figure 11.12 Schematic diagram of a combination glass electrode for measuring pH. 478 Modern Analytical Chemistry EXAMPLE 11. 4 The selectivity coefficient K H + /Na + for Corning 015 is approximately 10 –11 . What error in pH is expected for a solution of 0.05 M NaOH. SOLUTION A solution of 0.05 M NaOH has an actual H + , [H + ] act , concentration of 2 · 10 –13 M and a pH of 12.7. The electrode responds, however, to both H + and Na + , with the apparent concentration of H + , [H + ] app , given by [H + ] app =[H + ] act + K H + /Na + [Na + ]=2· 10 –13 + (10 –11 )(0.05) = 7 · 10 –13 The apparent concentration of H + corresponds to a pH of 12.2, giving an error of –0.5 pH units. Replacing Na 2 O and CaO with Li 2 O and BaO extends the useful pH range of glass membrane electrodes to pH levels greater than 12. Glass membrane pH electrodes are often available in a combination form that includes both the indicator and the reference electrode. The use of a single electrode greatly simplifies the measurement of pH. An example of a typical combination electrode is shown in Figure 11.12. Internal reference (Ag/AgCl) Salt bridge Sample reference (Ag/AgCl) 0.1 M HCl, AgCl (sat’d) pH-Sensitive membrane AgCl, KCl To meter 1400-CH11 9/9/99 2:06 PM Page 478 The response of the Corning 015 glass membrane to monovalent cations other than H + at high pH led to the development of glass membranes possessing a greater selectivity for other cations. For example, a glass membrane with a composition of 11% Na 2 O, 18% Al 2 O 3, and 71% SiO 2 is used as a Na + ion-selective electrode. Other glass electrodes have been developed for the analysis of Li + , K + , Rb + , Cs + , NH 4 + , Ag + , and Tl + . Several representative examples of glass membrane electrodes are listed in Table 11.1. Since the typical thickness of the glass membrane in an ion-selective electrode is about 50 m m, they must be handled carefully to prevent the formation of cracks or breakage. Before a glass electrode can be used it must be conditioned by soaking for several hours in a solution containing the analyte. Glass electrodes should not be allowed to dry out, as this destroys the membrane’s hydrated layer. If a glass elec- trode has been allowed to dry out, it must be reconditioned before it can be used. The composition of a glass membrane changes over time, affecting the electrode’s performance. The average lifetime for a glass electrode is several years. Crystalline Solid-State Ion-Selective Electrodes Solid-state ion-selective elec- trodes use membranes fashioned from polycrystalline or single-crystal inorganic salts. Polycrystalline ion-selective electrodes are made by forming a thin pellet of Ag 2 S, or a mixture of Ag 2 S and either a second silver salt or another metal sulfide. The pellet, which is 1–2 mm in thickness, is sealed into the end of a nonconducting plastic cylinder, and an internal solution containing the analyte and a reference electrode are placed in the cylinder. Charge is carried across the membrane by Ag + ions. The membrane potential for a Ag 2 S pellet develops as the result of a difference in the equilibrium position of the solubility reaction Ag 2 S(s) t 2Ag + (aq)+S 2– (aq) on the two sides of the membrane. When used to monitor the concentration of Ag + ions, the cell potential is E cell = K + 0.05916 log [Ag + ] The membrane also responds to the concentration of S 2– , with the cell potential given as EK cell 2 S=? ? 0 05916 2 . log[ ] Chapter 11 Electrochemical Methods of Analysis 479 Table 11.1 Representative Examples of Glass Membrane Ion-Selective Electrodes Analyte Membrane Composition Selectivity Coefficients a Na + 11% Na 2 O, 18% Al 2 O 3 , 71% SiO 2 K Na + /H + = 1000 K Na + /K + = K Na + /Li + =10 –3 Li + 15% Li 2 O, 25% Al 2 O 3 , 60% SiO 2 K Li + /Na + = 0.3 K Li + /K + =10 –3 K + 27% Na 2 O, 5% Al 2 O 3 , 68% SiO 2 K K + /Na + = 0.05 a Selectivity constants are approximate, and those found experimentally may vary substantially from the listed values. 3 solid-state ion-selective electrode An ion-selective electrode based on a sparingly soluble inorganic crystalline material. 1400-CH11 9/9/99 2:06 PM Page 479 480 Modern Analytical Chemistry If a mixture of an insoluble silver salt and Ag 2 S is used to make the membrane, then the membrane potential also responds to the concentration of the anion of the added silver salt. Thus, pellets made from a mixture of Ag 2 S and AgCl can serve as a Cl – ion-selective electrode, with a cell potential of E cell = K – 0.05916 log [Cl – ] Membranes fashioned from a mixture of Ag 2 S with CdS, CuS, or PbS are used to make ion-selective electrodes that respond to the concentration of Cd 2+ , Cu 2+ , or Pb 2+ . In this case the cell potential is where [M 2+ ] is the concentration of the appropriate metal ion. Several examples of polycrystalline, Ag 2 S-based ion-selective electrodes are listed in Table 11.2. The selectivity of these ion-selective electrodes is determined by solubility. Thus, a Cl – ion-selective electrode constructed using a Ag 2 S/AgCl mem- brane is more selective for Br – (K Cl – /Br – =10 2 ) and I – (K Cl – /l – =10 6 ) since AgBr and AgI are less soluble than AgCl. If the concentration of Br – is sufficiently high, the AgCl at the membrane–solution interface is replaced by AgBr, and the electrode’s response to Cl – decreases substantially. Most of the ion-selective electrodes listed in Table 11.2 can be used over an extended range of pH levels. The equilibrium be- tween S 2– and HS – limits the analysis for S 2– to a pH range of 13–14. Solutions of CN – , on the other hand, must be kept basic to avoid the release of HCN. The membrane of a F – ion-selective electrode is fashioned from a single crystal of LaF 3 that is usually doped with a small amount of EuF 2 to enhance the mem- brane’s conductivity. Since EuF 2 provides only two F – ions, compared with three for LaF 3 , each EuF 2 produces a vacancy in the crystal lattice. Fluoride ions move through the membrane by moving into adjacent vacancies. The LaF 3 membrane is sealed into the end of a nonconducting plastic tube, with a standard solution of F – , typically 0.1 M NaF, and a Ag/AgCl reference electrode. The membrane potential for a F – ion-selective electrode results from a difference in the solubility of LaF 3 on opposite sides of the membrane, with the potential given by E cell = K – 0.05916 log [F – ] One advantage of the F – ion-selective electrode is its freedom from interference. The only significant exception is OH – (K F – /OH – = 0.1), which imposes a maximum pH limit for a successful analysis. EXAMPLE 11. 5 What is the maximum pH that can be tolerated in the analysis of a 1 · 10 –5 M solution of F – if the error is to be less than 1%? SOLUTION To achieve an error of less than 1%, the difference between the actual, [F – ] act , and apparent, [F – ] app , concentrations of F – must be less than 1%; thus [F – ] app –[F – ] act ≤ 0.01[F – ] act ([F – ] act + K F – /OH –[OH – ])–[F – ]act ≤ 0.01[F – ] act K F – /OH –[OH – ] ≤ 0.01[F – ] act EK cell M=+ + 0 05916 2 2 . log[ ] 1400-CH11 9/9/99 2:06 PM Page 480 Solving for [OH – ], and making appropriate substitutions gives corresponding to a pH of less than 8. [] .[ ] (.)( ) . / OH F act FOH ? ?? ? ≤= × =× ?? 0 01 0 01 1 10 01 110 5 6 K Chapter 11 Electrochemical Methods of Analysis 481 Table 11.2 Representative Examples of Polycrystalline Ion-Selective Electrodes Analyte Membrane Composition Selectivity Coefficients a Ag + Ag 2 S K Ag + /Cu 2+ =10 –6 K Ag + /Pb 2+ =10 –10 Hg 2+ interferes Cd 2+ CdS/Ag 2 S K Cd 2+ /Fe 2+ = 200 K Cd 2+ /Pb 2+ =6 Ag + , Hg 2+ , Cu 2+ must be absent Cu 2+ CuS/Ag 2 S K Cu 2+ /Fe 3+ =10 K Cu 2+ /Cu + =1 Ag + , Hg 2+ must be absent Pb 2+ PbS/Ag 2 S K Pb 2+ /Fe 3+ =1 K Pb 2+ /Cd 2+ =1 Ag + , Hg 2+ , Cu 2+ must be absent Br – AgBr/Ag 2 S K Br – /I – = 5000 K Br – /CN – = 100 K Br – /Cl – =5· 10 –3 K Br – /OH – =1· 10 –5 S 2– must be absent Cl – AgCl/Ag 2 S K Cl – /I – =1· 10 6 K Cl – /CN – =1· 10 4 K Cl – /Br – = 100 K Cl – /OH – = 0.01 S 2– must be absent CN – AgI/Ag 2 S K CN – /I – = 100 K CN – /Br – =1· 10 –4 K CN – /Cl – =1· 10 –6 K CN – /OH – =1· 10 –8 S 2– must be absent I – AgI/Ag 2 S K I – /S 2– =30 K I – /CN – = 0.01 K I – /Br – =1· 10 –4 K I – /Cl – =1· 10 –6 K I – /OH – =1· 10 –7 SCN – AgSCN/Ag 2 S K SCN – /I – = 1000 K SCN – /Br – = 100 K SCN – /CN – = 100 K SCN – /Cl – = 0.1 K SCN – /OH – = 0.01 S 2– must be absent S 2– Ag 2 SHg 2+ interferes a Selectivity constants are approximate, and those found experimentally may vary substantially from the listed values. 3 1400-CH11 9/9/99 2:06 PM Page 481 Figure 11.13 Structure and formula of di-(n-decyl) phosphate. Below a pH of 4 the predominate form of fluoride in solution is HF, which, unlike F – , does not contribute to the membrane potential. For this reason, an analysis for total fluoride must be carried out at a pH greater than 4. Unlike ion-selective electrodes using glass membranes, crystalline solid-state ion-selective electrodes do not need to be conditioned before use and may be stored dry. The surface of the electrode is subject to poisoning, as described earlier for a Cl – ISE in contact with an excessive concentration of Br – . When this happens, the electrode can be returned to its original condition by sanding and polishing the crystalline membrane. Liquid-Based Ion-Selective Electrodes Another approach to constructing an ion-selective electrode is to use a hydrophobic membrane containing a selective, liquid organic complexing agent. Three types of organic liquids have been used: cation exchangers, anion exchangers, and neutral ionophores. When the ana- lyte’s concentration on the two sides of the membrane is different, a membrane potential is the result. Current is carried through the membrane by the analyte. One example of a liquid-based ion-selective electrode is that for Ca 2+ , which uses a porous plastic membrane saturated with di-(n-decyl) phosphate (Figure 11.13). As shown in Figure 11.14, the membrane is placed at the end of a noncon- ducting cylindrical tube and is in contact with two reservoirs. The outer reservoir contains di-(n-decyl) phosphate in di-n-octylphenylphosphonate, which soaks into the porous membrane. The inner reservoir contains a standard aqueous solution of Ca 2+ and a Ag/AgCl reference electrode. Calcium ion-selective electrodes are also available in which the di-(n-decyl) phosphate is immobilized in a polyvinyl chloride 482 Modern Analytical Chemistry – O—P — — O O O — — (C 10 H 21 O) 2 PO 2 – To meter Ag/AgCl reference electrode Di-(n-decyl) phosphate reservoir Standard Ca 2+ solution Membrane saturated with di-(n-decyl) phosphate Figure 11.14 Schematic diagram of a Ca 2+ liquid-based ion-selective electrode. ionophore A neutral ligand whose exterior is hydrophobic and whose interior is hydrophilic. liquid-based ion-selective electrode An ion-selective electrode in which a chelating agent is incorporated into a hydrophobic membrane. 1400-CH11 9/9/99 2:07 PM Page 482 (PVC) membrane, eliminating the need for a reservoir containing di-(n-decyl) phosphate. A membrane potential develops as the result of a difference in the equilibrium position of the complexation reaction Ca 2+ (aq) + 2(C 10 H 21 O) 2 PO 2 – (m) t Ca[(C 10 H 21 O) 2 PO 2 ] 2 (m) on the two sides of the membrane, where (m) indicates that the species is present in the membrane. The cell potential for the Ca 2+ ion-selective electrode is The selectivity of the electrode for Ca 2+ is very good, with only Zn 2+ showing greater selectivity. The properties of several representative liquid-based ion-selective electrodes are presented in Table 11.3. An electrode using a liquid reservoir can be stored in a dilute solution of analyte and needs no additional conditioning before use. The life- time of an electrode with a PVC membrane, however, is proportional to its expo- sure to aqueous solutions. For this reason these electrodes are best stored by cover- ing the membrane with a cap containing a small amount of wetted gauze to EK cell 2+ Ca=+ 0 05916 2 . log[ ] Chapter 11 Electrochemical Methods of Analysis 483 Table 11. 3 Representative Examples of Liquid-Based Ion-Selective Electrodes Analyte Membrane Composition Selectivity Coefficients a Ca 2+ di-(n-decyl) phosphate in PVC K Ca 2+ /Zn 2+ = 1–5 K Ca 2+ /Al 3+ = 0.90 K Ca 2+ /Mn 2+ = 0.38 K Ca 2+ /Cu 2+ = 0.070 K Ca 2+ /Mg 2+ = 0.032 K + Valinomycin in PVC K K + /Rb + = 1.9 K K + /Cs + = 0.38 K K + /Li + =1· 10 –4 K K + /Na + =1· 10 –5 Li + ETH 149 in PVC K Li + /H + =1 K Li + /Na + = 0.05 K Li + /K + =7· 10 –3 NH 4 + Nonactin and monactin in PVC K NH 4 + /K + = 0.12 K NH 4 + /H + = 0.016 K NH 4 + /Li + = 4.2 · 10 –3 K NH 4 + /Na + =2· 10 –3 ClO 4 – Fe(o-phen) 3 3+ in p-nitrocymene with K ClO 4 – /OH – =1 porous membrane K ClO 4 – /I – = 0.012 K ClO 4 – /NO 3 – = 1.5 · 10 –3 K ClO 4 – /Br –= 5.6 · 10 –4 K ClO 4 – /Cl – = 2.2 · 10 –4 NO 3 – tetradodecyl ammonium nitrate in PVC K NO 3 – /Cl – =6· 10 –3 K NO 3 – /F –=9· 10 –4 a Selectivity constants are approximate, and those found experimentally may vary substantially from the listed values. 3 1400-CH11 9/9/99 2:07 PM Page 483 Figure 11.15 Schematic diagram of a gas-sensing membrane electrode. maintain a humid environment. The electrode must then be conditioned be- fore use by soaking in a solution of analyte for 30–60 min. Gas-Sensing Electrodes A number of membrane electrodes have been devel- oped that respond to the concentration of dissolved gases. The basic design of these electrodes is shown in Figure 11.15 and consists of a thin membrane sepa- rating the sample from an inner solution containing an ion-selective electrode. The membrane is permeable to the gaseous analyte, but is not permeable to nonvolatile components in the sample matrix. Once the gaseous analyte passes through the membrane, it reacts in the inner solution, producing a species whose concentration can be monitored by an appropriate ion-selective elec- trode. For example, in the CO 2 electrode, CO 2 reacts in the inner solution to produce H 3 O + . CO 2 (aq)+2H 2 O(l) t HCO 3 – (aq)+H 3 O + (aq) 11.10 The change in the concentration of H 3 O + is monitored with a pH ion-selective electrode, for which the cell potential is given by equation 11.9. The relation- ship between the concentration of H 3 O + and CO 2 is given by rearranging the equilibrium constant expression for reaction 11.10; thus 11.11 where K is the equilibrium constant. If the amount of HCO 3 – in the internal solu- tion is sufficiently large, then its concentration is unaffected by the presence of CO 2 and remains constant. Substituting equation 11.11 into equation 11.9 gives E cell = K￠ + 0.05916 log [CO 2 ] where K￠ is a constant that includes the constant for the pH ion-selective electrode, the equilibrium constant for reaction 11.10, and the concentration of HCO 3 – . Gas-sensing electrodes have been developed for a variety of gases, the charac- teristics for which are listed in Table 11.4. The composition of the inner solution changes with use, and both it and the membrane must be replaced periodically. Gas-sensing electrodes are stored in a solution similar to the internal solution to minimize their exposure to atmospheric gases. Potentiometric Biosensors Potentiometric electrodes for the analysis of molecules of biochemical importance can be constructed in a fashion similar to that used for gas-sensing electrodes. The most common class of potentiometric biosensors are the so-called enzyme electrodes, in which an enzyme is trapped or immobilized at the surface of an ion-selective electrode. Reaction of the analyte with the enzyme produces a product whose concentration is monitored by the ion-selective elec- trode. Potentiometric biosensors have also been designed around other biologi- cally active species, including antibodies, bacterial particles, tissue, and hormone receptors. One example of an enzyme electrode is the urea electrode, which is based on the catalytic hydrolysis of urea by urease CO(NH 2 ) 2 (aq)+2H 2 O(l) t 2NH 4 + (aq)+CO 3 2– (aq) In one version of the urea electrode, shown in Figure 11.16, an NH 3 electrode is modified by adding a dialysis membrane that physically traps a pH 7.0 buffered so- lution of urease between the dialysis membrane and the gas-permeable [] [] [] HO CO HCO 3 + 2 3 = ? K 484 Modern Analytical Chemistry To meter Ag/AgCl external reference Inner solution Gas-permeable membrane Ion-selective electrode enzyme electrodes An electrode that responds to the concentration of a substrate by reacting the substrate with an immobilized enzyme, producing an ion that can be monitored with an ion-selective electrode. 1400-CH11 9/9/99 2:07 PM Page 484 Figure 11.16 Schematic diagram of an enzyme-based potentiometric biosensor for urea in which urease is trapped between two membranes. Chapter 11 Electrochemical Methods of Analysis 485 membrane. 4 When immersed in the sample, urea diffuses through the dialy- sis membrane, where it reacts with the enzyme urease. The NH 4 + that is pro- duced is in equilibrium with NH 3 NH 4 + (aq)+H 2 O(l) t H 3 O + (aq)+NH 3 (aq) which, in turn, diffuses through the gas-permeable membrane, where it is de- tected by a pH electrode. The response of the electrode to the concentration of urea is given by E cell = K – 0.05916 log [urea] 11.12 Another version of the urea electrode (Figure 11.17) immobilizes the enzyme in a polymer membrane formed directly on the tip of a glass pH electrode. 5 In this case, the electrode’s response is pH = K [urea] 11.13 Few potentiometric biosensors are commercially available. As shown in Figures 11.16 and 11.17, however, available ion-selective and gas-sensing elec- trodes may be easily converted into biosensors. Several representative exam- ples are described in Table 11.5, and additional examples can be found in sev- eral reviews listed in the suggested readings at the end of the chapter. 11B. 5 Quantitative Applications The potentiometric determination of an analyte’s concentration is one of the most common quantitative analytical techniques. Perhaps the most frequently employed, routine quantitative measurement is the potentiometric determination of a solu- tion’s pH, a technique considered in more detail in the following discussion. Other areas in which potentiometric applications are important include clinical chemistry, environmental chemistry, and potentiometric titrations. Before considering these applications, however, we must first examine more closely the relationship between cell potential and the analyte’s concentration, as well as methods for standardizing potentiometric measurements. Activity Versus Concentration In describing metallic and membrane indicator elec- trodes, the Nernst equation relates the measured cell potential to the concentration of analyte. In writing the Nernst equation, we often ignore an important detail—the Table 11. 4 Characteristics of Gas-Sensing Membrane Electrodes Analyte Reaction in Inner Solution Inner Solution Ion-Selective Electrode CO 2 CO 2 +2H 2 O t HCO 3 – +H 3 O + 0.01 M NaHCO 3 , 0.01 M NaCl glass pH electrode HCN HCN + H 2 O t CN – +H 3 O + 0.01 M KAg(CN) 2 Ag 2 S membrane electrode HF HF + H 2 O t F – +H 3 O + 1 M H 3 O + F – electrode H 2 SH 2 S+H 2 O t HS – +H 3 O + pH 5 citrate buffer Ag 2 S membrane electrode NH 3 NH 3 +H 2 O t NH 4 + +OH – 0.01 M NH 4 Cl, 0.1 M KNO 3 glass pH electrode NO 2 2NO 2 +3H 2 O t NO 3 – +NO 2 – +2H 3 O + 0.02 M NaNO 2 , 0.1 M KNO 3 glass pH electrode SO 2 SO 2 +2H 2 O t HSO 3 – +H 3 O + 0.001 M NaHSO 3 , pH 5 glass pH electrode Source: Data compiled from Cammann, K. Working with Ion-Selective Electrodes. Springer-Verlag: Berlin, 1977. To meter Ag/AgCl external reference Urease solution Inner solution Dialysis membrane NH 3 gas- permeable membrane pH electrode 1400-CH11 9/9/99 2:07 PM Page 485 486 Modern Analytical Chemistry potential of an electrochemical cell is a function of activity, not concen- tration. Thus, the Nernst equation for a metallic electrode of the first kind is more appropriately written as 11.14 where a M n+ is the activity of the metal ion. As described in Chapter 6, the activity of an ion is equal to the product of its concentration, [M n+ ], and a matrix-dependent activity coefficient, g M n+. a M n+ =[M n+ ]g M n+ 11.15 Substituting equation 11.15 into equation 11.14 and rearranging gives 11.16 Equation 11.16 can be solved for the metal ion’s concentration if its ac- tivity coefficient is known. This presents a serious complication since the activity coefficient may be difficult to determine. If, however, the standards and samples have an identical matrix, then g M n+ remains constant, and equation 11.16 simplifies to where K￠ includes the activity coefficient. Quantitative Analysis Using External Standards To determine the concentration of analyte in a sample, it is necessary to standardize the electrode. If the electrode’s response obeys the Nernst equation, EK n n cell M = ′ ? + 0 05916 1. log [] EK nnn n cell M M =? ? + + 0 05916 1 0 05916 1. log . log []γ EK nan cell M =? + 0 05916 1. log Internal reference (Ag/AgCl) Salt bridge Sample reference (Ag/AgCl) pH electrode Urease immobilized in polymer membrane To meter Figure 11.17 Schematic diagrams of a second enzyme- based potentiometric biosensor for urea in which urease is immobilized in a polymer matrix. Table 11. 5 Representative Examples of Potentiometric Biosensors Analyte Biologically Active Phase a Substance Determined 5￠ -adenosinemonophosphate (5￠ -AMP) AMP-deaminase (E) NH 3 L-arginine arginase + urease (E) NH 3 asparagine asparaginase (E) NH 4 + L-cysteine Proteus morganii (B) H 2 S L-glutamate yellow squash (T) CO 2 L-glutamine Sarcina flava (B) NH 3 oxalate oxalate decarboxylase (E) CO 2 penicillin penicillinase (E) H 3 O + L-phenylalanine L-amino acid oxidase and horseradish peroxidase (E) I – sugars bacteria from human dental plaque (B) H 3 O + urea urease (E) NH 3 or H 3 O + Source: Compiled from Cammann, K. Working with Ion-Selective Electrodes. Springer-Verlag: Berlin, 1977; and Lunte, C. E.; Heineman, W. R. “Electrochemical Techniques in Bioanalysis.” In Steckham, E., ed. Topics in Current Chemistry, Vol. 143, Springer-Verlag: Berlin, 1988, p. 8. 3,6 a Abbreviations: E = enzyme; B = bacterial particle; T = tissue. 1400-CH11 9/9/99 2:07 PM Page 486 Chapter 11 Electrochemical Methods of Analysis 487 total ionic strength adjustment buffer A solution containing a relatively high concentration of inert electrolytes such that its composition fixes the ionic concentration of all solutions to which it is added. then only the constant K need be determined, and standardizing with a single ex- ternal standard is possible. Since small deviations from the ideal “Nerstian” slope of – RT/nF or – RT/zF are frequently observed, standardization is usually accom- plished using two or more external standards. In most quantitative analyses we are interested in determining the concen- tration, not the activity, of the analyte. As noted earlier, however, the electrode’s response is a function of the analyte’s activity. In the absence of interferents, a calibration curve of potential versus activity is a straight line. A plot of poten- tial versus concentration, however, may be curved at higher concentrations of analyte due to changes in the analyte’s activity coefficient. A curved calibra- tion curve may still be used to determine the analyte’s concentration if the stan- dard’s matrix matches that of the sample. When the exact composition of the sample matrix is unknown, which often is the case, matrix matching becomes impossible. Another approach to matrix matching, which does not rely on knowing the exact composition of the sample’s matrix, is to add a high concentration of inert electrolyte to all samples and standards. If the concentration of added electrolyte is sufficient, any difference between the sample’s matrix and that of the standards becomes trivial, and the activity coefficient remains essentially constant. The solution of inert electrolyte added to the sample and standards is called a total ionic strength adjustment buffer (TISAB). EXAMPLE 11.6 The concentration of Ca 2+ in a water sample was determined by the method of external standards. The ionic strength of the samples and standards was maintained at a nearly constant level by making each solution 0.5 M in KNO 3 . The measured cell potentials for the external standards are shown in the following table. [Ca 2+ ] E cell (M) (V) 1.00 · 10 –5 –0.125 5.00 · 10 –5 –0.103 1.00 · 10 –4 –0.093 5.00 · 10 –4 –0.072 1.00 · 10 –3 –0.065 5.00 · 10 –3 –0.043 1.00 · 10 –2 –0.033 What is the concentration of Ca 2+ in a water sample if its cell potential is found to be –0.084 V? SOLUTION Linear regression gives the equation for the calibration curve as E cell = 0.027 + 0.0303 log [Ca 2+ ] Substituting the cell potential for the sample gives the concentration of Ca 2+ as 2.17 · 10 –4 M. Note that the slope of the calibration curve is slightly different from the ideal value of 0.05916/2 = 0.02958. 1400-CH11 9/9/99 2:07 PM Page 487 488 Modern Analytical Chemistry Quantitative Analysis Using the Method of Standard Additions Because of the dif- ficulty of maintaining a constant matrix for samples and standards, many quantita- tive potentiometric methods use the method of standard additions. A sample of vol- ume, V X , and analyte concentration, C X , is transferred to a sample cell, and the potential, (E cell ) X , measured. A standard addition is made by adding a small volume, V S , of a standard containing a known concentration of analyte, C S , to the sample, and the potential, (E cell ) S , measured. Provided that V S is significantly smaller than V X , the change in sample matrix is ignored, and the analyte’s activity coefficient re- mains constant. Example 11.7 shows how a one-point standard addition can be used to determine the concentration of an analyte. EXAMPLE 11. 7 The concentration of Ca 2+ in a sample of sea water is determined using a Ca ion-selective electrode and a one-point standard addition. A 10.00-mL sample is transferred to a 100-mL volumetric flask and diluted to volume. A 50.00-mL aliquot of sample is placed in a beaker with the Ca ion-selective electrode and a reference electrode, and the potential is measured as –0.05290 V. A 1.00-mL aliquot of a 5.00 · 10 –2 M standard solution of Ca 2+ is added, and a potential of –0.04417 V is measured. What is the concentration of Ca 2+ in the sample of sea water? SOLUTION To begin, we write Nernst equations for the two measured cell potentials. The cell potential for the sample is and that following the standard addition is where V T is the total volume (V S + V X ) after the standard addition. Subtracting the first equation from the second equation gives Replacing (E cell ) S –(E cell ) X with D E and rearranging yields Substituting known values for D E, V X , V S , V T , and C S, 2 0 00873 0 05916 50 00 1 00 0 2951 0 9804 9 804 10 2 4 (. ) . log .(. . log . . =+ × ? ? ? ? ? ? =+ × ? ? ? ? ? ? ? ? mL 51.00 mL mL)(5.00 10 M) (51.00 mL) X X C C 2 0 05916 ?EVVCVC C V V VC VC. log ()() log= + ? ? ? ? ? ? =+ ? ? ? ? ? ? XTX STS X X T SS TX ()() . log . logEE V V C V V CC cell S cell X X T X S T SX ?= + ? ? ? ? ? ? ? 0 05916 2 0 05916 2 () . logEK V V C V V C cell S X T X S T S =+ + ? ? ? ? ? ? 0 05916 2 () . logEK C cell X X =+ 0 05916 2 1400-CH11 9/9/99 2:07 PM Page 488 Chapter 11 Electrochemical Methods of Analysis 489 Representative Methods —Continued and taking the inverse log of both sides gives Finally, solving for C X gives the concentration of Ca 2+ as 9.88 · 10 –4 M. Since the original sample of sea water was diluted by a factor of 10, the concentration of Ca 2+ in the sea water sample is 9.88 · 10 –3 M. Free Ions Versus Complexed Ions In discussing the F – ion-selective electrode, we noted that the membrane potential is influenced by the concentration of F – , but not the concentration of HF. An analysis for fluoride, therefore, is pH-dependent. Below a pH of approximately 4, fluoride is present predominantly as HF, and a quantitative analysis for total fluoride is impossible. If the pH is increased to greater than 4, however, the equilibrium HF(aq)+H 2 O(l) t H 3 O + (aq)+F – (aq) shifts to the right, and a quantitative analysis for total fluoride is possible. Most potentiometric electrodes are selective for only the free, uncomplexed analyte and do not respond to complexed forms of the analyte. Solution condi- tions, therefore, must be carefully controlled if the purpose of the analysis is to de- termine the analyte’s total concentration. On the other hand, this selectivity pro- vides a significant advantage over other quantitative methods of analysis when it is necessary to determine the concentration of free ions. For example, calcium is present in urine both as free Ca 2+ ions and as protein-bound Ca 2+ ions. If a urine sample is analyzed by atomic absorption spectroscopy, the signal is proportional to the total concentration of Ca 2+ , since both free and bound calcium are atomized. Analysis with a Ca 2+ ISE, however, gives a signal that is a function of only free Ca 2+ ions since the protein-bound ions cannot interact with the electrode’s membrane. Representative Method Ion-selective electrodes find application in numerous quan- titative analyses, each of which has its own unique considerations. The following pro- cedure for the analysis of fluoride in toothpaste provides an instructive example. Method 11.1 Determination of Fluoride in Toothpaste 7 Description of the Method. The concentration of fluoride in toothpastes containing soluble F – may be determined with a F – ion-selective electrode, using a calibration curve prepared with external standards. Although the F – ISE is very selective (only OH – with K F – /OH – of 0.1 is a significant interferent), Fe 3+ and Al 3+ interfere with the analysis by forming soluble fluoride complexes that do not interact with the ion-selective electrode’s membrane. This interference is minimized by reacting any Fe 3+ and Al 3+ with a suitable complexing agent. Procedure. Prepare 1 L of a standard solution of 1.00% w/v SnF 2 , and transfer to a plastic bottle for storage. Using this solution, prepare 100 mL each of standards containing 0.32%, 0.36%, 0.40%, 0.44%, and 0.48% w/v SnF 2 , adding 1 973 0 9804 9 804 10 4 .. . =+ × ? C X 1400-CH11 9/9/99 2:07 PM Page 489 490 Modern Analytical Chemistry 400 mg of malic acid to each solution as a stabilizer. Transfer the standards to plastic bottles for storage. Prepare a total ionic strength adjustment buffer (TISAB) by mixing 500 mL of water, 57 mL of glacial acetic acid, 58 g of NaCl, and 4 g of the disodium salt of DCTA (trans-1,2-cyclohexanetetraacetic acid) in a 1-L beaker, stirring until dissolved. Cool the beaker in a water bath, and add 5 M NaOH until the pH is between 5 and 5.5. Transfer the contents of the beaker to a 1-L volumetric flask, and dilute to volume. Standards are prepared by placing approximately 1 g of a fluoride-free toothpaste, 30 mL of distilled water, and 1.00 mL of the standard into a 50-mL plastic beaker and stirring vigorously for 2 min with a stir bar. The resulting suspension is quantitatively transferred to a 100-mL volumetric flask along with 50 mL of TISAB and diluted to volume with distilled water. The entire standard solution is then transferred to a 250-mL plastic beaker until its potential is measured. Samples of toothpaste are prepared for analysis by using approximately 1-g portions and treating in the same manner as the standards. The cell potential for the standards and samples are measured using a F – ion-selective electrode and an appropriate reference electrode. The solution is stirred during the measurement, and 2–3 min is allowed for equilibrium to be reached. The concentration of F – in the toothpaste is reported as %w/w SnF 2 . Questions 1. The total ionic strength adjustment buffer serves several purposes in this procedure. Identify these purposes. The composition of the TISAB accomplishes three things: (1) The high concentration of NaCl (approximately 1 M) ensures that the ionic strength of the standards and samples are essentially identical. Since the activity coefficient for fluoride will be the same in all solutions, the Nernst equation can be written using the concentration of F – in place of its activity; (2) Glacial acetic acid and NaOH are used to prepare an acetic acid/acetate buffer of pH 5–5.5. The pH of this buffer is high enough to ensure that the predominant form of fluoride is F – instead of HF; and (3) DCTA is added as a complexing agent for any Fe 3+ or Al 3+ that might be present, preventing the formation of FeF 6 3– or AlF 6 3– . 2. Why is a fluoride-free toothpaste added to the standard solutions? Fluoride-free toothpaste is added as a precaution against any matrix effects that might influence the ion-selective electrode’s response. This assumes, of course, that the matrices of the two toothpastes are otherwise similar. 3. The procedure specifies that the standard and sample solutions should be stored in plastic containers. Why is it not a good idea to store the solutions in glass containers? The fluoride ion is capable of reacting with glass to form SiF 4 . 4. The slope of the calibration curve is found to be –57.98 mV per tenfold change in the concentration of F – , compared with the expected slope of –59.16 mV per tenfold change in concentration. What effect does this have on the quantitative analysis for %w/w SnF 2 in the toothpaste samples? No effect at all—this is the reason for preparing a calibration curve with multiple standards. Continued from page 489 1400-CH11 9/9/99 2:07 PM Page 490 Chapter 11 Electrochemical Methods of Analysis 491 Measurement of pH With the availability of inexpensive glass pH electrodes and pH meters, the determination of pH has become one of the most frequent quantita- tive analytical measurements. The potentiometric determination of pH, however, is not without complications, several of which are discussed in this section. One complication is the meaning of pH. 8,9 The conventional definition of pH as presented in most introductory texts is pH = –log [H + ] 11.17 The pH of a solution, however, is defined by the response of an electrode to the H + ion and, therefore, is a measure of its activity. pH = –log(a H +) 11.18 Calculating the pH of a solution using equation 11.17 only approximates the true pH. Thus, a solution of 0.1 M HCl has a calculated pH of 1.00 using equation 11.17, but an actual pH of 1.1 as defined by equation 11.18. 8 The difference between the two values occurs because the activity coefficient for H + is not unity in a matrix of 0.1 M HCl. Obviously the true pH of a solution is affected by the composition of its matrix. As an extreme example, the pH of 0.01 M HCl in 5 m LiCl is 0.8, a value that is more acidic than that of 0.1 M HCl! 8 A second complication in measuring pH results from uncertainties in the rela- tionship between potential and activity. For a glass membrane electrode, the cell po- tential, E X , for a solution of unknown pH is given as 11.19 where K includes the potential of the reference electrode, the asymmetry potential of the glass membrane and any liquid junction potentials in the electrochemical cell. All the contributions to K are subject to uncertainty and may change from day to day, as well as between electrodes. For this reason a pH electrode must be calibrated using a standard buffer of known pH. The cell potential for the standard, E S , is 11.20 where pH S is the pH of the standard. Subtracting equation 11.20 from equation 11.19 and solving for pH gives 11.21 which is the operational definition of pH adopted by the International Union of Pure and Applied Chemistry.* pH pH XS XS =? ?() . EEF RT2 303 EK RT F SS pH=? 2 303. EK RT Fa K RT F X H X + pH=? =?ln .1 2 303 *Equations 11.19–11.21 are defined for a potentiometric electrochemical cell in which the pH electrode is the cathode. In this case an increase in pH decreases the cell potential. Many pH meters are designed with the pH electrode as the anode so that an increase in pH increases the cell potential. The operational definition of pH then becomes This difference, however, does not affect the operation of a pH meter. pH pH XS XS =? ?() . EEF RT2 303 1400-CH11 9/9/99 2:07 PM Page 491 Calibrating the electrode presents a third complication since a standard with an accurately known activity for H + needs to be used. Unfortunately, it is not possible to calculate rigorously the activity of a single ion. For this reason pH electrodes are calibrated using a standard buffer whose composition is chosen such that the de- fined pH is as close as possible to that given by equation 11.18. Table 11.6 gives pH values for several primary standard buffer solutions accepted by the National Insti- tute of Standards and Technology. A pH electrode is normally standardized using two buffers: one near a pH of 7 and one that is more acidic or basic depending on the sample’s expected pH. The pH electrode is immersed in the first buffer, and the “standardize” or “calibrate” control is adjusted until the meter reads the correct pH. The electrode is placed in the second buffer, and the “slope” or “temperature” control is adjusted to the- buffer’s pH. Some pH meters are equipped with a temperature compensation fea- ture, allowing the pH meter to correct the measured pH for any change in tempera- ture. In this case a thermistor is placed in the sample and connected to the pH meter. The “temperature” control is set to the solution’s temperature, and the pH meter is calibrated using the “calibrate” and “slope” controls. If a change in the sample’s temperature is indicated by the thermistor, the pH meter adjusts the slope of the calibration based on an assumed Nerstian response of 2.303RT/F. Clinical Applications Perhaps the area in which ion-selective electrodes receive the widest use is in clinical analysis, where their selectivity for the analyte in a complex matrix provides a significant advantage over many other analytical methods. The most common analytes are electrolytes, such as Na + , K + , Ca 2+ , H + , and Cl – , and dis- solved gases, such as CO 2 . For extracellular fluids, such as blood and urine, the analy- sis can be made in vitro with conventional electrodes, provided that sufficient sample is available. Some clinical analyzers place a series of ion-selective electrodes in a flow 492 Modern Analytical Chemistry Table 11.6 pH Values for Selected NIST Primary Standard Buffers a Saturated 0.025 m 0.008695 m 0.025 m (25 °C) 0.05 m 0.05 m KH 2 PO 4 ,KH 2 PO 4 , 0.01 m NaHCO 3 , Temperature KHC 4 H 4 O 6 KH 2 C 6 H 5 O 7 KHC 8 H 4 O 4 0.025 m 0.03043 m Na 4 B 4 O 7 0.025 m (°C) (tartrate) (citrate) (phthalate) Na 2 HPO 4 Na 2 HPO 4 (borax) Na 2 CO 3 0 — 3.863 4.003 6.984 7.534 9.464 10.317 5 — 3.840 3.999 6.951 7.500 9.395 10.245 10 — 3.820 3.998 6.923 7.472 9.332 10.179 15 — 3.802 3.999 6.900 7.448 9.276 10.118 20 — 3.788 4.002 6.881 7.429 9.225 10.062 25 3.557 3.776 4.008 6.865 7.413 9.180 10.012 30 3.552 3.766 4.015 6.854 7.400 9.139 9.966 35 3.549 3.759 4.024 6.844 7.389 9.102 9.925 40 3.547 3.753 4.035 6.838 7.380 9.068 9.889 45 3.547 3.750 4.047 6.834 7.373 9.038 9.856 50 3.549 3.749 4.060 6.833 7.367 9.011 9.828 Source: Values taken from Bates, R. G. Determination of pH: Theory and Practice, 2nd ed. Wiley: New York, 1973. 10 a Concentrations are given in molality (moles solute per kilograms solvent). 1400-CH11 9/9/99 2:07 PM Page 492 Figure 11.18 Schematic diagram for the Kodak Ektachem analyzer for K + : (a) support base; (b) silver; (c) silver chloride; (d) potassium chloride film; (e) ion-selective membrane containing valinomycin; (f) paper salt bridge; (g) well for sample solution; (h) well for standard solution. Chapter 11 Electrochemical Methods of Analysis 493 cell, allowing several analytes to be monitored simultaneously. Standards, samples, and rinse solutions are pumped through the flow cell and across the surface of the electrodes. For smaller volumes of sample the analysis can be conducted using dis- posable ion-selective systems, such as the Kodak Ektachem analyzer for K + shown in Figure 11.18. The analyzer consists of separate electrodes for the sample and refer- ence solutions. Each electrode is constructed from several thin films, consisting of a Ag/AgCl reference electrode, a salt bridge and an ion-selective membrane, deposited on a support base. The two electrodes are connected by a paper salt bridge saturated with the sample and reference solutions. The overall dimensions of the analyzer are 2.8 cm · 2.4 cm with a thickness of 150 m m and require only 10 m L each of sample and reference solution. Similar analyzers are available for the determination of Na + , Cl – , and CO 2 . The analysis of intercellular fluids requires an ion-selective electrode that can be inserted directly into the desired cell. Liquid-based membrane microelec- trodes with tip diameters of less than 1 m m are constructed by heating and draw- ing out a hard-glass capillary tube with an initial diameter of approximately 1–2 mm (Figure 11.19). The tip of the microelectrode is made hydrophobic by dipping in dichlorodimethyl silane. An inner solution appropriate for the desired analyte and a Ag/AgCl wire reference electrode are placed within the microelec- trode. The tip of the microelectrode is then dipped into a solution containing the liquid complexing agent. The small volume of liquid complexing agent entering the microelectrode is retained within the tip by capillary action, eliminating the need for a solid membrane. Potentiometric microelectrodes have been developed for a number of clinically important analytes, including H + , K + , Na + , Ca 2+ , Cl – , and I – . gfh Potentiometer a b c d e 1400-CH11 9/9/99 2:07 PM Page 493 Figure 11.19 Schematic diagram of a liquid-based ion- selective microelectrode. Environmental Applications Although ion-selective electrodes find use in envi- ronmental analysis, their application is not as widespread as in clinical analysis. Standard methods have been developed for the analysis of CN – , F – , NH 3 , and NO 3 – in water and wastewater. Except for F – , however, other analytical methods are con- sidered superior. By incorporating the ion-selective electrode into a flow cell, the continuous monitoring of wastewater streams and other flow systems is possible. Such applications are limited, however, by the electrode’s response to the analyte’s activity, rather than its concentration. Considerable interest has been shown in the development of biosensors for the field screening and monitoring of environmental samples for a number of priority pollutants. 11 Potentiometric Titrations In Chapter 9 we noted that one method for determining the equivalence point of an acid–base titration is to follow the change in pH with a pH electrode. The potentiometric determination of equivalence points is feasible for acid–base, complexation, redox, and precipitation titrations, as well as for titrations in aqueous and nonaqueous solvents. Acid–base, complexation, and precipitation potentiometric titrations are usually monitored with an ion-selective electrode that is selective for the analyte, although an electrode that is selective for the titrant or a reaction product also can be used. A redox electrode, such as a Pt wire, and a refer- ence electrode are used for potentiometric redox titrations. More details about po- tentiometric titrations are found in Chapter 9. 11B.6 Evaluation Scale of Operation The working range for most ion-selective electrodes is from a maximum concentration of 0.1–1 M to a minimum concentration of 10 –5 –10 –10 M. This broad working range extends from major to ultratrace ana- lytes, and is significantly greater than many other analytical methods. For conven- tional ion-selective electrodes, macro-sized samples with minimum volumes of 0.05–10 mL are necessary. Microelectrodes and specially designed analyzers, such as the Kodak Ektachem analyzer for K + shown in Figure 11.18, may be used with ultramicro-sized samples provided that the sample taken for analysis is suffi- ciently large to be representative of the original sample. Accuracy The accuracy of a potentiometric analysis is limited by the measure- ment error for the cell’s potential. Several factors contribute to this measurement error, including the contribution to the potential from interfering ions, the finite current drawn through the cell while measuring the potential, differences in the analyte’s activity coefficient in the sample and standard solutions, and liquid junction potentials. Errors in accuracy due to interfering ions often can be elimi- nated by including a separation step before the potentiometric analysis. Modern high-impedance potentiometers minimize errors due to the passage of current through the electrochemical cell. Errors due to activity coefficients and liquid junction potentials are minimized by matching the matrix of the standards to that of the sample. Even in the best circumstances, however, a difference in po- tential of approximately – 1 mV is observed for samples and standards at equal concentration. 494 Modern Analytical Chemistry To meter Ag/AgCl reference electrode Inner solution Liquid complexing agent < 1 m m 1400-CH11 9/9/99 2:07 PM Page 494 Chapter 11 Electrochemical Methods of Analysis 495 The effect of an uncertainty in potential on the accuracy of a potentiometric method of analysis is evaluated using a propagation of uncertainty. For a membrane ion-selective electrode the general expression for potential is given as where z is the charge of the analyte. From Table 4.9 in Chapter 4, the error in the cell potential, D E cell is Rearranging and multiplying through by 100 gives the percent relative error in con- centration as 11.22 The relative measurement error in concentration, therefore, is determined by the magnitude of the error in measuring the cell’s potential and by the charge of the an- alyte. Representative values are shown in Table 11.7 for ions with charges of – 1 and – 2, at a temperature of 25 °C. Accuracies of 1–5% for monovalent ions and 2–10% for divalent ions are typical. Although equation 11.22 was developed for membrane electrodes, it also applies to metallic electrodes of the first and second kind when z is replaced by n. Precision The precision of a potentiometric measurement is limited by variations in temperature and the sensitivity of the potentiometer. Under most conditions, and with simple, general-purpose potentiometers, the potential can be measured with a repeatability of – 0.1 mV. From Table 11.7 this result corresponds to an un- certainty of – 0.4% for monovalent analytes, and – 0.8% for divalent analytes. The reproducibility of potentiometric measurements is about a factor of 10 poorer. Sensitivity The sensitivity of a potentiometric analysis is determined by the term RT/nF or RT/zF in the Nernst equation. Sensitivity is best for smaller values of n or z. % relative error A] [A] cell =×= × ??[ / 100 100 E RT zF ? ? E RT zF cell A] [A] = [ EK RT zF cell A]=+ ln[ Table 11. 7 Relationship Between Measurement Error in Potential and Relative Error in Concentration Relative Error in Concentration Error in Potential (%) (– mV) z =1 z =2 0.1 – 0.4 – 0.8 0.5 – 1.9 – 3.9 1.0 – 3.9 – 7.8 1.5 – 5.8 – 11.1 2.0 – 7.8 – 15.6 1400-CH11 9/9/99 2:07 PM Page 495 496 Modern Analytical Chemistry Selectivity As described earlier, most ion-selective electrodes respond to more than one analyte. For many ion-selective electrodes, however, the selectivity for the analyte is significantly greater than for most interfering ions. Published selectivity coefficients for ion-selective electrodes (representative values are found in Tables 11.1 through 11.3) provide a useful guide in helping the analyst determine whether a potentiometric analysis is feasible for a given sample. Time, Cost, and Equipment In comparison with competing methods, potentiome- try provides a rapid, relatively low-cost means for analyzing samples. Commercial instruments for measuring pH or potential are available in a variety of price ranges and include portable models for use in the field. 11C Coulometric Methods of Analysis In potentiometry, the potential of an electrochemical cell under static conditions is used to determine an analyte’s concentration. As seen in the preceding section, po- tentiometry is an important and frequently used quantitative method of analysis. Dynamic electrochemical methods, such as coulometry, voltammetry, and amper- ometry, in which current passes through the electrochemical cell, also are important analytical techniques. In this section we consider coulometric methods of analysis. Voltammetry and amperometry are covered in Section 11D. Coulometric methods of analysis are based on an exhaustive electrolysis of the analyte. By exhaustive we mean that the analyte is quantitatively oxidized or re- duced at the working electrode or reacts quantitatively with a reagent generated at the working electrode. There are two forms of coulometry: controlled-potential coulometry, in which a constant potential is applied to the electrochemical cell, and controlled-current coulometry, in which a constant current is passed through the electrochemical cell. The total charge, Q, in coulombs, passed during an electrolysis is related to the absolute amount of analyte by Faraday’s law Q = nFN 11.23 where n is the number of electrons transferred per mole of analyte, F is Faraday’s constant (96487 C mol –1 ), and N is the moles of analyte. A coulomb is also equiva- lent to an A s; thus, for a constant current, i, the charge is given as Q = it e 11.24 where t e is the electrolysis time. If current varies with time, as it does in controlled- potential coulometry, then the total charge is given by 11.25 In coulometry, current and time are measured, and equation 11.24 or equation 11.25 is used to calculate Q. Equation 11.23 is then used to determine the moles of analyte. To obtain an accurate value for N, therefore, all the current must result in the analyte’s oxidation or reduction. In other words, coulometry requires 100% current efficiency (or an accurately measured current efficiency established using a standard), a factor that must be considered in designing a coulometric method of analysis. Qitdt t tt = = = ∫ () 0 e Faraday’s law The current or charge passed in a redox reaction is proportional to the moles of the reaction’s reactants and products. current efficiency The percentage of current that actually leads to the analyte’s oxidation or reduction. coulometry An electrochemical method in which the current required to exhaustively oxidize or reduce the analyte is measured. 1400-CH11 9/9/99 2:07 PM Page 496 Figure 11.20 Current–time curve for controlled-potential coulometry. 11C.1 Controlled-Potential Coulometry The easiest method for ensuring 100% current efficiency is to maintain the working electrode at a constant potential that allows for the analyte’s quantitative oxidation or reduction, without simultaneously oxidizing or reducing an interfering species. The current flowing through an electrochemical cell under a constant potential is proportional to the analyte’s concentration. As electrolysis progresses the analyte’s concentration decreases, as does the current. The resulting current-versus-time pro- file for controlled-potential coulometry, which also is known as potentiostatic coulometry, is shown in Figure 11.20. Integrating the area under the curve (equa- tion 11.25), from t = 0 until t = t e , gives the total charge. In this section we consider the experimental parameters and instrumentation needed to develop a controlled- potential coulometric method of analysis. Selecting a Constant Potential In controlled-potential coulometry, the potential is selected so that the desired oxidation or reduction reaction goes to completion without interference from redox reactions involving other components of the sam- ple matrix. To see how an appropriate potential for the working electrode is se- lected, let’s develop a constant-potential coulometric method for Cu 2+ based on its reduction to copper metal at a Pt cathode working electrode. Cu 2+ (aq)+2e – t Cu(s) 11.26 A ladder diagram for a solution of Cu 2+ (Figure 11.21) provides a useful means for evaluating the solution’s redox properties. From the ladder diagram we can see that reaction 11.26 is favored when the working electrode’s potential is more negative than +0.342 V versus the SHE (+0.093 V versus the SCE). To maintain a 100% cur- rent efficiency, however, the potential must be selected so that the reduction of H 3 O + to H 2 does not contribute significantly to the total charge passed at the electrode. The potential needed for a quantitative reduction of Cu 2+ can be calculated using the Nernst equation 11.27 If we define a quantitative reduction as one in which 99.99% of the Cu 2+ is reduced to Cu, then the concentration of Cu 2+ at the end of the electrolysis must be [Cu 2+ ] ≤ 10 –4 [Cu 2+ ] 0 11.28 where [Cu 2+ ] 0 is the initial concentration of Cu 2+ in the sample. Substituting equa- tion 11.28 into equation 11.27 gives the desired potential electrode as If the initial concentration of Cu 2+ is 1.00 · 10 –4 M, for example, then the cathode’s potential must be more negative than +0.105 V versus the SHE (–0.139 V versus the SCE) to achieve a quantitative reduction of Cu 2+ to Cu. Note that at this potential H 3 O + is not reduced to H 2 , maintaining a 100% current efficiency. Many of the published procedures for the controlled-potential coulometric analysis of Cu 2+ call for potentials that are more negative than that shown for the reduction of H 3 O + in Figure 11.21. 12 Such potentials can be used, however, because the slow kinetics for reducing H 3 O + results in a significant overpotential that shifts the potential of the H 3 O + /H 2 redox couple to more negative potentials. EE=? ° ? Cu Cu 2+ 2+ Cu / . log [] 0 05916 2 1 10 4 0 EE=?+ ° + Cu Cu Cu 2 0 05916 2 1 2 / . log [] Chapter 11 Electrochemical Methods of Analysis 497 Current Timet = 0 t = t e Background current More (+) More (–) O 2 Cu 2+ Cu H 2 O H 2 H 3 O + E ° O 2 /H 2 O = 1.229 V E ° Cu 2+ /Cu = 0.342 V E ° H 3 O + /H 2 = 0.00 V E Figure 11.21 Ladder diagram for aqueous solutions of Cu 2+ . overpotential The difference between the potential actually required to initiate an oxidation or reduction reaction, and the potential predicted by the Nernst equation. 1400-CH11 9/9/99 2:07 PM Page 497 Figure 11.22 Charge–time curve obtained by integrating the current–time curve in Figure 11.20. Minimizing Electrolysis Time The current-time curve for controlled-potential coulometry in Figure 11.20 shows that the current decreases continuously throughout electrolysis. An exhaustive electrolysis, therefore, may require a long time. Since time is an important consideration in choosing and designing analyt- ical methods, the factors that determine the analysis time need to be considered. The change in current as a function of time in controlled-potential coulometry is approximated by an exponential decay; thus, the current at time t is i = i 0 e –kt 11.29 where i 0 is the initial current, and k is a constant that is directly proportional to the area of the working electrode and the rate of stirring and inversely proportional to the volume of the solution. For an exhaustive electrolysis in which 99.99% of the analyte is oxidized or reduced, the current at the end of the analysis, t e , may be ap- proximated as i ≤ (10 –4 )i 0 11.30 Substituting equation 11.30 into equation 11.29 and solving for t e gives the mini- mum time for an exhaustive electrolysis as From this equation we see that increasing k leads to a shorter analysis time. For this reason controlled-potential coulometry is carried out in small-volume electrochem- ical cells, using electrodes with large surface areas and with high stirring rates. A quantitative electrolysis typically requires approximately 30–60 min, although shorter or longer times are possible. Instrumentation The potential in controlled-potential coulometry is set using a three-electrode potentiostat. Two types of working electrodes are commonly used: a Pt electrode manufactured from platinum-gauze and fashioned into a cylindrical tube, and an Hg pool electrode. The large overpotential for reducing H 3 O + at mercury makes it the electrode of choice for analytes requiring negative potentials. For example, potentials more negative than –1 V versus the SCE are feasible at an Hg electrode (but not at a Pt electrode), even in very acidic solu- tions. The ease with which mercury is oxidized, however, prevents its use at po- tentials that are positive with respect to the SHE. Platinum working electrodes are used when positive potentials are required. The auxiliary electrode, which is often a Pt wire, is separated by a salt bridge from the solution containing the an- alyte. This is necessary to prevent electrolysis products generated at the auxiliary electrode from reacting with the analyte and interfering in the analysis. A satu- rated calomel or Ag/AgCl electrode serves as the reference electrode. The other essential feature of instrumentation for controlled-potential coulom- etry is a means of determining the total charge passed during electrolysis. One method is to monitor the current as a function of time and determine the area under the curve (see Figure 11.20). Modern instruments, however, use electronic integration to monitor charge as a function of time. The total charge at the end of the electrolysis then can be read directly from a digital readout or from a plot of charge versus time (Figure 11.22). t kk e =? = ? 1 10 921 4 ln( ) . 498 Modern Analytical Chemistry Char g e Time Total charge 1400-CH11 9/9/99 2:07 PM Page 498 Figure 11.23 Current–time curve for controlled-current coulometry. 11C.2 Controlled-Current Coulometry A second approach to coulometry is to use a constant current in place of a constant potential (Figure 11.23). Controlled-current coulometry, also known as amperostatic coulometry or coulometric titrimetry, has two advantages over controlled-potential coulometry. First, using a constant current makes for a more rapid analysis since the current does not decrease over time. Thus, a typical analysis time for controlled- current coulometry is less than 10 min, as opposed to approximately 30–60 min for controlled-potential coulometry. Second, with a constant current the total charge is simply the product of current and time (equation 11.24). A method for integrating the current–time curve, therefore, is not necessary. Using a constant current does present two important experimental problems that must be solved if accurate results are to be obtained. First, as electrolysis oc- curs the analyte’s concentration and, therefore, the current due to its oxidation or reduction steadily decreases. To maintain a constant current the cell potential must change until another oxidation or reduction reaction can occur at the working electrode. Unless the system is carefully designed, these secondary reac- tions will produce a current efficiency of less than 100%. The second problem is the need for a method of determining when the analyte has been exhaustively electrolyzed. In controlled-potential coulometry this is signaled by a decrease in the current to a constant background or residual current (see Figure 11.20). In controlled-current coulometry, however, a constant current continues to flow even when the analyte has been completely oxidized or reduced. A suitable means of determining the end-point of the reaction, t e , is needed. Maintaining Current Efficiency To illustrate why changing the working electrode’s potential can lead to less than 100% current efficiency, let’s consider the coulomet- ric analysis for Fe 2+ based on its oxidation to Fe 3+ at a Pt working electrode in 1 M H 2 SO 4 . Fe 2+ (aq) t Fe 3+ (aq)+e – The ladder diagram for this system is shown in Figure 11.24a. Initially the potential of the working electrode remains nearly constant at a level near the standard-state potential for the Fe 3+ /Fe 2+ redox couple. As the concentration of Fe 2+ decreases, however, the potential of the working electrode shifts toward more positive values until another oxidation reaction can provide the necessary current. Thus, in this case the potential eventually increases to a level at which the oxidation of H 2 O occurs. 6H 2 O(l) t O 2 (g)+4H 3 O + (aq)+4e – Since the current due to the oxidation of H 3 O + does not contribute to the oxidation of Fe 2+ , the current efficiency of the analysis is less than 100%. To maintain a 100% current efficiency the products of any competing oxidation reactions must react both rapidly and quantitatively with the remaining Fe 2+ . This may be accomplished, for example, by adding an excess of Ce 3+ to the analytical solution (Figure 11.24b). When the potential of the working electrode shifts to a more positive potential, the first species to be oxidized is Ce 3+ . Ce 3+ (aq) t Ce 4+ (aq)+e – The Ce 4+ produced at the working electrode rapidly mixes with the solution, where it reacts with any available Fe 2+ . Chapter 11 Electrochemical Methods of Analysis 499 Current Time i t Q = it 1400-CH11 9/9/99 2:07 PM Page 499 Figure 11.24 Ladder diagrams for the controlled-current coulometric analysis of Fe 2+ (a) without the addition of Ce 3+ , and (b) with the addition of Ce 3+ . The matrix is 1 M H 2 SO 4 in both cases. 500 Modern Analytical Chemistry Ce 4+ (aq)+Fe 2+ (aq) t Fe 3+ (aq)+Ce 3+ (aq) 11.31 Combining these reactions gives the desired overall reaction of Fe 2+ (aq) t Fe 3+ (aq)+e – In this manner, a current efficiency of 100% is maintained. Furthermore, since the concentration of Ce 3+ remains at its initial level, the potential of the working elec- trode remains constant as long as any Fe 2+ is present. This prevents other oxidation reactions, such as that for H 2 O, from interfering with the analysis. A species, such as Ce 3+ , which is used to maintain 100% current efficiency, is called a mediator. End Point Determination Adding a mediator solves the problem of maintaining 100% current efficiency, but does not solve the problem of determining when the analyte’s electrolysis is complete. Using the same example, once all the Fe 2+ has been oxidized current continues to flow as a result of the oxidation of Ce 3+ and, eventually, the oxidation of H 2 O. What is needed is a means of indicating when the oxidation of Fe 2+ is complete. In this respect it is convenient to treat a controlled- current coulometric analysis as if electrolysis of the analyte occurs only as a result of its reaction with the mediator. A reaction between an analyte and a mediator, such as that shown in reaction 11.31, is identical to that encountered in a redox titration. Thus, the same end points that are used in redox titrimetry (see Chapter 9), such as visual indicators, and potentiometric and conductometric measurements, may be used to signal the end point of a controlled-current coulometric analysis. For exam- ple, ferroin may be used to provide a visual end point for the Ce 3+ -mediated coulo- metric analysis for Fe 2+ . Instrumentation Controlled-current coulometry normally is carried out using a galvanostat and an electrochemical cell consisting of a working electrode and a counterelectrode. The working electrode, which often is constructed from Pt, is also More (+) More (–) O 2 H 2 O H 2 H 3 O + Fe 3+ Fe 2+ E ° Fe 3+ /Fe 2+ = + 0.68 V E More (+) More (–) O 2 H 2 O H 2 H 3 O + Fe 3+ Fe 2+ Ce 4+ Ce 3+ E ° Fe 3+ /Fe 2+ = + 0.68 V E ° Ce 4+ /Ce 3+ = + 1.44 V E (a) (b) mediator A species that transfers electrons from the electrode to the analyte. 1400-CH11 9/9/99 2:07 PM Page 500 Figure 11.25 Method for the external generation of oxidizing and reducing agents in coulometric titrations. called the generator electrode since it is where the mediator reacts to generate the species reacting with the analyte. The counterelectrode is isolated from the analyti- cal solution by a salt bridge or porous frit to prevent its electrolysis products from reacting with the analyte. Alternatively, oxidizing or reducing the mediator can be carried out externally, and the appropriate products flushed into the analytical solu- tion. Figure 11.25 shows one simple method by which oxidizing and reducing agents can be generated externally. A solution containing the mediator flows under the influence of gravity into a small-volume electrochemical cell. The products gen- erated at the anode and cathode pass through separate tubes, and the appropriate oxidizing or reducing reagent can be selectively delivered to the analytical solution. For example, external generation of Ce 4+ can be obtained using an aqueous solution of Ce 3+ and the products generated at the anode. The other necessary instrumental component for controlled-current coulometry is an accurate clock for measuring the electrolysis time, t e , and a switch for starting and stopping the electrolysis. Analog clocks can read time to the nearest – 0.01 s, but the need to frequently stop and start the electrolysis near the end point leads to a net uncertainty of – 0.1 s. Digital clocks provide a more accurate measurement of time, with errors of – 1 ms being possible. The switch must control the flow of current and the clock, so that an accurate determination of the electrolysis time is possible. Coulometric Titrations Controlled-current coulometric methods commonly are called coulometric titrations because of their similarity to conventional titrations. We already have noted, in discussing the controlled-current coulometric determi- nation of Fe 2+ , that the oxidation of Fe 2+ by Ce 4+ is identical to the reaction used in a redox titration. Other similarities between the two techniques also exist. Combin- ing equations 11.23 and 11.24 and solving for the moles of analyte gives 11.32 Compare this equation with the relationship between the moles of strong acid, N, titrated with a strong base of known concentration. N =(M base )(V base ) The titrant in a conventional titration is replaced in a coulometric titration by a constant-current source whose current is analogous to the titrant’s molarity. The time needed for an exhaustive electrolysis takes the place of the volume of titrant, and the switch for starting and stopping the electrolysis serves the same function as a buret’s stopcock. 11C. 3 Quantitative Applications Coulometry may be used for the quantitative analysis of both inorganic and organic compounds. Examples of controlled-potential and controlled-current coulometric methods are discussed in the following sections. Controlled-Potential Coulometry The majority of controlled-potential coulometric analyses involve the determination of inorganic cations and anions, including trace metals and halides. Table 11.8 provides a summary of several of these methods. The ability to control selectivity by carefully selecting the working electrode’s potential, makes controlled-potential coulometry particularly useful for the analysis of alloys. For example, the composition of an alloy containing Ag, Bi, Cd, and Sb N i nF t= ? ? ? ? ? ? () e Chapter 11 Electrochemical Methods of Analysis 501 Source of reducing agent Glass wool Source of oxidizing agent Mediator solution Cathode Anode coulometric titrations A titration in which the equivalence point is the time required for a constant current to completely oxidize or reduce the analyte. 1400-CH11 9/9/99 2:07 PM Page 501 can be determined by dissolving the sample and placing it in a matrix of 0.2 M H 2 SO 4 . A platinum working electrode is immersed in the solution and held at a constant potential of +0.40 V versus the SCE. At this potential Ag(I) deposits on the Pt electrode as Ag, and the other metal ions remain in solution. When electrolysis is complete, the total charge is used to determine the amount of silver in the alloy. The potential of the platinum electrode is then shifted to –0.08 V versus the SCE, depositing Bi on the working electrode. When the coulometric analysis for bismuth is complete, antimony is determined by shifting the working electrode’s potential to –0.33 V versus the SCE, depositing Sb. Finally, cadmium is determined following its electrodeposition on the Pt electrode at a potential of –0.80 V versus the SCE. Another area where controlled-potential coulometry has found application is in nuclear chemistry, in which elements such as uranium and polonium can be de- termined at trace levels. For example, microgram quantities of uranium in a medium of H 2 SO 4 can be determined by reducing U(VI) to U(IV) at a mercury working electrode. Controlled-potential coulometry also can be applied to the quantitative analy- sis of organic compounds, although the number of applications is significantly less than that for inorganic analytes. One example is the six-electron reduction of a nitro group, –NO 2 , to a primary amine, –NH 2 , at a mercury electrode. Solutions of picric acid, for instance, can be analyzed by reducing to triaminophenol. 502 Modern Analytical Chemistry Table 11.8 Representative Examples for the Controlled- Potential Coulometric Analysis of Inorganic Ions Analyte Electrolytic Reaction a Electrode Antimony Sb(III) + 3e – t Sb Pt Arsenic As(III) t As(V) + 2e – Pt Cadmium Cd(II) + 2e – t Cd Pt or Hg Cobalt Co(II) + 2e – t Co Pt or Hg Copper Cu(II) + 2e – t Cu Pt or Hg Halides Ag + X – t AgX + e – Ag Iron Fe(II) t Fe(III) + e – Pt Lead Pb(II) + 2e – t Pb Pt or Hg Nickel Ni(II) + 2e – t Ni Pt or Hg Plutonium Pu(III) t Pu(IV) + e – Pt Silver Ag(I) + e – t Ag Pt Tin Sn(II) + 2e – t Sn Pt Uranium U(VI) + 2e – t U(IV) Pt or Hg Zinc Zn(II) + 2e – t Zn Pt or Hg Source: Compiled from Rechnitz, G. A. Controlled-Potential Analysis. Macmillan: New York, 1963, p. 49. a Electrolytic reactions are written in terms of the change in oxidation state. The actual species in solution depend on the composition of the sample matrix. O 2 N NO 2 NO 2 OH + 18H 3 O + + 18e – + 24H 2 O H 2 N NH 2 NH 2 OH 1400-CH11 9/9/99 2:07 PM Page 502 Another example is the successive reduction of trichloroacetate to dichloroac- etate, and of dichloroacetate to monochloroacetate Cl 3 CCOO – (aq)+H 3 O + (aq)+2e – t Cl 2 HCCOO – (aq)+Cl – (aq)+H 2 O(l) Cl 2 HCCOO – (aq)+H 3 O + (aq)+2e – t ClH 2 CCOO – (aq)+Cl – (aq)+H 2 O(l) Mixtures of trichloroacetate and dichloroacetate are analyzed by selecting an initial potential at which only the more easily reduced trichloroacetate is reduced. When its electrolysis is complete, the potential is switched to a more negative potential at which dichloroacetate is reduced. The total charge for the first electrolysis is used to determine the amount of trichloroacetate, and the difference in total charge be- tween the first and second electrolyses gives the amount of dichloroacetate. Controlled-Current Coulometry The use of a mediator makes controlled-current coulometry a more versatile analytical method than controlled-potential coulome- try. For example, the direct oxidation or reduction of a protein at the working elec- trode in controlled-potential coulometry is difficult if the protein’s active redox site lies deep within its structure. The controlled-current coulometric analysis of the protein is made possible, however, by coupling its oxidation or reduction to a medi- ator that is reduced or oxidized at the working electrode. Controlled-current coulo- metric methods have been developed for many of the same analytes that may be de- termined by conventional redox titrimetry. These methods, several of which are summarized in Table 11.9, also are called coulometric redox titrations. Coupling the mediator’s oxidation or reduction to an acid–base, precipitation, or complexation reaction involving the analyte allows for the coulometric titration of analytes that are not easily oxidized or reduced. For example, when using H 2 O as a mediator, oxidation at the anode produces H 3 O + 6H 2 O(l) t 4H 3 O + (aq)+O 2 (g)+4e – while reduction at the cathode produces OH – . 2H 2 O(l)+2e – t 2OH – (aq)+H 2 (g) If the oxidation or reduction of H 2 O is carried out externally using the generator cell shown in Figure 11.25, then H 3 O + or OH – can be dispensed selectively into a solution containing a basic or acidic analyte. The resulting reaction is identical to that in an acid–base titration. Coulometric acid–base titrations have been used for Chapter 11 Electrochemical Methods of Analysis 503 Table 11. 9 Representative Examples of Coulometric Redox Titrations Electrochemically Generator-Electrode Mediator Generated Reagent Reaction Representative Application a Ag + Ag 2+ Ag + t Ag 2+ + e – H 2 C 2 O 4 + 2Ag 2+ +2H 2 O t 2CO 2 + 2Ag + +2H 3 O + Br – Br 2 2Br – t Br 2 +2e – H 2 S+Br 2 +2H 2 O t S + 2Br – +2H 3 O + Ce 3+ Ce 4+ Ce 3+ t Ce 4+ + e – Fe(CN) 6 4– +Ce 4+ t Fe(CN) 6 3– +Ce 3+ Cl – Cl 2 2Cl – t Cl 2 +2e – Tl(I) + Cl 2 t Tl(III) + 2Cl – Fe 3+ Fe 2+ Fe 3+ + e – t Fe 2+ Cr 2 O 7 2– + 6Fe 2+ + 14H 3 O + t 2Cr 3+ + 6Fe 3+ + 21H 2 O I – I 3 – 3I – t I 3 – +2e – 2S 2 O 3 2– +I 3 – t S 4 O 6 2– +3I – Mn 2+ Mn 3+ Mn 2+ t Mn 3+ + e – As(III) + 2Mn 3+ t As(V) + 2Mn 2+ a The analyte is the underlined species in each reaction. 1400-CH11 9/9/99 2:07 PM Page 503 504 Modern Analytical Chemistry Table 11.10 Representative Examples of Coulometric Titrations Using Acid–Base, Complexation, and Precipitation Reactions Type of Electrochemically Generator-Electrode Reaction Mediator Generated Reagent Reaction Representative Reaction a Acid–Base H 2 OH 3 O + 6H 2 O t 4H 3 O + +O 2 +4e – OH – +H 3 O + t 2H 2 O H 2 OOH – 2H 2 O+2e – t 2OH – +H 2 H 3 O + +OH – t 2H 2 O Complexation HgNH 3 Y 2– EDTA (HY 3– ) HgNH 3 Y 2– +NH 4 + +2e – Ca 2+ +HY 3– +H 2 O t CaY 2– +H 3 O + t HY 3– +Hg+2NH 3 Precipitation Ag Ag + Ag t Ag + + e – I – +Ag + t AgI Hg Hg 2 2+ 2Hg t Hg 2 2+ +2e – 2Cl – +Hg 2 2+ t Hg 2 Cl 2 Fe(CN) 6 3– Fe(CN) 6 4- Fe(CN) 6 3– +e – t Fe(CN) 6 4– 3Zn 2+ +2K + + 2Fe(CN) 6 4– t K 2 Zn 3 [Fe(CN) 6 ] 2 a The analyte is the underlined species in each reaction. the analysis of strong and weak acids and bases, in both aqueous and nonaqueous matrices. Examples of coulometric titrations involving acid–base, complexation, and precipitation reactions are summarized in Table 11.10. In comparison with conventional titrimetry, there are several advantages to the coulometric titrations listed in Tables 11.9 and 11.10. One advantage is that the electrochemical generation of a “titrant” that reacts immediately with the analyte al- lows the use of reagents whose instability prevents their preparation and storage as a standard solution. Thus, highly reactive reagents such as Ag 2+ and Mn 3+ can be used in coulometric titrations. Because it is relatively easy to measure small quantities of charge, coulometric titrations can be used to determine small quantities of analyte that cannot be measured accurately by a conventional titration. Quantitative Calculations The absolute amount of analyte in a coulometric analy- sis is determined by applying Faraday’s law (equation 11.23) with the total charge during the electrolysis given by equation 11.24 or equation 11.25. Example 11.8 shows the calculations for a typical coulometric analysis. EXAMPLE 11.8 The purity of a sample of Na 2 S 2 O 3 was determined by a coulometric redox titration using I – as a mediator, and I 3 – as the “titrant.” A sample weighing 0.1342 g is transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10.00-mL portion is transferred to an electrochemical cell along with 25 mL of 1 M KI, 75 mL of a pH 7.0 phosphate buffer, and several drops of a starch indicator solution. Electrolysis at a constant current of 36.45 mA required 221.8 s to reach the starch indicator end point. Determine the purity of the sample. SOLUTION From Table 11.9 we see that the coulometric titration of S 2 O 3 2– with I 3 – is 2S 2 O 3 2– (aq)+I 3 – (aq) t S 4 O 6 2– (aq)+3I – (aq) Oxidizing S 2 O 3 2– to S 4 O 6 2– requires one electron per S 2 O 3 2– (n = 1). Combining equations 11.23 and 11.24, and making an appropriate substitution for moles of Na 2 S 2 O 3 gives 1400-CH11 9/9/99 2:07 PM Page 504 Chapter 11 Electrochemical Methods of Analysis 505 Representative Methods —Continued Solving for the grams of Na 2 S 2 O 3 gives This represents the amount of Na 2 S 2 O 3 in a 10.00-mL portion of a 100-mL sample, thus 0.1325 g of Na 2 S 2 O 3 is present in the original sample. The purity of the sample, therefore, is Note that the calculation is worked as if S 2 O 3 2– is oxidized directly at the working electrode instead of in solution. Representative Method Every controlled-potential or controlled-current coulo- metric method has its own unique considerations. Nevertheless, the following pro- cedure for the determination of dichromate by a coulometric redox titration pro- vides an instructive example. Method 11.2 Determination of Dichromate by a Coulometric Redox Titration 13 Description of the Method. The concentration of Cr 2 O 7 2– in a sample is determined by a coulometric redox titration using Fe 3+ as a mediator and electrogenerated Fe 2+ as the “titrant.” The end point of the coulometric redox titration is determined potentiometrically. Procedure. The electrochemical cell consists of a Pt working electrode and a Pt counterelectrode that is maintained in a separate compartment connected to the analytical solution by a porous glass disk. The counter electrode’s compartment is filled with 0.2 M Na 2 SO 4 , the level of which must always be maintained above that of the solution in the electrochemical cell. Platinum and tungsten electrodes connected to a potentiometer are used to follow the change in potential during the analysis. A solution of approximately 0.3 M NH 4 Fe(SO 4 ) 2 is prepared for use as the mediator. The coulometric redox titration is carried out by adding 5.00 mL of the sample solution, 2 mL of 9 M H 2 SO 4 and 10–25 mL of the NH 4 Fe(SO 4 ) 2 mediator solution. Distilled water is added as needed until the electrodes are covered. Pure N 2 is bubbled through the solution for 15 min to remove traces of O 2 . A flow of N 2 is maintained during the electrolysis, but is turned off momentarily when measuring the potential. A magnetic stir bar is used to stir the solution. Adjust the current to 15–50 mA, and begin the titration. Periodically stop the titration, and measure the potential of the solution. In this manner a titration 0 1325 0 1342 100 98 73 . . .% g Na S O -g sample w/w Na S O 22 3 22 3 ×= g Na S O FW Na S O A)(221.8 s)(158.1 g/mol) (1 mol C/mol g Na S O 22 3 e223 22 3 = == ?? it nF ee () (. )( ) . 0 03645 96487 0 01325 nF it ()g Na S O FW Na S O 22 3 22 3 e = 1400-CH11 9/9/99 2:07 PM Page 505 506 Modern Analytical Chemistry Continued from page 505 curve of potential versus time is recorded. The time needed to reach the equivalence point is read from the titration curve. Questions 1. Does the platinum working electrode serve as the cathode or the anode in this analysis? Reduction of Fe 3+ to Fe 2+ occurs at the working electrode, making it the cathode in the electrochemical cell. 2. Pure N 2 is bubbled through the solution to remove any trace of dissolved O 2 . Why is this necessary? If O 2 is present, some of the electrogenerated Fe 2+ may be oxidized back to Fe 3+ by the reaction 4Fe 2+ (aq)+O 2 (g)+4H 3 O + (aq) t 4Fe 3+ (aq)+6H 2 O(l) Any Fe 2+ lost in this fashion must be replaced by the additional reduction of Fe 3+ , reducing the current efficiency and increasing the time needed to reach the titration’s end point. The net result is that the reported concentration of Cr 2 O 7 2– is too large. 3. What is the effect on the analysis if the NH 4 Fe(SO 4 ) 2 used to prepare the mediator solution is contaminated with trace amounts of Fe 2+ ? The Fe 2+ introduced when the mediator solution is first added will react with the Cr 2 O 7 2– before the electrolysis begins. As a result, the amount of Fe 2+ that must be generated is less than expected, decreasing the time needed to reach the end point of the titration. The current efficiency, therefore, is greater than 100%, and the reported concentration of Cr 2 O 7 2– is too small. Trace amount of Fe 2+ can be removed from the mediator solution by adding H 2 O 2 and heating at 50–70 °C until the evolution of O 2 ceases. Alternatively, a blank titration can be used to correct for any Fe 2+ initially present in the mediator. 11C. 4 Characterization Applications Studies aimed at characterizing the mechanisms of electrode reactions often make use of coulometry for determining the number of electrons involved in the reaction. To make such measurements a known amount of a pure compound is subject to a controlled-potential electrolysis. The coulombs of charge needed to complete the electrolysis are used to determine the value of n using Faraday’s law (equation 11.23). EXAMPLE 11. 9 A 0.3619-g sample of tetrachloropicolinic acid, C 6 HNO 2 Cl 4 , is dissolved in distilled water, transferred to a 1000-mL volumetric flask, and diluted to volume. An exhaustive controlled-potential electrolysis of a 10.00-mL portion of this solution at a spongy silver cathode requires 5.374 C of charge. What is the value of n for this reduction reaction? 1400-CH11 9/9/99 2:07 PM Page 506 SOLUTION The 10.00-mL portion of sample contains 3.619 mg, or 1.39 · 10 –5 mol of tetrachloropicolinic acid. Solving equation 11.23 for n and making appropriate substitutions gives Thus, reducing a molecule of tetrachloropicolinic acid requires four electrons. The overall reaction, which results in the selective formation of 3,6- dichloropicolinic acid, is 11C. 5 Evaluation Scale of Operation Coulometric methods of analysis can be used to analyze small absolute amounts of analyte. In controlled-current coulometry, for example, the moles of analyte consumed during an exhaustive electrolysis is given by equation 11.32. An electrolysis carried out with a constant current of 100 m A for 100 s, there- fore, consumes only 1 · 10 –7 mol of analyte if n = 1. For an analyte with a molecu- lar weight of 100 g/mol, 1 · 10 –7 mol corresponds to only 10 m g. The concentration of analyte in the electrochemical cell, however, must be sufficient to allow an accu- rate determination of the end point. When using visual end points, coulometric titrations require solution concentrations greater than 10 –4 M and, as with conven- tional titrations, are limited to major and minor analytes. A coulometric titration to a preset potentiometric end point is feasible even with solution concentrations of 10 –7 M, making possible the analysis of trace analytes. 14 Accuracy The accuracy of a controlled-current coulometric method of analysis is determined by the current efficiency, the accuracy with which current and time can be measured, and the accuracy of the end point. With modern instrumentation the maximum measurement error for current is about – 0.01%, and that for time is ap- proximately – 0.1%. The maximum end point error for a coulometric titration is at least as good as that for conventional titrations and is often better when using small quantities of reagents. Taken together, these measurement errors suggest that accu- racies of 0.1–0.3% are feasible. The limiting factor in many analyses, therefore, is current efficiency. Fortunately current efficiencies of greater than 99.5% are ob- tained routinely and often exceed 99.9%. In controlled-potential coulometry, accuracy is determined by current effi- ciency and the determination of charge. Provided that no interferents are present that are easier to oxidize or reduce than the analyte, current efficiencies of greater than 99.9% are easily obtained. When interferents are present, however, they can often be eliminated by applying a potential such that the exhaustive electrolysis of the interferents is possible without the simultaneous electrolysis of the analyte. Once the interferents have been removed the potential can be switched to a level at n Q FN e == × = ?? 5 374 139 10 401 5 . )( . ) . C (96478 C/mol mol C HNO Cl 624 Chapter 11 Electrochemical Methods of Analysis 507 Cl Cl Cl Cl Cl CO 2 – + 4e – + 2H 2 O + 2Cl – + 2OH – CO 2 – Cl NN 1400-CH11 9/9/99 2:07 PM Page 507 508 Modern Analytical Chemistry voltammetry An electrochemical method in which we measure current as a function of the applied potential. voltammogram A plot of current as a function of applied potential. which electrolysis of the analyte is feasible. The limiting factor in the accuracy of many controlled-potential coulometric methods of analysis is the determination of charge. With modern electronic integrators, the total charge can be determined with an accuracy of better than 0.5%. So what is to be done when an acceptable current efficiency is not feasible? If the analyte’s oxidation or reduction leads to its deposition on the working elec- trode, it may be possible to determine the analyte’s mass. In this case the working electrode is weighed before beginning the electrolysis and reweighed when electroly- sis of the analyte is complete. The difference in the electrode’s weight gives the ana- lyte’s mass. This technique is known as electrogravimetry. More details about this technique are provided in the suggested reading listed at the end of this chapter. Precision Precision is determined by the uncertainties of measuring current, time, and the end point in controlled-current coulometry and of measuring charge in controlled-potential coulometry. Precisions of – 0.1–0.3% are routinely obtained for coulometric titrations, and precisions of – 0.5% are typical for controlled-potential coulometry. Sensitivity For a coulometric method of analysis, the calibration sensitivity is equivalent to nF in equation 11.25. In general, coulometric methods in which the an- alyte’s oxidation or reduction involves a larger value of n show a greater sensitivity. Selectivity Selectivity in controlled-potential and controlled-current coulometry is improved by carefully adjusting solution conditions and by properly selecting the electrolysis potential. In controlled-potential coulometry the potential is fixed by the potentiostat, whereas in controlled-current coulometry the potential is deter- mined by the redox reaction involving the mediator. In either case, the ability to control the potential at which electrolysis occurs affords some measure of selectiv- ity. By adjusting pH or adding a complexing agent, it may be possible to shift the potential at which an analyte or interferent undergoes oxidation or reduction. For example, the standard-state reduction potential for Zn 2+ is –0.762 V versus the SHE, but shifts to –1.04 for Zn(NH 3 ) 4 2+ . This provides an additional means for controlling selectivity when an analyte and interferent undergo electrolysis at simi- lar potentials. Time, Cost, and Equipment Controlled-potential coulometry is a relatively time- consuming analysis, with a typical analysis requiring 30–60 min. Coulometric titra- tions, on the other hand, require only a few minutes and are easily adapted for au- tomated analysis. Commercial instrumentation for both controlled-potential and controlled-current coulometry is available and is relatively inexpensive. Low-cost potentiostats and constant-current sources are available for less than $1000. 11D Voltammetric Methods of Analysis In voltammetry a time-dependent potential is applied to an electrochemical cell, and the current flowing through the cell is measured as a function of that poten- tial. A plot of current as a function of applied potential is called a voltammogram and is the electrochemical equivalent of a spectrum in spectroscopy, provid- ing quantitative and qualitative information about the species involved in the oxidation or reduction reaction. 15 The earliest voltammetric technique to be introduced was polarography, which was developed by Jaroslav Heyrovsky 1400-CH11 9/9/99 2:07 PM Page 508 (1890–1967) in the early 1920s, for which he was awarded the Nobel Prize in chemistry in 1959. Since then, many different forms of voltammetry have been developed, a few of which are noted in Figure 11.1. Before examining these techniques and their applications in more detail, however, we must first consider the basic experimental design for making voltammetric measure- ments and the factors influencing the shape of the resulting voltammogram. 11D.1 Voltammetric Measurements Although early voltammetric methods relied on the use of only two electrodes, modern voltammetry makes use of a three-electrode po- tentiostat, such as that shown in Figure 11.4. A time-dependent po- tential excitation signal is applied to the working electrode, chang- ing its potential relative to the fixed potential of the reference electrode. The resulting current between the working and auxiliary electrodes is measured. The auxiliary electrode is generally a plat- inum wire, and the SCE and Ag/AgCl electrode are common refer- ence electrodes. Several different materials have been used as working elec- trodes, including mercury, platinum, gold, silver, and carbon. The earliest voltammetric techniques, including polarography, used mercury for the working electrode. Since mercury is a liquid, the working electrode often consists of a drop suspended from the end of a capillary tube (Figure 11.26). In the hanging mercury drop electrode, or HMDE, a drop of the desired size is formed by the ac- tion of a micrometer screw that pushes the mercury through a narrow capillary tube. In the dropping mercury electrode, or DME, mercury drops form at the end of the capillary tube as a result of gravity. Unlike the HMDE, the mercury drop of a DME grows continuously and has a finite lifetime of several seconds. At the end of its lifetime the mercury drop is dislodged, either manually or by gravity, and replaced by a new drop. The static mercury drop electrode, or SMDE, uses a solenoid-driven plunger to control the flow of mercury. The SMDE can be used as either a hanging mercury drop electrode or as a dropping mercury electrode. A single activation of the solenoid momentarily lifts the plunger, allowing enough mercury to flow through the capillary to form a single drop. To obtain a dropping mercury elec- trode the solenoid is activated repeatedly. A mercury film electrode consists of a thin layer of mercury deposited on the surface of a solid carbon, platinum, or gold elec- trode. The solid electrode is placed in a solution of Hg 2+ and held at a potential at which the reduction of Hg 2+ to Hg is favorable, forming a thin mercury film. Mercury has several advantages as a working electrode. Perhaps its most im- portant advantage is its high overpotential for the reduction of H 3 O + to H 2 , which allows for the application of potentials as negative as - 1 V versus the SCE in acidic solutions, and –2 V versus the SCE in basic solutions. A species such as Zn 2+ , which is difficult to reduce at other electrodes without simultaneously reducing H 3 O + , is easily reduced at a mercury working electrode. Other advantages include the ability of metals to dissolve in the mercury, resulting in the formation of an amalgam, and the ability to easily renew the surface of the electrode by extruding a new drop. One limitation to its use as a working electrode is the ease with which Hg is oxidized. For this reason, mercury electrodes cannot be used as at potentials more positive than –0.3 V to +0.4 V versus the SCE, depending on the composition of the solution. Chapter 11 Electrochemical Methods of Analysis 509 Micrometer assembly Capillary Capillary Hg Drop (a) (b) (c) Mercury reservoir Mercury reservoir Solenoid Solenoid Plunger Figure 11.26 Mercury electrodes: (a) hanging mercury drop electrode; (b) dropping mercury electrode; (c) static mercury drop electrode. hanging mercury drop electrode An electrode in which a drop of Hg is suspended from a capillary tube. dropping mercury electrode An electrode in which successive drops of Hg form at the end of a capillary tube as a result of gravity, with each drop providing a fresh electrode surface. static mercury drop electrode An electrode in which successive drops of Hg form at the end of a capillary tube as the result of a mechanical plunger, with each drop providing a fresh electrode surface. amalgam A metallic solution of mercury with another metal. 1400-CH11 9/9/99 2:07 PM Page 509 Figure 11.27 Schematic diagram of a solid electrode. Solid electrodes constructed using platinum, gold, silver, or carbon may be used over a range of potentials, including potentials that are negative and positive with respect to the SCE. For example, the potential range for a Pt electrode extends from approximately +1.2 V to –0.2 V versus the SCE in acidic solutions and from +0.7 V to - 1 V versus the SCE in basic solutions. Solid electrodes, therefore, can be used in place of mercury for many voltammetric analyses requiring negative po- tentials and for voltammetric analyses at positive potentials at which mercury elec- trodes cannot be used. Except for the carbon paste electrode, solid electrodes are fashioned into disks that are sealed into the end of an inert support and are in con- tact with an electrical lead (Figure 11.27). The carbon paste electrode is made by filling the cavity at the end of the inert support with a paste consisting of carbon particles and a viscous oil. Solid electrodes are not without problems, the most im- portant of which is the ease with which the electrode’s surface may be altered by the adsorption of solution species or the formation of oxide layers. For this reason solid electrodes need frequent reconditioning, either by applying an appropriate potential or by polishing. A typical arrangement for a voltammetric electrochemical cell is shown in Fig- ure 11.28. Besides the working, reference, and auxiliary electrodes, the cell also in- cludes a N 2 purge line for removing dissolved O 2 and an optional stir bar. Electro- chemical cells are available in a variety of sizes, allowing for the analysis of solution volumes ranging from more than 100 mL to as small as 50 m L. 11D.2 Current in Voltammetry When an analyte is oxidized at the working electrode, a current passes electrons through the external electric circuitry to the auxiliary electrode, where reduc- tion of the solvent or other components of the solution matrix occurs. Reducing an analyte at the working electrode requires a source of electrons, generating a current that flows from the auxiliary electrode to the cathode. In either case, a current resulting from redox reactions at the working and auxiliary electrodes is called a faradaic current. In this section we consider the factors affecting the magnitude of this faradaic current, as well as the source of any nonfaradaic currents. Sign Conventions Since the reaction of interest occurs at the working electrode, the classification of current is based on this reaction. A current due to the analyte’s reduction is called a cathodic current and, by convention, is considered positive. Anodic currents are due to oxidation reactions and carry a negative value. Influence of Applied Potential on the Faradaic Current As an example, let’s con- sider the faradaic current when a solution of Fe(CN) 6 3– is reduced to Fe(CN) 6 4– at the working electrode. The relationship between the concentrations of Fe(CN) 6 3– , Fe(CN) 6 4– , and the potential of the working electrode is given by the Nernst equa- tion; thus where +0.356 is the standard-state potential for the Fe(CN) 6 3– /Fe(CN) 6 4– redox couple, and x = 0 indicates that the concentrations of Fe(CN) 6 3– and Fe(CN) 6 4– are E x x =+ ? ? = ? = 0 356 0 05916 4 0 3 0 . . log [] [] Fe(CN) Fe(CN) 6 6 510 Modern Analytical Chemistry Electrical lead Solid disc electrode faradaic current Any current in an electrochemical cell due to an oxidation or reduction reaction. cathodic current A faradaic current due to a reduction reaction. anodic current A faradaic current due to an oxidation reaction. 1400-CH11 9/9/99 2:07 PM Page 510 Figure 11.28 Typical electrochemical cell for use in voltammetry. those at the surface of the electrode. Surface concentrations are used instead of bulk concentrations since the equilibrium position for the redox reaction Fe(CN) 6 3– (aq)+e – t Fe(CN) 6 4– (aq) can only be established electrochemically at the electrode’s surface. Let’s assume that we have a solution for which the concentration of Fe(CN) 6 3– is 1.0 mM and in which Fe(CN) 6 4– is absent. A ladder diagram for this redox exam- ple is shown in Figure 11.29. If a potential of +0.530 V is applied to the working electrode, the concentrations of Fe(CN) 6 3– and Fe(CN) 6 4– at the surface of the elec- trode are unaffected, and no faradaic current is observed. Switching the potential to +0.356 V, however, requires that [Fe(CN) 6 4– ] x=0 = [Fe(CN) 6 3– ] x=0 = 0.50 mM which is only possible if a portion of the Fe(CN) 6 3– at the electrode surface is re- duced to Fe(CN) 6 4– . If this was all that occurred after the potential was applied, the result would be a brief surge of faradaic current that would quickly return to zero. However, although the concentration of Fe(CN) 6 4– at the electrode surface is 0.50 mM, its concentration in the bulk of solution is zero. As a result, a concentration gradient exists between the solution at the electrode surface and the bulk solution. This concentration gradient creates a driving force that transports Fe(CN) 6 4– away from the electrode surface (Figure 11.30). The subsequent decrease in the concen- tration of Fe(CN) 6 4– at the electrode surface requires the further reduction of Fe(CN) 6 3– , as well as its transport from bulk solution to the electrode surface. Thus, a faradaic current continues to flow until there is no difference between the concen- trations of Fe(CN) 6 3– and Fe(CN) 6 4– at the electrode surface and their concentra- tions in the bulk of solution. * Although the applied potential at the working electrode determines if a faradaic current flows, the magnitude of the current is determined by the rate of the result- ing oxidation or reduction reaction at the electrode surface. Two factors contribute to the rate of the electrochemical reaction: the rate at which the reactants and prod- ucts are transported to and from the surface of the electrode, and the rate at which electrons pass between the electrode and the reactants and products in solution. Influence of Mass Transport on the Faradaic Current There are three modes of mass transport that influence the rate at which reactants and products are trans- ported to and from the electrode surface: diffusion, migration, and convection. Dif- fusion from a region of high concentration to a region of low concentration occurs whenever the concentration of an ion or molecule at the surface of the electrode is different from that in bulk solution. When the potential applied to the working electrode is sufficient to reduce or oxidize the analyte at the electrode surface, a con- centration gradient similar to that shown in Figure 11.31 is established. The volume of solution in which the concentration gradient exists is called the diffusion layer. Without other modes of mass transport, the width of the diffusion layer, d , in- creases with time as the concentration of reactants near the electrode surface de- creases. The contribution of diffusion to the rate of mass transport, therefore, is time-dependent. Chapter 11 Electrochemical Methods of Analysis 511 N 2 Purge line Auxiliary electrode Working electrode Reference electrode Stir bar Fe(CN) 6 3– E° = + 0.356 V Fe(CN) 6 4– E Figure 11.29 Ladder diagram for the Fe(CN) 6 3– /Fe(CN) 6 4– redox half-reaction. mass transport The movement of material toward or away from the electrode surface. diffusion The movement of material in response to a concentration gradient. diffusion layer The layer of solution adjacent to the electrode in which diffusion is the only means of mass transport. *In voltammetry the working electrode’s surface area is significantly smaller than that used in coulometry. Consequently, very little analyte undergoes electrolysis, and the analyte’s concentration in bulk solution remains essentially unchanged. 1400-CH11 9/9/99 2:07 PM Page 511 Figure 11.31 Concentration gradients for the analyte in the absence of convection, showing the time-dependent change in diffusion as a method of mass transport. Convection occurs when a mechanical means is used to carry reactants toward the electrode and to remove products from the electrode. The most common means of convection is to stir the solution using a stir bar. Other methods include rotating the electrode and incorporating the electrode into a flow cell. The final mode of mass transport is migration, which occurs when charged particles in solution are attracted or repelled from an electrode that has a positive or negative surface charge. Thus, when the electrode is positively charged, negatively charged particles move toward the electrode, while positively charged particles move toward the bulk solution. Unlike diffusion and convection, migration only af- fects the mass transport of charged particles. The flux of material to and from the electrode surface is a complex function of all three modes of mass transport. In the limit in which diffusion is the only signifi- cant means for the mass transport of the reactants and products, the current in a voltammetric cell is given by 11.33 where n is the number of electrons transferred in the redox reaction, F is Faraday’s constant, A is the area of the electrode, D is the diffusion coefficient for the reactant or product, C bulk and C x=0 are the concentration of the analyte in bulk solution and at the electrode surface, and d is the thickness of the diffusion layer. For equation 11.33 to be valid, convection and migration must not interfere with the formation of a diffusion layer between the electrode and the bulk of solu- tion. Migration is eliminated by adding a high concentration of an inert supporting electrolyte to the analytical solution. Ions of similar charge are equally attracted or repelled from the surface of the electrode and, therefore, have an equal probability of undergoing migration. The large excess of inert ions, however, ensures that few reactant and product ions will move as a result of migration. Although convection may be easily eliminated by not physically agitating the solution, in some situations it is desirable either to stir the solution or to push the solution through an electro- chemical flow cell. Fortunately, the dynamics of a fluid moving past an electrode re- sults in a small diffusion layer, typically of 0.001 – 0.01-cm thickness, in which the rate of mass transport by convection drops to zero (Figure 11.32). Influence of the Kinetics of Electron Transfer on the Faradaic Current The rate of mass transport is one factor influencing the current in a voltammetric experiment. The ease with which electrons are transferred between the electrode and the reac- tants and products in solution also affects the current. When electron transfer ki- netics are fast, the redox reaction is at equilibrium, and the concentrations of reac- tants and products at the electrode are those specified by the Nernst equation. Such systems are considered electrochemically reversible. In other systems, when electron transfer kinetics are sufficiently slow, the concentration of reactants and products at the electrode surface, and thus the current, differ from that predicted by the Nernst equation. In this case the system is electrochemically irreversible. Nonfaradaic Currents Faradaic currents result from a redox reaction at the elec- trode surface. Other currents may also exist in an electrochemical cell that are unre- lated to any redox reaction. These currents are called nonfaradaic currents and must be accounted for if the faradaic component of the measured current is to be determined. i nFAD C C x = ? = () bulk 0 δ 512 Modern Analytical Chemistry Fe(CN) 6 3– e – Fe(CN) 6 4– Moves toward electrode Moves away from electrode Electrode Figure 11.30 Schematic showing transport of Fe(CN) 6 3– toward the electrode and Fe(CN) 6 4– away from the electrode following the reduction of Fe(CN) 6 3– . [Anal yte] Distance from electrode Increasing time t = 0 migration The movement of a cation or anion in response to an applied potential. convection The movement of material in response to a mechanical force, such as stirring a solution. nonfaradaic current A current in an electrochemical cell that is not the result of a redox reaction. 1400-CH11 9/9/99 2:07 PM Page 512 Figure 11.32 Concentration gradient for the analyte showing the effects of diffusion and convection as methods of mass transport. The most important example of a nonfaradaic current occurs whenever the electrode’s potential is changed. In discussing migration as a means of mass trans- port, we noted that negatively charged particles in solution migrate toward a posi- tively charged electrode, and positively charged particles move away from the same electrode. When an inert electrolyte is responsible for migration, the result is a structured electrode–surface interface called the electrical double layer, or EDL, the exact structure of which is of no concern in the context of this text. The movement of charged particles in solution, however, gives rise to a short-lived, nonfaradaic charging current. Changing the potential of an electrode causes a change in the structure of the EDL, producing a small charging current. Residual Current Even in the absence of analyte, a small current inevitably flows through an electrochemical cell. This current, which is called the residual current, consists of two components: a faradaic current due to the oxidation or reduction of trace impurities, and the charging current. Methods for discriminating between the faradaic current due to the analyte and the residual current are discussed later in this chapter. 11D. 3 Shape of Voltammograms The shape of a voltammogram is determined by several experimental factors, the most important of which are how the current is measured and whether convection is included as a means of mass transport. Despite an abundance of different voltam- metric techniques, several of which are discussed in this chapter, only three shapes are common for voltammograms (Figure 11.33). The voltammogram in Figure 11.33a is characterized by a current that increases from the background residual current to a limiting current at potentials at which the analyte is oxidized or reduced. Since the magnitude of a faradaic current is in- versely proportional to d (equation 11.33), a limiting current implies that the thick- ness of the diffusion layer remains constant. The simplest method for obtaining a limiting current is to stir the solution (Figure 11.32), which can be accomplished with a magnetic stir bar, or by rotating the electrode. Voltammetric techniques that include convection by stirring are called hydrodynamic voltammetry. When con- vection is absent, the thickness of the diffusion layer increases with time (Figure 11.31), resulting in a peak current in place of a limiting current (Figure 11.33b). In the voltammograms in Figures 11.33a and 11.33b, the current is monitored as a function of the applied potential. Alternatively, the change in current following Chapter 11 Electrochemical Methods of Analysis 513 [Anal yte] Distance from electrode Convection Bulk solution Diffusion layer d E i i l E i i p E Di i p Figure 11.33 Three common shapes for voltammograms. (a) (b) (c) electrical double layer The interface between a positively or negatively charged electrode and the negatively or positively charged layer of solution in contact with the electrode. charging current A current in an electrochemical cell due to the electrical double layer’s formation. residual current The current in an electrochemical cell that is present even in the absence of the analyte. hydrodynamic voltammetry A form of voltammetry in which the solution is mechanically stirred. 1400-CH11 9/9/99 2:07 PM Page 513 a change in potential may be measured. The resulting voltammogram, which is shown in Figure 11.33c, also is characterized by a peak current. 11D. 4 Quantitative and Qualitative Aspects of Voltammetry Earlier we described a voltammogram as the electrochemical equivalent of a spec- trum in spectroscopy. In this section we consider how quantitative and qualitative information may be extracted from a voltammogram. Quantitative information is obtained by relating current to the concentration of analyte in the bulk solution. Qualitative information is obtained from the voltammogram by extracting the standard-state potential for the redox reaction. For simplicity we only consider voltammograms similar to that shown in Figure 11.33a. Determining Concentration Let’s assume that the redox reaction at the working electrode is O+ne – t R 11.34 and that initially only O is present in the bulk solution. The current is determined by the rate at which O diffuses through the fixed diffusion layer (see Figure 11.32), and is given by equation 11.33, or i = K O ([O] bulk – [O] x=0 ) 11.35 where K O is a constant equal to nFAD O /d . When the limiting current is reached, the concentration of O at the electrode surface is zero, and this equation simplifies to i lim = K O [O] bulk 11.36 Thus, the limiting current, i lim , is a linear function of the concentration of O in bulk solution, and a quantitative analysis is possible using any of the standardization methods discussed in Chapter 5. Equations similar to equation 11.35 can be devel- oped for other forms of voltammetry, in which peak currents are related to the ana- lyte’s concentration in bulk solution. Determining the Standard-State Potential To extract the standard-state potential, or formal potential, for reaction 11.34 from a voltammogram, it is necessary to rewrite the Nernst equation 11.37 in terms of current instead of the concentration of O and R. Substituting equation 11.36 into equation 11.35 and rearranging gives 11.38 To derive a similar equation for the concentration of R at the electrode surface we note that i = K R ([R] x=0 – [R] bulk ) Since the concentration of R in bulk solution is zero, this equation simplifies to i = K R [R] x=0 [O] lim O x ii K = = ? 0 EE n x x =? ° = = O/R R] O] 0 05916 0 0 . log [ [ 514 Modern Analytical Chemistry 1400-CH11 9/9/99 2:07 PM Page 514 Figure 11.34 Determination of limiting current and half- wave potential in linear scan hydrodynamic voltammetry. and 11.39 Substituting equations 11.39 and 11.38 into equation 11.37 and rearranging gives 11.40 When the current is half that of the limiting current equation 11.40 simplifies to 11.41 where E 1/2 is the half-wave potential (Figure 11.34). If K O is approximately equal to K R , which is often the case, then the half-wave potential is equal to the standard-state potential. Note that equation 11.41 is only valid if the redox reaction is electrochemically reversible. Voltammetric techniques giv- ing peak potentials also can be used to determine a redox reaction’s standard- state potential. 11D. 5 Voltammetric Techniques A number of voltammetric experiments are routinely used in quantitative and qualitative analyses. Several of these methods are briefly described in this section. Polarography The earliest voltammetric experiment was normal polarography at a dropping mercury electrode. In normal polarography the potential is linearly scanned, producing voltammograms such as that shown in Figure 11.35. Although polarography takes place in an unstirred solution, a limiting current is obtained because the falling Hg drops mix the solution. Each new Hg drop, therefore, grows in a solution whose composition is identical to that of the initial bulk solution. Oscillations in the current are due to the growth of the Hg drop, which leads to a EE n K K 12 0 05916 / . log=? ° O/R O R ii= 1 2 lim EE n K Kn i ii =? ? ? ° O/R O R lim 0 05916 0 05916. log . log [R] R x i K = = 0 Chapter 11 Electrochemical Methods of Analysis 515 Current Potential E 2 i = 2i lim i lim polarography A form of voltammetry using a dropping mercury electrode or a static mercury drop electrode. P otential Time Current Potential E 2 i max i avg Figure 11.35 Potential-excitation signal and voltammogram for normal polarography. (a) (b) 1400-CH11 9/9/99 2:07 PM Page 515 time-dependent change in the area of the working electrode. The limiting current, which is also called the diffusion current, may be measured from the maximum current, i max , or from the average current, i avg . The relationship between the con- centration of analyte, C A , and the limiting current is given by the Ilikovic equation (i lim ) max = 706nD 1/2 m 2/3 t 1/6 C A (i lim ) avg = 607nD 1/2 m 2/3 t 1/6 C A where n is the number of electrons transferred in the redox reaction, D is the analyte’s diffusion coefficient, m is the flow rate of the Hg, and t is the drop time. The half-wave potential, E 1/2 , provides qualitative information about the redox reaction (see Appendix 3E for a list of selected polarographic half-wave potentials). Normal polarography has been replaced by various forms of pulse polarogra- phy, 16 several examples of which are shown in Figure 11.36. Differential pulse po- larography (Figure 11.36b), for example, uses a series of potential pulses character- ized by a cycle of time t , a pulse time of t p , a potential pulse of D E p , and a potential step per cycle of D E s . Typical experimental conditions for differential pulse po- larography are t? 1 s, t p ? 50 ms, D E p ? 50 mV, D E s ? 2 mV. The current is mea- sured twice, for approximately 17 ms before the forward pulse and for approxi- mately 17 ms before the reverse pulse. The difference in the two currents gives rise to a peak-shaped voltammogram. Other forms of pulse polarography include nor- mal pulse polarography (Figure 11.36a), staircase polarography (Figure 11.36c), and square-wave polarography (Figure 11.36d). Limiting and peak currents are di- rectly proportional to the concentration of analyte, and half-wave and peak poten- tials can be used for qualitative purposes. The popularity of pulse polarography is due to a substantial improvement in sensitivity and detection limits from those in normal polarography. Polarography is used extensively for the analysis of metal ions and inorganic anions, such as IO 3 – and NO 3 – . Organic compounds containing easily reducible or oxidizable functional groups also can be studied polarographically. Functional groups that have been used include carbonyls, carboxylic acids, and carbon–carbon double bonds. Hydrodynamic Voltammetry In polarography a limiting current is obtained be- cause each falling drop of mercury returns the solution near the electrode to its ini- tial composition. As noted earlier, a limiting current is also obtained whenever the solution is stirred during the analysis. The simplest means of stirring the solution is with a magnetic stir bar. More commonly, however, stirring is achieved by rotating the electrode. In hydrodynamic voltammetry current is measured as a function of the po- tential applied to a solid working electrode. The same potential profiles used for polarography, such as a linear scan or a differential pulse, are used in hydrody- namic voltammetry. The resulting voltammograms are identical to those for po- larography, except for the lack of current oscillations resulting from the growth of the mercury drops. Because hydrodynamic voltammetry is not limited to Hg elec- trodes, it is useful for the analysis of analytes that are reduced or oxidized at more positive potentials. Stripping Voltammetry One of the most important quantitative voltammetric techniques is stripping voltammetry, which is composed of three related tech- niques: anodic, cathodic, and adsorptive stripping voltammetry. Since anodic strip- 516 Modern Analytical Chemistry stripping voltammetry A form of voltammetry in which the analyte is first deposited on the electrode and then removed, or “stripped,” electrochemically while monitoring the current as a function of the applied potential. 1400-CH11 9/9/99 2:07 PM Page 516 ping voltammetry has found the widest application, we consider it in the greatest detail. Anodic stripping voltammetry consists of two steps (Figure 11.37). The first is a controlled potential electrolysis in which the working electrode, usually a hanging mercury drop or mercury film, is held at a cathodic potential sufficient to deposit the metal ion on the electrode. For example, with Cu 2+ the deposition reaction is Cu 2+ (aq)+2e – t Cu(Hg) Chapter 11 Electrochemical Methods of Analysis 517 Figure 11.36 Potential-excitation signals and voltammograms for (a) normal pulse polarography, (b) differential pulse polarography, (c) staircase polarography, and (d) square-wave polarography. See text for an explanation of the symbols. Current is sampled at the time intervals indicated by the solid circles (?). Time t p D E p t P otential E i i l Time D E s D E p t P otential t p (a) (b) (c) (d) E Di i p Time t p D E s P otential E i i p Time D E s D E p t P otential E Di i p 1400-CH11 9/9/99 2:07 PM Page 517 Figure 11.37 Potential-excitation signal and voltammogram for anodic stripping voltammetry at a hanging mercury drop electrode. where Cu(Hg) indicates that the copper is amalgamated with the mercury. This step essentially serves as a means of preconcentrating the analyte from the larger volume of the solution to the smaller volume of the electrode. The solution is stirred during electrolysis to increase the rate of deposition. Near the end of the deposition time stirring is stopped, eliminating convection as a mode of mass transport. Deposition times of 1–30 min are common, with longer times being used for analytes at lower concentrations. In the second step, the potential is scanned anodically toward more positive potentials. When the potential of the working electrode is sufficiently positive the analyte is stripped from the electrode, returning to solution as its oxidized form Cu(Hg) t Cu 2+ (aq)+2e – The current during the stripping step is monitored as a function of potential, giving rise to peak-shaped voltammograms similar to that shown in Figure 11.37. The peak current is proportional to the analyte’s concentration in the solution. Anodic stripping voltammetry is very sensitive to experimental conditions, which must be carefully controlled if results are to be accurate and precise. Key vari- ables include the area of the mercury film electrode or the size of the Hg drop when using a hanging mercury drop electrode, the deposition time, the rest time, the rate of stirring, and the scan rate during the stripping step. Anodic stripping voltamme- try is best used for metals that form amalgams with mercury, several examples of which are listed in Table 11.11. The experimental design for cathodic stripping voltammetry is similar to that for anodic stripping voltammetry with two exceptions. First, the deposition step in- 518 Modern Analytical Chemistry E ° Cu 2+ /Cu = + 0.342 V Cu 2+ Cu More (+) More (–) Stirring No stirring Time Potential P otential Current 1400-CH11 9/9/99 2:07 PM Page 518 volves the oxidation of the Hg electrode to Hg 2 2+ , which then reacts with the analyte to form an insoluble film at the surface of the electrode. For example, when Cl – is the analyte the deposition step is 2Hg(l) + 2Cl – (aq) t Hg 2 Cl 2 (s)+2e – Second, stripping is accomplished by scanning cathodically toward a more negative potential, reducing Hg 2 2+ back to Hg and returning the analyte to solution. Hg 2 Cl 2 (s)+2e – t 2Hg(l) + 2Cl – (aq) Table 11.11 lists several analytes that have been analyzed successfully by cathodic stripping voltammetry. In adsorptive stripping voltammetry the deposition step occurs without elec- trolysis. Instead, the analyte adsorbs to the electrode’s surface. During deposition the electrode is maintained at a potential that enhances adsorption. For example, adsorption of a neutral molecule on a Hg drop is enhanced if the electrode is held at –0.4 V versus the SCE, a potential at which the surface charge of mercury is approx- imately zero. When deposition is complete the potential is scanned in an anodic or cathodic direction depending on whether we wish to oxidize or reduce the analyte. Examples of compounds that have been analyzed by absorptive stripping voltam- metry also are listed in Table 11.11. Amperometry The final voltammetric technique to be considered is amperome- try, in which a constant potential is applied to the working electrode, and current is measured as a function of time. Since the potential is not scanned, amperometry does not lead to a voltammogram. One important application of amperometry is in the construction of chemical sensors. One of the first amperometric sensors to be developed was for dissolved O 2 in blood, which was developed in 1956 by L. C. Clark. The design of the amper- ometric sensor is shown in Figure 11.38 and is similar to potentiometric mem- brane electrodes. A gas-permeable membrane is stretched across the end of the sensor and is separated from the working and counter electrodes by a thin solution of KCl. The working electrode is a Pt disk cathode, and an Ag ring anode is the Chapter 11 Electrochemical Methods of Analysis 519 Table 11.11 Representative Examples of Analytes Determined by Stripping Voltammetry Anodic Stripping Cathodic Stripping Absorptive Stripping Voltammetry Voltammetry Voltammetry bismuth bromide bilirubin cadmium chloride codeine copper iodide cocaine gallium mercaptans (RSH) digitoxin indium sulfide dopamine lead thiocyanate heme thallium monensin tin testosterone zinc Source: Compiled from Peterson, W. M.; Wong, R. V. Am. Lab. November 1981, 116–128; and Wang, J. Am. Lab. May 1985, 41–50. 17,18 amperometry A form of voltammetry in which we measure current as a function of time while maintaining a constant potential. 1400-CH11 9/9/99 2:07 PM Page 519 counter electrode. Although several gases can diffuse across the membrane, including O 2 , N 2 , and CO 2 , only oxygen is reduced at the cathode. O 2 (aq)+4H 3 O + (aq)+4e – t 6H 2 O(l) Another example of an amperometric sensor is the glucose sensor. In this case the single membrane in Figure 11.38 is replaced with three membranes. The outermost membrane is of polycarbonate, which is permeable to glucose and O 2 . The second membrane contains an immobilized preparation of glu- cose oxidase that catalyzes the oxidation of glucose to gluconolactone and hy- drogen peroxide. b -D-glucose(aq)+O 2 (g) +H 2 O(l) t gluconolactone(aq)+H 2 O 2 (aq) The hydrogen peroxide then diffuses through the innermost membrane of cel- lulose acetate, where it is oxidized at a Pt anode. H 2 O 2 (aq) + 2OH – (aq) t O 2 (g)+2H 2 O(l)+2e – Figure 11.39 summarizes the reactions taking place in this amperometric sen- sor. FAD is the oxidized form of flavin adenine nucleotide (the active site of the enzyme glucose oxidase), and FADH 2 is the active site’s reduced form. Note that O 2 serves as a mediator, carrying electrons to the electrode. Other mediators, such as Fe(CN) 6 3– , can be used in place of O 2 . By changing the enzyme and mediator, the amperometric sen- sor in Figure 11.39 is easily extended to the analysis of other sub- strates. Other bioselective materials may be incorporated into am- perometric sensors. For example, a CO 2 sensor has been developed using an amperometric O 2 sensor with a two-layer membrane, one of which contains an immobilized preparation of autotrophic bacte- ria. 19 As CO 2 diffuses through the membranes, it is converted to O 2 by the bacteria, increasing the concentration of O 2 at the Pt cathode. 11D.6 Quantitative Applications Quantitative voltammetry has been applied to a wide variety of sample types, including environmental samples, clinical samples, pharmaceu- tical formulations, steels, gasoline, and oil. Selecting the Voltammetric Technique The choice of which voltammetric tech- nique to use depends on the sample’s characteristics, including the analyte’s ex- pected concentration and the location of the sample. Amperometry is best suited for use as a detector in flow systems or as a selective sensor for the rapid analysis of a single analyte. The portability of amperometric sensors, which are similar to po- tentiometric sensors, make them ideal for field studies. Pulse polarography and stripping voltammetry can frequently be used inter- changeably, although each has its advantages and disadvantages. Pulse polarography is better for analyzing a wider range of inorganic and organic analytes because the need to preconcentrate the analyte at the electrode surface restricts the application of anodic and cathodic stripping voltammetry. When either pulse polarography or anodic stripping voltammetry can be used, the selection is often based on the analyte’s expected concentration and the desired 520 Modern Analytical Chemistry To potentiostat Pt Disk electrode Ag Ring electrode Electrolyte solution Membrane Glucose Gluconolactone H 2 O 2 O 2 Electrode e – FADH 2 FAD Figure 11.38 Clark amperometric sensor for the determination of dissolved O 2 . Figure 11.39 Schematic showing the reactions by which an amperometric biosensor responds to glucose. 1400-CH11 9/9/99 2:07 PM Page 520 accuracy and precision. Detection limits for normal pulse polarography generally are on the order of 10 –6 –10 –7 M, whereas those for differential pulse polarography, staircase, and square-wave polarography are between 10 –7 M and 10 –8 M. Precon- centrating the analyte in stripping voltammetry lowers the detection limit for many analytes to as little as 10 –10 M. On the other hand, the current in stripping voltam- metry is much more sensitive than pulse polarography to changes in experimental conditions, which may lead to poorer precision and accuracy. Anodic stripping voltammetry also suffers from occasional interferences when two metals, such as Cu and Zn, combine to form an intermetallic com- pound in the mercury amalgam. The deposition potential for Zn 2+ is sufficiently negative that any Cu 2+ present in the sample is also deposited. After deposition, intermetallic compounds such as CuZn and CuZn 2 form within the mercury amalgam. During the stripping step, the zinc in the intermetallic compounds strips at potentials near that of copper, decreasing the current for zinc and in- creasing the current for copper. This problem can often be overcome by adding a third element that forms a stronger intermetallic compound with the interfering metal. Thus, adding Ga 3+ minimizes this problem by forming an intermetallic compound of Cu and Ga. Correcting for Residual Current In any quantitative analysis the signal due to the analyte must be corrected for signals arising from other sources. The total measured current in any voltammetric experiment, i tot , consists of two parts: that due to the analyte’s oxidation or reduction, i a , and a background, or residual, current, i r . i tot = i a + i r The residual current, in turn, has two sources. One source is a faradaic current due to the oxidation or reduction of trace impurities in the sample, i i . The other source is the charging current, i ch , that is present whenever the working electrode’s poten- tial changes. i r = i i + i ch Faradaic currents due to impurities can usually be minimized by carefully preparing the sample. For example, one important impurity is dissolved O 2 , which is reduced first to H 2 O 2 and then to H 2 O. Dissolved O 2 is removed by bubbling an inert gas such as N 2 through the sample before the analysis. Two methods are commonly used to correct for the residual current. One method is to extrapolate the total measured current when the analyte’s faradaic current is zero. This is the method shown in the voltammograms included in this chapter. The advantage of this method is that it does not require any additional data. On the other hand, extrapolation assumes that changes in the residual cur- rent with potential are predictable, which often is not the case. A second, and more rigorous, approach is to obtain a voltammogram for an appropriate blank. The blank’s residual current is then subtracted from the total current obtained with the sample. Analysis for Single Components The analysis of samples containing only a single electroactive analyte is straightforward. Any of the standardization methods dis- cussed in Chapter 5 can be used to establish the relationship between current and the concentration of analyte. Chapter 11 Electrochemical Methods of Analysis 521 1400-CH11 9/9/99 2:07 PM Page 521 522 Modern Analytical Chemistry EXAMPLE 11.10 The concentration of As(III) in water can be determined by differential pulse polarography in 1 M HCl. The initial potential is set to –0.1 V versus the SCE, and is scanned toward more negative potentials at a rate of 5 mV/s. Reduction of As(III) to As(0) occurs at a potential of approximately - 0.44 V versus the SCE. The peak currents, corrected for the residual current, for a set of standard solutions are shown in the following table. [As(III)] i p (M) (m A) 1.00 · 10 –6 0.298 3.00 · 10 –6 0.947 6.00 · 10 –6 1.83 9.00 · 10 –6 2.72 What is the concentration of As(III) in a sample of water if the peak current under the same conditions is 1.37 m A? SOLUTION Linear regression gives the equation for the calibration curve as i p (m A) = 0.0176 + 3.01 · 10 5 [As(III)] Substituting the sample’s peak current into the regression equation gives the concentration of As(III) as 4.49 · 10 –6 M. EXAMPLE 11.11 The concentration of copper in a sample of sea water is determined by anodic stripping voltammetry using the method of standard additions. When a 50.0-mL sample is analyzed, the peak current is 0.886 m A. A 5.00-m L spike of 10.0-ppm Cu 2+ is added, giving a peak current of 2.52 m A. Calculate the parts per million of copper in the sample of sea water. SOLUTION Peak currents in anodic stripping voltammetry are a linear function of concentration i p = k(ppm Cu 2+ ) where k is a constant. Thus, for the sample we write 0.886 = k(ppm Cu 2+ ) and for the standard addition 252 0 0500 500 10 500 10 10 0 6 6 6 . . . () . (. = +× + ? ? ? × × ? ? ? ? ? ? k L 0.0500 L L ppm Cu L 0.0500 L + 5.00 10 L ppm) 2+ 1400-CH11 9/9/99 2:07 PM Page 522 Figure 11.40 Voltammograms showing the independent analysis of two components. Chapter 11 Electrochemical Methods of Analysis 523 Solving the first equation for k, substituting into the second equation, and simplifying gives Solving gives the concentration of Cu 2+ as 5.42 · 10 –4 ppm, or 0.542 ppb. Multicomponent Analysis One advantage of voltammetry as a quantitative method of analysis is its capability for analyzing two or more analytes in a single sample. As long as the components behave independently, the resulting voltammo- gram for a multicomponent mixture is a summation of their respective individual voltammograms. If the separation between the half-wave potentials or peak poten- tials is sufficient, each component can be determined independently as if it were the only component in the sample (Figure 11.40). The minimum separation between the half-wave potentials or peak potentials for the independent analysis of two com- ponents depends on several factors, including the type of electrode and the poten- tial-excitation signal. For normal polarography the separation must be at least – 0.2–0.3 V, and for differential pulse voltammetry a minimum separation of – 0.04–0.05 V is needed. When the overlap between the voltammograms for two components pre- vents their independent analysis, a simultaneous analysis similar to that used in spectrophotometry may be possible. An example of this approach is outlined in Example 11.12. EXAMPLE 11.12 The differential pulse polarographic analysis of mixtures of indium and cadmium in 0.1 M HCl is complicated by the overlap of their respective voltammograms. 20 The peak potential for indium is at –0.557 V, and that for cadmium occurs at a potential of –0.597 V. When a 0.800-ppm indium standard is analyzed, the peak current (in arbitrary units) is found to be 200.5 at –0.557 V and 87.5 at –0.597 V. A standard solution of 0.793-ppm cadmium gives peak currents of 58.5 at –0.557 V and 128.5 at –0.597 V. What is the concentration of indium and cadmium in a sample if the peak current is 167.0 at a potential of –0.557 V and 99.5 at a potential of –0.597 V? SOLUTION Peak currents in differential pulse polarography are a linear function of the concentration of analyte; thus i p = k(ppm analyte) where k is a constant that depends on the analyte and the applied potential. To determine the concentrations of indium and cadmium in the sample, we must first find the values of k for each analyte at each potential. For simplicity we will identify the potential of –0.557 V as E1, and that for –0.597 V as E2. The values of k are kk EEIn, 1 In, ppm ppm ppm = 109.4 ppm== = ?? 200 5 0 800 250 6 87 5 0 800 1 2 1 . . . . . 2 52 0 8859 8 859 10 10 0 5 .. ( . )( . ) =+ × ? ppm) (ppm Cu 2+ Current Potential (i lim ) 1 (i lim ) 2 Analyte 2 Analyte 1 Current Potential (i peak ) 1 (i peak ) 2 Analyte 2Analyte 1 (a) (b) 1400-CH11 9/9/99 2:07 PM Page 523 Next, we write simultaneous equations for the current at the two potentials i E1 = 250.6 ppm –1 (ppm In) + 73.8 ppm –1 (ppm Cd) = 167.0 i E2 = 109.4 ppm –1 (ppm In) + 162.0 ppm –1 (ppm Cd) = 99.5 Solving the simultaneous equations, which is left as an exercise, gives the concentration of indium as 0.606 ppm and the concentration of cadmium as 0.205 ppm. Environmental Samples One area in which quantitative voltammetry has had a significant effect is in the analysis of trace metals in environmental samples. The most common samples are natural waters, including groundwater, lakes, rivers and streams, sea water, rain, and snow. Concentrations of trace metals at the parts-per- billion level can be determined using differential pulse polarography, whereas with anodic stripping voltammetry the determination of trace metals at the pptr (parts- per-trillion) level is possible. The combination of low detection limits and the capa- bility for the simultaneous analysis of several analytes makes differential pulse po- larography and anodic stripping voltammetry ideally suited for such samples. One interesting application of anodic stripping voltammetry to the analysis of natural waters is the determination of the speciation, or chemical form, of the trace metals. The speciation of a trace metal is important because its bioavailability, toxi- city, and ease of transport through the environment often depend on its chemical form. For example, trace metals strongly bound to colloidal particles are generally not available to aquatic lifeforms and, therefore, are not toxic. Unfortunately, an- odic stripping voltammetry cannot distinguish the exact chemical form of a trace metal as closely related species, such as Pb 2+ and PbCl + , yield only a single stripping peak. Instead, trace metals are divided into several “operationally defined” cate- gories that have environmental significance. Several speciation schemes have been developed, 21 but we consider only the speciation scheme proposed by Batley and Florence, 22 which uses a combination of anodic stripping voltammetry with ion-exchange and UV irradiation to divide solu- ble trace metals into seven classes. Anodic stripping voltammetry in a pH 4.8 acetic acid buffer is used to distinguish labile metals present as hydrated ions, weakly bound complexes, or weakly adsorbed on colloidal surfaces from those metals that are bound in stronger complexes or strongly adsorbed. Only those metals that are hydrated, weakly bound, or weakly adsorbed deposit at the electrode. Ion exchange and UV irradiation are used to further subdivide the trace metals. A Chelex-100 ion-exchange resin is used to distinguish between ionic metals and strongly bound metals, whereas UV radiation is used to separate metals bound to organic and inor- ganic phases. Table 11.12 shows how trace metals are divided into seven classes using these three experimental techniques. The analysis of sea water samples, for example, showed that cadmium, copper, and lead were primarily present as labile organic complexes or as labile adsorbates on organic colloids (see group II, Table 11.12). Differential pulse polarography and stripping voltammetry have been applied to the analysis of trace metals in airborne particulates, incinerator fly ash, rocks, kk EECd, Cd, ppm ppm ppm ppm 1 1 2 1 58 5 0 793 73 8 128 5 0 793 162 0== == ?? . . . . . . 524 Modern Analytical Chemistry 1400-CH11 9/9/99 2:07 PM Page 524 minerals, and sediments. The trace metals, of course, must be brought into solution by digesting or extracting before the voltammetric analysis. Amperometric sensors also are used to analyze environmental samples. For ex- ample, the dissolved O 2 sensor described earlier is routinely used for the determina- tion of dissolved oxygen and biochemical oxygen demand, or BOD, in waters and wastewaters. The latter test, which is a measure of the amount of oxygen required by aquatic bacteria during the decomposition of organic matter, is of importance in evaluating the efficiency of wastewater treatment plants and in monitoring organic pollution in natural waters. A high BOD corresponds to a high concentration of or- ganic material that may seriously deplete the level of dissolved oxygen in the water. Other amperometric sensors have been developed to monitor anionic surfactants in water, and CO 2 , H 2 SO 4 , and NH 3 in atmospheric gases. 19 Clinical Samples Differential pulse polarography and stripping voltammetry have been used to determine the concentration of trace metals in a variety of matrices, including blood, urine, and tissue samples. The determination of lead in blood is of considerable interest due to concerns about lead poisoning. Because the concen- tration of lead in blood is so small, anodic stripping voltammetry frequently is the method of choice. The analysis is complicated, however, by the presence of pro- teins that may adsorb at the surface of the mercury electrode, inhibiting either the deposition or stripping of lead. In addition, proteins may prevent the electrodepo- sition of lead through the formation of stable, nonlabile complexes. For these rea- sons samples of whole blood must be digested or ashed before the analysis. Differ- ential pulse polarography is one of the few techniques that can be used for the routine quantitative analysis of drugs in biological fluids at concentrations of less than 10 –6 M. 23 Amperometric sensors based on enzyme catalysts also have wide ap- plicability. 24 Table 11.13 provides a partial list of enzymatic amperometric sensors. Miscellaneous Samples Besides environmental and clinical samples, differential pulse polarography and stripping voltammetry have been used for the analysis of trace metals in other samples, including food, steels and other alloys, gasoline, gun- powder residues, and pharmaceuticals. Voltammetry is also an important tool for Chapter 11 Electrochemical Methods of Analysis 525 Table 11.12 Speciation of Soluble Trace Metals According to the Scheme of Batley and Florence 22 Method Speciation of Soluble Metals ASV Labile Metals Nonlabile or Bound Metals Ion-Exchange Removed Not Removed Removed Not Removed UV Irradiation Released Not Released Released Not Released Released Not Released Description free metal ions, labile organic labile nonlabile nonlabile nonlabile nonlabile labile organic complexes inorganic organic inorganic organic inorganic complexes, and labile complexes complexes complexes complexes complexes and labile metals and labile and nonlabile and nonlabile inorganic absorbed on metals metals metals complexes organics absorbed on absorbed on absorbed on inorganics organics inorganics Group I II III IV V VI VII 1400-CH11 9/9/99 2:07 PM Page 525 526 Modern Analytical Chemistry Table 11.1 3 Representative Examples of Amperometric Biosensors Analyte Enzyme Species Detected choline choline oxidase H 2 O 2 ethanol alcohol oxidase H 2 O 2 formaldehyde formaldehyde dehydrogenase NADH a glucose glucose oxidase H 2 O 2 glutamine glutaminase, glutamate oxidase H 2 O 2 glycerol glycerol dehydrogenase NADH, O 2 lactate lactate oxidase H 2 O 2 phenol polyphenol oxidase quinone inorganic P nucleoside phosphorylase O 2 Source: Compiled from Cammann, K.; Lemke, U.; Rohen, A.; et al. Angew. Chem. Int. Ed. Engl. 1991, 30, 516–539. a NADH is the reduced form of nicotinamide adenine dinucleotide. Representative Methods —Continued the quantitative analysis of organics, particularly in the pharmaceutical industry, in which it is used to determine the concentration of drugs and vitamins in formula- tions. 23 For example, voltammetric methods have been developed for the quantita- tive analysis of vitamin A, niacinamide, and riboflavin. When the compound of in- terest is not electroactive, it often can be derivatized to an electroactive form. One example is the differential pulse polarographic determination of sulfanilamide, in which it is converted into an electroactive azo dye by coupling with sulfamic acid and 1-naphthol. Method 11. 3 Determination of Chlorpromazine 25 Description of Method. The amount of chlorpromazine in a pharmaceutical formulation is determined voltammetrically at a graphite working electrode in a nonstirred solution. Calibration is achieved using the method of standard additions. Procedure. Place 10.00 mL of a solution consisting of 0.01 M HCl and 0.1 M KCl in the electrochemical cell. Place a graphite working electrode, a Pt auxiliary electrode, and a SCE reference electrode in the cell, and record the voltammogram from +0.2 V to 2.0 V at a scan rate of 50 mV/s. Weigh out an appropriate amount of the pharmaceutical formulation, and dissolve it in a small amount of the electrolyte. Transfer the solution to a 100-mL volumetric flask, and dilute to volume with the electrolyte. Filter a small amount of the diluted solution, and transfer 1.00 mL of the filtrate to the voltammetric cell. Mix the contents of the voltammetric cell, and allow the solution to sit for 10 s before recording the voltammogram. Return the potential to +0.2 V, add 1.00 mL of a chlorpromazine standard, and record the voltammogram. Report the %w/w chlorpromazine in the formulation. Questions 1. Is chlorpromazine oxidized or reduced at the graphite working electrode? Since the direction of the scan is toward more positive potentials, the chlorpromazine is oxidized at the graphite working electrode. 1400-CH11 9/9/99 2:07 PM Page 526 Chapter 11 Electrochemical Methods of Analysis 527 2. Why is it not necessary to remove dissolved O 2 from the solution before recording the voltammogram? Dissolved O 2 is a problem when scanning toward more negative potentials where its reduction can lead to a significant cathodic current. In this procedure we are scanning toward more positive potentials and generating anodic currents; thus, dissolved O 2 is not an interferent. 3. What is the purpose of waiting 10 s after mixing the contents of the electrochemical cell before recording the voltammogram? Mixing the solution leads to convection, but the delay allows convection to cease. 4. In preparing the sample for analysis the initial solution is filtered. Why is it not necessary to collect the entire filtrate before proceeding? This analysis is an example of a concentration technique. Once the original sample is brought to volume in the 100-mL volumetric flask, any portion of the sample solution, even that obtained on filtering, may be used for the analysis. 11D. 7 Characterization Applications In the previous section we saw how voltammetry can be used to determine the con- centration of an analyte. Voltammetry also can be used to obtain additional infor- mation, including verifying electrochemical reversibility, determining the number of electrons transferred in a redox reaction, and determining equilibrium constants for coupled chemical reactions. Our discussion of these applications is limited to the use of voltammetric techniques that give limiting currents, although other voltammetric techniques also can be used to obtain the same information. Electrochemical Reversibility and Determination of n In deriving a relationship between E 1/2 and the standard-state potential for a redox couple (11.41), we noted that the redox reaction must be reversible. How can we tell if a redox reaction is re- versible from its voltammogram? For a reversible reaction, equation 11.40 describes the voltammogram. A plot of E versus log(i/i lim – i) for a reversible reaction, therefore, should be a straight line with a slope of –0.05916/n. In addition, the slope should yield an inte- ger value for n. EXAMPLE 11.1 3 The following data were obtained from the linear scan hydrodynamic voltammogram of a reversible reduction reaction Potential Current (V vs. SCE) (m A) -0.358 0.37 -0.372 0.95 -0.382 1.71 -0.400 3.48 -0.410 4.20 -0.435 4.97 EE n K Kn i ii =? ? ? ° O/R O R lim 0 05916 0 05916. log . log 1400-CH11 9/9/99 2:07 PM Page 527 The limiting current was 5.15 m A. Show that the reduction reaction is reversible, and determine values for n and E 1/2 . SOLUTION If the reaction is reversible, then a plot of E versus log(i/i lim – i) will be a straight line with a slope that yields an integer value for n. As shown in Figure 11.41, plotting the data in this manner gives a straight line. Linear regression gives the equation for the straight line as Knowing that gives a value for n of 1.97, or 2 electrons. Combining equations 11.40 and 11.41 shows that the y-intercept for a plot of E versus log(i/i lim – i) is equal to the half- wave potential. Thus, E 1/2 for the reduction reaction is –0.391 V versus the SCE. Slope =? =?0 02999 0 05916 . . n E i ii =? ? ? 0 391 0 02999. . log lim 528 Modern Analytical Chemistry log ( ) i i lim – i E (V vs SCE) –2 –1 0 1 –0.44 –0.34 –0.36 –0.38 –0.40 –0.42 2 Figure 11.41 Graphical determination of electrochemical reversibility, n, and half-wave potential in linear scan hydrodynamic voltammetry. Determining Equilibrium Constants for Coupled Chemical Reactions An- other important application of voltammetry is the determination of equilib- rium constants for solution reactions that are coupled to a redox reaction oc- curring at the electrode. The presence of the solution reaction affects the ease of electron transfer, shifting the potential to more negative or more positive potentials. Consider, for example, the reduction of O to R O+ne – t R the voltammogram for which is shown in Figure 11.42. If a ligand, L, capable of forming a strong complex with O, is present, then the reaction O + pL t OL p also must be considered. The overall reaction, therefore, is OL p + ne – t R+pL Current PotentialMore (+) More (–) O + ne – R OL p + ne – R + pL Increasing [L] Figure 11.42 Effect of complexation on voltammograms. 1400-CH11 9/9/99 2:07 PM Page 528 Because of its stability, reduction of the OL p complex is less favorable than the re- duction of O. As shown in Figure 11.42, the voltammograms for OL p occur at po- tentials more negative than those for O. Furthermore, the shift in the voltammo- gram depends on the ligand’s concentration. The shift in the voltammogram for a metal ion in the presence of a ligand may be used to determine both the metal–ligand complex’s stoichiometry and its forma- tion constant. To derive a relationship between the relevant variables we begin with two equations: the Nernst equation for the reduction of O and the stability constant for the complex at the electrode surface 11.42 In the absence of ligand the half-wave potential occurs when [R] x=0 and [O] x=0 are equal; thus, from the Nernst equation we have 11.43 where the subscript “nc” signifies that no complex is present. When ligand is present we must account for its effect on the concentration of O. Solving equation 11.42 for [O] x=0 and substituting into the Nernst equation gives 11.44 If the ligand is present in excess and the formation constant is sufficiently large, such that all of O is present as the complex, then [R] x=0 and [OL p ] x=0 are equal at the half-wave potential, and equation 11.44 simplifies to 11.45 where the subscript “c” indicates that the complex is present. Defining D E 1/2 as D E 1/2 =(E 1/2 ) c – (E 1/2 ) nc 11.46 and substituting equations 11.44 and 11.45 gives, after rearranging A plot of D E 1/2 versus log [L], therefore, gives a straight line. The slope of the line is used to determine the stoichiometric coefficient, p, and the y-intercept gives the value for the formation constant b p . ?E n p n p12 0 05916 0 05916 / . log . log[=? ?β L] () . log[ / EE n x p p12 0 0 05916 cO/R L]=? ° = β EE n R x x p p px =? ° = = = O/R L] OL 0 05916 0 0 0 . log [] [ [] β () // EE 12 nc OR = ° β p px x x p = = = = [] [[ OL O] L] 0 0 0 EE n x x =? ° = = O/R R] [O] 0 05916 0 0 . log [ Chapter 11 Electrochemical Methods of Analysis 529 1400-CH11 9/9/99 2:08 PM Page 529 530 Modern Analytical Chemistry EXAMPLE 11.1 4 A voltammogram for the two-electron reduction of M has a half-wave potential of - 0.226 V versus the SCE. In the presence of an excess of the ligand L, the following half-wave potentials are recorded [L] (E 1/2 ) c (M) (V vs. SCE) 0.020 –0.494 0.040 –0.512 0.060 –0.523 0.080 –0.530 0.100 –0.536 Determine values for the stoichiometry of the complex and the formation constant of the complex. SOLUTION First we must calculate values of D E 1/2 using equation 11.46; thus [L] DE 1/2 (M) (V vs. SCE) 0.020 –0.268 0.040 –0.286 0.060 –0.297 0.080 –0.304 0.100 –0.310 The plot of D E 1/2 as a function of the log of the ligand concentration is shown in Figure 11.43. Linear regression gives the equation for the straight line as D E 1/2 = –0.370 – 0.0601 log [L] From the slope and intercept we know that and Solving these equations gives the stoichiometry of the complex as ML 2 (p = 2.03), with a formation constant of b 2 = 3.5 · 10 12 . ?=?0 371 0 05916 2 . . log β p ?=?0 0601 0 05916 2 . . p log[L] D E 2 –1.8 –1.6 –1.4 –1.2 –1.0 –0.8 –0.32 –0.26 –0.27 –0.28 –0.29 –0.30 –0.31 Figure 11.43 Graphical determination of the stoichiometry and formation constant for a complexation reaction. 1400-CH11 9/9/99 2:08 PM Page 530 11D.8 Evaluation Scale of Operation Voltammetry is routinely used to analyze samples at the parts-per-million level and, in some cases, can be used to detect analytes at the parts-per-billion or parts-per-trillion level. Most analyses are carried out in con- ventional electrochemical cells using macro samples; however, microcells are available that require as little as 50 m L of sample. Microelectrodes, with diame- ters as small as 2 m m, allow voltammetric measurements to be made on even smaller samples. For example, the concentration of glucose in 200-m m pond snail neurons has been successfully monitored using a 2-m m amperometric glu- cose electrode. 26 Accuracy The accuracy of a voltammetric analysis often is limited by the ability to correct for residual currents, particularly those due to charging. For analytes at the parts-per-million level, accuracies of – 1–3% are easily obtained. As expected, a de- crease in accuracy is experienced when analyzing samples with significantly smaller concentrations of analyte. Precision Precision is generally limited by the uncertainty in measuring the limit- ing or peak current. Under most experimental conditions, precisions of – 1–3% can be reasonably expected. One exception is the analysis of ultratrace analytes in com- plex matrices by stripping voltammetry, for which precisions as poor as – 25% are possible. Sensitivity In many voltammetric experiments, sensitivity can be improved by ad- justing the experimental conditions. For example, in stripping voltammetry, sensi- tivity is improved by increasing the deposition time, by increasing the rate of the linear potential scan, or by using a differential-pulse technique. One reason for the popularity of potential pulse techniques is an increase in current relative to that ob- tained with a linear potential scan. Selectivity Selectivity in voltammetry is determined by the difference between half-wave potentials or peak potentials, with minimum differences of – 0.2–0.3 V re- quired for a linear potential scan, and – 0.04–0.05 V for differential pulse voltamme- try. Selectivity can be improved by adjusting solution conditions. As we have seen, the presence of a complexing ligand can substantially shift the potential at which an analyte is oxidized or reduced. Other solution parameters, such as pH, also can be used to improve selectivity. Time, Cost, and Equipment Commercial instrumentation for voltammetry ranges from less than $1000 for simple instruments to as much as $20,000 for more sophisticated instruments. In general, less expensive instrumentation is limited to linear potential scans, and the more expensive instruments allow for more complex potential-excitation signals using potential pulses. Except for stripping voltammetry, which uses long deposition times, voltammetric analyses are relatively rapid. Chapter 11 Electrochemical Methods of Analysis 531 1400-CH11 9/9/99 2:08 PM Page 531 532 Modern Analytical Chemistry 11E KEY TERMS amalgam (p. 509) amperometry (p. 519) anode (p. 466) anodic current (p. 510) asymmetry potential (p. 476) auxiliary electrode (p. 463) cathode (p. 466) cathodic current (p. 510) charging current (p. 513) convection (p. 512) coulometric titrations (p. 501) coulometry (p. 496) counter electrode (p. 462) current efficiency (p. 496) diffusion (p. 511) diffusion layer (p. 511) dropping mercury electrode (p. 509) electrical double layer (p. 513) electrode of the first kind (p. 474) electrode of the second kind (p. 475) enzyme electrodes (p. 484) faradaic current (p. 510) Faraday’s law (p. 496) galvanostat (p. 464) glass electrode (p. 477) hanging mercury drop electrode (p. 509) hydrodynamic voltammetry (p. 513) indicator electrode (p. 462) ionophore (p. 482) ion-selective electrode (p. 475) liquid-based ion-selective electrode (p. 482) liquid junction potential (p. 470) mass transport (p. 511) mediator (p. 500) membrane potential (p. 475) migration (p. 512) nonfaradaic current (p. 512) Ohm’s law (p. 463) overpotential (p. 497) polarography (p. 515) potentiometer (p. 464) potentiostat (p. 465) redox electrode (p. 475) reference electrode (p. 462) residual current (p. 513) salt bridge (p. 466) saturated calomel electrode (p. 472) silver/silver chloride electrode (p. 473) solid-state ion-selective electrode (p. 479) standard hydrogen electrode (p. 471) static mercury drop electrode (p. 509) stripping voltammetry (p. 516) total ionic strength adjustment buffer (p. 487) voltammetry (p. 508) voltammogram (p. 508) Electrochemical methods covered in this chapter include poten- tiometry, coulometry, and voltammetry. Potentiometric methods are based on the measurement of an electrochemical cell’s poten- tial when only a negligible current is allowed to flow. In principle the Nernst equation can be used to calculate the concentration of species in the electrochemical cell by measuring its potential and solving the Nernst equation; the presence of liquid junction po- tentials, however, necessitates the use of an external standardiza- tion or the use of standard additions. Potentiometric electrodes are divided into two classes: metallic electrodes and membrane electrodes. The smaller of these classes are the metallic electrodes. Electrodes of the first kind respond to the concentration of their cation in solution; thus the potential of an Ag wire is determined by the concentration of Ag + in solution. When another species is present in solution and in equilibrium with the metal ion, then the electrode’s potential will respond to the concentration of that ion. For example, an Ag wire in contact with a solution of Cl – will respond to the concentration of Cl – since the relative concentrations of Ag + and Cl – are fixed by the solubility product for AgCl. Such electrodes are called electrodes of the second kind. The potential of a membrane electrode is determined by a dif- ference in the composition of the solution on either side of the membrane. Electrodes using a glass membrane respond to ions that bind to negatively charged sites on the glass membrane’s sur- face. A pH electrode is one example of a glass membrane elec- trode. Other kinds of membrane electrodes include those using in- soluble crystalline solids and liquid ion exchangers incorporated in a hydrophobic membrane. The F – ion-selective electrode, which uses a single crystal of LaF 3 as the ion-selective membrane, is an example of a solid-state electrode. The Ca 2+ ion-selective elec- trode, in which a chelating ligand such as di-(n-decyl)phosphate is immobilized in a PVC membrane, is an example of a liquid-based ion-selective electrode. Potentiometric electrodes also can be designed to respond to molecules by incorporating a reaction producing an ion whose con- centration can be determined using a traditional ion-selective elec- trode. Gas-sensing electrodes, for example, include a gas-permeable membrane that isolates the ion-selective electrode from the solution containing the analyte. Diffusion of a dissolved gas across the mem- brane alters the composition of the inner solution in a manner that can be followed with an ion-selective electrode. Enzyme electrodes operate in the same way. Coulometric methods are based on Faraday’s law that the total charge or current passed during an electrolysis is proportional to the amount of reactants and products in the redox reaction. If the electrolysis is 100% efficient, in that only the analyte is oxidized or reduced, then the total charge or current can be used to determine 11F SUMMARY 1400-CH11 9/9/99 2:08 PM Page 532 the amount of analyte in a sample. In controlled-potential coulometry, a constant potential is applied and the current is mea- sured as a function of time, whereas in controlled-current coulometry, the current is held constant and the time required to completely oxidize or reduce the analyte is measured. In voltammetry we measure the current in an electrochemical cell as a function of the applied potential. Individual voltammetric methods differ in terms of the type of electrode used, how the ap- plied potential is changed, and whether the transport of material to the electrode’s surface is enhanced by stirring. Polarography is a voltammetric experiment conducted at a mercury electrode under conditions in which the solution is not stirred. Normal polarography uses a dropping mercury electrode (or a static mercury drop electrode) and a linear potential scan in an unstirred solution. Other forms of polarography include nor- mal pulse polarography, differential pulse polarography, staircase polarography, and square-wave polarography, all of which apply a series of potential pulses to the mercury electrode. In hydrodynamic voltammetry the solution is stirred either by using a magnetic stir bar or by rotating the electrode. Because the solution is stirred, a dropping mercury electrode cannot be used and is replaced with a solid electrode. Both linear potential scans or potential pulses can be applied. In stripping voltammetry the analyte is first deposited on the electrode, usually as the result of an oxidation or reduction reac- tion. The potential is then scanned, either linearly or by using po- tential pulses, in a direction that removes the analyte by a reduc- tion or oxidation reaction. Amperometry is a voltammetric method in which a constant potential is applied to the electrode and the resulting current is measured. Amperometry is most often used in the construction of chemical sensors that, as with potentiometric sensors, are used for the quantitative analysis of single analytes. One important exam- ple, for instance, is the Clark O 2 electrode, which responds to the concentration of dissolved O 2 in solutions such as blood and water. Chapter 11 Electrochemical Methods of Analysis 533 Chan, W. H.; Wong, M. S.; Yip, C. W. “Ion-Selective Electrode in Organic Analysis: A Salicylate Electrode,” J. Chem. Educ. 1986, 63, 915–916. The preparation of an ion-selective electrode for salicylate is described. The electrode incorporates an ion-pair of crystal violet and salicylate in a PVC matrix as the ion-selective membrane. Its use for the determination of acetylsalicylic acid in aspirin tablets is described. A similar experiment is described by Creager, S. E.; Lawrence, K. D.; Tibbets, C. R. in “An Easily Constructed Salicylate-Ion-Selective Electrode for Use in the Instructional Laboratory,” J. Chem. Educ. 1995, 72, 274–276. Christopoulos, T. K.; Diamandis, E. P. “Use of a Sintered Glass Crucible for Easy Construction of Liquid-Membrane Ion-Selective Electrodes,” J. Chem. Educ. 1988, 65, 648. Liquid-membrane ion-selective electrodes are prepared by removing the bottom below the glass filter, cleaning the filter, and making it hydrophobic using trimethylchlorosilane. Suitable liquid ion exchangers are added and allowed to penetrate the hydrophobic glass membrane. Adding an appropriate internal reference solution and reference electrode completes the ion-selective electrode. Fricke, G. H.; Kuntz, M. J. “Inexpensive Solid-State Ion- Selective Electrodes for Student Use,” J. Chem. Educ. 1977, 54, 517–520. Instructions are provided in this experiment for the preparation of solid-state electrodes for the quantitative analysis of Ag + , Cl – , Cu 2+ , and Pb 2+ . All electrodes are based on AgCl, AgCl/Ag 2 S, CuS/Ag 2 S, and PbS/Ag 2 S preparations. Electrodes are prepared by (a) mixing the solid with silicon grease and coating on an Ag wire; (b) mixing the solid with silicone grease and packing into the end of a glass tube; or (c) preparing a pressed pellet. Lloyd, B. W.; O’Brien, F. L.; Wilson, W. D. “Student Preparation and Analysis of Chloride and Calcium Ion Selective Electrodes,” J. Chem. Educ. 1976, 53, 328–330. This experiment describes the preparation of liquid ion- exchange electrodes for Cl – and Ca 2+ . The liquid ion- exchange solutions are incorporated into PVC membranes and fixed to the end of glass tubing. The internal solutions are either NaCl or CaCl 2 , and a Ag/AgCl reference electrode is situated in the internal solution. Martínez-Fàbregas, E.; Alegret, S. “A Practical Approach to Chemical Sensors through Potentiometric Transducers: Determination of Urea in Serum by Means of a Biosensor,” J. Chem. Educ. 1994, 71, A67–A70. Directions are provided for constructing and characterizing an ammonium ion-selective electrode. The electrode is then modified to respond to urea by adding a few milligrams of urease and covering with a section of dialysis membrane. Directions for determining urea in serum also are provided. 11G Suggested EXPERIMENTS The following set of suggested experiments describes the preparation of solid-state and liquid ion-exchange ion- selective electrodes, as well as potentiometric biosensors. Experiments —Continued 1400-CH11 9/9/99 2:08 PM Page 533 534 Modern Analytical Chemistry Continued from page 533 Experiments The following four experiments describe useful and accessible applications of coulometry. Mifflin, T. E.; Andriano, K. M.; Robbins, W. B. “Determination of Penicillin Using an Immobilized Enzyme Electrode,” J. Chem. Educ. 1984, 61, 638–639. Directions for preparing a potentiometric biosensor for penicillin are provided in this experiment. The enzyme penicillinase is immobilized in a polyacrylamide polymer formed on the surface of a glass pH electrode. The electrode shows a linear response to penicillin G over a concentration range of 10 –5 M to 10 –3 M. Palanivel, A.; Riyazuddin, P. “Fabrication of an Inexpensive Ion-Selective Electrode,” J. Chem. Educ. 1984, 61, 290. This experiment describes the preparation of a Ag 2 S/CuS electrode using the graphite rod from a dry cell D-type battery. After obtaining the graphite rod, it is inserted and sealed in a polyethylene tube, leaving a 2-cm portion exposed. The protruding portion of the graphite rod, is placed in a saturated solution of Na 2 S for 1 h, and in a saturated solution of CuS for an additional hour. In a related experiment (Riyazuddin, P.; Devika, D. “Potentiometric Acid–Base Titrations with Activated Graphite Electrodes,” J. Chem. Educ. 1997, 74, 1198–1199.), activated graphite electrodes are used for the potentiometric titration of HCl with NaOH. Ramaley, L; Wedge, P. J.; Crain, S. M. “Inexpensive Instrumental Analysis: Part 1. Ion-Selective Electrodes,” J. Chem. Educ. 1994, 71, 164–167. This experiment describes the preparation and evaluation of two liquid-membrane Na + ion-selective electrodes, using either the sodium salt of monensin or a hemisodium ionophore as ion exchangers incorporated into a PVC matrix. Electrodes prepared using monensin performed poorly, but those prepared using hemisodium showed a linear response over a range of 0.1 M to 3 · 10 –5 M Na + with slopes close to the theoretical value. Bertotti, M.; Vaz, J. M.; Telles, R. “Ascorbic Acid Determination in Natural Orange Juice,” J. Chem. Educ. 1995, 72, 445–447. The titration of ascorbic acid using coulometrically generated I 2 and Br 2 is described in this experiment. Details are also given for the polarographic analysis of ascorbic acid. Kalbus, G. E.; Lieu, V. T. “Dietary Fat and Health: An Experiment on the Determination of Iodine Number of Fats and Oils by Coulometric Titration,” J. Chem. Educ. 1991, 68, 64–65. The iodine number of fats and oils provides a quantitative measurement of the degree of unsaturation. A solution containing a 100% excess of ICl is added to the sample, reacting across the double-bonded sites of unsaturation. The excess ICl is converted to I 2 by adding KI. The resulting I 2 is reacted with a known excess of Na 2 S 2 O 3 . To complete the analysis the excess S 2 O 3 2– is back titrated with coulometrically generated I 2 . L?tz, A. “A Variety of Electrochemical Methods in a Coulo- metric Titration Experiment,” J. Chem. Educ. 1998, 75, 775–777. This experiment describes three coulometric titrations for the quantitative analysis of mixtures containing KI and HCl. Iodide is determined by titrating with coulometrically generated Br 2 . Chloride is determined indirectly by the coulometric titration of H + . Finally, iodide and chloride are determined together by precipitating AgI and AgCl following the anodic dissolution of an Ag wire. Swim, J.; Earps, E.; Reed, L. M.; et al. “Constant-Current Coulometric Titration of Hydrochloric Acid,” J. Chem. Educ. 1996, 73, 679–683. Directions are provided for constructing an inexpensive constant-current source and demonstrate its use in determining the concentration of HCl coulometrically. —Continued The following set of experiments illustrate several applications of potentiometric electrodes. Harris, T. M. “Potentiometric Measurement in a Freshwater Aquarium,” J. Chem. Educ. 1993, 70, 340–341. This experiment describes a semester-long project in which the concentration of several ions in a fresh water aquarium are monitored. Ions that are monitored using potentiometric electrodes include H + (pH electrode), Cl – (chloride electrode), HCO 3 – (CO 2 electrode), NH 4 + (NH 3 electrode), and NO 3 – (NH 3 electrode). Nitrate concentrations were determined following its conversion to ammonia. Radic, N.; Komijenovic, J. “Potentiometric Determination of an Overall Formation Constant Using an Ion-Selective Membrane Electrode,” J. Chem. Educ. 1993, 70, 509–511. The titration of Al 3+ with F – in a water–acetonitrile mixture is followed potentiometrically using a fluoride ion-selective electrode. The cumulative formation constant for AlF 6 3– is determined from the titration curve. Selig, W. S. “Potentiometric Titrations Using Pencil and Graphite Sensors,” J. Chem. Educ. 1984, 61, 80–81. This experiment describes the use of coated graphite electrodes for the potentiometric monitoring of precipitation, acid–base, complexation, and redox titrations. 1400-CH11 9/9/99 2:08 PM Page 534 Chapter 11 Electrochemical Methods of Analysis 535 Experiments The final set of experiments provide examples of voltammetric and amperometric analyses. García-Armada, P.; Losada, J.; de Vicente-Pérez, S. “Cation Analysis Scheme by Differential Pulse Polarography,” J. Chem. Educ. 1996, 73, 544–547. Directions are provided for the simultaneous qualitative and quantitative analysis of Bi(III), Cd(II), Co(II), Cu(II), Cr(III), Mn(II), Ni(II), Pb(II), Sb(III), and Zn(II). The analysis is accomplished using differential pulse polarography in three separate supporting electrolytes. Herrera-Melián, J. A.; Do?a-Rodríguez, J. M.; Hernández- Brito, J.; et al. “Voltammetric Determination of Ni and Co in Water Samples,” J. Chem. Educ. 1997, 74, 1444–1445. This experiment describes the application of cathodic stripping voltammetry to the analysis of Ni and Co in fresh water and sea water. Marin, D.; Mendicuti, F. “Polarographic Determination of Composition and Thermodynamic Stability Constant of a Complex Metal Ion,” J. Chem. Educ. 1988, 65, 916–918. This experiment describes the determination of the stability (cumulative formation) constant for the formation of Pb(OH) 3 – by measuring the shift in the half-wave potential for the reduction of Pb 2+ as a function of the concentration of OH – . The influence of ionic strength is also considered, and results are extrapolated to zero ionic strength to determine the thermodynamic formation constant. Sittampalam, G.; Wilson, G. S. “Amperometric Determination of Glucose at Parts Per Million Levels with Immobilized Glucose Oxidase,” J. Chem. Educ. 1982, 59, 70–73. This experiment describes the use of a commercially available amperometric biosensor for glucose that utilizes the enzyme glucose oxidase. The concentration of glucose in artificial unknowns, serum, or plasma can be determined using a normal calibration curve. Town, J. L.; MacLaren, F.; Dewald, H. D. “Rotating Disk Voltammetry Experiment,” J. Chem. Educ. 1991, 68, 352–354. This experiment introduces hydrodynamic voltammetry using a rotating working electrode. Its application for the quantitative analysis of K 4 Fe(CN) 6 is demonstrated. Wang, J. “Sensitive Electroanalysis Using Solid Electrodes,” J. Chem. Educ. 1982, 59, 691–692. The technique of hydrodynamic modulation voltammetry (HMV), in which the rate of stirring is pulsed between high and low values, is demonstrated in this experiment. The application of HMV for the quantitative analysis of ascorbic acid in vitamin C tablets using the method of standard additions also is outlined. Wang, J. “Anodic Stripping Voltammetry,” J. Chem. Educ. 1983, 60, 1074–1075. The application of anodic stripping voltammetry for the quantitative analysis of Cd, Pb, and Cu in natural waters is described in this experiment. Wang, J.; Maccà, C. “Use of Blood-Glucose Test Strips for Introducing Enzyme Electrodes and Modern Biosensors,” J. Chem. Educ. 1996, 73, 797–800. Commercially available kits for monitoring blood-glucose use an amperometric biosensor incorporating the enzyme glucose oxidase. This experiment describes how such monitors can be adapted to the quantitative analysis of glucose in beverages. 1. Identify the anode and cathode for the following electrochemical cells, and write the oxidation or reduction reaction occurring at each electrode. (a) Pt | FeCl 2 (0.015 M), FeCl 3 (0.045 M) || AgNO 3 (0.1 M) | Ag (b) Ag | AgBr(s) | NaBr (1.0 M) || CdCl 2 (0.05 M) | Cd (c) Pb, PbSO 4 |H 2 SO 4 (1.5 M) || H 2 SO 4 (2.0 M) | PbSO 4 , PbO 2 2. Calculate the potential for the electrochemical cells in problem 1. 3. Calculate the molar concentration for the underlined component in the following cell if the cell potential is measured at +0.294 V Ag | AgCl (sat’d) | NaCl (0.1 M) || KI (x M)|I 2 (s)|Pt 4. What reaction prevents Zn from being used as an electrode of the first kind in acidic solution? Which other electrodes of the first kind would you expect to behave in the same manner as Zn when immersed in acidic solutions? 5. Creager and colleagues designed a salicylate ion-selective electrode using a PVC membrane impregnated with tetraalkylammonium salicylate. 27 To determine the ion- selective electrode’s selectivity coefficient for benzoate, a set of salicylate calibration standards was prepared in which the concentration of benzoate was held constant at 0.10 M. Using the following data, determine the value of the selectivity coefficient. 11H PROBLEMS 1400-CH11 9/9/99 2:08 PM Page 535 [Salicylate] Potential (M) (mV) 1.0 20.2 1.0 · 10 –1 73.5 1.0 · 10 –2 126 1.0 · 10 –3 168 1.0 · 10 –4 182 1.0 · 10 –5 182 1.0 · 10 –6 177 What is the maximum acceptable concentration of benzoate if this ion-selective electrode is to be used for analyzing samples containing as little as 10 –5 M salicylate with an accuracy of better than 1%? 6. Watanabe and co-workers described a new membrane electrode for the determination of cocaine, which is a weak base alkaloid with a pK a of 8.64. 28 The response of the electrode for a fixed concentration of cocaine was found to be independent of pH in the range of 1–8, but decreased sharply above a pH of 8. Offer an explanation for the source of this pH dependency. 7. Show that equation 11.12 for the urea electrode is correct. 8. The pH-based urea electrode has the response shown in equation 11.13. Rewrite the equation to show the relationship between potential, E, and the concentration of urea. Comment on your result. 9. Mifflin and associates described a membrane electrode for the quantitative analysis of penicillin in which the enzyme penicillinase is immobilized in a polyacrylamide gel that is coated on a glass pH electrode. 29 The following data were collected for a series of penicillin standards. [Penicillin] Potential (M) (mV) 1.0 · 10 –2 220 2.0 · 10 –3 204 1.0 · 10 –3 190 2.0 · 10 –4 153 1.0 · 10 –4 135 1.0 · 10 –5 96 1.0 · 10 –6 80 Construct a calibration curve for the electrode, and report (a) the range of concentrations in which a linear response is observed, (b) the equation for the calibration curve in this range, and (c) the concentration of penicillin in a sample that yields a potential of 142 mV. 10. Ion-selective electrodes can be incorporated in flow cells to monitor the concentration of an analyte in standards and samples that are pumped through the flow cell. As the analyte passes through the cell, a potential spike is recorded instead of a steady-state potential. The concentration of K + in serum has been determined in this fashion, using standards prepared in a matrix of 0.014 M NaCl. 30 [K + ] Potential (mM) (arb. units) 0.1 25.5 0.2 37.2 0.4 50.8 0.6 58.7 0.8 64.0 1.0 66.8 A 1.00-mL sample of serum is diluted to volume in a 10-mL volumetric flask and analyzed, giving a potential of 51.1. Report the concentration of K + in the sample of serum. 11. Wang and Taha described an interesting application of potentiometry called batch injection. 31 As shown in the following figure, an ion-selective electrode is placed in an inverted position in a large-volume tank, and a fixed volume of a sample or standard solution is injected toward the electrode’s surface using a micropipet. The response of the electrode is a peak whose height is proportional to the analyte’s concentration. The following data were collected using a pH electrode Potential pH (mV) 2 +300 3 +240 4 +168 5 +81 6 +35 8 –92 9 –168 10 –235 11 –279 Determine the pH for each of the following samples, given the recorded peak potential: (a) tomato juice, +167 mV; (b) tap water, –27 mV; (c) coffee, 122 mV. 536 Modern Analytical Chemistry Reference electrode Ion-selective electrode Stir bar Micropipet tip 1400-CH11 9/9/99 2:08 PM Page 536 12. The concentration of NO 3 – in a water sample is determined by a one-point standard addition using an NO 3 – ion-selective electrode. A 25.00-mL sample is placed in a beaker, and a potential of +0.102 V is measured. A 1.00-mL aliquot of a 200.0 ppm standard solution of NO 3 – is added, after which the potential is found to be +0.089 V. Report the concentration of NO 3 – in parts per million. 13. The following data were collected for the analysis of fluoride in tap water and in toothpaste. (a) For the analysis of tap water, three 25.0-mL samples were each mixed with 25.0 mL of TISAB, and the potential was measured with an F – ISE relative to a saturated calomel electrode. Five 1.00-mL additions of a standard solution of 100.0-ppm F – were added to each, measuring the potential following each addition. Potential Milliliters of (mV) standard F – added Trial 1 Trial 2 Trial 3 0.00 –79 –82 –81 1.00 –119 –119 –118 2.00 –133 –133 –133 3.00 –142 –142 –142 4.00 –149 –148 –148 5.00 –154 –153 –153 Determine the parts per million of F – in the tap water. (b) For the analysis of toothpaste a 0.3619-g sample was transferred to a 100-mL volumetric flask along with 50.0 mL of TISAB and diluted to volume with distilled water. Three 20.0-mL aliquots were removed, and the potential was measured with an F – ion-selective electrode using a saturated calomel electrode as a reference. Five separate 1.00-mL additions of a 100.0-ppm solution of F – were added to each, measuring the potential following each addition. Potential Milliliters of (mV) standard F – added Trial 1 Trial 2 Trial 3 0.00 –55 –54 –55 1.00 –82 –82 –83 2.00 –94 –94 –94 3.00 –102 –103 –102 4.00 –108 –108 –109 5.00 –112 –112 –113 Report the parts per million of F – in the sample of toothpaste. 14. You are responsible for determining the amount of KI in iodized salt and decide to use an I – ion-selective electrode. Describe how you would perform this analysis using (a) external standards and (b) the method of standard additions. 15. The purity of a sample of picric acid, C 6 H 3 N 3 O 7 , is determined by controlled-potential coulometry, converting the picric acid to triaminophenol, C 6 H 9 N 3 O. A 0.2917-g sample of picric acid is placed in a 1000-mL volumetric flask and diluted to volume. A 10.00-mL portion of this solution is transferred to a coulometric cell and diluted till the Pt cathode is immersed. The exhaustive electrolysis of the sample requires 21.67 C of charge. Report the purity of the picric acid. 16. The concentration of H 2 S in the drainage from an abandoned mine can be determined by a coulometric titration using KI as a mediator and I 3 – as the “titrant.” H 2 S(aq)+I 3 – (aq)+2H 2 O(l) t 2H 3 O + (aq)+3I – (aq)+S(s) A 50.00-mL sample of water is placed in a coulometric cell, along with an excess of KI and a small amount of starch as an indicator. Electrolysis is carried out at a constant current of 84.6 mA, requiring 386 s to reach the starch end point. Report the concentration of H 2 S in the sample in parts per million. 17. One method for the determination of H 3 AsO 3 is by a coulometric titration using I 3 – as a “titrant.” The relevant reactions and standard-state potentials are summarized as follows. H 3 AsO 4 (aq)+2H 3 O + (aq)+2e – t H 3 AsO 3 (aq)+3H 2 O(l) E° = +0.559 I 3 – (aq)+2e – t 3I – E° = +0.536 Explain why the coulometric titration must be done in neutral solutions (pH ? 7), instead of in strongly acidic solutions (pH < 0). 18. The reduction of acrylonitrile, CH 2 =CHCN, to adiponitrile, NC(CH 2 ) 4 CN, is an important industrial process. A 0.594-g sample of acrylonitrile is placed in a 1-L volumetric flask and diluted to volume. An exhaustive controlled-potential electrolysis of a 1.00-mL portion of the diluted acrylonitrile requires 1.080 C of charge. What is the value of n for this reduction? 19. A linear-potential scan hydrodynamic voltammogram for a mixture of Fe 2+ and Fe 3+ is shown in the figure, where i l,a and Chapter 11 Electrochemical Methods of Analysis 537 Current Potential i l,c i l,a 0 i l,c are the anodic and cathodic limiting currents. (a) Show that the potential is given by (b) What is the potential when i = 0 for a solution that is 0.1 mM Fe 3+ and 0.05 mM Fe 2+ ? You may assume that K Fe 3+ ? K Fe 2+ . EE K K ii ii = ° ?? ? ? Fe Fe Fe Fe 3+ 2+ 3+ 2+ / , , . log . log0 05916 0 05916 1 1 a c 1400-CH11 9/9/99 2:08 PM Page 537 20. The amount of sulfur in aromatic monomers can be determined by differential pulse polarography. Standard solutions are prepared for analysis by dissolving 1.000 mL of the purified monomer in 25.00 mL of an electrolytic solvent, adding a known amount of S, deaerating, and measuring the peak current. The following results were obtained for a set of calibration standards Micrograms Peak current of added S (m A) 0 0.14 28 0.70 56 1.23 112 2.41 168 3.42 Analysis of a 1.000-mL sample, treated in the same manner as the standards, gives a peak current of 1.77 m A. Report the amount of sulfur present in the sample in milligrams per milliliter. 21. The purity of a sample of K 3 Fe(CN) 6 was determined using linear-potential scan hydrodynamic voltammetry at a glassy carbon electrode using the method of external standards. The following data were obtained for a set of calibration standards. [K 3 Fe(CN) 6 ] Limiting current (mM) (m A) 2.0 127 4.0 252 6.0 376 8.0 500 10.0 624 A sample was prepared for analysis by diluting a 0.246-g sample to volume in a 100-mL volumetric flask. The limiting current for the sample was found to be 444 m A. Report the purity of this sample of K 3 Fe(CN) 6 . 22. Anodic stripping voltammetry at a mercury film electrode can be used to determine whether an individual has recently fired a gun by looking for traces of antimony in residue collected from the individual’s hands. In a typical analysis a sample is collected with a cotton-tipped swab that had been wetted with 5% v/v HNO 3 . When returned to the lab, the swab is placed in a vial containing 5.00 mL of 4 M HCl that is 0.02 M in hydrazine sulfate. After allowing the swab to soak overnight, 4.00 mL of the solution is transferred to an electrochemical cell, and 100 m L of 1.00 · 10 –2 M HgCl 2 is added. Anodic stripping voltammetry of the sample gives a peak current of 0.38 m A. After a standard addition of 100 m L of 5.00 · 10 –2 ppb Sb was added, anodic stripping voltammetry is repeated, giving a peak current of 1.14 m A. How many nanograms of Sb is collected from the individual’s hand? 23. Zinc can be used as an internal standard in the analysis of thallium by differential pulse polarography. A standard solution containing 5.00 · 10 –5 M Zn 2+ and 2.50 · 10 –5 M Tl + gave peak currents of 5.71 m A and 3.19 m A, respectively. An 8.713-g sample of an alloy known to be free of zinc was dissolved in acid, transferred to a 500-mL volumetric flask, and diluted to volume. A 25.0-mL portion of this solution was mixed with 25.0 mL of a 5.00 · 10 –4 M solution of Zn 2+ . Analysis of this solution gave a peak current for Zn 2+ of 12.3 m A, and for Tl + of 20.2 m A. Report the Tl% w/w in the alloy. 24. Differential pulse voltammetry at a carbon working electrode can be used to determine the concentrations of ascorbic acid and caffeine in drug formulations. 32 In a typical analysis, a 0.9183-g tablet is crushed and ground into a fine powder. A 0.5630-g sample of this powder is transferred to a 100-mL volumetric flask, brought into solution, and diluted to volume. A 0.500-mL portion is then transferred to a voltammetric cell containing 20.00 mL of a suitable supporting electrolyte. The resulting voltammogram gives peak currents of 1.40 m A and 3.88 m A for ascorbic acid and caffeine, respectively. A 0.500-mL aliquot of a standard solution containing 250.0 ppm ascorbic acid and 200.0 ppm caffeine is then added. A voltammogram of this solution gives peak currents of 2.80 m A and 8.02 m A for ascorbic acid and caffeine, respectively. Report the number of milligrams of ascorbic acid and of caffeine in the tablet. 25. Ratana-ohpas and co-workers describe a stripping analysis method for determining levels of tin in canned fruit juices. 33 Standards containing 50.0 ppb Sn 4+ , 100.0 ppb Sn 4+ , and 150.0 ppb Sn 4+ were analyzed, giving signals of 83.0, 171.6, and 260.2, respectively. A 2.00-mL sample of lychee juice was mixed with 20.00 mL of 1:1 HCl/HNO 3 . A 0.500-mL portion of this mixture was added to 10 mL of 6 M HCl and the volume adjusted to 30.00 mL. Analysis of this diluted sample gave a signal of 128.2. Report the parts per million of Sn 4+ in the original sample of lychee juice. 26. Sittampalam and Wilson described the preparation and use of an amperometric sensor for glucose. 34 The sensor is calibrated by measuring the steady-state current when it is immersed in standard solutions of glucose. A typical set of calibration data is shown in the following table. Milligrams of Current glucose/100 mL (arb. units) 2.0 17.2 4.0 32.9 6.0 52.1 8.0 68.0 10.0 85.8 A 2.00-mL sample of a solution containing an unknown amount of glucose is diluted to 10 mL in a volumetric flask, and a steady-state current of 23.6 is measured. What is the concentration of glucose in the sample in milligrams per 100 mL? 27. Differential pulse polarography is used to determine the concentrations of lead, thallium, and indium in a mixture. 20 538 Modern Analytical Chemistry 1400-CH11 9/9/99 2:08 PM Page 538 Since the polarographic peaks for lead and thallium, and for thallium and indium overlap, a simultaneous analysis is necessary. Peak currents (in arbitrary units) at potentials of –0.385 V, –0.455 V, and –0.557 V were measured for a single standard solution and for a sample, giving the results shown in the following table. Concentration i peak at i peak at i peak at Analyte (m g/mL) –0.385 V –0.455V –0.557 V Pb 2+ 1.0 26.1 2.9 0 Tl + 2.0 7.8 23.5 3.2 In 3+ 0.4 0 0 22.9 sample 60.6 28.8 54.1 Report the concentrations of Pb 2+ , Tl + , and In 3+ in the sample. 28. Abass and colleagues developed an amperometric biosensor for NH 4 + that uses the enzyme glutamate dehydrogenase to catalyze the following reaction. 2-oxyglutarate(aq)+NH 4 + (aq) + NADH(aq) fi glutamate(aq) + NAD + (aq)+H 2 O(l) where NADH is the reduced form of nicotinamide adenine dinucleotide. 35 The biosensor actually responds to the concentration of NADH; however, the rate of the reaction depends on the concentration of NH 4 + . If the initial concentrations of 2-oxyglutarate and NADH are the same for all samples and standards, then the signal is proportional to the concentration of NH 4 + . As shown in the following table, the sensitivity of the method is dependent on pH. Sensitivity pH (nA s –1 M –1 ) 6.2 1.67 · 10 3 6.75 5.00 · 10 3 7.3 9.33 · 10 3 7.7 1.03 · 10 4 8.3 1.27 · 10 4 9.3 2.67 · 10 3 Two possible explanations for the effect of pH on the sensitivity of this analysis are the acid–base chemistry of NH 4 + , and, the acid–base chemistry of the enzyme. Given that the pK a for NH 4 + is 9.244, explain the source of this pH- dependent sensitivity. 29. The speciation scheme of Batley and Florence 22 requires eight measurements on four samples. After removing insoluble particulates by filtration, the solution is analyzed for the concentration of anodic stripping voltammetry (ASV) labile metal and the total concentration of metal. A portion of the filtered solution is passed through an ion-exchange column, and the concentrations of ASV metal and total metal are determined. A second portion of the filtered solution is irradiated with UV light, and the concentrations of ASV metal and total metal are measured. Finally, a third portion of the filtered solution is irradiated with UV light, passed through an ion-exchange column, and the concentrations of ASV-labile metal and total metal are again determined. The groups that are included in each measurement are summarized in the following table (see Table 11.12). Groups Groups Contributing Groups Contributing Separation Remaining to Concentration to Concentration Step in Solution of ASV- Labile Metal of Total Metal 1. filtration I, II, III, IV, I, II, III I, II, III, IV, V, VI, V, VI, VII VII 2. ion- exchange II, III, VI, VII II, III II, III, VI, VII 3. UV I, II, III, IV, I, II, III, IV, VI I, II, III, IV, V, VI, irradiation V, VI, VII VII 4. UV irradiation/ ion-exchange III, VII III III, VII (a) Show how these eight measurements can be used to determine the concentration of metal present in each group. (b) Batley and Florence 22a report the following results for the speciation of cadmium, lead, and copper in a sample of sea water Measurement ppb Cd 2+ ppb Pb 2+ ppb Cu 2+ filtration: ASV-labile 0.24 0.39 0.26 filtration: total 0.28 0.50 0.40 ion-exchange: ASV-labile 0.21 0.33 0.17 ion-exchange: total 0.26 0.43 0.24 UV: ASV-labile 0.26 0.37 0.33 UV: total 0.28 0.50 0.43 ion-exchange/UV: labile 0 0 0 ion-exchange/UV: total 0.02 0.12 0.10 Determine the speciation of each metal in this sample of sea water. 30. The polarographic half-wave potentials (versus the SCE) for Pb 2+ and Tl + in 1 M HCl are, respectively, –0.44 V and –0.45 V. In an electrolyte of 1 M NaOH, however, the half-wave potentials are –0.76 V for Pb 2+ and –0.48 V for Tl + . Why does the change in electrolyte have such a great effect on the half-wave potential for Pb 2+ , but not on the half-wave potential for Tl + ? 31. The following data were collected for the reduction of Pb 2+ by normal pulse polarography Potential Current (V vs. SCE) (m A) –0.345 0.16 –0.370 0.98 –0.383 2.05 –0.393 3.13 –0.409 4.62 –0.420 5.16 Chapter 11 Electrochemical Methods of Analysis 539 1400-CH11 9/9/99 2:08 PM Page 539 The limiting current was 5.67 m A. Verify that the reduction reaction is reversible, and determine values for n and E 1/2 . The half-wave potentials for the normal pulse polarograms of Pb 2+ in the presence of several different concentrations of OH – are shown in the following table. [OH – ] E 1/2 (M) (V vs. SCE) 0.050 -0.646 0.100 -0.673 0.150 -0.689 0.300 -0.715 Determine the stoichiometry of the Pb–hydroxide complex and its formation constant. 32. The concentration of Cu 2+ in sea water may be determined by anodic stripping voltammetry at a hanging mercury drop electrode. To determine the total amount of Cu 2+ in a sample of sea water, it is necessary to free any copper bound in organic complexes. To a 20.00-mL sample of sea water is added 1 mL of 0.05 M HNO 3 and 1 mL of 0.1% H 2 O 2 . The sample is irradiated with UV light for 8 h and then diluted to volume in a 25-mL volumetric flask. Deposition of Cu 2+ takes place at –0.3 V for 10 min, producing a peak current of 26.1 (arbitrary units). A second 20.00-mL sample of the sea water is treated identically, except that 0.1 mL of a 5.00 m M solution of Cu 2+ is added, producing a peak current of 38.4. Report the concentration of Cu 2+ in the sea water in parts per billion. 33. Thioamide drugs, such as that shown here, can be determined by cathodic stripping analysis. Deposition occurs at +0.05 V versus the SCE. During the stripping step, the potential is scanned cathodically, and a stripping peak is observed at - 0.55 V. In a typical application a 2.00-mL sample of urine was mixed with 2.00 mL of a pH 4.78 buffer. Following a 2-minute deposition, a peak current of 0.562 m A was measured. A 0.10-mL addition of a 5.00 m M solution of the thioamide drug was added to the same solution. A peak current of 0.837 m A was recorded following the same 2-min deposition time. Report the drug’s molar concentration in the urine sample. 34. The concentration of vanadium (V) in sea water can be determined by adsorptive stripping voltammetry after forming a complex with catechol. 36 The catechol-V(V) complex is deposited on a hanging mercury drop electrode at a potential of - 0.1 V (vs. Ag/AgCl). A cathodic potential scan gives a stripping peak that is proportional to the concentration of V(V) in the sample of sea water. The following standard additions were used to analyze a sample of sea water. [V (V)] add i peak (M) (nA) 2.0 · 10 –8 24 4.0 · 10 –8 33 8.0 · 10 –8 52 1.2 · 10 –7 69 1.8 · 10 –7 97 2.8 · 10 –7 140 Determine the molar concentration of V (V) in the sample of sea water, assuming that the standard additions result in a negligible change in the sample’s volume. 540 Modern Analytical Chemistry CH 3 CSNH 2 N The following general references provide a broad introduction to electrochemical methods of analysis. Bard, A. J.; Faulkner, L. R. Electrochemical Methods. Wiley: New York, 1980. Faulkner, L. R. “Electrochemical Characterization of Chemical Systems.” In Kuwana, T. E., ed., Physical Methods in Modern Chemical Analysis, Vol. 3. Academic Press: New York, 1983, pp. 137–248. Kissinger, P. T.; Heineman, W. R. Laboratory Techniques in Electroanalytical Chemistry. 2nd ed. Marcel Dekker: New York, 1996. Lingane, J. J. Electroanalytical Chemistry. 2nd ed. Interscience: New York, 1958. Sawyer, D. T.; Roberts, J. L., Jr. Experimental Electrochemistry for Chemists. Wiley-Interscience: New York, 1974. Vassos, B. H.; Ewing, G. W. Electroanalytical Chemistry. Wiley- Interscience: New York, 1983. Three short articles providing a good understanding of important general principles of electrochemistry follow. Faulkner, L. R. “Understanding Electrochemistry: Some Distinctive Concepts,” J. Chem. Educ. 1983, 60, 262–264. Maloy, J. T. “Factors Affecting the Shape of Current-Potential Curves,” J. Chem. Educ. 1983, 60, 285–289. Zoski, C. G. “Charging Current Discrimination in Analytical Voltammetry,” J. Chem. Educ. 1986, 63, 910–914. 11I SUGGESTED READINGS 1400-CH11 9/9/99 2:08 PM Page 540 Additional information on potentiometry and ion-selective electrodes can be found in the following sources. Bates, R. G. Determination of pH: Theory and Practice. 2nd ed. Wiley: New York, 1973. Buck, R. P. “Potentiometry: pH Measurements and Ion Selective Electrodes.” In Weissberger, A., ed., Physical Methods of Organic Chemistry, Vol. 1, Part IIA. Wiley: New York, 1971, pp. 61–162. Cammann, K. Working with Ion-Selective Electrodes. Springer-Verlag: Berlin, 1977. Evans, A. Potentiometry and Ion-Selective Electrodes. Wiley: New York, 1987. Frant, M. S. “Where Did Ion Selective Electrodes Come From?” J. Chem. Educ. 1997, 74, 159–166. Light, T. S. “Industrial Use and Application of Ion-Selective Electrodes,” J. Chem. Educ. 1997, 74, 171–177. Rechnitz, G. A. “Ion and Bio-Selective Membrane Electrodes,” J. Chem. Educ. 1983, 60, 282–284. Ruzicka, J. “The Seventies—Golden Age for Ion-Selective Electrodes,” J. Chem. Educ. 1997, 74, 167–170. Young, C. C. “Evolution of Blood Chemistry Analyzers Based on Ion Selective Electrodes,” J. Chem. Educ. 1997, 74, 177–182. The following sources provide additional information on electrochemical biosensors. Alvarez-Icasa, M.; Bilitewski, U. “Mass Production of Biosensors,” Anal. Chem. 1993, 65, 525A–533A. Meyerhoff, M. E.; Fu, B.; Bakker, E.; et al. “Polyion-Sensitive Membrane Electrodes for Biomedical Analysis,” Anal. Chem. 1996, 68, 168A–175A. Nicolini, C.; Adami, M; Antolini, F.; et al. “Biosensors: A Step to Bioelectronics,” Phys. World, May 1992, 30–34. Schultz, J. S. “Biosensors,” Sci. Am. August 1991, 64–69. Thompson, M.; Krull, U. “Biosensors and the Transduction of Molecular Recognition,” Anal. Chem. 1991, 63, 393A–405A. Vadgama, P. “Designing Biosensors,” Chem. Brit. 1992, 28, 249–252. The following is a good source covering the clinical application of electrochemistry. Wang, J. Electroanalytical Techniques in Clinical Chemistry and Laboratory Medicine. VCH: New York, 1998. Coulometry is covered in the following texts. Milner, G. W. C.; Phillips, G. Coulometry in Analytical Chemistry. Pergamon: New York, 1967. Rechnitz, G. A. Controlled-Potential Analysis. Macmillan: New York, 1963. For a description of electrogravimetry, see the following resource. Tanaka, N. “Electrodeposition,” In Kolthoff, I. M.; Elving, P. J., eds. Treatise on Analytical Chemistry, Part I: Theory and Practice, Vol. 4. Interscience: New York, 1963. The following sources provide additional information on polarography and pulse polarography. Flato, J. B. “The Renaissance in Polarographic and Voltammetric Analysis,” Anal. Chem. 1972, 44(11), 75A–87A. Kolthoff, I. M.; Lingane, J. J. Polarography. Interscience: New York, 1952. Osteryoung, J. “Pulse Voltammetry,” J. Chem. Educ. 1983, 60, 296–298. Additional information on stripping voltammetry is available in the following text. Wang, J. Stripping Analysis. VCH Publishers: Deerfield Beach, FL, 1985, pp. 116–124. Chapter 11 Electrochemical Methods of Analysis 541 1. Stork, J. T. Anal. Chem. 1993, 65, 344A–351A. 2. Sawyer, D. T.; Roberts, J. L., Jr. Experimental Electrochemistry for Chemists. Wiley-Interscience: New York, 1974, p. 22. 3. Cammann, K. Working with Ion-Selective Electrodes. Springer-Verlag: Berlin, 1977. 4. (a) Papastathopoulos, D. S.; Rechnitz, G. A. Anal. Chim. Acta 1975, 79, 17; (b) Riechel, T. L. J. Chem. Educ. 1984, 61, 640–642. 5. Tor, R.; Freeman, A. Anal. Chem. 1986, 58, 1042–1046. 6. Lunte, C. E.; Heineman, W. R. “Electrochemical Techniques in Bioanalysis” in Steckhan, E., ed. Topics in Current Chemistry. Vol. 143. Springer-Verlag: Berlin, 1988, p. 8. 7. Procedure adapted from Kennedy, J. H. Analytical Chemistry— Practice. Harcourt Brace Jovanovich: San Diego, 1984, p. 117–118. 8. Hawkes, S. J. J. Chem. Educ. 1994, 71, 747–749. 9. Kristensen, H. B.; Saloman, A.; Kokholm, G. Anal. Chem. 1991, 63, 885A–891A. 10. Bates, R. G. Determination of pH: Theory and Practice. 2nd ed. Wiley: New York, 1973. 11. Rogers, K. R.; Williams, L. R. Trends Anal. Chem. 1995, 14, 289–294. 12. Rechnitz, G. A. Controlled-Potential Analysis. Macmillan: New York, 1963, p. 49. 13. Procedure adapted from Bassett, J.; Denney, R. C.; Jeffery, G. H.; et al. Vogel’s Textbook of Quantitative Inorganic Analysis. Longman: London, 1978, pp. 559–560. 14. Curran, D. J. “Constant-Current Coulometry.” Chapter 20 in Kissinger, P. T.; Heineman, W. R., eds. Laboratory Techniques in Electroanalytical Chemistry. Marcel Dekker, Inc.: New York, 1984, pp. 539–568. 15. Maloy, J. T. J. Chem. Educ. 1983, 60, 285–289. 16. Osteryoung, J. J. Chem. Educ. 1983, 60, 296–298. 17. Peterson, W. M.; Wong, R. V. Am. Lab. November 1981, 116–128. 18. Wang, J. Am. Lab. May 1985, 41–50. 19. Karube, I.; Nomura, Y.; Arikawa, Y. Trends in Anal. Chem. 1995, 14, 295–299. 20. Lanza, P. J. Chem. Educ. 1990, 67, 704–705. 21. Wang, J. Stripping Analysis. VCH Publishers: Deerfield Beach, FL, 1985, pp. 116–124. 22. (a) Batley, G. E.; Florence, T. M. Anal. Lett. 1976, 9, 379–388; (b) Batley, G. E.; Florence, T. M. Talanta 1977, 24, 151; (c) Batley, G. E.; Florence, T. M. Anal. Chem. 1980, 52, 1962–1963; (d) Florence, T. M.; Batley, G. E. CRC Crit. Rev. Anal. Chem. 1980, 9, 219–296. 23. Brooks, M. A. “Application of Electrochemistry to Pharmaceutical Analysis” Chapter 21. In Kissinger, P. T.; Heineman, W. R., eds. 11J REFERENCES 1400-CH11 9/9/99 2:08 PM Page 541 Laboratory Techniques in Electroanalytical Chemistry. Marcel Dekker, Inc.: New York, 1984, pp. 569–607. 24. Cammann, K.; Lemke, U.; Rohen, A.; et al. Angew. Chem. Int. Ed. Engl. 1991, 30, 516–539. 25. Procedure adapted from Pungor, E. A Practical Guide to Instrumental Analysis. CRC Press: Boca Raton, FL, 1995, pp. 34–37. 26. Abe, T.; Lau, L. L.; Ewing, A. G. J. Am. Chem. Soc. 1991, 113, 7421–7423. 27. Creager, S. E.; Lawrence, K. D.; Tibbets, C. R. J. Chem. Educ. 1995, 72, 274–276. 28. Watanabe, K.; Okada, K.; Oda, H.; et al. Anal. Chim. Acta 1995, 316, 371–375. 29. Mifflin, T. E.; Andriano, K. M.; Robbins, W. B. J. Chem. Educ. 1984, 61, 638–639. 30. Meyerhoff, M. E.; Kovach, P. M. J. Chem. Educ. 1983, 9, 766–768. 31. Wang, J.; Taha, Z. Anal. Chim. Acta 1991, 252, 215–221. 32. Lau, O.; Luk, S.; Cheung, Y. Analyst, 1989, 114, 1047–1051. 33. Ratana-ohpas, R; Kanatharana, P.; Ratana-ohpas, W.; et al. Anal. Chim. Acta 1996, 333, 115–118. 34. Sittampalam, G; Wilson, G. S. J. Chem. Educ. 1982, 59, 70–73. 35. Abass, A. K.; Hart, J. P.; Cowell, D. C.; et al. Anal. Chim. Acta 1998, 373, 1–8. 36. van der Berg, C. M. G.; Huang, Z. Q. Anal. Chem. 1984, 56, 2383–2386. 542 Modern Analytical Chemistry 1400-CH11 9/9/99 2:08 PM Page 542 Chapter 12 543 Chromatographic and Electrophoretic Methods Drawing from an arsenal of analytical techniques, many of which were the subject of the preceding four chapters, analytical chemists have designed methods for the analysis of analytes at increasingly lower concentrations and in increasingly more complex matrices. Despite the power of these techniques, they often suffer from a lack of selectivity. For this reason, many analytical procedures include a step to separate the analyte from potential interferents. Several separation methods, such as liquid—liquid extractions and solid-phase microextractions, were discussed in Chapter 7. In this chapter we consider two additional separation methods that combine separation and analysis: chromatography and electrophoresis. 1400-CH12 9/8/99 4:28 PM Page 543 544 Modern Analytical Chemistry 12A Overview of Analytical Separations In Chapter 7 we examined several methods for separating an analyte from potential interferents. For example, in a liquid–liquid extraction the analyte and interferent are initially present in a single liquid phase. A second, immiscible liquid phase is in- troduced, and the two phases are thoroughly mixed by shaking. During this process the analyte and interferents partition themselves between the two phases to differ- ent extents, affecting their separation. Despite the power of these separation tech- niques, there are some significant limitations. 12A.1 The Problem with Simple Separations Suppose we have a sample containing an analyte in a matrix that is incompatible with our analytical method. To determine the analyte’s concentration we first sepa- rate it from the matrix using, for example, a liquid–liquid extraction. If there are additional analytes, we may need to use additional extractions to isolate them from the analyte’s matrix. For a complex mixture of analytes this quickly becomes a te- dious process. Furthermore, the extent to which we can effect a separation depends on the distribution ratio of each species in the sample. To separate an analyte from its ma- trix, its distribution ratio must be significantly greater than that for all other com- ponents in the matrix. When the analyte’s distribution ratio is similar to that of an- other species, then a separation becomes impossible. For example, let’s assume that an analyte, A, and a matrix interferent, I, have distribution ratios of 5 and 0.5, re- spectively. In an attempt to separate the analyte from its matrix, a simple liquid– liquid extraction is carried out using equal volumes of sample and a suitable extrac- tion solvent. Following the treatment outlined in Chapter 7, it is easy to show that a single extraction removes approximately 83% of the analyte and 33% of the inter- ferent. Although it is possible to remove 99% of A with three extractions, 70% of I is also removed. In fact, there is no practical combination of number of extractions or volume ratio of sample and extracting phases that produce an acceptable separa- tion of the analyte and interferent by a simple liquid–liquid extraction. 12A.2 A Better Way to Separate Mixtures The problem with a simple extraction is that the separation only occurs in one di- rection. In a liquid–liquid extraction, for example, we extract a solute from its ini- tial phase into the extracting phase. Consider, again, the separation of an analyte and a matrix interferent with distribution ratios of 5 and 0.5, respectively. A single liquid–liquid extraction transfers 83% of the analyte and 33% of the interferent to the extracting phase (Figure 12.1). If the concentrations of A and I in the sample were identical, then their concentration ratio in the extracting phase after one ex- traction is Thus, a single extraction improves the separation of the solutes by a factor of 2.5. As shown in Figure 12.1, a second extraction actually leads to a poorer separation. After combining the two portions of the extracting phase, the concentration ratio decreases to [. . . A] [I] == 097 055 18 [. . . A] [I] == 083 033 25 1400-CH12 9/8/99 4:28 PM Page 544 Figure 12.1 Progress of a liquid–liquid extraction using two identical extractions of a sample (initial phase) with fresh portions of the extracting phase. All numbers are fractions of solute in the phases; A = analyte, I = interferent. We can improve the separation by first extracting the solutes into the extracting phase, and then extracting them back into a fresh portion of the initial phase (Fig- ure 12.2). Because solute A has the larger distribution ratio, it is extracted to a greater extent during the first extraction and to a lesser extent during the second ex- traction. In this case the final concentration ratio of [. . . A] [I] == 069 011 63 Chapter 12 Chromatographic and Electrophoretic Methods 545 0.83 0.17 0.33 0.83 0.33 0.67 0 1 0 AI AI AI AI 0.97 0.55 AI 1 0 0.17 0 0.67 0.14 0.03 0.22 0.14 0.22 0.45 Extract Extract Separate Separate Add new extracting phase Extracting phase Initial phase Combine 0.83 0.17 0.33 0.83 0.33 0.67 0 1 0 AI AI 1 0.83 0 0.33 0 Extract Extract Separate AI AI 0.69 0.14 0.11 0.69 0.11 0.22 Separate Add new initial phase Extracting phase Initial phase Figure 12.2 Progress of a liquid–liquid extraction in which the solutes are first extracted into the extracting phase and then extracted back into a fresh portion of the initial phase. All numbers are fractions of solute in the phases; A = analyte, I = interferent. 1400-CH12 9/8/99 4:28 PM Page 545 in the extracting phase is significantly greater. The process of extracting the solutes back and forth between fresh portions of the two phases, which is called a counter- current extraction, was developed by Craig in the 1940s. 1* The same phenomenon forms the basis of modern chromatography. Chromatographic separations are accomplished by continuously passing one sample-free phase, called a mobile phase, over a second sample-free phase that re- mains fixed, or stationary. The sample is injected, or placed, into the mobile phase. As it moves with the mobile phase, the sample’s components partition themselves between the mobile and stationary phases. Those components whose distribution ratio favors the stationary phase require a longer time to pass through the system. Given sufficient time, and sufficient stationary and mobile phase, solutes with simi- lar distribution ratios can be separated. The history of modern chromatography can be traced to the turn of the cen- tury when the Russian botanist Mikhail Tswett (1872–1919) used a column packed with a stationary phase of calcium carbonate to separate colored pigments from plant extracts. The sample was placed at the top of the column and carried through the stationary phase using a mobile phase of petroleum ether. As the sample moved through the column, the pigments in the plant extract separated into individual col- ored bands. Once the pigments were adequately separated, the calcium carbonate was removed from the column, sectioned, and the pigments recovered by extrac- tion. Tswett named the technique chromatography, combining the Greek words for “color” and “to write.” There was little interest in Tswett’s technique until 1931 when chromatography was reintroduced as an analytical technique for biochemical separations. Pioneering work by Martin and Synge in 1941 2 established the impor- tance of liquid–liquid partition chromatography and led to the development of a theory for chromatographic separations; they were awarded the 1952 Nobel Prize in chemistry for this work. Since then, chromatography in its many forms has become the most important and widely used separation technique. Other separation meth- ods, such as electrophoresis, effect a separation without the use of a stationary phase. 12A. 3 Classifying Analytical Separations Analytical separations may be classified in three ways: by the physical state of the mobile phase and stationary phase; by the method of contact between the mobile phase and stationary phase; or by the chemical or physical mechanism responsible for separating the sample’s constituents. The mobile phase is usually a liquid or a gas, and the stationary phase, when present, is a solid or a liquid film coated on a solid surface. Chromatographic techniques are often named by listing the type of mobile phase, followed by the type of stationary phase. Thus, in gas–liquid chro- matography the mobile phase is a gas and the stationary phase is a liquid. If only one phase is indicated, as in gas chromatography, it is assumed to be the mobile phase. Two common approaches are used to bring the mobile phase and stationary phase into contact. In column chromatography, the stationary phase is placed in a narrow column through which the mobile phase moves under the influence of gravity or pressure. The stationary phase is either a solid or a thin, liquid film coating on a solid particulate packing material or the column’s walls. In planar chromatography the stationary phase coats a flat glass, metal, or plastic plate 546 Modern Analytical Chemistry countercurrent extraction A liquid–liquid extraction in which solutes are extracted back and forth between fresh portions of two extracting phases. mobile phase In chromatography, the extracting phase that moves through the system. stationary phase In chromatography, the extracting phase that remains in a fixed position. chromatography A separation in which solutes partition between a mobile and stationary phase. column chromatography A form of chromatography in which the stationary phase is retained in a column. *The theory behind countercurrent extractions is outlined in Appendix 6. planar chromatography A form of chromatography in which the stationary phase is immobilized on a flat surface. 1400-CH12 9/8/99 4:28 PM Page 546 Chapter 12 Chromatographic and Electrophoretic Methods 547 – – – – – – + + + + + + + + + ++ + (a) (b) (c) (d) (e) Figure 12.3 Schematics showing the basis of separation in (a) adsorption chromatography, (b) partition chromatography, (c) ion-exchange chromatography, (d) size- exclusion chromatography, and (e) electrophoresis. For the separations in (a), (b), and (d) the solute represented by the solid circle ( ? ) is the more strongly retained. and is placed in a developing chamber. A reservoir containing the mobile phase is placed in contact with the stationary phase, and the mobile phase moves by capillary action. The mechanism by which solutes separate provides a third means for charac- terizing a separation (Figure 12.3). In adsorption chromatography, solutes sepa- rate based on their ability to adsorb to a solid stationary phase. In partition chro- matography, a thin liquid film coating a solid support serves as the stationary phase. Separation is based on a difference in the equilibrium partitioning of solutes between the liquid stationary phase and the mobile phase. Stationary phases consisting of a solid support with covalently attached anionic (e.g., –SO 3 – ) or cationic (e.g., –N(CH 3 ) 3 + ) functional groups are used in ion-exchange chro- matography. Ionic solutes are attracted to the stationary phase by electrostatic forces. Porous gels are used as stationary phases in size-exclusion chromatogra- phy, in which separation is due to differences in the size of the solutes. Large solutes are unable to penetrate into the porous stationary phase and so quickly pass through the column. Smaller solutes enter into the porous stationary phase, increasing the time spent on the column. Not all separation methods require a stationary phase. In an electrophoretic separation, for example, charged solutes migrate under the influence of an applied potential field. Differences in the mo- bility of the ions account for their separation. 12B General Theory of Column Chromatography Of the two methods for bringing the stationary and mobile phases into contact, the more important is column chromatography. In this section we develop a general theory that we may apply to any form of column chromatogra- phy. With appropriate modifications, this theory also can be applied to planar chromatography. A typical column chromatography experiment is outlined in Figure 12.4. Al- though the figure depicts a liquid–solid chromatographic experiment similar to that first used by Tswett, the design of the column and the physical state of the 1400-CH12 9/8/99 4:28 PM Page 547 Figure 12.6 Typical chromatogram of detector response as a function of retention time. Figure 12.4 Progress of a column chromatographic separation showing the separation of two solute bands. stationary and mobile phases may vary. The sample is introduced at the top of the column as a narrow band. Ideally, the solute’s initial concentration profile is rec- tangular (Figure 12.5a). As the sample moves down the column the solutes begin to separate, and the individual solute bands begin to broaden and develop a Gaussian profile (Figures 12.5b,c). If the strength of each solute’s interaction with the stationary phase is sufficiently different, then the solutes separate into individ- ual bands (Figure 12.5d). The progress of a chromatographic separation is moni- tored with a suitable detector situated at the end of the column. A plot of the de- tector’s signal as a function of time or volume of eluted mobile phase is known as a chromatogram (Figure 12.6) and consists of a peak for each of the separated solute bands. A chromatographic peak may be characterized in many ways, two of which are shown in Figure 12.7. The retention time, t r , is the elapsed time from the introduc- tion of the solute to the peak maximum. The retention time also can be measured indirectly as the volume of mobile phase eluting between the solute’s introduction and the appearance of the solute’s peak maximum. This is known as the retention volume, V r . Dividing the retention volume by the mobile phase’s flow rate, u, gives the retention time. The second important parameter is the chromatographic peak’s width at the baseline, w. As shown in Figure 12.7, baseline width is determined by the inter- section with the baseline of tangent lines drawn through the inflection points on either side of the chromatographic peak. Baseline width is measured in units of time or volume, depending on whether the retention time or retention volume is of interest. 548 Modern Analytical Chemistry chromatogram A plot of the detector’s signal as function of elution time or volume. retention time The time a solute takes to move from the point of injection to the detector (t r ). retention volume The volume of mobile phase needed to move a solute from its point of injection to the detector (V r ). baseline width The width of a solute’s chromatographic band measured at the baseline (w). Distance down column (a) (b) (c) (d) Concentration of solute Retention time Detector signal Figure 12.5 Another view of the progress of a column chromatographic separation showing the separation of two solute bands. 1400-CH12 9/8/99 4:28 PM Page 548 Chapter 12 Chromatographic and Electrophoretic Methods 549 Detector signal Retention time Injection t r t m w Figure 12.7 Measurement of the column’s void time, t m , and the retention time, t r , and baseline width, w, for a solute. void time The time required for unretained solutes to move from the point of injection to the detector (t m ). void volume The volume of mobile phase needed to move an unretained solute from the point of injection to the detector. resolution The separation between two chromatographic bands (R). Besides the solute peak, Figure 12.7 also shows a small peak eluted soon after the sample is injected into the mobile phase. This peak results from solutes that move through the column at the same rate as the mobile phase. Since these solutes do not interact with the stationary phase, they are considered nonretained. The time or volume of mobile phase required to elute nonretained components is called the column’s void time, t m , or void volume. 12B.1 Chromatographic Resolution The goal of chromatography is to separate a sample into a series of chromato- graphic peaks, each representing a single component of the sample. Resolution is a quantitative measure of the degree of separation between two chromatographic peaks, A and B, and is defined as 12.1 As shown in Figure 12.8, the degree of separation between two chromatographic peaks improves with an increase in R. For two peaks of equal size, a resolution of 1.5 corresponds to an overlap in area of only 0.13%. Because resolution is a quanti- tative measure of a separation’s success, it provides a useful way to determine if a change in experimental conditions leads to a better separation. EXAMPLE 12.1 In a chromatographic analysis of lemon oil a peak for limonene has a retention time of 8.36 min with a baseline width of 0.96 min. g -Terpinene elutes at 9.54 min, with a baseline width of 0.64 min. What is the resolution between the two peaks? SOLUTION Using equation 12.1, we find that the resolution is R t ww = + = ? + = 22954836 064 096 148 ? r BA (. . ) .. . R tt ww t ww = ? + = + r,B r, A BA r BA 05 2 .( ) ? Figure 12.8 Three examples of chromatographic resolution. Concentration of solute Distance down column R = 0.75 R = 1.25 R = 1.50 1400-CH12 9/8/99 4:28 PM Page 549 550 Modern Analytical Chemistry From equation 12.1 it is clear that resolution may be improved either by in- creasing D t r or by decreasing w A or w B (Figure 12.9). We can increase D t r by en- hancing the interaction of the solutes with the column or by increasing the col- umn’s selectivity for one of the solutes. Peak width is a kinetic effect associated with the solute’s movement within and between the mobile phase and stationary phase. The effect is governed by several factors that are collectively called column effi- ciency. Each of these factors is considered in more detail in the following sections. 12B.2 Capacity Factor The distribution of a solute, S, between the mobile phase and stationary phase can be represented by an equilibrium reaction S m t S s and its associated partition coefficient, K D , and distribution ratio, D, 12.2 where the subscripts m and s refer to the mobile phase and stationary phase, respec- tively. As long as the solute is not involved in any additional equilibria in either the mobile phase or stationary phase, the equilibrium partition coefficient and the dis- tribution ratio will be the same. Conservation of mass requires that the total moles of solute remain constant throughout the separation, thus (moles S) tot = (moles S) m + (moles S) s 12.3 Solving equation 12.3 for the moles of solute in the stationary phase and substitut- ing into equation 12.2 gives D = [] [] S S s tot m tot K D s m S S = [] [] (a) (b) (c) Figure 12.9 Two methods for improving chromatographic resolution: (a) Original separation showing a pair of poorly resolved solutes; (b) Improvement in resolution due to an increase in column efficiency; (c) Improvement in resolution due to a change in column selectivity. 1400-CH12 9/8/99 4:28 PM Page 550 where V m and V s are the volumes of the mobile and stationary phases. Rearranging and solving for the fraction of solute in the mobile phase, f m , gives 12.4 Note that this equation is identical to that describing the extraction of a solute in a liquid–liquid extraction (equation 7.25 in Chapter 7). Since the volumes of the sta- tionary and mobile phase may not be known, equation 12.4 is simplified by dividing both the numerator and denominator by V m ; thus 12.5 where 12.6 is the solute’s capacity factor. A solute’s capacity factor can be determined from a chromatogram by measur- ing the column’s void time, t m , and the solute’s retention time, t r (see Figure 12.7). The mobile phase’s average linear velocity, u, is equal to the length of the column, L, divided by the time required to elute a nonretained solute. 12.7 By the same reasoning, the solute’s average linear velocity, v, is 12.8 The solute can only move through the column when it is in the mobile phase. Its average linear velocity, therefore, is simply the product of the mobile phase’s aver- age linear velocity and the fraction of solute present in the mobile phase. v = uf m 12.9 Substituting equations 12.5, 12.7, and 12.8 into equation 12.9 gives Finally, solving this equation for k￠ gives 12.10 where t r ￠ is known as the adjusted retention time. ′ = ? = ′ k tt t t t rm m r m L t L tk rm = + ′ ? ? ? ? ? ? 1 1 v L t = r u L t = m ′ =kD V V s m f DV V k m Sm = + = + ′ 1 1 1 1(/ ) f V VDV m m tot m ms moles S) moles S) == + ( ( D V V VV V = ? = ?{( ( } ( ( ( moles S) moles S) (moles S) moles S) moles S) moles S) tot m s mm tot m m m ms Chapter 12 Chromatographic and Electrophoretic Methods 551 capacity factor A measure of how strongly a solute is retained by the stationary phase (k￠ ). adjusted retention time The difference between a solute’s retention time and column’s void time (t r ￠ ). 1400-CH12 9/8/99 4:28 PM Page 551 552 Modern Analytical Chemistry EXAMPLE 12.2 In a chromatographic analysis of low-molecular-weight acids, butyric acid elutes with a retention time of 7.63 min. The column’s void time is 0.31 min. Calculate the capacity factor for butyric acid. SOLUTION 12B. 3 Column Selectivity The relative selectivity of a chromatographic column for a pair of solutes is given by the selectivity factor, a , which is defined as 12.11 The identities of the solutes are defined such that solute A always has the smaller retention time. Accordingly, the selectivity factor is equal to 1 when the solutes elute with identical retention times, and is greater than 1 when t r,B is greater than t r,A . EXAMPLE 12. 3 In the same chromatographic analysis for low-molecular-weight acids considered in Example 12.2, the retention time for isobutyric acid is 5.98 min. What is the selectivity factor for isobutyric acid and butyric acid? SOLUTION First we must calculate the capacity factor for isobutyric acid. Using the void time from Example 12.2, this is The selectivity factor, therefore, is 12B. 4 Column Efficiency At the beginning of a chromatographic separation the solute occupies a narrow band of finite width. As the solute passes through the column, the width of its band α= ′ ′ == k k buty iso 23 6 18 3 129 . . . ′ = ? = ? =k tt t rm m min min min 598 031 031 18 3 .. . . α= ′ ′ = ? ? k k tt tt B A r,B m r, A m ′ = ? = ? =k tt t rm m min min 763 031 031 23 6 . min . . . selectivity factor The ratio of capacity factors for two solutes showing the column’s selectivity for one of the solutes (a ). 1400-CH12 9/8/99 4:28 PM Page 552 continually increases in a process called band broadening. Column efficiency pro- vides a quantitative measure of the extent of band broadening. In their original theoretical model of chromatography, Martin and Synge 2 treated the chromatographic column as though it consists of discrete sections at which partitioning of the solute between the stationary and mobile phases occurs. They called each section a theoretical plate and defined column efficiency in terms of the number of theoretical plates, N, or the height of a theoretical plate, H; where 12.12 A column’s efficiency improves with an increase in the number of theoretical plates or a decrease in the height of a theoretical plate. Assuming a Gaussian profile, the extent of band broadening is measured by the variance or standard deviation of a chromatographic peak. The height of a theoreti- cal plate is defined as the variance per unit length of the column 12.13 where the variance, s 2 , has units of distance squared. Because retention time and peak width are usually measured in seconds or minutes, it is more convenient to ex- press the standard deviation in units of time, t , by dividing s by the mobile phase’s average linear velocity. 12.14 When a chromatographic peak has a Gaussian shape, its width at the baseline, w, is four times its standard deviation, t . w =4t 12.15 Combining equations 12.13 through 12.15 gives the height of a theoretical plate in terms of the easily measured chromatographic parameters t r and w. 12.16 The number of theoretical plates in a chromatographic column is obtained by com- bining equations 12.12 and 12.16. 12.17 Alternatively, the number of theoretical plates can be approximated as where w 1/2 is the width of the chromatographic peak at half its height. N t w = ? ? ? ? ? ? 5 545 12 2 . / r N t w r = ? ? ? ? ? ? 16 2 H Lw t = 2 16 r 2 τ σσ == u t L r H L = σ 2 N L H = Chapter 12 Chromatographic and Electrophoretic Methods 553 band broadening The increase in a solute’s baseline width as it moves from the point of injection to the detector. theoretical plate A quantitative means of evaluating column efficiency that treats the column as though it consists of a series of small zones, or plates, in which partitioning between the mobile and stationary phases occurs. 1400-CH12 9/8/99 4:28 PM Page 553 554 Modern Analytical Chemistry EXAMPLE 12. 4 A chromatographic analysis for the chlorinated pesticide Dieldrin gives a peak with a retention time of 8.68 min and a baseline width of 0.29 min. How many theoretical plates are involved in this separation? Given that the column used in this analysis is 2.0 meters long, what is the height of a theoretical plate? SOLUTION Using equation 12.17, the number of theoretical plates is Solving equation 12.12 for H gives the average height of a theoretical plate as It is important to remember that a theoretical plate is an artificial construct and that no such plates exist in a chromatographic column. In fact, the number of theo- retical plates depends on both the properties of the column and the solute. As a re- sult, the number of theoretical plates for a column is not fixed and may vary from solute to solute. 12B. 5 Peak Capacity Another important consideration is the number of solutes that can be baseline re- solved on a given column. An estimate of a column’s peak capacity, n c , is 12.18 where V min and V max are the smallest and largest volumes of mobile phase in which a solute can be eluted and detected. 3 A column with 10,000 theoretical plates, for example, can resolve no more than if the minimum and maximum volumes of mobile phase in which the solutes can elute are 1 mL and 30 mL. This estimate provides an upper bound on the number of solutes that might be separated and may help to exclude from con- sideration columns that do not have enough theoretical plates to separate a complex mixture. Just because a column’s theoretical peak capacity is larger than the number of solutes to be separated, however, does not mean that the separation will be feasible. In most situations the peak capacity obtained is less n c mL mL solutes=+ =1 10 000 4 30 1 86 , ln n NV V c max min =+1 4 ln H L N == = (. . 20 014 m)(1000 mm/m) 14, 300 plates mm/plate N t w = ? ? ? ? ? ? = ? ? ? ? ? ? =16 16 868 029 14 300 22 r min min plates . . , peak capacity The maximum number of solutes that can be resolved on a particular column (n c ). 1400-CH12 9/8/99 4:28 PM Page 554 than the estimated value because the retention characteristics of some solutes are too similar to effect their separation. Nevertheless, columns with more theo- retical plates, or a greater range of possible elution volumes, are more likely to separate a complex mixture. 12B.6 Nonideal Behavior The treatment of chromatography outlined in Section 12B assumes that a solute elutes as a symmetrical band, such as that shown in Figure 12.7. This ideal behavior occurs when the solute’s partition coefficient, K D , is constant for all concentrations of solute. In some situations, chromatographic peaks show nonideal behavior, lead- ing to asymmetrical peaks, similar to those shown in Figure 12.10. The chromato- graphic peak in Figure 12.10a is an example of “fronting” and is most often the re- sult of overloading the column with sample. Figure 12.10b, which is an example of “tailing,” occurs when some sites on the stationary phase retain the solutes more strongly than other sites. Chapter 12 Chromatographic and Electrophoretic Methods 555 Figure 12.10 Nonideal asymmetrical chromatographic bands showing (a) fronting and (b) tailing. Also depicted are the corresponding sorption isotherms showing the relationship between the concentration of solute in the stationary phase as a function of its concentration in the mobile phase. [S] m [S] s Ideal [S] m [S] s Ideal (a) (b) fronting A tail at the beginning of a chromatographic peak, usually due to injecting too much sample. tailing A tail at the end of a chromatographic peak, usually due to the presence of highly active sites in the stationary phase. 1400-CH12 9/8/99 4:28 PM Page 555 556 Modern Analytical Chemistry 12C Optimizing Chromatographic Separations Now that we have defined capacity factor, selectivity, and column efficiency we con- sider their relationship to chromatographic resolution. Since we are only interested in the resolution between solutes eluting with similar retention times, it is safe to as- sume that the peak widths for the two solutes are approximately the same. Equation 12.1, therefore, is written as 12.19 Solving equation 12.17 for w B and substituting into equation 12.19 gives 12.20 The retention times for solutes A and B are replaced with their respective capacity factors by rearranging equation 12.10 t r = k￠ t m + t m and substituting into equation 12.20. Finally, solute A’s capacity factor is eliminated using equation 12.11. After rearrang- ing, the equation for the resolution between the chromatographic peaks for solutes A and B is 12.21 Besides resolution, another important factor in chromatography is the amount of time required to elute a pair of solutes. The time needed to elute solute B is 12.22 Equations 12.21 and 12.22 contain terms corresponding to column efficiency, column selectivity, and capacity factor. These terms can be varied, more or less in- dependently, to obtain the desired resolution and analysis time for a pair of solutes. The first term, which is a function of the number of theoretical plates or the height of a theoretical plate, accounts for the effect of column efficiency. The second term is a function of a and accounts for the influence of column selectivity. Finally, the third term in both equations is a function of k B ￠ , and accounts for the effect of solute B’s capacity factor. Manipulating these parameters to improve resolution is the sub- ject of the remainder of this section. 12C.1 Using the Capacity Factor to Optimize Resolution One of the simplest ways to improve resolution is to adjust the capacity factor for solute B. If all other terms in equation 12.21 remain constant, increasing k B ￠ im- proves resolution. As shown in Figure 12.11, however, the effect is greatest when the t RH u k k r,B B B = ? ? ? ? ? ? ? + ′ ′ 16 1 1 2 2 3 2 α α () () RN k k = ?? ? ? ? ? ? ′ + ′ ? ? ? ? ? ? 1 4 1 1 B B B α α RN kk k = ′ ? ′ + ′ ? ? ? ? ? ? 1 41 B BA B RN tt t = ?? ? ? ? ? ? 1 4 B r,B r, A r,B R tt w = ? r,B r, A B 1400-CH12 9/8/99 4:28 PM Page 556 original capacity factor is small. Furthermore, large increases in k B ￠ do not lead to proportionally larger increases in resolution. For example, when the original value of k B ￠ is 1, increasing its value to 10 gives an 82% improvement in resolution; a fur- ther increase to 15 provides a net improvement in resolution of only 87.5%. Any improvement in resolution obtained by increasing k B ￠ generally comes at the expense of a longer analysis time. This is also indicated in Figure 12.11, which shows the relative change in retention time as a function of the new capacity factor. Note that a minimum in the retention time curve occurs when k B ￠ is equal to 2, and that retention time increases in either direction. Increasing k B ￠ from 2 to 10, for ex- ample, approximately doubles solute B’s retention time. The relationship between capacity factor and analysis time can be advantageous when a separation produces an acceptable resolution with a large k B ￠ . In this case it may be possible to decrease k B ￠ with little loss in resolution while significantly short- ening the analysis time. A solute’s capacity factor is directly proportional to its distribution ratio (equa- tion 12.6), which, in turn, is proportional to the solute’s equilibrium distribution constant. To increase k B ￠ without significantly changing a , which also is a function of k B ￠ , it is necessary to alter chromatographic conditions in a way that leads to a general, nonselective increase in the capacity factor for both solutes. In gas chro- matography, this is usually accomplished by decreasing the column’s temperature. At a lower temperature a solute’s vapor pressure decreases, ensuring that it spends more time in the stationary phase increasing its capacity factor. In liquid chro- matography, changing the mobile phase’s solvent strength is the easiest way to change a solute’s capacity factor. When the mobile phase has a low solvent strength, solutes spend proportionally more time in the stationary phase, thereby increasing their capacity factors. Additionally, equation 12.6 shows that the capacity factor is proportional to the volume of stationary phase. Increasing the volume of stationary phase, therefore, also leads to an increase in k B ￠ . Adjusting the capacity factor to improve resolution between one pair of solutes may lead to an unacceptably long retention time for other solutes. For ex- ample, improving resolution for solutes with short retention times by increasing Chapter 12 Chromatographic and Electrophoretic Methods 557 Figure 12.11 Effect of a change in k B ￠ on resolution and retention time. The original value of k B ￠ is assumed to be 1. Relative impr o vement in resolution New capacity factor 010155 202530 Relative c hang e in retention time 2 1.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 5 4.5 0 0.5 1 1.5 2 2.5 3 3.5 4 1400-CH12 9/8/99 4:28 PM Page 557 Figure 12.12 The general elution problem in chromatography. Improving the resolution of the overlapping bands in chromatogram (a) results in a longer analysis time for chromatogram (b). k B ￠ may substantially increase the retention times for later eluting solutes. On the other hand, decreasing k B ￠ as a means of shortening the overall analysis time may lead to a loss of resolution for solutes eluting with shorter retention times. This difficulty is encountered so frequently that it is known as the general elution problem (Figure 12.12). One solution to the general elution problem is to make incremental adjustments to the capacity factor over time. Thus, initial chromato- graphic conditions are adjusted to enhance the resolution for solutes with short retention times. As the separation progresses, chromatographic conditions are changed in a manner that increases the elution rate (decreases the retention time) for later eluting solutes. In gas chromatography this is accomplished by tempera- ture programming. The column’s initial temperature is selected such that the first solutes to elute are fully resolved. The temperature is then increased, either continuously or in steps, to bring off later eluting components with both an ac- ceptable resolution and a reasonable analysis time. In liquid chromatography the same effect can be obtained by increasing the solvent’s eluting strength. This is known as a gradient elution. 12C.2 Using Column Selectivity to Optimize Resolution A second approach to improving resolution is to adjust alpha, a . In fact, when a is nearly 1, it usually is not possible to improve resolution by adjusting k B ￠ or N. Changes in a often have a more dramatic effect on resolution than k B ￠ . For example, changing a from 1.1 to 1.5 improves resolution by 267%. A change in a is possible if chromatographic conditions are altered in a man- ner that is more selective for one of the solutes. If a solute participates in a sec- ondary equilibrium reaction in either the stationary or mobile phase, then it may be possible to alter that phase in a way that selectively changes the solute’s capacity fac- tor. For example, Figure 12.13a shows how the pH of an aqueous mobile phase can be used to control the retention times, and thus the capacity factors, for two substi- tuted benzoic acids. The resulting change in a is shown in Figure 12.13b. In gas chromatography, adjustments in a are usually accomplished by changing the sta- tionary phase, whereas changing the composition of the mobile phase is used in liq- uid chromatography. 558 Modern Analytical Chemistry Concentration of solute Retention time Concentration of solute Retention time (a) (b) temperature programming The process of changing the column’s temperature to enhance the separation of both early and late eluting solutes. gradient elution The process of changing the mobile phase’s solvent strength to enhance the separation of both early and late eluting solutes. 1400-CH12 9/8/99 4:28 PM Page 558 Figure 12.13 Use of column selectivity to improve chromatographic resolution showing: (a) the variation in retention time with mobile phase pH, and (b) the resulting change in alpha with mobile phase pH. 12C. 3 Using Column Efficiency to Optimize Resolution If the capacity factor and a are known, then equation 12.21 can be used to calculate the number of theoretical plates needed to achieve a desired resolution (Table 12.1). For example, given a = 1.05 and k B ￠ = 2.0, a resolution of 1.25 requires approxi- mately 24,800 theoretical plates. If the column only provides 12,400 plates, half of what is needed, then the separation is not possible. How can the number of theoret- ical plates be doubled? The easiest way is to double the length of the column; how- ever, this also requires a doubling of the analysis time. A more desirable approach is to cut the height of a theoretical plate in half, providing the desired resolution with- out changing the analysis time. Even better, if H can be decreased by more than Chapter 12 Chromatographic and Electrophoretic Methods 559 16 14 12 10 8 6 4 2 0 Retention time (min) pH 3 3.5 4 4.5 5 5.5 p-Aminobenzoic acid p-Hydroxybenzoic acid 1.6 1.5 1.4 1.3 1.2 1.1 1 Alpha pH 3 3.5 4 4.5 5 5.5 (a) (b) 1400-CH12 9/8/99 4:28 PM Page 559 Figure 12.14 Schematics illustrating the contributions to band broadening due to (a) multiple paths, (b) longitudinal diffusion, and (c) mass transfer. 50%, it also may be possible to achieve the desired resolution with an even shorter analysis time by decreasing k B ￠ or a . To determine how the height of a theoretical plate can be decreased, it is neces- sary to understand the experimental factors contributing to the broadening of a solute’s chromatographic band. Several theoretical treatments of band broadening have been proposed. We will consider one approach in which the height of a theo- retical plate is determined by four contributions: multiple paths, longitudinal diffu- sion, mass transfer in the stationary phase, and mass transfer in the mobile phase. Multiple Paths Solute molecules passing through a chromatographic column travel separate paths that may differ in length. Because of these differences in path length, solute molecules injected simultaneously elute at different times. The princi- pal factor contributing to this variation in path length is a nonhomogeneous pack- ing of the stationary phase in the column. Differences in particle size and packing consistency cause solute molecules to travel paths of different length. Some solute molecules follow relatively straight paths through the column, but others follow a longer, more tortuous path (Figure 12.14a). The contribution of multiple paths to the height of a theoretical plate, H p , is H p =2l d p 12.23 where d p is the average diameter of the particulate packing material, and l is a con- stant accounting for the consistency of the packing. A smaller range of particle sizes and a more consistent packing produce a smaller value for l . Note that for an open tubular column, which does not contain packing material, H p is 0. Longitudinal Diffusion The second contribution to band broadening is the result of the solute’s longitudinal diffusion in the mobile phase. Even if the mobile phase velocity is 0, solute molecules are constantly in motion, diffusing through the mo- bile phase. Since the concentration of solute is greatest at the center of a chromato- graphic band, more solute diffuses toward the band’s forward and rear edges than diffuses toward the band’s center. The net result is an increase in the band’s width (Figure 12.14b). The contribution of longitudinal diffusion to the height of a theo- retical plate, H d , is 560 Modern Analytical Chemistry Table 12.1 Number of Theoretical Plates Needed to Achieve Desired Resolution for Selected Values of k B ￠ and a R = 1.00 R = 1.25 R = 1.50 k B ￠ ‘= 1.05 ‘= 1.10 ‘= 1.05 ‘= 1.10 ‘= 1.05 ‘= 1.10 0.5 63,500 17,400 99,200 27,200 143,000 39,200 1.0 28,200 7,740 44,100 12,100 63,500 17,400 1.5 19,600 5,380 30,600 8,400 44,100 12,100 2.0 15,900 4,360 24,800 6,810 35,700 9,800 3.0 12,500 3,440 19,600 5,380 28,200 7,740 5.0 10,200 2,790 15,900 4,360 22,900 6,270 10.0 8,540 2,340 13,300 3,660 19,200 5,270 A B Mobile phase Stationary phase Interface S S (a) (b) (c) longitudinal diffusion One contribution to band broadening in which solutes diffuse from areas of high concentration to areas of low concentration. 1400-CH12 9/8/99 4:28 PM Page 560 12.24 where D m is the solute’s diffusion coefficient in the mobile phase, u is the mobile phase velocity, and g is a constant related to the column packing. The effect of H d on the height of a theoretical plate is minimized by a high mobile-phase velocity. Because a solute’s diffusion coefficient is larger in a gaseous mobile phase than in a liquid mobile phase, longitudinal diffusion is a more serious problem in gas chromatography. Mass Transfer The final two contributions to band broadening result from the fi- nite time required for a solute molecule to diffuse through the stationary phase and mobile phase. A chromatographic separation occurs because solutes move between the stationary and mobile phases. For a solute to move from one phase to the other, it must first diffuse to the interface between the two phases (Figure 12.14c)—a process called mass transfer. A contribution to band broadening occurs whenever the solute’s movement to the interface is not fast enough to maintain a true equilib- rium distribution of solute between the two phases. Thus, solute molecules in the mobile phase move farther down the column than expected before passing into the stationary phase. Solute molecules in the stationary phase, on the other hand, take longer than expected to cross into the mobile phase. The contributions of mass transfer in the stationary phase, H s , and mass transfer in the mobile phase, H m , are given by 12.25 12.26 where d f is the thickness of the stationary phase, d c is the column’s diameter, D s is the solute’s diffusion coefficient in the stationary phase, q is a constant related to the column packing material, and the remaining terms are as previously defined. As in- dicated in equation 12.26, the exact form of H m is unknown, although it is a func- tion of particle size and column diameter. The contribution of mass transfer to the height of a theoretical plate is smallest for slow mobile-phase velocities, smaller di- ameter packing materials, and thinner films of stationary phase. Putting It All Together The net height of a theoretical plate is a summation of the contributions from each of the terms in equations 12.23–12.26; thus, H = H p+ H d + H s + H m 12.27 An alternative form of this equation is the van Deemter equation 12.28 which emphasizes the importance of the mobile phase’s flow rate. In the van Deemter equation, A accounts for multiple paths (H p ), B/u for longitudinal diffu- sion (H d ), and Cu for the solute’s mass transfer in the stationary and mobile phases (H s and H m ). HA B u Cu=++ H fn d d D u m p 2 c 2 m = (, ) H qk d kD u s f 2 s = ′ + ′()1 2 H D u d m = 2γ Chapter 12 Chromatographic and Electrophoretic Methods 561 van Deemter equation An equation showing the effect of the mobile phase’s flow rate on the height of a theoretical plate. mass transfer One contribution to band broadening due to the time required for a solute to move from the mobile phase or the stationary phase to the interface between the two phases. 1400-CH12 9/8/99 4:28 PM Page 561 Figure 12.15 Plot of the height of a theoretical plate as a function of mobile-phase velocity using the van Deemter equation. The contributions to the terms A, B/u, and Cu also are shown. There is some disagreement on the correct equation for describing the relation- ship between plate height and mobile-phase velocity. 4 In addition to the van Deemter equation (equation 12.28), another equation is that proposed by Hawkes where C s and C m are the mass transfer terms for the stationary and mobile phases respectively. A third equation was devised by Knox. All three equations, and others, have been used to characterize chromatographic systems, with no single equation providing the best explanation in every case. 5 To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. The remaining parameters affecting the height of a theoretical plate are deter- mined by the construction of the column and suggest how the column’s design may be used to improve efficiency. For example, both H p and H m are a function of the size of the particles used for the packing material. Decreasing particle size, therefore, is one approach to improving efficiency. A decrease in particle size is limited, however, by the need for a greater pressure to push the mobile phase through the column. One of the most important advances in column construction has been the de- velopment of open tubular, or capillary columns that contain no packing material (d p = 0). Instead, the interior wall of a capillary column is coated with a thin film of the stationary phase. The absence of packing material means that the mobile phase H B u Cu=++Au 1/3 H B u CCu=+ +() sm 562 Modern Analytical Chemistry 10.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 Height of plate (mm) Flow rate (mL/min) 0 2040608010120 Optimum flow rate Total Cu A B/u capillary column A narrow bored column that usually does not contain a particulate packing material. 1400-CH12 9/8/99 4:28 PM Page 562 Figure 12.16 Schematic diagram for a typical gas chromatograph. Chapter 12 Chromatographic and Electrophoretic Methods 563 SeptumInjector block Detector block Oven Column Flow meter Mobile phase gas supply Flow controller Pressure regulator can move through the column with substantially less pressure. As a result, capillary columns can be manufactured with much greater lengths than is possible with a packed column. Furthermore, plate height is reduced because the H p term in equa- tion 12.27 disappears and the H m term becomes smaller. The combination of a smaller height for a theoretical plate and a longer column leads to an approximate 100-fold increase in the number of theoretical plates. Capillary columns are not without disadvantages. Because capillary columns are much narrower than packed columns, they require a significantly smaller amount of sample. Difficulties with re- producibly injecting small samples complicates the use of capillary chromatography for quantitative work. Another approach to improving resolution is to use thin films of stationary phase. Capillary columns used in gas chromatography and the bonded phases com- monly used in HPLC provide a significant decrease in plate height due to the reduc- tion of the H s term in equation 12.27. 12D Gas Chromatography In gas chromatography (GC) the sample, which may be a gas or liquid, is injected into a stream of an inert gaseous mobile phase (often called the carrier gas). The sample is carried through a packed or capillary column where the sample’s compo- nents separate based on their ability to distribute themselves between the mobile and stationary phases. A schematic diagram of a typical gas chromatograph is shown in Figure 12.16. 12D.1 Mobile Phase The most common mobile phases for GC are He, Ar, and N 2 , which have the ad- vantage of being chemically inert toward both the sample and the stationary phase. The choice of which carrier gas to use is often determined by the instrument’s de- tector. With packed columns the mobile-phase velocity is usually within the range of 25–150 mL/min, whereas flow rates for capillary columns are 1–25 mL/min. Ac- tual flow rates are determined with a flow meter placed at the column outlet. gas chromatography A chromatographic technique in which the mobile phase is a gas. 1400-CH12 9/8/99 4:28 PM Page 563 12D.2 Chromatographic Columns A chromatographic column provides a location for physically retaining the station- ary phase. The column’s construction also influences the amount of sample that can be handled, the efficiency of the separation, the number of analytes that can be eas- ily separated, and the amount of time required for the separation. Both packed and capillary columns are used in gas chromatography. Packed Columns A packed column is constructed from glass, stainless steel, copper or aluminum and is typically 2–6 m in length, with an internal diameter of 2–4 mm. The column is filled with a particulate solid support, with particle diam- eters ranging from 37–44 m m to 250–354 m m. The most widely used particulate support is diatomaceous earth, which is com- posed of the silica skeletons of diatoms. These particles are quite porous, with sur- face areas of 0.5–7.5 m 2 /g, which provides ample contact between the mobile phase and stationary phase. When hydrolyzed, the surface of a diatomaceous earth con- tains silanol groups (–SiOH), providing active sites that absorb solute molecules in gas–solid chromatography. In gas–liquid chromatography (GLC), separation is based on the partitioning of solutes between a gaseous mobile phase and a liquid stationary phase coated on the solid packing material. To avoid the adsorption of solute molecules on exposed packing material, which degrades the quality of the separation, surface silanols are deactivated by silanizing with dimethyldichlorosilane and washing with an alcohol (typically methanol) before coating with stationary phase. More recently, solid supports made from glass beads or fluorocarbon polymers have been introduced. These supports have the advantage of being more inert than di- atomaceous earth. To minimize the multiple path and mass transfer contributions to plate height (equations 12.23 and 12.26), the packing material should be of as small a diameter as is practical and loaded with a thin film of stationary phase (equation 12.25). Compared with capillary columns, which are discussed in the next section, packed columns can handle larger amounts of sample. Samples of 0.1–10 m L are routinely analyzed with a packed column. Column efficiencies are typically several hundred to 2000 plates/m, providing columns with 3000–10,000 theoretical plates. Assuming V max /V min is approximately 50, 3 a packed column with 10,000 theoretical plates has a peak capacity (equation 12.18) of Capillary Columns Capillary, or open tubular columns are constructed from fused silica coated with a protective polymer. Columns may be up to 100 m in length with an internal diameter of approximately 150–300 m m (Figure 12.17). Larger bore columns of 530 m m, called megabore columns, also are available. n c =+ ≈1 10 000 4 50 100 , ln( ) 564 Modern Analytical Chemistry Si(CH 3 ) 2 Cl 2 Si(CH 3 ) 2 Cl + HCl ROH Si OH O O O O Si O Si(CH 3 ) 2 OR + HCl O O Si O open tubular column A capillary column that does not contain a particulate packing material. gas–liquid chromatography A chromatographic technique in which the mobile phase is a gas and the stationary phase is a liquid coated either on a solid packing material or on the column’s walls. packed column A wide-bore column containing a particulate packing material. 1400-CH12 9/8/99 4:28 PM Page 564 Figure 12.17 Photo of a capillary column. Courtesy of Alltech Associates, Inc., Deerfield, Illinois. Capillary columns are of two principal types. Wall-coated open tubular columns (WCOT) contain a thin layer of stationary phase, typically 0.25 m m thick, coated on the capillary’s inner wall. In support-coated open tubular columns (SCOT), a thin layer of a solid support, such as a diatomaceous earth, coated with a liquid stationary phase is attached to the capillary’s inner wall. Capillary columns provide a significant improvement in separation efficiency. The pressure needed to move the mobile phase through a packed column limits its length. The absence of packing material allows a capillary column to be longer than a packed column. Although most capillary columns contain more theoretical plates per meter than a packed column, the more important contribution to their greater efficiency is the ability to fashion longer columns. For example, a 50-m capillary column with 3000 plates/m has 150,000 theoretical plates and, assuming V max /V min is approximately 50, 3 a peak capacity of almost 380. On the other hand, packed columns can handle larger samples. Due to its smaller diameter, capillary columns require smaller samples; typically less than 10 –2 m L. 12D. 3 Stationary Phases Selectivity in gas chromatography is influenced by the choice of stationary phase. Elution order in GLC is determined primarily by the solute’s boiling point and, to a lesser degree, by the solute’s interaction with the stationary phase. Solutes with significantly different boiling points are easily separated. On the other hand, two solutes with similar boiling points can be separated only if the stationary phase se- lectively interacts with one of the solutes. In general, nonpolar solutes are more easily separated with a nonpolar stationary phase, and polar solutes are easier to separate using a polar stationary phase. The main criteria for selecting a stationary phase are that it should be chemi- cally inert, thermally stable, of low volatility, and of an appropriate polarity for the solutes being separated. Although hundreds of stationary phases have been devel- oped, many of which are commercially available, the majority of GLC separations are accomplished with perhaps five to ten common stationary phases. Several of Chapter 12 Chromatographic and Electrophoretic Methods 565 wall-coated open tubular column An open tubular column in which the stationary phase is coated on the column’s walls. support-coated open tubular column An open tubular column in which the stationary phase is coated on a solid support that is attached to the column’s walls. 1400-CH12 9/8/99 4:28 PM Page 565 these are listed in Table 12.2, in order of increasing polarity, along with their physi- cal properties and typical applications. Many stationary phases have the general structure shown in Figure 12.18a. A stationary phase of polydimethyl siloxane, in which all the –R groups are methyl groups (–CH 3 ), is nonpolar and often makes a good first choice for a new separa- tion. The order of elution when using polydimethyl siloxane usually follows the boiling points of the solutes, with lower boiling solutes eluting first. Replacing some of the methyl groups with other substituents increases the stationary phase’s polar- ity, providing greater selectivity. Thus, in 50% methyl-50% phenyl polysiloxane, 50% of the –R groups are phenyl groups (–C 6 H 5 ), producing a slightly polar sta- tionary phase. Increasing polarity is provided by substituting trifluoropropyl (–C 3 H 6 CF 3 ) and cyanopropyl (–C 3 H 6 CN) functional groups or using a stationary phase based on polyethylene glycol (Figure 12.18b). An important problem with all liquid stationary phases is their tendency to “bleed” from the column. The temperature limits listed in Table 12.2 are those that minimize the loss of stationary phase. When operated above these limits, a col- umn’s useful lifetime is significantly shortened. Capillary columns with bonded or 566 Modern Analytical Chemistry Table 12.2 Selected Stationary Phases for Gas–Liquid Chromatography Temperature Limit Stationary Phase Polarity Trade Names (°C) Applications squalane nonpolar squalane 150 low-boiling aliphatic hydrocarbons Apezion L nonpolar Apezion L 300 amides fatty acid methyl esters high-boiling aliphatic hydrocarbons terpenoids polydimethyl siloxane slightly polar SE-30 300–350 alkaloids amino acid derivatives drugs pesticides phenols steroids 50% methyl-50% phenyl moderately polar OV-17 375 alkaloids polysiloxane drugs pesticides polyaromatic hydrocarbons polychlorinated biphenyls 50% trifluoropropyl-50% moderately polar OV-210 275 alkaloids methyl polysiloxane amino acid derivatives drugs halogenated compounds ketones phenols 50% cyanopropyl-50% polar OV-225 275 nitriles phenylmethyl polysiloxane pesticides steroids polyethylene glycol polar Carbowax 20M 225 aldehydes esters ethers phenols bleed The tendency of a stationary phase to elute from the column. 1400-CH12 9/8/99 4:28 PM Page 566 cross-linked stationary phases provide superior stability. Bonded stationary phases are attached to the capillary’s silica surface. Cross- linking, which is done after the stationary phase is placed in the capillary column, links together separate polymer chains, thereby providing greater stability. Another important characteristic of a gas chromatographic column is the thickness of the stationary phase. As shown in equa- tion 12.25, separation efficiency improves with thinner films. The most common film thickness is 0.25 m m. Thicker films are used for highly volatile solutes, such as gases, because they have a greater capacity for retaining such solutes. Thinner films are used when separating solutes of low volatility, such as steroids. A few GLC stationary phases rely on chemical selectivity. The most notable are stationary phases containing chiral functional groups, which can be used for sepa- rating enantiomers. 6 12D. 4 Sample Introduction Three considerations determine how samples are introduced to the gas chromato- graph. First, all constituents injected into the GC must be volatile. Second, the ana- lytes must be present at an appropriate concentration. Finally, injecting the sample must not degrade the separation. Preparing a Volatile Sample Gas chromatography can be used to separate analytes in complex matrices. Not every sample that can potentially be analyzed by GC, however, can be injected directly into the instrument. To move through the col- umn, the sample’s constituents must be volatile. Solutes of low volatility may be re- tained by the column and continue to elute during the analysis of subsequent sam- ples. Nonvolatile solutes condense on the column, degrading the column’s performance. Volatile analytes can be separated from a nonvolatile matrix using any of the extraction techniques described in Chapter 7. Liquid–liquid extractions, in which analytes are extracted from an aqueous matrix into methylene chloride or other or- ganic solvent, are commonly used. Solid-phase extractions also are used to remove unwanted matrix constituents. An attractive approach to isolating analytes is a solid-phase microextraction (SPME). In one approach, which is illustrated in Figure 12.19, a fused silica fiber is placed inside a syringe needle. The fiber, which is coated with a thin organic film, such as polydimethyl siloxane, is lowered into the sample by depressing a plunger and is exposed to the sample for a predetermined time. The fiber is then withdrawn into the needle and transferred to the gas chromatograph for analysis. Volatile analytes also can be separated from a liquid matrix using a purge and trap or by headspace sampling. In a purge and trap (see Figure 7.19 in Chapter 7), an inert gas, such as He or N 2 , is bubbled through the sample, purging the volatile compounds. These compounds are swept through a trap packed with an absorbent material, such as Tenax, where they are collected. Heating the trap and back flush- ing with carrier gas transfers the volatile compounds to the gas chromatograph. In headspace sampling the sample is placed in a closed vial with an overlying air space. After allowing time for the volatile analytes to equilibrate between the sample and the overlying air, a portion of the vapor phase is sampled by syringe and in- jected into the gas chromatograph. Chapter 12 Chromatographic and Electrophoretic Methods 567 RR R n OSi R R R R R Si O Si HO CH 2 OHCH 2 CH 2 CH 2 ) n (O (a) (b) Figure 12.18 General structures of common stationary phases for gas chromatography. solid-phase microextraction A solid-phase extraction in which the solid adsorbent is coated on a fused- silica fiber held within a syringe needle. headspace sampling The sampling of the vapor phase overlying a liquid phase. Syringe barrel Syringe needle Retractable rod Fused-silica fiber coated with stationary phase Figure 12.19 Schematic diagram of a device for solid- phase microextractions. 1400-CH12 9/8/99 4:28 PM Page 567 Thermal desorption is used to release volatile analytes from solids. A portion of the solid is placed in a glass-lined, stainless steel tube and held in place with plugs of glass wool. After purging with carrier gas to remove O 2 (which could lead to oxida- tion reactions when heating the sample), the sample is heated. Volatile analytes are swept from the tube by the carrier gas and carried to the GC. To maintain efficiency the solutes often are concentrated at the top of the column by cooling the column inlet below room temperature, a process known as cryogenic focusing. Nonvolatile analytes must be chemically converted to a volatile derivative before analysis. For example, amino acids are not sufficiently volatile to analyze directly by gas chromatography. Reacting an amino acid with 1-butanol and acetyl chloride produces an esterfied amino acid. Subsequent treatment with trifluoroacetic acid gives the amino acid’s volatile N-trifluoroacetyl-n-butyl ester derivative. Adjusting the Analyte s Concentration Analytes present at concentrations too small to give an adequate signal need to be concentrated before analyzing. A side benefit of many of the extraction methods outlined earlier is that they often concen- trate the analytes. Volatile organic materials isolated from aqueous samples by a purge and trap, for example, can be concentrated by as much as 1000-fold. When an analyte is too concentrated, it is easy to overload the column, thereby seriously degrading the separation. In addition, the analyte may be present at a con- centration level that exceeds the detector’s linear response. Dissolving the sample in a volatile solvent, such as methylene chloride, makes its analysis feasible. Injecting the Sample To avoid any precolumn loss in resolution due to band broadening, a sample of sufficient size must be introduced in a small volume of mo- bile phase. An example of a simple injection port for a packed column is shown in Figure 12.20. Injections are made through a rubber septum using a microliter sy- ringe. The injector block is heated to a temperature that is at least 50 °C above the sample component with the highest boiling point. In this way rapid vaporization of the entire sample is ensured. Capillary columns require the use of a special injector to avoid overloading the column with sample. Several capillary injectors are available, the most common of which is a split/splitless injector. 7 When used for a split injection only about 0.1–1% of the sample enters the column, with the remainder carried off as waste. In a splitless injection, which is useful for trace analysis, the column temperature is held 20–25 °C below the solvent’s boiling point. As the solvent enters the column, it condenses, forming a barrier that traps the solutes. After allowing time for the solutes to concentrate, the column’s temperature is increased, and the separation begins. A splitless injection allows a much higher percentage of the solutes to enter the chromatographic column. For samples that decompose easily, an on-column injection may be necessary. In this method the sample is injected on the column without heating. The column temperature is then increased, volatilizing the sample with as low a temperature as is practical. 12D. 5 Temperature Control As noted earlier, control of the column’s temperature is critical to attaining a good separation in gas chromatography. For this reason the column is located inside a thermostated oven. In an isothermal separation the column is maintained at a con- stant temperature, the choice of which is dictated by the solutes. Normally, the tem- 568 Modern Analytical Chemistry cryogenic focusing The process of concentrating volatile solutes by cooling the column’s inlet below room temperature. split injection A technique for injecting samples onto a capillary column in which only a small portion of the sample enters the column. splitless injection A technique for injecting a sample onto a capillary column that allows a higher percentage of the sample to enter the column. on-column injection The direct injection of thermally unstable samples onto a capillary column. 1400-CH12 9/8/99 4:28 PM Page 568 Figure 12.20 Schematic diagram of an injector for packed column gas chromatography. perature is set slightly below that for the lowest boiling solute so as to increase the solute’s interaction with the stationary phase. One difficulty with an isothermal separation is that a temperature favoring the separation of low-boiling solutes may cause unacceptably long retention times for higher boiling solutes. Ovens capable of temperature programming provide a solu- tion to this problem. The initial temperature is set below that for the lowest boiling solute. As the separation progresses, the temperature is slowly increased at either a uniform rate or in a series of steps. 12D.6 Detectors for Gas Chromatography The final part of a gas chromatograph is the detector. The ideal detector has several desirable features, including low detection limits, a linear response over a wide range of solute concentrations (which makes quantitative work easier), responsive- ness to all solutes or selectivity for a specific class of solutes, and an insensitivity to changes in flow rate or temperature. Thermal Conductivity Detector One of the earliest gas chromatography detectors, which is still widely used, is based on the mobile phase’s thermal conductivity (Fig- ure 12.21). As the mobile phase exits the column, it passes over a tungsten–rhenium wire filament. The filament’s electrical resistance depends on its temperature, which, in turn, depends on the thermal conductivity of the mobile phase. Because of its high thermal conductivity, helium is the mobile phase of choice when using a thermal conductivity detector (TCD). When a solute elutes from the column, the thermal conductivity of the mobile phase decreases and the temperature of the wire filament, and thus its resistance, in- creases. A reference cell, through which only the mobile phase passes, corrects for any time-dependent variations in flow rate, pressure, or electrical power, all of which may lead to a change in the filament’s resistance. A TCD detector has the advantage of universality, since it gives a signal for any solute whose thermal conductivity differs from that of helium. Another ad- vantage is that it gives a linear response for solute concentrations over a range of 10 4 –10 5 orders of magnitude. The detector also is nondestructive, making it pos- sible to isolate solutes with a postdetector cold trap. Unfortunately, the thermal Chapter 12 Chromatographic and Electrophoretic Methods 569 Needle Syringe barrel Septum Septum cap Column Carrier gas thermal conductivity detector A universal GC detector in which the signal is a change in the thermal conductivity of the mobile phase. Heated detector block Carrier gas in Wire filament Carrier gas out Figure 12.21 Schematic diagram of a thermal conductivity detector for gas chromatography. 1400-CH12 9/8/99 4:28 PM Page 569 conductivity detector’s detection limit is poor in comparison with other popular detectors. Flame Ionization Detector Combustion of an organic compound in an H 2 /air flame results in a flame rich in electrons and ions. If a po- tential of approximately 300 V is applied across the flame, a small cur- rent of roughly 10 –9 –10 –12 A develops. When amplified, this current provides a useful analytical signal. This is the basis of the popular flame ionization detector (FID), a schematic of which is shown in Figure 12.22. Most carbon atoms, except those in carbonyl and carboxylic groups, generate a signal, making the FID an almost universal detector for organic compounds. Most inorganic compounds and many gases, such as H 2 O and CO 2 , cannot be detected, making the FID detector ideal for the analysis of atmospheric and aqueous environmental sam- ples. Advantages of the FID include a detection limit that is approxi- mately two to three orders of magnitude smaller than that for a thermal conductiv- ity detector and a linear response over 10 6 –10 7 orders of magnitude in the amount of analyte injected. The sample, of course, is destroyed when using a flame ioniza- tion detector. Electron Capture Detector The electron capture detector is an example of a selec- tive detector. The detector consists of a beta emitter (a beta particle is an electron) such as 63 Ni. The emitted electrons ionize the mobile phase, which is usually N 2 , re- sulting in the production of additional electrons that give rise to an electric current between a pair of electrodes (Figure 12.23). When a solute with a high cross section for the capture of electrons elutes from the column, the electric current decreases. This decrease in electric current serves as the signal. The ECD is highly selective to- ward solutes with electronegative functional groups, such as halogens, and nitro groups and is relatively insensitive to amines, alcohols, and hydrocarbons. Although its detection limit is excellent, its linear range extends over only about two orders of magnitude. Other Detectors Two additional detectors are similar in design to a flame ioniza- tion detector. In the flame photometric detector optical emission from phospho- rus and sulfur provides a detector selective for compounds containing these ele- ments. The thermionic detector responds to compounds containing nitrogen or phosphorus. Two common detectors, which also are independent instruments, are Fourier transform infrared spectrophotometers (FT–IR) and mass spectrometers (MS). In GC–FT–IR, effluent from the column flows through an optical cell constructed 570 Modern Analytical Chemistry flame ionization detector A nearly universal GC detector in which the solutes are combusted in an H 2 /air flame, producing a measurable current. Collector Air H 2 Column Carrier gas Power supply – + Figure 12.22 Schematic diagram of a flame ionization detector for gas chromatography. electron capture detector A detector for GC that provides selectivity for solutes with halogen and nitro functional groups. + Electrode Carrier gas in – Electrode Carrier gas out b –Emitter e – Figure 12.23 Schematic diagram of an electron capture detector for gas chromatography. 1400-CH12 9/8/99 4:28 PM Page 570 Figure 12.24 (a) Total ion chromatogram for a ten-component mixture; (b) Chromatogram recorded using selective ion monitoring for mass-to-charge ratios of 93 and 95, which are characteristic ions for the monoterpenes a -pinene (t r = 5.08 min), b -pinene (t r = 5.81 min), camphor (t r = 8.51 min), and menthol (t r = 8.93 min). (Chromatograms courtesy of Bryan Hanson and Sara Peters, DePauw University). from a 10–40-cm Pyrex tube with an internal diameter of 1–3 mm. The cell’s inte- rior surface is coated with a reflecting layer of gold. Multiple reflections of the source radiation as it is transmitted through the cell increase the optical path length through the sample. In GC–MS effluent from the column is introduced directly into the mass spec- trometer’s ionization chamber in a manner that eliminates the majority of the car- rier gas. In the ionization chamber all molecules (remaining carrier gas, solvent, and solutes) are ionized, and the ions are separated by their mass-to-charge ratio. Be- cause each solute undergoes a characteristic fragmentation into smaller ions, its mass spectrum of ion intensity as a function of mass-to-charge ratio provides qual- itative information that can be used to identify the solute. As a GC detector, the total ion current for all ions reaching the detector is usu- ally used to obtain the chromatogram (Figure 12.24a). Selectivity can be achieved by monitoring only specific mass-to-charge ratios (Figure 12.24b), a process called selective ion monitoring. A mass spectrometer provides excellent detection limits, typically 25 fg to 100 pg, with a linear range spanning five orders of magnitude. 12D. 7 Quantitative Applications Gas chromatography is widely used for the analysis of a diverse array of samples in environmental, clinical, pharmaceutical, biochemical, forensic, food science, and petrochemical laboratories. Examples of these applications are discussed in the fol- lowing sections. Environmental Analysis One of the most important environmental applications of gas chromatography is for the analysis of numerous organic pollutants in air, water, and wastewater. The analysis of volatile organics in drinking water, for example, is accomplished by a purge and trap, followed by their separation on a capillary col- umn with a nonpolar stationary phase. A flame ionization, electron capture, or Chapter 12 Chromatographic and Electrophoretic Methods 571 100% 17% TOT (a) (b) 95 + 93 250 4:10 300 5:00 350 5:50 400 6:40 450 7:30 500 8:20 550 9:10 mass spectrum A plot of ion intensity as a function of the ion’s mass-to-charge ratios. 1400-CH12 9/8/99 4:28 PM Page 571 Figure 12.25 Examples of the application of gas chromatography to the analysis of (a) chlorinated pesticides in water, (b) blood alcohols, (c) Scotch whiskey, and (d) unleaded gasoline. (Chromatograms courtesy of Alltech Associates, Inc. Deerfield, IL). mass spectrometer can be used as a detector. Figure 12.25a shows a typical chro- matogram for the analysis of chlorinated pesticides in water. Clinical Analysis Clinical, pharmaceutical, and forensic labs make frequent use of gas chromatography for the analysis of drugs. Because the sample’s matrix is often incompatible with the GC column, analytes generally must be isolated by extrac- tion. Figure 12.25b shows how gas chromatography can be used in monitoring blood alcohol levels. Consumer Goods Many flavors, spices, and fragrances are readily analyzed by GC, using headspace analysis or thermal desorption. Foods and beverages are analyzed either directly or following a suitable extraction. Volatile materials, such as those found in spices and fragrances, often can be obtained by headspace sampling. Fig- ure 12.25c shows a typical analysis of a sample of Scotch whiskey. Petroleum Industry Gas chromatography is ideally suited for the analysis of petro- leum products, including gasoline, diesel fuel, and oil. A typical chromatogram for the analysis of unleaded gasoline is shown in Figure 12.25d. Quantitative Calculations In a quantitative analysis, the height or area of an ana- lyte’s chromatographic peak is used to determine its concentration. Although peak height is easy to measure, its utility is limited by the inverse relationship between the height and width of a chromatographic peak. Unless chromatographic condi- tions are carefully controlled to maintain a constant column efficiency, variations in 572 Modern Analytical Chemistry 048 1. Propachlor 2. Trifluralin 3.a-HCH 4. Hexachlorobenzene 5.b-HCH 6.g-HCH 7.d-HCH 8. Heptachlor 9. Aldrin 10. DCPA 11. Heptachlor Epoxide 12.g-Chlordane Column: Econo-Cap? EC-5, 30m · 0.32mm ID · 0.25μm (Part No. 19646) Temp: 60°C to 300°C at 4°C/min Carrier Gas: Helium, 1.45mL/min Detector: ECD at 320°C Injector: 250°C 13.a-Chlordane 14. Dieldrin 15. p,p'-DDE 16. Endrin 17. p,p'-DDD 18. Endrin Aldehyde 19. Endosulfan Sulfate 20. p,p'-DDT 21. Methoxychlor 22. cis-Permethrin 23. trans-Permethrin Chlorinated pesticides in water 12 16 20 24 28 32 36 40 44 52 Min.48 1 2 3 4 5 6 7 8 9 10 11 13 12 14 15 16 17 18 19 20 21 22 23 (a) 1400-CH12 9/8/99 4:28 PM Page 572 Figure 12.25 Continued peak height may decrease the accuracy and precision of the quantitative analysis. A better choice is to measure the area under the chromatographic peak with an inte- grating recorder. Since peak area is directly proportional to the amount of analyte that was injected, changes in column efficiency will not affect the accuracy or preci- sion of the analysis. Calibration curves are usually constructed by analyzing a series of external standards and plotting the detector’s signal as a function of their known concentra- tions. As long as the injection volume is identical for every standard and sample, calibration curves prepared in this fashion give both accurate and precise results. Unfortunately, even under the best of conditions, replicate injections may have vol- umes that differ by as much as 5% and often may be substantially worse. For this Chapter 12 Chromatographic and Electrophoretic Methods 573 1 2 3 4 5 6 0 1 2 3 Min. 1. Methanol 2. Ethanol 3. Acetone 4. 2-Propanol 5. 1-Propanol 6. Dioxane (I.S.) CHROM 1252 Column: Heliflex AT-1, 10m · 0.53mm ID · 5μm (Part No. 16842) Temp: 35°C (1min) to 130°C at 30°C/min Carrier Gas: Helium, 6mL/min Detector: FID Blood alcohols 0 1 2 3 4 5 6 7 8 2 4 6 8 10 12 Min. 1. Acetaldehyde 2. Ethyl Acetate 3. Methanol 4. Ethanol 5. n-Propanol 6. Isobutanol 7. Amyl Alcohol/Isoamyl Alcohol 8. Acetic Acid Column: Econo-Cap EC-WAX, 30m · 0.45mm · 1.0μm (Part No. 19663) Temp: 50°C (2min) to 200°C at 15°C/min Carrier Gas: Helium, 34cm/sec Split Ratio: 20:1 Detector: FID/250°C Injector: 250°C Scotch whiskey CHROM 2124 1 13 234 5 678 9 10 11 12 14 15 16 17 18 19 20 21 22 23 24 25 0 90 Min.30 6010 20 40 50 70 805 1525354555657585 1. Isobutane 2. n-Butane 3. Isopentane 4. Pentane 5. 2,3-Dimethylbutane 6. 2-Methlypentane 7. 3-Methylpentane 8. Hexane 9. 2,4-Dimethylpentane 10. Benzene 11. 2-Methylhexane 12. 3-Methylhexane 13. n-Heptane 14. Toluene 15. Ethylbenzene 16. m-Xylene 17. p-Xylene 18. o-Xylene 19. 1-Methyl-3-Ethylbenzene 20. 1,3,5-Trimethylbenzene 21. 1,2,4-Trimethylbenzene 22. 1,2,3-Trimethylbenzene 23. Naphthalene 24. 2-Methylnaphthalene 25. 1-Methylnaphthalene Column: AT-Petro, 100m · 0.25mm ID · 0.50μm (Part No. 13948) Temp: 35°C (15min) to 200°C at 2°C/min Carrier Gas: Helium, 0.65mL/min Detector: FID at 275°C Unleaded gasoline analysis CHROM 1859 (b) (c) (d) 1400-CH12 9/8/99 4:28 PM Page 573 574 Modern Analytical Chemistry reason, quantitative work requiring high accuracy and precision is accomplished using an internal standard. EXAMPLE 12. 5 Marriott and Carpenter 8 report the following data for five replicate injections of a mixture of 1% v/v methylisobutylketone (peak 1) and 1% v/v p-xylene (peak 2). Injection Peak Peak Area A 1 49075 2 78112 B 1 85829 2 135404 C 1 84136 2 132332 D 1 71681 2 112889 E 1 58054 2 91287 Assume that p-xylene is the analyte and that methylisobutylketone is the internal standard. Determine the 95% confidence interval for a single-point standardization, with and without using the internal standard. SOLUTION For a single-point external standard (omitting the internal standard) the relationship between peak area, A 2 , and the concentration, C 2 , of p-xylene is A 2 = kC 2 Substituting the known concentration for the p-xylene standard and the appropriate peak areas, gives the following values for the constant k. 78112 135404 132332 112889 91287 The average value for k is 110,000, with a standard deviation of 25,100 (a relative standard deviation of 22.8%). The 95% confidence interval is 110,000 – 31,200. For an internal standardization, the relationship between the peak areas for the analyte, A 2 , and the internal standard, A 1 , and their respective concentrations, C 1 and C 2 , is Substituting the known concentrations and the appropriate peak areas gives the following values for the constant k. 1.5917 1.5776 1.5728 1.5749 1.5724 The average value for k is 1.5779, with a standard deviation of 0.0080 (a relative standard deviation of 0.507%). The 95% confidence interval is 1.5779 – 0.0099. As this example clearly shows, the variation in individual peak areas between injections is substantial. The use of an internal standard, however, corrects for these variations, providing a means for accurate and precise calibration. A A k C C 2 1 2 1 = 1400-CH12 9/8/99 4:28 PM Page 574 Chapter 12 Chromatographic and Electrophoretic Methods 575 Kovat’s retention index A means for normalizing retention times by comparing a solute’s retention time with those for normal alkanes. 12D.8 Qualitative Applications Gas chromatography also can be used for qualitative purposes. When using an FT–IR or a mass spectrometer as the detector, the available spectral information often can be used to identify individual solutes. With conventional nonspectroscopic detectors, other methods must be used to identify the solutes. One approach is to spike the sample by adding an aliquot of a suspected analyte and looking for an increase in peak height. Retention times also can be compared with values measured for standards, provided that the operating conditions are identical. Because of the difficulty of exactly matching such condi- tions, tables of retention times are of limited utility. Kovat’s retention index provides one solution to the matching of retention times. Under isothermal conditions, the adjusted retention times of normal alkanes increase logarithmically. Kovat defined the retention index, I, for a normal alkane as 100 times the number of carbon atoms; thus, the retention index is 400 for butane and 500 for pentane. To determine the retention index for another compound, its adjusted retention time is measured relative to that for the normal alkanes eluting just before and after. For example, a compound eluting between butane and pen- tane has a retention index between 400 and 500. The exact value for the com- pound’s retention index, I cpd , is given as 12.29 where x is the normal alkane eluting before the compound, and x + 1 is the normal alkane eluting just after the compound. EXAMPLE 12.6 In a separation of a mixture of hydrocarbons, the following adjusted retention times were measured. propane 2.23 min isobutane 5.71 min butane 6.67 min What is the Kovat’s retention index for each of these hydrocarbons? SOLUTION Kovat’s retention index for a normal alkane is 100 times the number of carbons; thus I propane = 100 · 3 = 300 I butane = 100 · 4 = 400 To find Kovat’s retention index for isobutane, we use equation 12.29. I tt I isobutane r isobutane r propane r butane r propane propane = ′ ? ′ ′ ? ′ ? ? ? ? ? ? + = ? ? ? ? ? ? ? ? += 100 100 571 223 667 223 300 386 (log ) (log ) (log ) (log ) log( . ) log( . ) log( . ) log( . ) I tt I x xx cpd r cpd r rr x = ′ ? ′ ′ ? ′ ? ? ? ? ? ? + + 100 1 (log ) (log ) (log ) (log ) 1400-CH12 9/8/99 4:28 PM Page 575 576 Modern Analytical Chemistry Representative Methods —Continued 12D. 9 Representative Method Although each gas chromatographic method has its own unique considerations, the following description of the determination of trihalomethanes in drinking water provides an instructive example of a typical procedure. Method 12.1 Determination of Trihalomethanes in Drinking Water 9 Description of Method. Trihalomethanes, such as chloroform (CHCl 3 ) and bromoform (CHBr 3 ), are found in most chlorinated waters. Since chloroform is a suspected carcinogen, the determination of trihalomethanes in public drinking water supplies is of considerable importance. In this method the trihalomethanes CHCl 3 , CHBrCl 2 , CHBr 2 Cl, and CHBr 3 are isolated by a liquid–liquid extraction with pentane and determined by gas chromatography using an electron capture detector. Because of its volatility and ubiquitous presence in most labs, chloroform from other sources is a significant interferent. Procedure. Samples are collected in 40-mL vials with screw-caps lined with a Teflon septum. Fill the vial to overflowing, ensuring that there are no air bubbles. Add a reducing agent of ascorbic acid (25 mg/40 mL) to quench the further production of trihalomethanes, and seal the vial. Store samples at 4 °C, and analyze within 14 days. Prepare a standard stock solution for each trihalomethane by placing 9.8 mL of methanol in a 10-mL volumetric flask. Let the volumetric flask stand for 10 min, or until all surfaces wetted with methanol are dry. Weigh the volumetric flask to the nearest – 0.1 mg. Using a 100-m L syringe, add 2 or 3 drops of the trihalomethane to the volumetric flask, allowing it to drop directly into the methanol. Reweigh the flask before diluting to volume and mixing. Transfer to a 15-mL screw-cap vial with Teflon liner, and report the concentration in micrograms per milliliter. Standard stock solutions are stable for 4 weeks when stored at 4 °C. Prepare a single multicomponent working standard from the stock standards by making appropriate dilutions with methanol. Concentrations in the working standards should be at such a level that a 20-m L sample added to 100 mL of water gives a calibration standard whose response for each trihalomethane is within – 25% of that for the samples to be analyzed. Samples and calibration standards are prepared for analysis using a 10-mL syringe. Add 10.00 mL of each sample and standard to separate 14-mL screw-cap vials containing 2.00 mL of pentane. Shake vigorously for 1 min to effect the separation. Wait 60 s for the phases to separate. Inject 3.0-m L aliquots of the pentane layer into a GC equipped with a 2-mm internal diameter, 2-m long glass column packed with a stationary phase of 10% squalane on a packing material of 80/100 mesh Chromosorb WAW. Operate the column at 67 °C and a flow rate of 25 mL/min. Questions 1. A simple liquid–liquid extraction rarely extracts 100% of the analyte. How does this method account for incomplete extractions? Both the samples and standards are treated identically so their relative concentrations are unaffected by an incomplete extraction. 2. This method uses a short, packed column that generally produces a poor resolution of chromatographic peaks. The liquid–liquid extraction used to extract the trihalomethanes is nonselective. Besides the trihalomethanes, a wide range of nonpolar and polar organic constituents, such as benzene and 1400-CH12 9/8/99 4:28 PM Page 576 Chapter 12 Chromatographic and Electrophoretic Methods 577 phenol, also are extracted. Why does the presence of these other compounds not interfere with this analysis? An electron capture detector is relatively insensitive to nonhalogenated compounds, providing the additional selectivity. 3. Although chloroform is an analyte, it also can be interferent. Due to its volatility, chloroform present in the laboratory air may diffuse through the sample vial’s Teflon septum, contaminating the samples. How can we determine whether samples have been contaminated in this manner? A sample blank of trihalomethane-free water can be kept with the samples at all times. If the sample blank shows no evidence for chloroform, then we can safely assume that the samples also are free from contamination. 4. Why is it necessary to collect samples such that there is no headspace (layer of air overlying the liquid) in the sample vial? Due to the volatility of trihalomethanes, the presence of a headspace allows for the possible loss of analyte. 12D.10 Evaluation Scale of Operation Analytes present at levels from major to ultratrace compo- nents have been successfully determined by gas chromatography. Depending on the choice of detector, samples with major and minor analytes may need to be di- luted before analysis. The thermal conductivity and flame ionization detectors can handle larger amounts of analyte; other detectors, such as the electron capture de- tector or a mass spectrometer, require substantially smaller amounts of analyte. Although the volume of sample injected is quite small (often less than a micro- liter), the amount of available material from which the injection volume is taken must be sufficient to be a representative sample. For trace analytes, the actual amount of analyte injected is often in the picogram range. Using the tri- halomethane analysis described in Method 12.1 as an example, a 3.0-m L injection of a water sample containing 1 m g/L of CHCl 3 corresponds to 15 pg of CHCl 3 (as- suming a complete extraction of CHCl 3 ). Accuracy The accuracy of a gas chromatographic method varies substantially from sample to sample. For routine samples, accuracies of 1–5% are common. For analytes present at very low concentration levels, for samples with complex matrices, or for samples requiring significant processing before analysis, accu- racy may be substantially poorer. In the analysis for trihalomethanes described in Method 12.1, for example, determinate errors as large as – 25% are possible. 9 Precision The precision of a gas chromatographic analysis includes contribu- tions from sampling, sample preparation, and the instrument. The relative stan- dard deviation due to the gas chromatographic portion of the analysis is typically 1–5%, although it can be significantly higher. The principal limitations to preci- sion are detector noise and the reproducibility of injection volumes. In quantita- tive work, the use of an internal standard compensates for any variability in injec- tion volumes. 1400-CH12 9/8/99 4:28 PM Page 577 578 Modern Analytical Chemistry Sensitivity In a gas chromatographic analysis, sensitivity (the slope of a calibra- tion curve) is determined by the detector’s characteristics. Of greater interest for quantitative work is the detector’s linear range; that is, the range of concentrations over which a calibration curve is linear. Detectors with a wide linear range, such as a thermal conductivity detector and flame ionization detector, can be used to analyze samples of varying concentration without adjusting operating conditions. Other de- tectors, such as the electron capture detector, have a much narrower linear range. Selectivity Because it combines separation with analysis, gas chromatography provides excellent selectivity. By adjusting conditions it is usually possible to design a separation such that the analytes elute by themselves. Additional selectivity can be provided by using a detector, such as the electron capture detector, that does not re- spond to all compounds. Time, Cost, and Equipment Analysis time can vary from several minutes for sam- ples containing only a few constituents to more than an hour for more complex samples. Preliminary sample preparation may substantially increase the analysis time. Instrumentation for gas chromatography ranges in price from inexpensive (a few thousand dollars) to expensive (more than $50,000). The more expensive mod- els are equipped for capillary columns and include a variety of injection options and more sophisticated detectors, such as a mass spectrometer. Packed columns typi- cally cost $50–$200, and the cost of a capillary column is typically $200–$1000. 12E High-Performance Liquid Chromatography Although gas chromatography is widely used, it is limited to samples that are ther- mally stable and easily volatilized. Nonvolatile samples, such as peptides and carbo- hydrates, can be analyzed by GC, but only after they have been made more volatile by a suitable chemical derivatization. For this reason, the various techniques in- cluded within the general scope of liquid chromatography are among the most commonly used separation techniques. Although simple column chromatography, first introduced by Tswett, is still used in large-scale preparative work, the focus of this section is on high-performance liquid chromatography (HPLC). In HPLC, a liquid sample, or a solid sample dissolved in a suitable solvent, is carried through a chromatographic column by a liquid mobile phase. Separation is determined by solute/stationary-phase interactions, including liquid–solid adsorp- tion, liquid–liquid partitioning, ion exchange and size exclusion, and by solute/mobile-phase interactions. In each case, however, the basic instrumentation is essentially the same. A schematic diagram of a typical HPLC instrument is shown in Figure 12.26. The remainder of this section deals exclusively with HPLC separations based on liquid–liquid partitioning. Other forms of liquid chromatog- raphy receive consideration later in this chapter. 12E.1 HPLC Columns An HPLC typically includes two columns: an analytical column responsible for the separation and a guard column. The guard column is placed before the analytical column, protecting it from contamination. Analytical Columns The most commonly used columns for HPLC are constructed from stainless steel with internal diameters between 2.1 mm and 4.6 mm, and high-performance liquid chromatography A chromatographic technique in which the mobile phase is a liquid. 1400-CH12 9/8/99 4:28 PM Page 578 Figure 12.26 Schematic diagram of a high-performance liquid chromatograph. lengths ranging from approximately 30 mm to 300 mm. These columns are packed with 3–10 m m porous silica particles that may have an irregular or spherical shape. Typical column efficiencies are 40,000–60,000 theoretical plates/m. Assuming V max /V min is approximately 50, 3 a 25-cm column with 50,000 plates/m has 12,500 theoretical plates and a peak capacity (equation 12.18) of 110. Microcolumns use less solvent and, because the sample is diluted to a lesser extent, produce larger signals at the detector. These columns are made from fused silica capil- laries with internal diameters of 44–200 m m and lengths of up to several meters. Micro- columns packed with 3–5-m m particles have been prepared with column efficiencies of up to 250,000 theoretical plates. 10 Open tubular microcolumns also have been developed, with internal diameters of 1–50 m m and lengths of approximately 1 m. These columns, which contain no packing material, may be capable of obtaining column efficiencies of up to 1 million theoretical plates. 11 The development of open tubular columns, however, has been limited by the difficulty of preparing columns with internal diameters less than 10 m m. Guard Columns Two problems tend to shorten the lifetime of an analytical column. First, solutes binding irreversibly to the stationary phase degrade the column’s per- formance by decreasing the available stationary phase. Second, particulate material injected with the sample may clog the analytical column. To minimize these prob- lems, a guard column is placed before the analytical column. Guard columns usually contain the same particulate packing material and stationary phase as the analytical column, but are significantly shorter and less expensive; a length of 7.5 mm and a cost one-tenth of that for the corresponding analytical column is typical. Because they are intended to be sacrificial, guard columns are replaced regularly. 12E.2 Stationary Phases In liquid–liquid chromatography the stationary phase is a liquid film coated on a packing material consisting of 3–10 m m porous silica particles. The stationary phase may be partially soluble in the mobile phase, causing it to “bleed” from the column Chapter 12 Chromatographic and Electrophoretic Methods 579 Pump Solvent proportioning valve Mobile phase reservoirs Pulse damper Loop injector Guard column Column Detector guard column An inexpensive column used to protect a more expensive analytical column. 1400-CH12 9/8/99 4:28 PM Page 579 over time. To prevent this loss of stationary phase, it is covalently bound to the sil- ica particles. Bonded stationary phases are attached by reacting the silica particles with an organochlorosilane of the general form Si(CH 3 ) 2 RCl, where R is an alkyl or substituted alkyl group. To prevent unwanted interactions between the solutes and any unreacted –SiOH groups, the silica frequently is “capped” by reacting it with Si(CH 3 ) 3 Cl; such columns are designated as end-capped. The properties of a stationary phase are determined by the nature of the organosilane’s alkyl group. If R is a polar functional group, then the stationary phase will be polar. Examples of polar stationary phases include those for which R contains a cyano (–C 2 H 4 CN), diol (–C 3 H 6 OCH 2 CHOHCH 2 OH), or amino (–C 3 H 6 NH 2 ) functional group. Since the stationary phase is polar, the mobile phase is a nonpolar or moderately polar solvent. The combination of a polar stationary phase and a nonpolar mobile phase is called normal-phase chromatography. In reverse-phase chromatography, which is the more commonly encountered form of HPLC, the stationary phase is nonpolar and the mobile phase is polar. The most common nonpolar stationary phases use an organochlorosilane for which the R group is an n-octyl (C 8 ) or n-octyldecyl (C 18 ) hydrocarbon chain. Most reverse- phase separations are carried out using a buffered aqueous solution as a polar mo- bile phase. Because the silica substrate is subject to hydrolysis in basic solutions, the pH of the mobile phase must be less than 7.5. 12E. 3 Mobile Phases The elution order of solutes in HPLC is governed by polarity. In a normal-phase separation the least polar solute spends proportionally less time in the polar station- ary phase and is the first solute to elute from the column. Retention times are con- trolled by selecting the mobile phase, with a less polar mobile phase leading to longer retention times. If, for example, a separation is poor because the solutes are eluting too quickly, switching to a less polar mobile phase leads to longer retention times and more opportunity for an acceptable separation. When two solutes are ad- equately resolved, switching to a more polar mobile phase may provide an accept- able separation with a shorter analysis time. In a reverse-phase separation the order of elution is reversed, with the most polar solute being the first to elute. Increasing the polarity of the mobile phase leads to longer retention times, whereas shorter re- tention times require a mobile phase of lower polarity. Choosing a Mobile Phase Several indices have been developed to assist in selecting a mobile phase, the most useful of which is the polarity index. 12 Table 12.3 provides values for the polarity index, P￠ , of several commonly used mobile phases, in which larger values of P￠ correspond to more polar solvents. Mobile phases of intermedi- ate polarity can be fashioned by mixing together two or more of the mobile phases in Table 12.3. For example, a binary mobile phase made by combining solvents A and B has a polarity index, P ￠ AB , of 580 Modern Analytical Chemistry Si OH O O O O Si O Si(CH 3 ) 2 R + HCl Si(CH 3 ) 2 RCl normal-phase chromatography Liquid chromatography using a polar stationary phase and a nonpolar mobile phase. reverse-phase chromatography Liquid chromatography using a nonpolar stationary phase and a polar mobile phase. polarity index A quantitative measure of a solvent’s polarity. bonded stationary phase A liquid stationary phase that is chemically bonded to a particulate packing material. 1400-CH12 9/8/99 4:28 PM Page 580 Chapter 12 Chromatographic and Electrophoretic Methods 581 Table 12. 3 Properties of HPLC Mobile Phases Polarity Index UV Cutoff Mobile Phase (P￠ ) (nm) cyclohexane 0.04 210 n-hexane 0.1 210 carbon tetrachloride 1.6 265 i-propyl ether 2.4 220 toluene 2.4 286 diethyl ether 2.8 218 tetrahydrofuran 4.0 220 ethanol 4.3 210 ethyl acetate 4.4 255 dioxane 4.8 215 methanol 5.1 210 acetonitrile 5.8 190 water 10.2 — P￠ AB = f A P￠ A + f B P￠ B 12.30 where P￠ A and P￠ B are the polarity indexes for solvents A and B, and f A and f B are the volume fractions of the two solvents. EXAMPLE 12. 7 A reverse-phase HPLC separation is carried out using a mobile-phase mixture of 60% v/v water and 40% v/v methanol. What is the mobile phase’s polarity index? SOLUTION From Table 12.3 we find that the polarity index is 10.2 for water and 5.1 for methanol. Using equation 12.30, the polarity index for a 60:40 water–methanol mixture is P ￠ AB = (0.60)(10.2) + (0.40)(5.1) = 8.2 A useful guide when using the polarity index is that a change in its value of 2 units corresponds to an approximate tenfold change in a solute’s capacity factor. Thus, if k￠ is 22 for the reverse-phase separation of a solute when using a mobile phase of water (P￠ = 10.2), then switching to a 60:40 water–methanol mobile phase (P￠ = 8.2) will decrease k￠ to approximately 2.2. Note that the capacity factor de- creases because we are switching from a more polar to a less polar mobile phase in a reverse-phase separation. Changing the mobile phase’s polarity index, by changing the relative amounts of two solvents, provides a means of changing a solute’s capacity factor. Such 1400-CH12 9/8/99 4:29 PM Page 581 Figure 12.27 Solvent triangle for optimizing reverse-phase HPLC separations. Binary and ternary mixtures contain equal volumes of each of the aqueous mobile phases making up the vertices of the triangle. changes, however, are not very selective; thus, two solutes that significantly overlap may continue to be poorly resolved even after making a significant change in the mobile phase’s polarity. To effect a better separation between two solutes it is often necessary to im- prove the selectivity factor, a . Two approaches are commonly used to accomplish this improvement. When a solute is a weak acid or a weak base, adjusting the pH of the aqueous mobile phase can lead to significant changes in the solute’s retention time. This is shown in Figure 12.13a for the reverse-phase separation of p- aminobenzoic acid and p-hydroxybenzoic acid on a nonpolar C 18 column. At more acidic pH levels, both weak acids are present as neutral molecules. Because they par- tition favorably into the stationary phase, the retention times for the solutes are fairly long. When the pH is made more basic, the solutes, which are now present as their conjugate weak base anions, are less soluble in the stationary phase and elute more quickly. Similar effects can be achieved by taking advantage of metal–ligand complexation and other equilibrium reactions. 13 A second approach to changing the selectivity factor for a pair of solutes is to change one or more of the mobile-phase solvents. In a reverse-phase separation, for example, this is accomplished by changing the solvent mixed with water. Besides methanol, other common solvents for adjusting retention times are acetonitrile and tetrahydrofuran (THF). A common strategy for finding the best mobile phase is to use the solvent triangle shown in Figure 12.27. The separation is first optimized using an aqueous mobile phase of acetonitrile to produce the best separation within the desired analysis time (methanol or THF also could be chosen first). Table 12.4 is used to estimate the composition of methanol/H 2 O and THF/H 2 O mobile phases that will produce similar analysis times. These mobile phases are then adjusted, if necessary, establishing the three points of the solvent triangle. Four additional mo- bile phases are prepared using the binary and ternary mobile phases indicated in Figure 12.27. From these seven mobile phases it is possible to estimate how a change in the mobile-phase composition might affect the separation. Isocratic Versus Gradient Elution When a separation uses a single mobile phase of fixed composition it is called an isocratic elution. It is often difficult, however, to find a single mobile-phase composition that is suitable for all solutes. Recalling the general elution problem, a mobile phase that is suitable for early eluting solutes may lead to unacceptably long retention times for later eluting solutes. Optimizing con- 582 Modern Analytical Chemistry CH 3 CN/H 2 O + CH 3 OH/H 2 O CH 3 CN/H 2 O + CH 3 OH/H 2 O + THF/H 2 O CH 3 CN/H 2 O + THF/H 2 O CH 3 OH/H 2 O + THF/H 2 O CH 3 CN/H 2 O THF/H 2 OCH 3 OH/H 2 O isocratic elution The use of a mobile phase whose composition remains constant throughout the separation. 1400-CH12 9/8/99 4:29 PM Page 582 ditions for late eluting solutes, on the other hand, may provide an inadequate sepa- ration of early eluting solutes. Changing the composition of the mobile phase with time provides a solution to this problem. For a reverse-phase separation the initial mobile-phase composition is relatively polar. As the separation progresses, the mo- bile phase’s composition is made less polar. Such separations are called gradient elutions. 12E. 4 HPLC Plumbing An important feature of HPLC instrumentation (see Figure 12.26) is the presence of several solvent reservoirs. As discussed in the previous section, controlling the mo- bile phase’s polarity plays an important role in improving a liquid chromatographic separation. The availability of several solvent reservoirs allows the mobile phase’s composition to be quickly and easily varied. This is essential when using a gradient elution, for which the mobile-phase composition is systematically changed from a weaker solvent to a stronger solvent. Before they are used, mobile-phase solvents must be treated to remove dis- solved gases, such as N 2 and O 2 , and small particulate matter, such as dust. Dis- solved gases often lead to the formation of gas bubbles when the mobile phase en- ters the detector, resulting in a distortion of the detector’s signal. Degassing is accomplished in several ways, but the most common are the use of a vacuum pump or sparging with an inert gas, such as He, which has a low solubility in the mobile phase. Particulate material capable of clogging the HPLC tubing or column is re- moved by filtering. If the instrument is not designed to do so, degassing and filter- ing can be completed before the solvents are placed in their reservoirs. The mobile-phase solvents are pulled from their reservoirs by the action of a pump. Most HPLC instruments use a reciprocating pump consisting of a piston whose back-and-forth movement is capable both of maintaining a constant flow rate of up to several milliliters per minute and of obtaining the high output pressure needed to push the mobile phase through the chromatographic column. A solvent proportioning valve controls the mobile phase’s composition, making possible the Chapter 12 Chromatographic and Electrophoretic Methods 583 Table 12. 4 Organic Solvent/H 2 O Mobile- Phase Compositions Having Approximately Equal Solvent Strength %v/v CH 3 OH %v/v CH 3 CN %v/v THF 00 0 10 6 4 20 14 10 30 22 16 40 32 24 50 40 30 60 50 36 70 60 44 80 72 52 90 87 62 100 99 71 1400-CH12 9/8/99 4:29 PM Page 583 Figure 12.28 Schematic diagram of a loop injector in the (a) load and (b) inject positions. necessary change in the mobile phase’s composition when using a gradient elution. The back and forth movement of a reciprocating pump results in a pulsed flow that contributes noise to the chromatogram. To eliminate this problem a pulse damper is placed at the outlet of the pump. 12E. 5 Sample Introduction The typical operating pressure of an HPLC is sufficiently high that it is impossible to inject the sample in the same manner as in gas chromatography. Instead, the sample is introduced using a loop injector (Figure 12.28). Sampling loops are inter- changeable, and available with volumes ranging from 0.5 m L to 2 mL. In the load position the sampling loop is isolated from the mobile phase and is open to the atmosphere. A syringe with a capacity several times that of the sampling loop is used to place the sample in the loop. Any extra sample beyond that needed to fill the sample loop exits through the waste line. After loading the sample, the in- jector is turned to the inject position. In this position the mobile phase is directed through the sampling loop, and the sample is swept onto the column. 12E.6 Detectors for HPLC As with gas chromatography, numerous detectors have been developed for use in monitoring HPLC separations. 14 To date, the majority of HPLC detectors are not unique to the method, but are either stand-alone instruments or modified versions of the same. Spectroscopic Detectors The most popular HPLC detectors are based on spectro- scopic measurements, including UV/Vis absorption, and fluorescence. These detec- tors range from simple designs, in which the analytical wavelength is selected using appropriate filters, to essentially a modified spectrophotometer equipped with a flow cell. When using a UV/Vis detector, the resulting chromatogram is a plot of absorbance as a function of elution time. Instruments utilizing a diode array spec- trophotometer record entire spectra, giving a three-dimensional chromatogram showing absorbance as a function of wavelength and elution time. Figure 12.29a shows a typical flow cell for HPLC when using a UV/Vis spectrophotometer as a de- tector. The flow cell has a volume of 1–10 m L and a path length of 0.2–1 cm. One limitation to using absorbance is that the mobile phase must not absorb strongly at 584 Modern Analytical Chemistry Mobile phase in Mobile phase out Loop Sample outSample in Mobile phase in Mobile phase out Loop Sample outSample in (a) (b) loop injector A means for injecting samples in which the sample is loaded into a short section of tubing and injected onto the column by redirecting the mobile phase through the loop. Colorplate 12 shows a photo of an HPLC equipped with a diode array detector. 1400-CH12 9/8/99 4:29 PM Page 584 Figure 12.29 Schematic diagrams of flow cell detectors for HPLC using (a) UV/Vis absorption spectrophotometry and (b) amperometry. the chosen wavelength. Table 12.3 lists the wavelengths below which UV/Vis ab- sorbance cannot be used for different mobile phases. Detectors based on ab- sorbance provide detection limits of as little as 100 pg–1 ng of injected analyte. Fluorescence detectors provide additional selectivity since fewer solutes are capable of fluorescing. The resulting chromatogram is a plot of fluorescence intensity as a function of time. Detection limits are as little as 1–10 pg of injected analyte. Electrochemical Detectors Another common group of HPLC detectors are those based on electrochemical measurements such as amperometry, voltammetry, coulometry, and conductivity. Figure 12.29b, for example, shows an amperometric flow cell. Effluent from the column passes over the working electrode, which is held at a potential favorable for oxidizing or reducing the analytes. The potential is held constant relative to a downstream reference electrode, and the current flowing be- tween the working and auxiliary electrodes is measured. Detection limits for amper- ometric electrochemical detection are 10 pg–1 ng of injected analyte. Other Detectors Several other detectors have been used in HPLC. Measuring a change in the mobile phase’s refractive index is analogous to monitoring the mobile phase’s thermal conductivity in gas chromatography. A refractive index detector is nearly universal, responding to almost all compounds, but has a poorer detection limit of 100 ng–1 m g of injected analyte. Furthermore, a refractive index detector is not useful for a gradient elution unless the mobile-phase components have identical refractive indexes. Another useful detector is a mass spectrometer. The advantages of using a mass spectrometer in HPLC are the same as for gas chromatography. De- tection limits are quite good, typically 100 pg–1 ng of injected analyte, with values Chapter 12 Chromatographic and Electrophoretic Methods 585 To waste Window Window From column DetectorSource Auxiliary electrode Reference electrode electrodeWorking To waste From column Potentiostat (a) (b) 1400-CH12 9/8/99 4:29 PM Page 585 586 Modern Analytical Chemistry as low as 1–10 pg in some situations. In addition, a mass spectrometer provides qualitative, structural information that can help identify the analytes. The interface between the HPLC and mass spectrometer is technically more difficult than that in a GC–MS because of the incompatibility of a liquid mobile phase with the mass spectrometer’s high vacuum requirement. Recent developments in mass spectrome- try, however, have led to a growing interest in LC–MS. 12E. 7 Quantitative Applications HPLC is routinely used for both qualitative and quantitative analyses of environ- mental, pharmaceutical, industrial, forensic, clinical, and consumer product sam- ples. Figure 12.30 shows several representative examples. Preparing Samples for Analysis Samples in liquid form can be analyzed di- rectly, after a suitable clean-up to remove any particulate materials or after a suitable extraction to remove matrix interferents. In determining polyaromatic hydrocarbons (PAH) in wastewater, for example, an initial extraction with CH 2 Cl 2 serves the dual purpose of concentrating the analytes and isolating them from matrix interferents. Solid samples must first be dissolved in a suitable sol- vent, or the analytes of interest must be brought into solution by extraction. For example, an HPLC analysis for the active ingredients and degradation products in a pharmaceutical tablet often begins by extracting the powdered tablet with a portion of mobile phase. Gases are collected by bubbling through a trap contain- ing a suitable solvent. Organic isocyanates in industrial atmospheres can be de- termined in this manner by bubbling the air through a solution of 1-(2- methoxyphenyl)piperazine in toluene. Reacting the isocyanates with 1-(2-methoxyphenyl)piperazine serves the dual purposes of stabilizing them against degradation before the HPLC analysis while also forming a derivative that can be monitored by UV absorption. Quantitative Calculations Quantitative analyses are often easier to conduct with HPLC than GC because injections are made with a fixed-volume injection loop in- stead of a syringe. As a result, variations in the amount of injected sample are mini- mized, and quantitative measurements can be made using external standards and a normal calibration curve. EXAMPLE 12.8 The concentration of PAHs in soil can be determined by first extracting the PAHs with methylene chloride. The extract is then diluted, if necessary, and the PAHs are separated by HPLC using a UV/Vis or fluorescence detector. Calibration is achieved using one or more external standards. In a typical analysis, a 2.013-g sample of dried soil is extracted with 20.00 mL of methylene chloride. After filtering to remove the soil, a 1-mL portion of the extract is removed and diluted to 10 mL with acetonitrile. Injecting 5 m L of the diluted extract into an HPLC gives a signal of 0.217 (arbitrary units) for the PAH fluoranthene. When 5 m L of a 20.0-ppm fluoranthene standard is analyzed using the same conditions, a signal of 0.258 is measured. Report the parts per million of fluoranthene in the soil. 1400-CH12 9/8/99 4:29 PM Page 586 Figure 12.30 Examples of the application of HPLC to the analysis of (a) acetaminophen, salicylic acid, and caffeine; (b) chlorinated pesticides; (c) tricyclic antidepressants; and (d) peptides. (Chromatograms courtesy of Alltech Associates, Inc. Deerfield, IL). 587 0 3 6 9 Min. 1 2 3 4 1. Acetaminophen (0.43mg/mL) 2. Caffeine (0.30mg/mL) 3. Benzoic Acid (Internal Std.) (1.45mg/mL) 4. Salicylic Acid (1.52mg/mL) Column: Adsorbosil C18, 5μm, 100 · 4.6mm Mobile Phase: Water:Methanol:Glacial Acetic Acid (69:28:3) Flowrate: 1.5mL/min Column Temp: 45°C Detector: UV at 275nm Acetaminophen, salicylic acid and caffeine CHROM 7227 1 3 5 2 4 6 0 2 4 6 8 10 Min. 1. Aldrin 2. Heptachlor 3. Endrin 4. Dieldrin 5. TDE 6. Methoxychlor Column: Adsorbosphere PAC. 5μm, 250 · 4.6mm Mobile Phase: Isooctane:Ethyl Acetate (97.3:2.7) Flowrate: 1.0mL/min Detector: UV at 254nm Chlorinated pesticides CHROM 5006 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 Min. 1. Trimipramine 2. Doxepin 3. Amitriptyline 4. Imipramine 5. Nordoxepin 6. Nortriptyline 7. Desipramine 8. Protriptyline Column: Adsorbosphere CN-AQ, 5μm, 150 · 4.6mm Mobile Phase: Acetonitrile:Methanol:0.01M K 2 HPO 4 , pH 7.0 (60:15:25) Flowrate: 2.0mL/min Detector: UV at 215nm Tricyclic antidepressants CHROM 7170 Column: Alltima C18, 3μm, 100 · 4.6mm Mobile Phase: A: 0.1%v/v TFA B: 0.085%v/v TFA in Acetonitrile:Water (90:10) Gradient: Time: 020 %B: 15 45 Flowrate: 1.0mL/min Detector: UV at 220nm 02 1 2 3 4 5 4681012 Min. 1. Gly-Tyr 2. Val-Tyr-Val 3. Methionine Enkephalin 4. Angiotensin II 5. Leu-Enkephalin Peptides CHROM 8761 (a) (b) (c) (d) 1400-CH12 9/8/99 4:29 PM Page 587 588 Modern Analytical Chemistry Representative Methods —Continued SOLUTION For a single-point external standard, the relationship between the signal, S, and the concentration, C, of fluoranthene is S = kC Substituting the known values for the standard’s signal and concentration gives the value of k as Using this value for k and the signal obtained for the extracted and diluted soil sample gives a fluoranthene concentration of This, of course, is the concentration of fluoranthene in the sample injected into the HPLC. The concentration of fluoranthene in the soil is 12E.8 Representative Method Although each HPLC method has its own unique considerations, the following de- scription of the determination of the fluoxetine in serum provides an instructive ex- ample of a typical procedure. Method 12.2 Determination of Fluoxetine in Serum 15 Description of Method. Fluoxetine, whose structure is shown in Figure 12.31a, is another name for the antidepressant drug Prozac. The determination of fluoxetine and its metabolite norfluoxetine, Figure 12.31b, in serum is an important part of monitoring its therapeutic use. The analysis is complicated by the complex matrix of serum samples. A solid-phase extraction followed by an HPLC analysis using a fluorescence detector provides the necessary selectivity and detection limits. (.16 8 1670 g/mL) (10 mL/1 mL) 20 mL 2.013 g ppm μ× × = C S k == = ? 0 217 0 0129 16 8 1 . . . ppm ppm k S C == = ? 0 258 20 0 0 0129 1 . . . ppm ppm F 3 CCH 2 CH 3 CH 2 OCH NH F 3 CCH 2 CH 2 OCH NH 2 (a) (b) Figure 12.31 Structures of (a) fluoxetine and (b) norfluoxetine. 1400-CH12 9/8/99 4:29 PM Page 588 Chapter 12 Chromatographic and Electrophoretic Methods 589 Procedure. A known amount of the antidepressant protriptyline is added to a serum sample, serving as an internal standard. A 0.5-mL aliquot of the serum is passed through a solid-phase extraction cartridge containing silica particles with a bonded C 18 phase. After washing to remove interfering constituents from the sample matrix, the remaining constituents, including both analytes and the internal standard, are removed by washing the cartridge with 0.25 mL of a 25:75 v/v mixture of 0.1 M HClO 4 and acetonitrile. A 20-m L aliquot is injected onto a 15-cm · 4.6-mm column packed with a 5-m m C 8 -bonded stationary phase. An isocratic mobile-phase mixture of 37.5:62.5 v/v acetonitrile and water (containing 1.5 g of tetramethylammonium perchlorate and 0.1 mL of 70% v/v HClO 4 ) is used to elute the sample. Detection is with a fluorescence detector using an excitation wavelength of 235 nm and an emission wavelength of 310 nm. Questions 1. What is the purpose of including an initial solid-phase extraction? A direct injection of serum is not advisable since the presence of particulate material in the sample matrix may clog the column. In addition, some of the sample’s constituents may absorb too strongly to the stationary phase, thus degrading the column’s performance. Finally, although an HPLC is capable of separating and analyzing complex mixtures, an analysis may still be difficult if there are too many constituents to provide adequate resolution of the analytes. The solid-phase extraction serves the purpose of cleaning up the sample before using the HPLC. 2. One advantage of an HPLC analysis is that a loop injector often eliminates the need for an internal standard. Why is an internal standard used in this analysis? What assumption(s) must we make about the internal standard? An internal standard is used because of difficulties introduced in the solid-phase extraction. For example, the volume of serum taken for the solid-phase extraction and the volume of solvent used to remove the analyte and internal standard are quite small (0.5 mL and 0.25 mL, respectively). The precision and accuracy with which these volumes can be measured are not as good as when using larger volumes. Using an internal standard compensates for any variations in these volumes. To be useful, the analytes and internal standard must be assumed to be retained to the same extent during the initial loading and washing of the cartridge, and they must be assumed to be extracted to the same extent during the final elution. 3. If the peaks for fluoxetine and protriptyline are insufficiently resolved, how might you alter the mobile phase to improve their separation? Decreasing the amount of acetonitrile and increasing the amount of water in the mobile will increase retention times, thereby providing a better resolution. 12E. 9 Evaluation When compared with gas chromatography, HPLC has only a few differences in the scale of operation; accuracy; precision; sensitivity; selectivity; and time, cost, and equipment necessary. Injection volumes in HPLC are usually significantly larger than in GC because of the greater capacity of HPLC columns. Precision in HPLC is often better due to the routine use of loop injectors. Because HPLC is not limited to volatile analytes, the range of compounds that can be analyzed is somewhat larger than for GC. Capillary GC columns, on the other hand, have more theoretical plates, providing greater resolving power for complex mixtures. 1400-CH12 9/8/99 4:29 PM Page 589 Figure 12.32 Example of the application of liquid–solid chromatography to the analysis of amphetamines. (Chromatogram courtesy of Alltech Associates, Inc. Deerfield, IL). 590 Modern Analytical Chemistry 1 2 0 2 4 6 8 10 Min. 1. 3,4-Methylenedioxy- amphetamine (MDA) 2. 3,4-Methylenedioxy- methylamphetamine (MDMA) Column: Adsorbosphere silica, 3 μm, 150 · 4.6 mm Mobile Phase: Methylene chloride: Methanol with 1% NH 4 OH (80:20) Flowrate: 1.25 mL/min Detector: UV at 280 nm MDA CHROM 5445 ion-exchange chromatography A form of liquid chromatography in which the stationary phase is an ion- exchange resin. liquid–solid adsorption chromatography A form of liquid chromatography in which the stationary phase is a solid adsorbent. 12F Liquid—Solid Adsorption Chromatography In liquid–solid adsorption chromatography (LSC) the column packing also serves as the stationary phase. In Tswett’s original work the stationary phase was finely di- vided CaCO 3 , but modern columns employ porous 3–10-m m particles of silica or alumina. Since the stationary phase is polar, the mobile phase is usually a nonpolar or moderately polar solvent. Typical mobile phases include hexane, isooctane, and methylene chloride. The usual order of elution, from shorter to longer retention times, is olefins < aromatic hydrocarbons < ethers < esters, aldehydes, ketones < alcohols, amines < amides < carboxylic acids For most samples liquid–solid chromatography does not offer any special advan- tages over liquid–liquid chromatography (LLC). One exception is for the analysis of isomers, where LLC excels. Figure 12.32 shows a typical LSC separation of two am- phetamines on a silica column using an 80:20 mixture of methylene chloride and methanol containing 1% NH 4 OH as a mobile phase. Nonpolar stationary phases, such as charcoal-based absorbents, also may be used. 12G Ion-Exchange Chromatography In ion-exchange chromatography (IEC) the stationary phase is a cross-linked poly- mer resin, usually divinylbenzene cross-linked polystyrene, with covalently attached ionic functional groups (Figure 12.33). The counterions to these fixed charges are mobile and can be displaced by ions that compete more favorably for the exchange sites. Ion-exchange resins are divided into four categories: strong acid cation ex- changers; weak acid cation exchangers; strong base anion exchangers; and weak base anion exchangers. Table 12.5 provides a list of several common ion-exchange resins. 1400-CH12 9/8/99 4:29 PM Page 590 Figure 12.33 Structures of styrene, divinylbenzene, and a styrene-divinylbenzene co-polymer modified for use as an ion-exchange resin. The ion- exchange sites, indicated by R, are mostly in the para position and are not necessarily bound to all styrene units. Strong acid cation exchangers include a sulfonic acid functional group that re- tains its anionic form, and thus its capacity for ion-exchange, in strongly acidic so- lutions. The functional groups for a weak acid cation exchanger, however, are fully protonated at pH levels less then 4, thereby losing their exchange capacity. The strong base anion exchangers are fashioned using a quaternary amine, therefore re- taining a positive charge even in strongly basic solutions. Weak base anion exchang- ers, however, remain protonated only at pH levels that are moderately basic. Under more basic conditions, a weak base anion exchanger loses its positive charge and, therefore, its exchange capacity. Chapter 12 Chromatographic and Electrophoretic Methods 591 Ion-exchange site Cross-linking CH 2 styrene CH CH 2 divinylbenzene CH CH 2 CH CH 2 CH R CH 2 CHCH 2 CHCH 2 CH R CH 2 CH R CH 2 CH CH 2 CHCH 2 CHCH 2 CH R CH 2 CH R CH 2 CH CH 2 CHCH 2 CHCH 2 CH R Table 12. 5 Examples of Common Ion-Exchange Resins Type Functional Group Examples strong acid cation exchanger sulfonic acid –SO 3 – –CH 2 CH 2 SO 3 – weak acid cation exchanger carboxylic acid –COO – –CH 2 COO – strong base anion exchanger quaternary amine –CH 2 N(CH 3 ) 3 + –CH 2 CH 2 N(CH 2 CH 3 ) 3 + weak base anion exchanger amine –NH 3 + –CH 2 CH 2 NH(CH 2 CH 3 ) 2 + 1400-CH12 9/8/99 4:29 PM Page 591 The ion-exchange reaction of a monovalent cation, M + , at a strong acid ex- change site is –SO 3 – –H + (s) + M + (aq) t –SO 3 – –M + (s) + H + (aq) The equilibrium constant for this ion-exchange reaction, which is also called the se- lectivity coefficient, is 12.31 where the brackets { } indicate a surface concentration. Rearranging equation 12.31 shows that the distribution ratio for the exchange reaction is a function of the concentration of H + and, therefore, the pH of the mobile phase. Ion-exchange resins are incorporated into HPLC columns either as micron- sized porous polymer beads or by coating the resin on porous silica particles. Selec- tivity is somewhat dependent on whether the resin includes a strong or weak ex- change site and on the extent of cross-linking. The latter is particularly important because it controls the resin’s permeability and, therefore, the accessibility of the ex- change sites. An approximate order of selectivity for a typical strong acid cation ex- change resin, in order of decreasing D, is Al 3+ > Ba 2+ > Pb 2+ > Ca 2+ > Ni 2+ > Cd 2+ > Cu 2+ > Co 2+ > Zn 2+ > Mg 2+ > Ag + > K + > NH 4 + >Na + > H + > Li + Note that highly charged ions bind more strongly than ions of lower charge. Within a group of ions of similar charge, those ions with a smaller hydrated radius (Table 6.1 in Chapter 6) or those that are more polarizable bind more strongly. For a strong base anion exchanger the general order is SO 4 2– > I – > HSO 4 – > NO 3 – > Br – > NO 2 – > Cl – > HCO 3 – > CH 3 COO – > OH – > F – Again, ions of higher charge and smaller hydrated radius bind more strongly than ions with a lower charge and a larger hydrated radius. The mobile phase in IEC is usually an aqueous buffer, the pH and ionic com- position of which determines a solute’s retention time. Gradient elutions are possi- ble in which the ionic strength or pH of the mobile phase is changed with time. For example, an IEC separation of cations might use a dilute solution of HCl as the mo- bile phase. Increasing the concentration of HCl speeds the elution rate for more strongly retained cations, since the higher concentration of H + allows it to compete more successfully for the ion-exchange sites. Ion-exchange columns can be substituted into the general HPLC instrument shown in Figure 12.26. The most common detector measures the conductivity of the mobile phase as it elutes from the column. The high concentration of electrolyte in the mobile phase is a problem, however, because the mobile-phase ions dominate the conductivity. For example, if a dilute solution of HCl is used as the mobile phase, the presence of large concentrations of H 3 O + and Cl – produces a background conductivity that may prevent the detection of analytes eluting from the column. To minimize the mobile phase’s contribution to conductivity, an ion-suppressor column is placed between the analytical column and the detector. This column se- lectively removes mobile-phase electrolyte ions without removing solute ions. For example, in cation ion-exchange chromatography using a dilute solution of HCl as DK== ?? = ??amount of M in stationary phase amount of M in mobile phase SO M M SO H H + + 3 – + + 3 – + + {} [] {} [] K = ? ?? {}[]SO –M H SO H M 3 – ++ 3 – ++ 592 Modern Analytical Chemistry ion-suppressor column A column used to minimize the conductivity of the mobile phase in ion- exchange chromatography. 1400-CH12 9/8/99 4:29 PM Page 592 Chapter 12 Chromatographic and Electrophoretic Methods 593 single-column ion chromatography Ion-exchange chromatography in which conditions are adjusted so that an ion- suppressor column is not needed. size-exclusion chromatography A form of liquid chromatography in which the stationary phase is a porous material and in which separations are based on the size of the solutes. inclusion limit In size-exclusion chromatography, the smallest solute that can be separated from other solutes; all smaller solutes elute together. the mobile phase, the suppressor column contains an anion-exchange resin. The ex- change reaction H + (aq) + Cl – (aq) + Resin + –OH – t Resin + –Cl – +H 2 O(l) replaces the ionic HCl with H 2 O. Analyte cations elute as hydroxide salts instead of as chloride salts. A similar process is used in anion ion-exchange chromatography in which a cation ion-exchange resin is placed in the suppressor column. If the mo- bile phase contains Na 2 CO 3 , the exchange reaction 2Na + (aq) + CO 3 2– (aq) + 2Resin – –H + t 2Resin – –Na + +H 2 CO 3 (aq) replaces a strong electrolyte, Na 2 CO 3 , with a weak electrolyte, H 2 CO 3 . Ion suppression is necessary when using a mobile phase containing a high con- centration of ions. Single-column ion chromatography, in which an ion-suppressor column is not needed, is possible if the concentration of ions in the mobile phase can be minimized. Typically this is done by using a stationary phase resin with a low capacity for ion exchange and a mobile phase with a small concentration of ions. Because the background conductivity due to the mobile phase is sufficiently small, it is possible to monitor a change in conductivity as the analytes elute from the column. A UV/Vis absorbance detector can also be used if the solute ions absorb ultravi- olet or visible radiation. Alternatively, solutions that do not absorb in the UV/Vis range can be detected indirectly if the mobile phase contains a UV/Vis-absorbing species. In this case, when a solute band passes through the detector, a decrease in absorbance is measured at the detector. Ion-exchange chromatography has found important applications in water analysis and in biochemistry. For example, Figure 12.34a shows how ion-exchange chromatography can be used for the simultaneous analysis of seven common an- ions in approximately 12 min. Before IEC, a complete analysis of the same set of anions required 1–2 days. Ion-exchange chromatography also has been used for the analysis of proteins, amino acids, sugars, nucleotides, pharmaceuticals, con- sumer products, and clinical samples. Several examples are shown in Figure 12.34. 12H Size-Exclusion Chromatography Two classes of micron-sized stationary phases have been encountered in this sec- tion: silica particles and cross-linked polymer resin beads. Both materials are porous, with pore sizes ranging from approximately 50 to 4000 ? for silica particles and from 50 to 1,000,000 ? for divinylbenzene cross-linked polystyrene resins. In size-exclusion chromatography, also called molecular-exclusion or gel-permeation chromatography, separation is based on the solute’s ability to enter into the pores of the column packing. Smaller solutes spend proportionally more time within the pores and, consequently, take longer to elute from the column. The size selectivity of a particular packing is not infinite, but is limited to a moderate range. All solutes significantly smaller than the pores move through the column’s entire volume and elute simultaneously, with a retention volume, V r , of V r = V i + V o 12.32 where V i is the volume of mobile phase occupying the packing material’s pore space, and V o is volume of mobile phase in the remainder of the column. The maxi- mum size for which equation 12.32 holds is the packing material’s inclusion limit, 1400-CH12 9/8/99 4:29 PM Page 593 Figure 12.34 Examples of the application of ion-exchange chromatography to the analysis of (a) inorganic anions, (b) inorganic cations, (c) antifreeze, and (d) vitamins. (Chromatograms courtesy of Alltech Associates, Inc. Deerfield, IL). 594 04 1 2 3 4 5 6 7 8 12 Min. 1. Fluoride, (2 ppm) 2. Chloride, (4 ppm) 3. Nitrite, (4 ppm) 4. Bromide, (4 ppm) 5. Nitrate, (4 ppm) 6. Phosphate, (6 ppm) 7. Sulfate, (6 ppm) Column: Allsep? Anion, 100 · 4.6 mm Mobile Phase: 0.7 mM NaHCO 3 :1.2 mM Na 2 CO 3 Flowrate: 1.0 mL/min Temperature: 40°C Detector: Suppressed conductivity Anion standards 0 25 Min.2015105 6 5 4 3 2 1 7 8 9 1. Lithium (0.5 ppm) 2. Sodium (0.5 ppm) 3. Ammonium (0.5 ppm) 4. Potassium (0.8 ppm) 5. Nickel (5 ppm) 6. Zinc (5 ppm) 7. Cobalt (5 ppm) 8. Magnesium (0.35 ppm) and Manganese (0.35 ppm) 9. Calcium (0.7 ppm) Column: Universal cation 100 · 4.6 mm Mobile Phase: 2 mM Tartaric acid/1 mM oxalic acid Flowrate: 1.0 mL/min Detector: Conductivity Monovalent and divalent cations, and transition metals CHROM 6355 1 2 3 4 5 6 7 8 9 10 14 Min.121086420 1. Glycolate 2. Phosphate 3. Formate 4. Chloride 5. Nitrite 6. Bromide 7. Chlorate 8. Nitrate 9. Benzoate 10. Sulfate Column: Wescan Anion/S, 250 · 4.6 mm Mobile Phase: 4 mM Phthalic Acid, pH3.9 Flowrate: 3.4 mL/min Detector: Conductivity Antifreeze analysis CHROM 5885 1 2 3 4 0 10 20 30 Min. 1. Sodium (1 ppm) 2. L-Carnitine (3 ppm) 3. Choline (3 ppm) 4. Calcium, trace Column: Universal cation, 100 · 4.6 mm Mobile Phase: 5 mM HCl Flowrate: 1.0 mL/min Detector: Conductivity Carnitine and choline in vitamins CHROM 6291 (a) (b) (c) (d) 1400-CH12 9/8/99 4:29 PM Page 594 Figure 12.35 Examples of the application of size-exclusion chromatography to the analysis of proteins. The separation in (a) uses a single column; that in (b) uses three columns, providing a wider range of size selectivity. (Chromatograms courtesy of Alltech Associates, Inc. Deerfield, IL). or permeation limit. All solutes too large to enter the pores elute simultaneously with a retention volume of V r = V o 12.33 Equation 12.33 defines the packing material’s exclusion limit. In between the inclusion limit and the exclusion limit, each solute spends an amount of time in the pore space proportional to its size. The retention volume for a solute is V r = V o + DV i 12.34 where D is the solute’s distribution ratio, which ranges from 0 at the exclusion limit to 1 at the inclusion limit. The validity of equation 12.34 requires that size exclusion be the only interaction between the solute and the stationary phase responsible for the separation. To this end, silica particles used for size exclusion are deactivated as described earlier, and polymer resins are synthesized without exchange sites. Size-exclusion chromatography provides a rapid means for separating larger mol- ecules, including polymers and biomolecules. Figure 12.35 shows the application of size-exclusion chromatography for the analysis of protein mixtures. In Figure 12.35a, a column packing with 300 ? pores, with an inclusion limit of 7500 g/mol and an exclu- sion limit of 1.2 · 10 6 g/mol, is used to separate a mixture of three proteins. Mixtures containing a wider range of formula weights can be separated by joining together sev- eral columns in series. Figure 12.35b shows an example spanning an inclusion limit of 4000 g/mol and an exclusion limit of 7.5 · 10 6 g/mol. Another important application is for the determination of formula weights. Calibration curves of log(formula weight) versus V r are prepared between the exclu- sion limit and inclusion limit (Figure 12.36). Since the retention volume is, to some Chapter 12 Chromatographic and Electrophoretic Methods 595 024 6 8 10 Min. 1 2 3 1. Thyroglobulin 2. Holo-transferrin 3. Lysozyme Column: Macrosphere GPC, 7.0μm, 300?, 250 · 4.6mm Mobile Phase: 0.05M Potassium Phosphate Dibasic & 0.15M Sodium Sulfate, pH7.0 Flowrate: 0.5mL/min Detector: UV at 280nm Proteins CHROM 8273 1 2 3 4 5 6 0 5 10 15 20 25 Min. 1. Thyroglobulin 2. Ferritin 3. Transferrin 4.b-Lactoglobulin 5. Myoglobin 6. Glycyl-L-Tyrosine Column: Macrosphere GPC 150/300/500?, 7μm, 250 · 4.6mm (3) Mobile Phase: 0.05M KH 2 PO 4 & 0.15M Na 2 SO 4 , pH7.0 Flowrate: 0.3mL/min Detector: UV at 280nm Proteins CHROM 5192 (a) (b) exclusion limit In size-exclusion chromatography, the largest solute that can be separated from other solutes; all larger solutes elute together. 1400-CH12 9/8/99 4:29 PM Page 595 Figure 12.36 Calibration curve for the determination of formula weight by size-exclusion chromatography. 596 Modern Analytical Chemistry log(Form ula weight) Retention volume Inclusion limit Exclusion limit Supercritical fluid Gas Triple point Liquid Solid Critical point Pressure Temperature Figure 12.37 Phase diagram for a supercritical fluid. supercritical fluid chromatography A separation technique in which the mobile phase is a supercritical fluid. Table 12.6 Typical Properties of Gases, Liquids, and Supercritical Fluids a Density Viscosity Diffusion coefficient Phase (g cm –3 ) (g cm –1 s –1 ) (cm 2 s –1 ) gas ? 10 –3 ? 10 –4 ? 10 –1 supercritical fluid ? 0.1–1 ? 10 –4 –10 –3 ? 10 –4 –10 –3 liquid ? 1 ? 10 –2 < 10 –5 a Values are reported to the nearest factor of 10. degree, a function of a solute’s size and shape, reasonably accurate determinations of formula weight are possible only if the standards are carefully chosen to mini- mize the effect of shape. Size-exclusion chromatography can be carried out using conventional HPLC instrumentation, replacing the HPLC column with an appropriate size-exclusion column. A UV/Vis detector is the most common means for obtaining the chromatogram. 12I Supercritical Fluid Chromatography Despite their importance, gas chromatography and liquid chromatography cannot be used to separate and analyze all types of samples. Gas chromatography, particu- larly when using capillary columns, provides for rapid separations with excellent resolution. Its application, however, is limited to volatile analytes or those analytes that can be made volatile by a suitable derivatization. Liquid chromatography can be used to separate a wider array of solutes; however, the most commonly used de- tectors (UV, fluorescence, and electrochemical) do not respond as universally as the flame ionization detector commonly used in gas chromatography. Supercritical fluid chromatography (SFC) provides a useful alternative to gas chromatography and liquid chromatography for some samples. The mobile phase in supercritical fluid chromatography is a gas held at a temperature and pressure ex- ceeding its critical point (Figure 12.37). Under these conditions the mobile phase is neither a gas nor a liquid. Instead, the mobile phase is a supercritical fluid whose properties are intermediate between those of a gas and a liquid (Table 12.6). Specifi- cally, supercritical fluids have viscosities that are similar to those of gases, which means that they can move through either capillary or packed columns without the need for the high pressures encountered in HPLC. Analysis time and resolution, al- though not as good as in GC, are usually better than that obtainable with conven- tional HPLC. The density of a supercritical fluid, however, is much closer to that of a liquid, accounting for its ability to function as a solvent. The mobile phase in SFC, therefore, behaves more like the liquid mobile phase in HPLC than the gaseous mo- bile phase in GC. The most common mobile phase for supercritical fluid chromatography is CO 2 . Its low critical temperature, 31 °C, and critical pressure, 72.9 atm, are rela- tively easy to achieve and maintain. Although supercritical CO 2 is a good solvent for nonpolar organics, it is less useful for polar solutes. The addition of an organic modifier, such as methanol, improves the mobile phase’s elution strength. Other common mobile phases and their critical temperatures and pressures are listed in Table 12.7. Colorplate 11 shows the phase transition of liquid CO 2 to supercritical CO 2 . 1400-CH12 9/8/99 4:29 PM Page 596 Chapter 12 Chromatographic and Electrophoretic Methods 597 electrophoresis A separation technique based on a solute’s ability to move through a conductive medium under the influence of an electric field. Figure 12.38 Example of the application of supercritical fluid chromatography to the analysis of triglycerides. (Chromatogram courtesy of Alltech Associates, Inc. Deerfield, IL). capillary electrophoresis Electrophoresis taking place in a capillary tube. electropherogram The equivalent of a chromatogram in electrophoresis. The instrumentation necessary for supercritical fluid chromatography is essentially the same as that for a stan- dard GC or HPLC. The only important addition is the need for a pressure restrictor to maintain the critical pressure. Gradient elutions, similar to those in HPLC, are accom- plished by changing the applied pressure over time. The re- sulting change in the density of the mobile phase affects its solvent strength. Detection can be accomplished using stan- dard GC detectors or HPLC detectors. Supercritical fluid chromatography has found many applications in the analysis of polymers, fossil fuels, waxes, drugs, and food products. Its application in the analysis of triglycerides is shown in Figure 12.38. 12J Electrophoresis Thus far all the separations we have considered involve a mobile phase and a stationary phase. Separation of a com- plex mixture of analytes occurs because each analyte has a different ability to partition between the two phases. An analyte whose distribution ratio favors the stationary phase is retained on the column for a longer time, thereby eluting with a longer retention time. Although the meth- ods described in the preceding sections involve different types of stationary and mobile phases, all are forms of chromatography. Electrophoresis is another class of separation tech- niques in which analytes are separated based on their ability to move through a conductive medium, usually an aqueous buffer, in response to an applied electric field. In the absence of other effects, cations migrate toward the electric field’s negatively charged cathode, and anions mi- grate toward the positively charged anode. More highly charged ions and ions of smaller size, which means they have a higher charge-to-size ratio, migrate at a faster rate than larger ions, or ions of lower charge. Neutral species do not experi- ence the electric field and remain stationary. As we will see shortly, under normal conditions even neutral species and anions migrate toward the cathode. In either case, differences in their rate of migration allow for the separation of complex mix- tures of analytes. There are several forms of electrophoresis. In slab gel electrophoresis the con- ducting buffer is retained within a porous gel of agarose or polyacrylamide. Slabs are formed by pouring the gel between two glass plates separated by spacers. Typical thicknesses are 0.25–1 mm. Gel electrophoresis is an important technique in bio- chemistry, in which it is frequently used for DNA sequencing. Although it is a pow- erful tool for the qualitative analysis of complex mixtures, it is less useful for quanti- tative work. In capillary electrophoresis the conducting buffer is retained within a capillary tube whose inner diameter is typically 25–75 m m. Samples are injected into one end of the capillary tube. As the sample migrates through the capillary, its components separate and elute from the column at different times. The resulting electrophero- gram looks similar to the chromatograms obtained in GC or HPLC and provides Table 12. 7 Critical Point Properties for Selected Supercritical Fluids Critical Temperature Critical Pressure Compound (°C) (atm) carbon dioxide 31.3 72.9 ethane 32.4 48.3 nitrous oxide 36.5 71.4 ammonia 132.3 111.3 diethyl ether 193.6 36.3 isopropanol 235.3 47.0 methanol 240.5 78.9 ethanol 243.4 63.0 water 374.4 226.8 024 6 1 2 3 4 5 6 1. iso-Octane 2. Tripalmitin (16:0) 3. Tristearin (18:0) 4. Triarachidin (20:0) 5. Tierucin (22:1) 6. Tribehenin (22:0) Column: Deltabond C8, 250 · 4.6 Mobile Phase: 7% (v/v) Methanol-Modified CO 2 Flowrate: 2.0mL/min Column Temp: 40°C Detector: ELSD Triglycerides by SFC 8 Min. CHROM 8139 1400-CH12 9/8/99 4:29 PM Page 597 598 Modern Analytical Chemistry both qualitative and quantitative information. Only capillary electrophoretic meth- ods receive further consideration in this text. 12 J .1 Theory of Capillary Electrophoresis In capillary electrophoresis the sample is injected into a buffered solution retained within a capillary tube. When an electric field is applied to the capillary tube, the sample’s components migrate as the result of two types of mobility: elec- trophoretic mobility and electroosmotic mobility. Electrophoretic mobility is the solute’s response to the applied electric field. As described earlier, cations move toward the negatively charged cathode, anions move toward the positively charged anode, and neutral species, which do not respond to the electric field, re- main stationary. The other contribution to a solute’s migration is electroosmotic flow, which occurs when the buffer solution moves through the capillary in re- sponse to the applied electric field. Under normal conditions the buffer solution moves toward the cathode, sweeping most solutes, even anions, toward the nega- tively charged cathode. Electrophoretic Mobility The velocity with which a solute moves in response to the applied electric field is called its electrophoretic velocity, n ep ; it is defined as n ep = m ep E 12.35 where m ep is the solute’s electrophoretic mobility, and E is the magnitude of the ap- plied electric field. A solute’s electrophoretic mobility is defined as 12.36 where q is the solute’s charge, h is the buffer solvent’s viscosity, and r is the solute’s radius. Using equations 12.35 and 12.36, we can make several important conclu- sions about a solute’s electrophoretic velocity. Electrophoretic mobility, and, there- fore, electrophoretic velocity, is largest for more highly charged solutes and solutes of smaller size. Since q is positive for cations and negative for anions, these species migrate in opposite directions. Neutral species, for which q is 0, have an elec- trophoretic velocity of 0. Electroosmotic Mobility When an electric field is applied to a capillary filled with an aqueous buffer, we expect the buffer’s ions to migrate in response to their elec- trophoretic mobility. Because the solvent, H 2 O, is neutral, we might reasonably ex- pect it to remain stationary. What is observed under normal conditions, however, is that the buffer solution moves toward the cathode. This phenomenon is called the electroosmotic flow. Electroosmosis occurs because the walls of the capillary tubing are electrically charged. The surface of a silica capillary contains large numbers of silanol groups (Si–OH). At pH levels greater than approximately 2 or 3, the silanol groups ionize to form negatively charged silanate ions (Si–O – ). Cations from the buffer are at- tracted to the silanate ions. As shown in Figure 12.39, some of these cations bind tightly to the silanate ions, forming an inner, or fixed, layer. Other cations are more loosely bound, forming an outer, or mobile, layer. Together these two layers are called the double layer. Cations in the outer layer migrate toward the cathode. Be- cause these cations are solvated, the solution is also pulled along, producing the electroosmotic flow. μ piη ep = q r6 electrophoretic mobility A measure of a solute’s ability to move through a conductive medium in response to an applied electric field (m ep ). electroosmotic flow The movement of the conductive medium in response to an applied electric field. electrophoretic velocity The velocity with which a solute moves through the conductive medium due to its electrophoretic mobility (n ep ). 1400-CH12 9/8/99 4:29 PM Page 598 Figure 12.40 Schematic showing a comparison of the flow profiles for (a) GC and HPLC, and (b) electrophoresis. Electroosmotic flow velocity, n eof , is a function of the magnitude of the applied electric field and the buffer solution’s electroosmotic mobility, m eof . n eof = m eof E 12.37 Electroosmotic mobility is defined as 12.38 where e is the buffer solution’s dielectric constant, z is the zeta potential, and h is the buffer solution’s viscosity. Examining equations 12.37 and 12.38 shows that the zeta potential plays an important role in determining the electroosmotic flow velocity. Two factors deter- mine the zeta potential and, therefore, the electroosmotic velocity. First, the zeta potential is directly proportional to the charge on the capillary walls, with a greater density of silanate ions corresponding to a larger zeta potential. Below a pH of 2, for example, there are few silanate ions; thus, the zeta potential and electroosmotic flow velocity are 0. As the pH level is increased, both the zeta potential and the electroos- motic flow velocity increase. Second, the zeta potential is proportional to the thick- ness of the double layer. Increasing the buffer solution’s ionic strength provides a higher concentration of cations, decreasing the thickness of the double layer. The electroosmotic flow profile is very different from that for a phase moving under forced pressure. Figure 12.40 compares the flow profile for electroosmosis with that for hydrodynamic pressure. The uniform, flat profile for electroosmosis helps to minimize band broadening in capillary elec- trophoresis, thus improving separation efficiency. Total Mobility A solute’s net, or total velocity, n tot , is the sum of its elec- trophoretic velocity and the electroosmotic flow velocity; thus, n tot = n ep + n eof and m tot = m ep + m eof μ εζ piη eof = 4 Chapter 12 Chromatographic and Electrophoretic Methods 599 Figure 12.39 Schematic diagram showing the origin of electroosmotic flow. Bulk solution Mobile layer Fixed layer Capillary wall Doub le la y e r Direction of electroosmotic flow + + + + + + + + + + + + + + + + + + ++ + + + + + + + + + + + + – – – – – – – – – – – – – – – – –– – –––––––– Anode + Cathode – zeta potential The change in potential across a double layer (z ). electroosmotic flow velocity The velocity with which the solute moves through the capillary due to the electroosmotic flow (n eof ). Anode + Cathode – Hydrodynamic flow profile Electroosmotic flow profile (a) (b) 1400-CH12 9/8/99 4:29 PM Page 599 Under normal conditions the following relationships hold (n tot ) cations > n eof (n tot ) anions < n eof (n tot ) neutrals = n eof Thus, cations elute first in an order corresponding to their electrophoretic mobili- ties, with small, highly charged cations eluting before larger cations of lower charge. Neutral species elute as a single band, with an elution rate corresponding to the electroosmotic flow velocity. Finally, anions are the last components to elute, with smaller, highly charged anions having the longest elution time. Migration Time A solute’s total velocity is given by where l is the distance the solute travels between its point of injection and the detec- tor, and t m is the migration time. Since n tot = m tot E =(m ep + m eof )E we have, after rearranging, 12.39 Finally, the magnitude of the electric field is 12.40 where V is the applied potential, and L is the length of the capillary tube. Substitut- ing equation 12.40 into equation 12.39 gives 12.41 Examining equation 12.41 shows that we can decrease a solute’s migration time (and thus the total analysis time) by applying a higher voltage or by using a shorter capillary tube. Increasing the electroosmotic flow also shortens the analysis time, but, as we will see shortly, at the expense of resolution. Efficiency The efficiency of capillary electrophoresis is characterized by the num- ber of theoretical plates, N, just as it is in GC or HPLC. In capillary electrophoresis, the number of theoretic plates is determined by 12.42 where D is the solute’s diffusion coefficient. From equation 12.42 it is easy to see that the efficiency of a capillary electrophoretic separation increases with higher voltages. Again, increasing the electroosmotic flow velocity improves efficiency, but at the expense of resolution. Two additional observations deserve comment. N V D = +()μμ ep eof 2 t lL V m ep eof = +()μμ E V L = t l E m ep eof = +()μμ ν tot m = l t 600 Modern Analytical Chemistry 1400-CH12 9/8/99 4:29 PM Page 600 Figure 12.41 Schematic diagram for capillary electrophoresis. The sample and source reservoir are switched when making injections. Figure 12.42 Schematic diagram showing a cross section of a capillary column for capillary electrophoresis. Chapter 12 Chromatographic and Electrophoretic Methods 601 First, solutes with larger electrophoretic mobilities (in the same direction as the electroosmotic flow) have greater efficiencies; thus, smaller, more highly charged solutes are not only the first solutes to elute, but do so with greater efficiency. Sec- ond, efficiency in capillary electrophoresis is independent of the capillary’s length. Typical theoretical plate counts are approximately 100,000–200,000 for capillary electrophoresis. Selectivity In chromatography, selectivity is defined as the ratio of the capacity factors for two solutes (equation 12.11). In capillary electrophoresis, the analogous expression for selectivity is where m ep,1 and m ep,2 are the electrophoretic mobilities for solutes 1 and 2, respec- tively, chosen such that a ? 1. Selectivity often can be improved by adjusting the pH of the buffer solution. For example, NH 4 + is a weak acid with a pK a of 9.24. At a pH of 9.24 the concentrations of NH 4 + and NH 3 are equal. Decreasing the pH below 9.24 increases its electrophoretic mobility because a greater fraction of the solute is present as the cation NH 4 + . On the other hand, raising the pH above 9.24 increases the proportion of the neutral NH 3 , decreasing its electrophoretic mobility. Resolution The resolution between two solutes is 12.43 where m avg is the average electrophoretic mobility for the two solutes. Examining equation 12.43 shows that increasing the applied voltage and decreasing the elec- troosmotic flow velocity improves resolution. The latter effect is particularly impor- tant because increasing electroosmotic flow improves analysis time and efficiency while decreasing resolution. 12 J .2 Instrumentation The basic instrumentation for capillary electrophoresis is shown in Figure 12.41 and includes a power supply for applying the electric field, anode and cathode compart- ments containing reservoirs of the buffer solution, a sample vial containing the sample, the capillary tube, and a detector. Each part of the instrument receives fur- ther consideration in this section. Capillary Tubes Figure 12.42 shows a cross section of a typical capillary tube. Most capillary tubes are made from fused silica coated with a 20–35-m m layer of poly- imide to give it mechanical strength. The inner diameter is typically 25–75 m m, which is smaller than that for a capillary GC column, with an outer diameter of 200–375 m m. The narrow bore of the capillary column and the relative thickness of the capil- lary’s walls are important. When an electric field is applied to a capillary containing a conductive medium, such as a buffer solution, current flows through the capillary. This current leads to Joule heating, the extent of which is proportional to the capil- lary’s radius and the magnitude of the electric field. Joule heating is a problem be- cause it changes the buffer solution’s viscosity, with the solution at the center of the R V D = ? + 0 177 12 .( ) () / μμ μμ ep,2 ep,1 avg eof α μ μ = ep,1 ep,2 Anode (+) Cathode (–) Capillary tube Detector Destination reservoir Source reservoir Sample Power supply Capillary opening Polyimide coating Fused silica capillary Joule heating The heating of a conductive solution due to the passage of an electric current through the solution. 1400-CH12 9/8/99 4:29 PM Page 601 602 Modern Analytical Chemistry capillary being less viscous than that near the capillary walls. Since the solute’s elec- trophoretic mobility depends on the buffer’s viscosity (see equation 12.36), solutes in the center of the capillary migrate at a faster rate than solutes near the capillary walls. The result is additional band broadening that degrades the separation. Capil- laries with smaller inner diameters generate less Joule heating, and those with larger outer diameters are more effective at dissipating the heat. Capillary tubes may be placed inside a thermostated jacket to control heating, in which case smaller outer diameters allow a more rapid dissipation of thermal energy. Injecting the Sample The mechanism by which samples are introduced in capil- lary electrophoresis is quite different from that used in GC or HPLC. Two types of injection are commonly used: hydrodynamic injection and electrokinetic injec- tion. In both cases the capillary tube is filled with buffer solution. One end of the capillary tube is placed in the destination reservoir, and the other is placed in the sample vial. Hydrodynamic injection uses pressure to force a small portion of the sample into the capillary tubing. To inject a sample hydrodynamically a difference in pres- sure is applied across the capillary by either pressurizing the sample vial or by ap- plying a vacuum to the destination reservoir. The volume of sample injected, in liters, is given by the following equation 12.44 where D P is the pressure difference across the capillary in pascals, d is the capillary’s inner diameter in meters, t is the amount of time that the pressure is applied in sec- onds, h is the buffer solution’s viscosity in kilograms per meter per second (kg m –1 s –1 ), and L is the length of the capillary tubing in meters. The factor of 10 3 changes the units from cubic meters to liters. EXAMPLE 12. 9 A hydrodynamic injection is made by applying a pressure difference of 2.5 · 10 3 Pa (approximately 0.02 atm) for 2 s to a 75-cm long capillary tube with an internal diameter of 50 m m. Assuming that the buffer solution’s viscosity is 10 –3 kg m –1 s –1 , what volume of sample is injected? SOLUTION Making appropriate substitutions into equation 12.44 gives the volume of injected sample as Since the injected sample plug is cylindrical, its length, l plug , is easily calculated using the equation for the volume of a cylinder. V = p r 2 l plug Thus, l V r L plug 3 2 L)(10 m m m mm== × × =× = ?? ? ? pi 2 93 6 4 110 314 25 10 510 05 (/) ( . )( ) . V inj 4 Pa)(50 10 m) s) (128)(0.001 kg m s m) L = 1 nL= ×× ×=× ? ?? ? (. (. )( )( . 25 10 314 2 075 10 1 10 36 11 39 V Pd t L inj =× ? 4 3 128 10 pi η hydrodynamic injection An injection technique in capillary electrophoresis in which pressure is used to inject sample into the capillary column. 1400-CH12 9/8/99 4:29 PM Page 602 Figure 12.43 Schematic diagram demonstrating stacking. Electrokinetic injections are made by placing both the capillary and the anode into the sample vial and briefly applying an electric field. The moles of solute in- jected into the capillary, n inj , are determined using 12.45 where C is the solute’s concentration in the sample, t is the amount of time that the electric field is applied, r is the capillary’s radius, m ep is the solute’s electrophoretic mobility, m eof is the electroosmotic mobility, E is the applied electric field, and k buf and k samp are the conductivities of the buffer solution and sample, respectively. An important consequence of equation 12.45 is that it is inherently biased toward sam- pling solutes with larger electrophoretic mobilities. Those solutes with the largest electrophoretic mobilities (smaller, more positively charged ions) are injected in greater numbers than those with the smallest electrophoretic mobilities (smaller, more negatively charged ions). When a solute’s concentration in the sample is too small to reliably analyze, it may be possible to inject the solute in a manner that increases its concentration in the capillary tube. This method of injection is called stacking. Stacking is accom- plished by placing the sample in a solution whose ionic strength is significantly less than that of the buffering solution. Because the sample plug has a lower concentra- tion of ions than the buffering solution, its resistance is greater. Since the electric current passing through the capillary is fixed, we know from Ohm’s law E = iR that the electric field in the sample plug is greater than that in the buffering solu- tion. Electrophoretic velocity is directly proportional to the electric field (see equa- tion 12.35); thus, ions in the sample plug migrate with a greater velocity. When the solutes reach the boundary between the sample plug and the buffering solution, the electric field decreases and their electrophoretic velocity slows down, “stacking” to- gether in a smaller sampling zone (Figure 12.43). Applying the Electric Field Migration in electrophoresis occurs in response to the applied electric field. The ability to apply a large electric field is important because n Ctr E inj ep eof buf samp =+piμμ κ κ 2 () Chapter 12 Chromatographic and Electrophoretic Methods 603 Sample plug Anode + Cathode – Stacked cations Sample plug Anode + Cathode – Sample plug Anode + Cathode – stacking A means of concentrating solutes in capillary electrophoresis after their injection onto the capillary column. electrokinetic injection An injection technique in capillary electrophoresis in which an electric field is used to inject sample into the capillary column. 1400-CH12 9/8/99 4:29 PM Page 603 Figure 12.44 Schematic diagrams of two approaches to on-column detection using UV/Vis absorption spectroscopy. higher voltages lead to shorter analysis times (see equation 12.41), more efficient separations (see equation 12.42), and better resolution (see equation 12.43). Be- cause narrow-bore capillary tubes dissipate Joule heating so efficiently, voltages of up to 40 kV can be applied. Detectors Most of the detectors used in HPLC also find use in capillary elec- trophoresis. Among the more common detectors are those based on the absorption of UV/Vis radiation, fluorescence, conductivity, amperometry, and mass spectrom- etry. Whenever possible, detection is done “on-column” before the solutes elute from the capillary tube and additional band broadening occurs. UV/Vis detectors are among the most popular. Because absorbance is directly proportional to path length, the capillary tubing’s small diameter leads to signals that are smaller than those obtained in HPLC. Several approaches have been used to increase the path length, including a Z-shaped sample cell or multiple reflections (Figure 12.44). Detection limits are about 10 –7 M. Better detection limits are obtained using fluorescence, particularly when using a laser as an excitation source. When using fluorescence detection, a small portion of the capillary’s protective coating is removed and the laser beam is focused on the inner portion of the capillary tubing. Emission is measured at an angle of 90° to the laser. Because the laser provides an intense source of radiation that can be focused to a narrow spot, detection limits are as low as 10 –16 M. Solutes that do not absorb UV/Vis radiation or undergo fluorescence can be detected by other detectors. Table 12.8 provides a list of detectors used in capillary electrophoresis along with some of their important characteristics. 12 J . 3 Capillary Electrophoresis Methods There are several different forms of capillary electrophoresis, each of which has its particular advantages. Several of these methods are briefly described in this section. Capillary Zone Electrophoresis The simplest form of capillary electrophoresis is capillary zone electrophoresis (CZE). In CZE the capillary tube is filled with a buffer solution and, after loading the sample, the ends of the capillary tube are placed in reservoirs containing additional buffer solution. Under normal conditions, the end of the capillary containing the sample is the anode, and solutes migrate toward 604 Modern Analytical Chemistry hn hn Polyimide coating Polyimide coating Capillary opening Fused silica capillary Reflective coating Reflective coating (a) (b) capillary zone electrophoresis A form of capillary electrophoresis in which separations are based on differences in the solutes’ electrophoretic mobilities. 1400-CH12 9/8/99 4:29 PM Page 604 Figure 12.45 Schematic diagram showing the reversal of electroosmotic flow. the cathode at a velocity determined by their electrophoretic mobility and the elec- troosmotic flow. Cations elute first, with smaller, more highly charged cations elut- ing before larger cations with smaller charges. Neutral species elute as a single band. Finally, anions are the last species to elute, with smaller, more negatively charged an- ions being the last to elute. The direction of electroosmotic flow and, therefore, the order of elution in CZE can be reversed. This is accomplished by adding an alkylammonium salt to the buffer solution. As shown in Figure 12.45, the positively charged end of the alkyl- ammonium ion binds to the negatively charged silanate ions on the capillary’s walls. The alkylammonium ion’s “tail” is hydrophobic and associates with the tail of an- other alkylammonium ion. The result is a layer of positive charges to which anions in the buffer solution are attracted. The migration of these solvated anions toward Chapter 12 Chromatographic and Electrophoretic Methods 605 Table 12.8 Characteristics of Selected Detectors for Capillary Electrophoresis Detection Limit On-Column Detector Selectivity Moles Injected Molarity a Detection? UV/Vis absorbance solute must have UV/Vis absorbing chromophore 10 –13 –10 –16 10 –5 –10 –7 yes indirect absorbance universal 10 –12 –10 –15 10 –4 –10 –6 yes fluorescence solute must have favorable fluorescent quantum 10 –15 –10 –17 10 –7 –10 –9 yes efficiency laser fluorescence solute must have favorable fluorescent quantum 10 –18 –10 –20 10 –13 –10 –16 yes efficiency mass spectrometer universal when monitoring all ions; selective when 10 –16 –10 –17 10 –8 –10 –10 no monitoring single ion amperometry solute must undergo oxidation or reduction 10 –18 –10 –19 10 –7 –10 –10 no conductivity universal 10 –15 –10 –16 10 –7 –10 –9 no radiometric solutes must be radioactive 10 –17 –10 –19 10 –10 –10 –12 yes Source: Adapted from Baker, D. R. Capillary Electrophoresis. Wiley-Interscience: New York, 1995. 16 a Concentration depends on the volume of sample injected. – + + – + + – + + – + + – + + – + + – + + – + + Bulk solution Mobile layer Direction of electroosmotic flow Capillary wall + + + + + + + + + + + + + + + + + + + – – – – – – – – – – – – – – – – – – – – – – – – – – – Anode + Cathode – 1400-CH12 9/8/99 4:29 PM Page 605 Figure 12.46 (a) Structure of sodium dodecylsulfate; (b) structure of a micelle. the anode reverses the electroosmotic flow’s direction. The order of elution in this case is exactly the opposite of that observed under normal conditions. Capillary zone electrophoresis also can be accomplished without an electroos- motic flow by coating the capillary’s walls with a nonionic reagent. In the absence of electroosmotic flow only cations migrate from the anode to the cathode. Anions elute into the source reservoir while neutral species remain stationary. Capillary zone electrophoresis provides effective separations of any charged species, including inorganic anions and cations, organic acids and amines, and large biomolecules such as proteins. For example, CZE has been used to separate a mix- ture of 36 inorganic and organic ions in less than 3 minutes. 17 Neutral species, of course, cannot be separated. Micellar Electrokinetic Capillary Chromatography One limitation to CZE is its in- ability to separate neutral species. Micellar electrokinetic chromatography (MEKC) overcomes this limitation by adding a surfactant, such as sodium dodecyl- sulfate (Figure 12.46a) to the buffer solution. Sodium dodecylsulfate, (SDS) has a long-chain hydrophobic “tail” and an ionic functional group, providing a negatively charged “head.” When the concentration of SDS is sufficiently large, a micelle forms. A micelle consists of an agglomeration of 40–100 surfactant molecules in which the hydrocarbon tails point inward, and the negatively charged heads point outward (Figure 12.46b). Because micelles are negatively charged, they migrate toward the cathode with a velocity less than the electroosmotic flow velocity. Neutral species partition them- selves between the micelles and the buffer solution in much the same manner as they do in HPLC. Because there is a partitioning between two phases, the term “chromatography” is used. Note that in MEKC both phases are “mobile.” The elution order for neutral species in MEKC de- pends on the extent to which they partition into the mi- celles. Hydrophilic neutrals are insoluble in the micelle’s hydrophobic inner environment and elute as a single band as they would in CZE. Neutral solutes that are ex- tremely hydrophobic are completely soluble in the mi- celle, eluting with the micelles as a single band. Those neutral species that exist in a partition equilibrium be- tween the buffer solution and the micelles elute between the completely hydrophilic and completely hydrophobic neutrals. Those neutral species favoring the buffer solution elute before those favoring the micelles. Micellar electrokinetic chromatography has been used to separate a wide variety of samples, including mixtures of pharmaceutical compounds, vitamins, and explosives. Capillary Gel Electrophoresis In capillary gel electrophoresis (CGE) the capil- lary tubing is filled with a polymeric gel. Because the gel is porous, solutes mi- grate through the gel with a velocity determined both by their electrophoretic mobility and their size. The ability to effect a separation based on size is useful when the solutes have similar electrophoretic mobilities. For example, fragments of DNA of varying length have similar charge-to-size ratios, making their separa- tion by CZE difficult. Since the DNA fragments are of different size, a CGE sepa- ration is possible. The capillary used for CGE is usually treated to eliminate electroosmotic flow, thus preventing the gel’s extrusion from the capillary tubing. Samples are injected 606 Modern Analytical Chemistry capillary gel electrophoresis A form of capillary electrophoresis in which the capillary column contains a gel enabling separations based on size. micellar electrokinetic capillary chromatography A form of capillary electrophoresis in which neutral solutes are separated based on their ability to partition into a charged micelle. micelle An agglomeration of molecules containing ionic “heads” and hydrophobic “tails,” which form into a structure with a hydrophobic interior and a hydrophilic exterior. H 3 C (CH 2 ) 11 O – Na + SO O Represented as O – – – – – – – – – – – – – – – – – – – – – – – – – (a) (b) 1400-CH12 9/8/99 4:29 PM Page 606 Chapter 12 Chromatographic and Electrophoretic Methods 607 capillary electrochromatography A form of capillary electrophoresis in which a stationary phase is included within the capillary column. Representative Methods —Continued electrokinetically because the gel provides too much resistance for hydrodynamic sampling. The primary application of CGE is the separation of large biomolecules, including DNA fragments, proteins, and oligonucleotides. Capillary Electrochromatography Another approach to separating neutral species is capillary electrochromatography (CEC). In this technique the capillary tubing is packed with 1.5–3-m m silica particles coated with a bonded, nonpolar stationary phase. Neutral species separate based on their ability to partition between the sta- tionary phase and the buffer solution (which, due to electroosmotic flow, is the mo- bile phase). Separations are similar to the analogous HPLC separation, but without the need for high-pressure pumps. Furthermore, efficiency in CEC is better than in HPLC, with shorter analysis times. 12J. 4 Representative Method Although each capillary electrophoretic method has its own unique considerations, the following description of the determination of a vitamin B complex provides an instructive example of a typical procedure. Method 12. 3 Determination of a Vitamin B Complex by CZE or MEKC 18 Description of Method. The water-soluble vitamins B 1 (thiamine hydrochloride), B 2 (riboflavin), B 3 (niacinamide), and B 6 (pyridoxine hydrochloride) may be determined by CZE using a pH 9 sodium tetraborate/sodium dihydrogen phosphate buffer or by MEKC using the same buffer with the addition of sodium dodecyl- sulfate. Detection is by UV absorption at 200 nm. An internal standard of o-ethoxybenzamide is used to standardize the method. Procedure. A vitamin B complex tablet is crushed and placed in a beaker with 20.00 mL of a 50% v/v methanol solution that is 20 mM in sodium tetraborate and contains 100.0 ppm of o-ethoxybenzamide. After mixing for 2 min to ensure that the B vitamins are dissolved, a 5.00-mL portion is passed through a 0.45-m m filter to remove insoluble binders. An approximately 4-nL sample is loaded into a 50-m m internal diameter capillary column. For CZE the capillary column contains a 20 mM pH 9 sodium tetraborate/sodium dihydrogen phosphate buffer. For MEKC the buffer is also 150 mM in sodium dodecylsulfate. A 40-kV/m electric field is used to effect both the CZE and MEKC separations. Questions 1. Methanol, which elutes at 4.69 min, is included as a neutral species to indicate the electroosmotic flow. When using standard solutions of each vitamin, CZE peaks are found at 3.41 min, 4.69 min, 6.31 min, and 8.31 min. Examine the structures and pK a information in Figure 12.47, and determine the order in which the four B vitamins elute. Vitamin B 1 is a cation and must, therefore, elute before the neutral species methanol; thus it elutes first at 3.41 min. Vitamin B 3 is a neutral species and should elute with methanol at 4.69 min. The remaining two B vitamins are weak acids that partially ionize in the pH 9 buffer. Of the two, vitamin B 6 is the stronger acid and is ionized (as the anion) to a greater extent. Vitamin B 6 , therefore, is the last of the vitamins to elute. 1400-CH12 9/8/99 4:29 PM Page 607 608 Modern Analytical Chemistry 2. The order of elution when using MEKC is vitamin B 3 (5.58 min), vitamin B 6 (6.59 min), vitamin B 2 (8.81 min), and vitamin B 1 (11.21 min). What conclusions can you make about the solubility of the B vitamins in the sodium dodecylsulfate micelles? The elution time for vitamin B 1 shows the greatest change, increasing from 3.41 min to 11.21 min. Clearly vitamin B 1 has the greatest solubility in the micelles. Vitamins B 2 and B 3 have a more limited solubility in the micelles, showing slightly longer elution times. Interestingly, the elution time for vitamin B 6 decreases in the presence of the micelles. 3. A quantitative analysis for vitamin B 1 was carried out using this procedure. When a solution of 100.0 ppm B 1 and 100.0 ppm o-ethoxybenzamide was analyzed, the peak area for vitamin B 1 was 71% of that for the internal standard. The analysis of a 0.125-g vitamin B complex tablet gave a peak area for vitamin B 1 that was 1.82 times as great as that for the internal standard. How many milligrams of vitamin B 1 are in the tablet? For an internal standardization the relevant equation is where S A and S IS are, respectively, the signals for the analyte and internal standard, and C A and C IS are their respective concentrations. Making appropriate substitutions for the standard solution gives k as 0.71. Substituting values for the sample gives the concentration of vitamin B 1 as 256 ppm. This is the concentration in the sample as injected. To determine the number of milligrams of vitamin B 1 , we must account for the sample’s dissolution; thus 256 0 0200 5 1 mg L L mg vitamin B 1 ×=.. 182 1 071 100 0 . . . =× C A ppm 71 100 100 0 =×k . ppm 100.0 ppm S S k C C A IS A IS = N CONH 2 B 3 pK a = 9.7 B 6 N OH CH 3 CH 2 OH O O HOH 2 C pK a = 9.0 B 2 N N N NH H 3 C H 3 C CH 2 CH 2 OH CHHO CHHO CHHO H 3 C CH 3 CH 2 CH 2 OH N N NH 2 B 1 N + S Continued from page 607 Figure 12.47 Structures of the vitamins B 1 , B 2 , B 3 , and B 6 . 1400-CH12 9/8/99 4:29 PM Page 608 12 J . 5 Evaluation When compared with GC and HPLC, capillary electrophoresis provides similar lev- els of accuracy, precision, and sensitivity and a comparable degree of selectivity. The amount of material injected into a capillary electrophoretic column is significantly smaller than that for GC and HPLC; typically 1 nL versus 0.1 m L for capillary GC and 1–100 m L for HPLC. Detection limits for capillary electrophoresis, however, are 100–1000 times poorer than those for GC and HPLC. The most significant advan- tages of capillary electrophoresis are improvements in separation efficiency, time, and cost. Capillary electrophoretic columns contain substantially more theoretical plates (? 10 6 plates/m) than that found in HPLC (? 10 5 plates/m) and capillary GC columns (? 10 3 plates/m), providing unparalleled resolution and peak capacity. Sep- arations in capillary electrophoresis are fast and efficient. Furthermore, the capillary column’s small volume means that a capillary electrophoresis separation requires only a few microliters of buffer solution, compared with 20–30 mL of mobile phase for a typical HPLC separation. Chapter 12 Chromatographic and Electrophoretic Methods 609 12K KEY TERMS adjusted retention time (p. 551) band broadening (p. 553) baseline width (p. 548) bleed (p. 566) bonded stationary phase (p. 580) capacity factor (p. 551) capillary column (p. 562) capillary electrochromatography (p. 607) capillary electrophoresis (p. 597) capillary gel electrophoresis (p. 606) capillary zone electrophoresis (p. 604) chromatogram (p. 548) chromatography (p. 546) column chromatography (p. 546) countercurrent extraction (p. 546) cryogenic focusing (p. 568) electrokinetic injection (p. 603) electron capture detector (p. 570) electroosmotic flow (p. 598) electroosmotic flow velocity (p. 599) electropherogram (p. 597) electrophoresis (p. 597) electrophoretic mobility (p. 598) electrophoretic velocity (p. 598) exclusion limit (p. 595) flame ionization detector (p. 570) fronting (p. 555) gas chromatography (p. 563) gas–liquid chromatography (p. 564) gradient elution (p. 558) guard column (p. 579) headspace sampling (p. 567) high-performance liquid chromatography (p. 578) hydrodynamic injection (p. 602) inclusion limit (p. 593) ion-exchange chromatography (p. 590) ion-suppressor column (p. 592) isocratic elution (p. 582) Joule heating (p. 601) Kovat’s retention index (p. 575) liquid–solid adsorption chromatography (p. 590) longitudinal diffusion (p. 560) loop injector (p. 584) mass spectrum (p. 571) mass transfer (p. 561) micellar electrokinetic capillary chromatography (p. 606) micelle (p. 606) mobile phase (p. 546) normal-phase chromatography (p. 580) on-column injection (p. 568) open tubular column (p. 564) packed column (p. 564) peak capacity (p. 554) planar chromatography (p. 546) polarity index (p. 580) resolution (p. 549) retention time (p. 548) retention volume (p. 548) reverse-phase chromatography (p. 580) selectivity factor (p. 552) single-column ion chromatography (p. 593) size-exclusion chromatography (p. 593) solid-phase microextraction (p. 567) split injection (p. 568) splitless injection (p. 568) stacking (p. 603) stationary phase (p. 546) supercritical fluid chromatography (p. 596) support-coated open tubular column (p. 565) tailing (p. 555) temperature programming (p. 558) theoretical plate (p. 553) thermal conductivity detector (p. 569) van Deemter equation (p. 561) void time (p. 549) void volume (p. 549) wall-coated open tubular column (p. 565) zeta potential (p. 599) 1400-CH12 9/8/99 4:29 PM Page 609 610 Modern Analytical Chemistry Chromatography and electrophoresis are powerful analytical tech- niques that can separate a sample into its components while pro- viding a means for determining their concentration. Chromato- graphic separations utilize the selective partitioning of the sample’s components between a stationary phase that is immobi- lized within a column and a mobile phase that passes through the column. The effectiveness of a separation is described by the resolution between the chromatographic bands for two components and is a function of the component’s capacity factor, the column’s effi- ciency, and the column’s selectivity. A component’s capacity fac- tor is a measure of the degree to which it successfully partitions into the stationary phase, with larger capacity factors correspond- ing to more strongly retained components. The column’s selectiv- ity for two components is the ratio of the component’s capacity factors, providing a relative measure of the column’s ability to re- tain the two components. Column efficiency accounts for those factors that cause a component’s chromatographic band to in- crease in width during the separation. Column efficiency is de- fined in terms of the number of theoretical plates and the height of a theoretical plate, the latter of which is a function of a number of parameters, most notably the mobile phase’s flow rate. Chromato- graphic separations are optimized by increasing the number of theoretical plates, increasing the column’s selectivity, or increasing the component’s capacity factors. In gas chromatography (GC) the mobile phase is an inert gas, and the stationary phase is a nonpolar or polar organic liquid that is either coated on a particulate material and packed into a wide- bore column or coated on the walls of a narrow-bore capillary col- umn. Gas chromatography is useful for the analysis of volatile components. In high-performance liquid chromatography (HPLC) the mo- bile phase is either a nonpolar solvent (normal phase) or a polar solvent (reverse phase). A stationary phase of opposite polarity, which is bonded to a particulate material, is packed into a wide- bore column. HPLC can be applied to a wider range of samples than GC; however, the separation efficiency for HPLC is not as good as that for GC. Together, GC and HPLC account for the largest number of chromatographic separations. Other separation techniques, how- ever, find specialized applications. Of particular importance are: ion-exchange chromatography, which is useful for separating an- ions and cations; size-exclusion chromatography, which is useful for separating large molecules; and supercritical fluid chromatog- raphy, which combines many of the advantages of GC and HPLC for the analysis of materials that are not easily analyzed by either of these methods. In capillary zone electrophoresis a sample’s components are separated based on their ability to move through a conductive medium under the influence of an applied electric field. Because of the effect of electroosmotic flow, positively charged solutes elute first, with smaller, more highly charged cationic solutes eluting be- fore larger cations of lower charge. Neutral species elute without undergoing further separation. Finally, anions elute last, with smaller, more negatively charged anions being the last to elute. By adding a surfactant, neutral species also can be separated by micel- lar electrokinetic capillary chromatography. Electrophoretic sepa- rations also can take advantage of the ability of polymeric gels to separate solutes by size (capillary gel electrophoresis) and the abil- ity of solutes to partition into a stationary phase (capillary elec- trochromatography). In comparison to GC and HPLC, capillary electrophoresis provides faster and more efficient separations. 12L SUMMARY The first set of experiments describes the application of gas chromatography. These experiments encompass a variety of different types of samples, columns, and detectors. Most experiments may be easily modified to use available equipment and detectors. Elderd, D. M.; Kildahl, N. K.; Berka, L. H. “Experiments for Modern Introductory Chemistry: Identification of Arson Accelerants by Gas Chromatography,” J. Chem. Educ. 1996, 73, 675–677. Although aimed at the introductory class, this simple experiment provides a nice demonstration of the use of GC for a qualitative analysis. Students obtain chromatograms for several possible accelerants using headspace sampling and then analyze the headspace over a sealed sample of charred wood to determine the accelerant used in burning the wood. Separations are carried out using a wide-bore capillary column with a stationary phase of methyl 50% phenyl silicone and a flame ionization detector. 12M Suggested EXPERIMENTS The following experiments may be used to illustrate the application of chromatography and electrophoresis to a number of different types of samples. Experiments are grouped by the type of technique, and each is briefly annotated. Experiments —Continued 1400-CH12 9/8/99 4:29 PM Page 610 Chapter 12 Chromatographic and Electrophoretic Methods 611 Experiments Graham, R. C.; Robertson, J. K. “Analysis of Trihalomethanes in Soft Drinks,” J. Chem. Educ. 1988, 65, 735–737. Trihalomethanes are extracted from soft drinks using a liquid–liquid extraction with pentane. Samples are analyzed using a packed column containing 20% OV-101 on 80/100 mesh Gaschrom Q equipped with an electron capture detector. Kegley, S. E.; Hansen, K. J.; Cunningham, K. L. “Determination of Polychlorinated Biphenyls (PCBs) in River and Bay Sediments,” J. Chem. Educ. 1996, 73, 558–562. This somewhat lengthy experiment provides a thorough introduction to the use of GC for the analysis of trace-level environmental pollutants. Sediment samples are extracted by sonicating with 3 · 100-mL portions of 1:1 acetone:hexane. The extracts are then filtered and concentrated before bringing to a final volume of 10 mL. Samples are analyzed with a capillary column using a stationary phase of 5% phenylmethyl silicone, a splitless injection, and an ECD detector. Quach, D. T.; Ciszkowski, N. A.; Finlayson-Pitts, B. J. “A New GC-MS Experiment for the Undergraduate Instrumental Analysis Laboratory in Environmental Chemistry: Methyl-t- butyl Ether and Benzene in Gasoline,” J. Chem. Educ. 1998, 75, 1595–1598. This experiment describes the determination of methyl-t- butyl ether and benzene in gasoline using the method of standard additions. Two compounds naturally present at high concentration (o-xylene and toluene) are used as internal standards to correct for variations in the amount of sample injected into the GC. Because of the complexity of gasoline, single-ion monitoring is used to determine the signals for the analytes and internal standards. Separations are carried out using a capillary column with a stationary phase of 5% diphenyl/95% dimethylsiloxane. Rice, G. W. “Determination of Impurities in Whiskey Using Internal Standard Techniques,” J. Chem. Educ. 1987, 64, 1055–1056. An internal standard of 1-butanol is used to determine the concentrations of one or more of the following impurities commonly found in whiskey: acetaldehyde, methanol, ethyl acetate, 1-propanol, 2-methyl-1-propanol, acetic acid, 2-methyl-1-butanol and 3-methyl-1-butanol. A packed column using 5% Carbowax 20m on 80/120 Carbopak B and an FID detector were used. Rubinson, J. F.; Neyer-Hilvert, J. “Integration of GC-MS Instrumentation into the Undergraduate Laboratory: Separation and Identification of Fatty Acids in Commercial Fats and Oils,” J. Chem. Educ. 1997, 74, 1106–1108. Fatty acids from commercial fats and oils, such as peanut oil, are extracted with methanolic NaOH and made volatile by derivatizing with a solution of methanol/BF 3 . Separations are carried out using a capillary 5% phenylmethyl silicone column with MS detection. By searching the associated spectral library students are able to identify the fatty acids present in their sample. Quantitative analysis is by external standards. Rudzinski, W. E.; Beu, S. “Gas Chromatographic Determination of Environmentally Significant Pesticides,” J. Chem. Educ. 1982, 59, 614–615. Students analyze samples of orange juice that have been spiked with diazinon, malathion, and ethion. Samples are extracted with acetonitrile and then extracted with pet ether. The pesticide residues are then purified using an activated magnesium silicate (Florisil) column, eluting the pesticides with mixtures of pet ether and ethyl ether. After removing most of the solvent, samples are analyzed by GC using a packed glass column containing 1.5% SP-2250/1.95% SP- 2401 on 100/120 Supelcoport. Both electron capture and flame ionization detection are used. Welch, W. C.; Greco, T. G. “An Experiment in Manual Multiple Headspace Extraction for Gas Chromatography,” J. Chem. Educ. 1993, 70, 333–335. The principle of headspace sampling is introduced in this experiment using a mixture of methanol, chloroform, 1,2- dichloroethane, 1,1,1-trichloroethane, benzene, toluene, and p-xylene. Directions are given for evaluating the distribution coefficient for the partitioning of a volatile species between the liquid and vapor phase and for its quantitative analysis in the liquid phase. Both packed (OV-101) and capillary (5% phenyl silicone) columns were used. The GC is equipped with a flame ionization detector. Another experiment with the same focus is Ramachandran, B. R.; Allen, J. M.; Halpern, A. M. “Air-Water Partitioning of Environmentally Important Organic Compounds,” J. Chem. Educ. 1996, 73, 1058–1061. This experiment provides an alternative approach to measuring the partition coefficient (Henry’s law constant) for volatile organic compounds in water. A OV-101 packed column and flame ionization detector are used. Williams, K. R.; Pierce, R. E. “The Analysis of Orange Oil and the Aqueous Solubility of d-Limonene,” J. Chem. Educ. 1998, 75, 223–226. Two experiments are described in this paper. In the first experiment students determine the %w/w orange oil in a prepared sample by analyzing for d-limonene using anisole as an internal standard. Separations are accomplished using —Continued 1400-CH12 9/8/99 4:29 PM Page 611 612 Modern Analytical Chemistry Continued from page 611 Experiments a megabore open tubular column with a 5% phenylmethyl silicone bonded stationary phase and a thermal conductivity detector. In the second experiment the solubility of d-limonene is determined by equilibrating different volumes of d-limonene with water and measuring the amount of d-limonene in the overlying vapor phase using headspace sampling. Wong, J. W.; Ngim, K. K.; Shibamoto, T.; et al. “Determination of Formaldehyde in Cigarette Smoke,” J. Chem. Educ. 1997, 74, 1100–1103. Formaldehyde from cigarette smoke is collected by trapping the smoke in a 1-L separatory funnel and extracting into an aqueous solution. To aid in its detection, cysteamine is included in the aqueous extracting solution, leading to the formation of a thiazolidine derivative. Samples are analyzed using a DB-1 capillary column with a thermionic or flame photometric detector. Directions also are given for using an HPLC. Formaldehyde is derivatized using 2,4- dinitrophenylhydrazine, and samples are analyzed using a C 18 column with a UV detector set to 365 nm. Yang, M. J.; Orton, M. L., Pawliszyn, J. “Quantitative Determination of Caffeine in Beverages Using a Combined SPME-GC/MS Method,” J. Chem. Educ. 1997, 74, 1130–1132. Caffeine in coffee, tea, and soda is determined by a solid- phase microextraction using an uncoated silica fiber, followed by a GC analysis using a capillary SPB-5 column with an MS detector. Standard solutions are spiked with 13 C 3 caffeine as an internal standard. The second set of experiments describes the application of high-performance liquid chromatography. These experiments encompass a variety of different types of samples and a variety of common detectors. Bidlingmeyer, B. A.; Schmitz, S. “The Analysis of Artificial Sweeteners and Additives in Beverages by HPLC,” J. Chem. Educ. 1991, 68, A195–A200. The concentrations of benzoic acid, aspartame, caffeine, and saccharin in a variety of beverages are determined in this experiment. A C 18 column and a mobile phase of 80% v/v acetic acid (pH = 4.2) and 20% v/v methanol are used to effect the separation. A UV detector set to 254 nm is used to measure the eluent’s absorbance. The ability to adjust retention times by changing the mobile phase’s pH is also explored. DiNunzio, J. E. “Determination of Caffeine in Beverages by High Performance Liquid Chromatography,” J. Chem. Educ. 1985, 62, 446–447. The concentration of caffeine in a typical serving of coffee and soda is determined in this experiment. Separations are achieved using a C 18 column with a mobile phase of 30% v/v methanol in water, with UV detection at a wavelength of 254 nm. Ferguson, G. K. “Quantitative HPLC Analysis of an Analgesic/Caffeine Formulation: Determination of Caffeine,” J. Chem. Educ. 1998, 75, 467–469. The %w/w caffeine in an analgesic formulation is determined in this experiment. The separation uses a C 18 column with a mobile phase of 94.1% v/v water, 5.5% v/v acetonitrile, 0.2% v/v triethylamine, and 0.2% v/v acetic acid. A UV detector is set to 254 nm. Ferguson, G. K. “Quantitative HPLC Analysis of a Psychotherapeutic Medication: Simultaneous Determination of Amitriptyline Hydrochloride and Perphenazine,” J. Chem. Educ. 1998, 75, 1615–1618. This experiment describes a quantitative analysis for the active ingredients in a prescription antipsychotic medication. The separation makes use of a cyanopropyl derivatized column and a mobile phase of 70% v/v acetonitrile, 5% v/v methanol, and 25% v/v 0.1 M aqueous KH 2 PO 4 . A UV detector set to 215 nm is used to measure the eluent’s absorbance. Haddad, P.; Hutchins, S.; Tuffy, M. “High Performance Liquid Chromatography of Some Analgesic Compounds,” J. Chem. Educ. 1983, 60, 166–168. This experiment focuses on developing an HPLC separation capable of distinguishing acetylsalicylic acid, paracetamol, salicylamide, caffeine, and phenacetin. A C 18 column and UV detection are used to obtain chromatograms. Solvent parameters used to optimize the separation include the pH of the buffered aqueous mobile phase, the %v/v methanol added to the aqueous mobile phase, and the use of tetrabutylammonium phosphate as an ion-pairing reagent. Mueller, B. L.; Potts, L. W. “HPLC Analysis of an Asthma Medication,” J. Chem. Educ. 1988, 65, 905–906. This experiment describes the quantitative analysis of the asthma medication Quadrinal for the active ingredients theophylline, salicylic acid, phenobarbital, ephedrine HCl, and potassium iodide. Separations are carried out using a C 18 column with a mobile phase of 19% v/v acetonitrile, 80% v/v water, and 1% acetic acid. A small amount of triethylamine (0.03% v/v) is included to ensure the elution of ephedrine HCl. A UV detector set to 254 nm is used to record the chromatogram. —Continued 1400-CH12 9/8/99 4:29 PM Page 612 Chapter 12 Chromatographic and Electrophoretic Methods 613 Experiments Remcho, V. T.; McNair, H. M.; Rasmussen, H. T. “HPLC Method Development with the Photodiode Array Detector,” J. Chem. Educ. 1992, 69, A117–A119. A mixture of methyl paraben, ethyl paraben, propyl paraben, diethyl phthalate, and butyl paraben is separated by HPLC. This experiment emphasizes the development of a mobile- phase composition capable of separating the mixture. A photodiode array detector demonstrates the coelution of the two compounds. Siturmorang, M.; Lee, M. T. B.; Witzeman, L. K.; et al. “Liquid Chromatography with Electrochemical Detection (LC-EC): An Experiment Using 4-Aminophenol,” J. Chem. Educ. 1998, 75, 1035–1038. The use of an amperometric detector is emphasized in this experiment. Hydrodynamic voltammetry (see Chapter 11) is first performed to identify a potential for the oxidation of 4-aminophenol without an appreciable background current due to the oxidation of the mobile phase. The separation is then carried out using a C 18 column and a mobile phase of 50% v/v pH 5, 20 mM acetate buffer with 0.02 M MgCl 2 , and 50% v/v methanol. The analysis is easily extended to a mixture of 4-aminophenol, ascorbic acid, and catechol, and to the use of a UV detector. Tran, C. D.; Dotlich, M. “Enantiomeric Separation of Beta- Blockers by High Performance Liquid Chromatography,” J. Chem. Educ. 1995, 72, 71–73. This experiment introduces the use of a chiral column (a b -cyclodextrin-bonded C 18 column) to separate the beta-blocker drugs Inderal LA (S-propranolol and R-propranolol), Tenormim (DL-atenolol) and Lopressor (DL-metaprolol). The mobile phase was 90:10 (v/v) acetonitrile and water. A UV detector set to 254 nm is used to obtain the chromatogram. Van Arman, S. A.; Thomsen, M. W. “HPLC for Undergraduate Introductory Laboratories,” J. Chem. Educ. 1997, 74, 49–50. In this experiment students analyze an artificial RNA digest consisting of cytidine, uridine, thymidine, guanosine, and adenosine using a C 18 column and a mobile phase of 0.4% v/v triethylammonium acetate, 5% v/v methanol, and 94.6% v/v water. The chromatogram is recorded using a UV detector at a wavelength of 254 nm. Wingen, L. M.; Low, J. C.; Finlayson-Pitts, B. J. “Chromatography, Absorption, and Fluorescence: A New Instrumental Analysis Experiment on the Measurement of Polycyclic Aromatic Hydrocarbons in Cigarette Smoke,” J. Chem. Educ. 1998, 75, 1599–1603. The analysis of cigarette smoke for 16 different polyaromatic hydrocarbons is described in this experiment. Separations are carried out using a polymeric bonded silica column with a mobile phase of 50% v/v water, 40% v/v acetonitrile, and 10% v/v tetrahydrofuran. A notable feature of this experiment is the evaluation of two means of detection. The ability to improve sensitivity by selecting the optimum excitation and emission wavelengths when using a fluorescence detector is demonstrated. A comparison of fluorescence detection with absorbance detection shows that better detection limits are obtained when using fluorescence. The third set of experiments provides a few representative applications of ion chromatography. Bello, M. A.; Gustavo González, A. “Determination of Phosphate in Cola Beverages Using Nonsuppressed Ion Chromatography,” J. Chem. Educ. 1996, 73, 1174–1176. In this experiment phosphate is determined by single- column, or nonsuppressed, ion chromatography using an anionic column and a conductivity detector. The mobile phase is a mixture of n-butanol, acetonitrile, and water (containing sodium gluconate, boric acid, and sodium tetraborate). Kieber, R. J.; Jones, S. B. “An Undergraduate Laboratory for the Determination of Sodium, Potassium, and Chloride,” J. Chem. Educ. 1994, 71, A218–A222. Three techniques, one of which is ion chromatography, are used to determine the concentrations of three ions in solution. The combined concentrations of Na + and K + are determined by an ion exchange with H + , the concentration of which is subsequently determined by an acid–base titration using NaOH. Flame atomic absorption is used to measure the concentration of Na + , and K + is determined by difference. The concentration of Cl – is determined by ion- exchange chromatography on an anionic column using a mobile phase of HCO 3 – and CO 3 2– with ion suppression. A conductivity detector is used to record the chromatogram. Koubek, E.; Stewart, A. E. “The Analysis of Sulfur in Coal,” J. Chem. Educ. 1992, 69, A146–A148. Sulfur in coal is converted into a soluble sulfate by heating to 800 °C in the presence of MgO and Na 2 CO 3 . After dissolving in water, the concentration of sulfate is determined by single-column ion chromatography, using an anionic column and a mobile phase of 1 mM potassium hydrogen phthalate. A conductivity detector is used to record the chromatogram. Luo, P.; Luo, M. A.; Baldwin, R. P. “Determination of Sugars in Food Products,” J. Chem. Educ., 1993, 70, 679–681. —Continued 1400-CH12 9/8/99 4:29 PM Page 613 614 Modern Analytical Chemistry Continued from page 613 Experiments The concentrations of nine sugars (fucose, methylglucose, arabinose, glucose, fructose, lactose, sucrose, cellobiose, and maltose) in beer, milk, and soda are determined using an anionic column and a mobile phase of 0.1 M NaOH. Detection is by amperometry at a Cu working electrode. The last set of experiments provides examples of the application of capillary electrophoresis. These experiments encompass a variety of different types of samples and include examples of capillary zone electrophoresis and micellar electrokinetic chromatography. Conradi, S.; Vogt, C.; Rohde, E. “Separation of Enatiomeric Barbiturates by Capillary Electrophoresis Using a Cyclodextrin-Containing Run Buffer,” J. Chem. Educ. 1997, 74, 1122–1125. In this experiment the enantiomers of cyclobarbital and thiopental, and phenobarbital are separated using MEKC with cyclodextran as a chiral selector. By adjusting the pH of the buffer solution and the concentration and type of cyclodextran, students are able to find conditions in which the enantiomers of cyclobarbital and thiopental are resolved. Conte, E. D.; Barry, E. F.; Rubinstein, H. “Determination of Caffeine in Beverages by Capillary Zone Electrophoresis,” J. Chem. Educ. 1996, 73, 1169–1170. Caffeine in tea and coffee is determined by CZE using nicotine as an internal standard. The buffer solution is 50 mM sodium borate adjusted to pH 8.5 with H 3 PO 4 . A UV detector set to 214 nm is used to record the electropherograms. Hage, D. S.; Chattopadhyay, A.; Wolfe, C. A. C.; et al. “Determination of Nitrate and Nitrite in Water by Capillary Electrophoresis,” J. Chem. Educ. 1998, 75, 1588–1590. In this experiment the concentrations of NO 2 – and NO 3 – are determined by CZE using IO 4 – as an internal standard. The buffer solution is 0.60 M sodium acetate buffer adjusted to a pH of 4.0. A UV detector set to 222 nm is used to record the electropherogram. Janusa, M. A.; Andermann, L. J.; Kliebert, N. M.; et al. “Determination of Chloride Concentration Using Capillary Electrophoresis,” J. Chem. Educ. 1998, 75, 1463–1465. Directions are provided for the determination of chloride in samples using CZE. The buffer solution includes pyromellitic acid which allows the indirect determination of chloride by monitoring absorbance at 250 nm. McDevitt, V. L.; Rodríguez, A.; Williams, K. R. “Analysis of Soft Drinks: UV Spectrophotometry, Liquid Chromatography, and Capillary Electrophoresis,” J. Chem. Educ. 1998, 75, 625–629. Caffeine, benzoic acid, and aspartame in soft drinks are analyzed by three methods. Using several methods to analyze the same sample provides students with the opportunity to compare results with respect to accuracy, volume of sample required, ease of performance, sample throughput, and detection limit. Thompson, L.; Veening, H.; Strain, T. G. “Capillary Electrophoresis in the Undergraduate Instrumental Analysis Laboratory: Determination of Common Analgesic Formulations,” J. Chem. Educ. 1997, 74, 1117–1121. Students determine the concentrations of caffeine, acetaminophen, acetylsalicylic acid, and salicylic acid in several analgesic preparations using both CZE (70 mM borate buffer solution, UV detection at 210 nm) and HPLC (C 18 column with 3% v/v acetic acid mixed with methanol as a mobile phase, UV detection at 254 nm). Vogt, C.; Conradi, S.; Rhode, E. “Determination of Caffeine and Other Purine Compounds in Food and Pharmaceuticals by Micellar Electrokinetic Chromatography,” J. Chem. Educ. 1997, 74, 1126–1130. This experiment describes a quantitative analysis for caffeine, theobromine, and theophylline in tea, pain killers, and cocoa. Separations are accomplished by MEKC using a pH 8.25 borate–phosphate buffer with added SDS. A UV detector set to 214 nm is used to record the electropherogram. An internal standard of phenobarbital is included for quantitative work. Weber, P. L.; Buck, D. R. “Capillary Electrophoresis: A Fast and Simple Method for the Determination of the Amino Acid Composition of Proteins,” J. Chem. Educ. 1994, 71, 609–612. This experiment describes a method for determining the amino acid composition of cyctochrome c and lysozyme. The proteins are hydrolyzed in acid, and an internal standard of a -aminoadipic acid is added. Derivatization with naphthalene-2,3-dicarboxaldehyde gives derivatives that absorb at 420 nm. Separation is by MEKC using a buffer solution of 50 mM SDS in 20 mM sodium borate. 1400-CH12 9/8/99 4:29 PM Page 614 Chapter 12 Chromatographic and Electrophoretic Methods 615 1. The following data were obtained for four compounds separated on a 20-m capillary column. t r w Compound (min) (min) A 8.04 0.15 B 8.26 0.15 C 8.43 0.16 (a) Calculate the number of theoretical plates for each compound and the average number of theoretical plates for the column. (b) Calculate the average height of a theoretical plate. (c) Explain why it is possible for each compound to have a different number of theoretical plates. 2. Using the data from Problem 1, calculate the resolution and selectivity factors for each pair of adjacent compounds. For resolution, use both equations 12.1 and 12.21, and compare your results. Discuss how you might improve the resolution between compounds B and C. The retention time for an unretained solute is 1.19 min. 3. Using the chromatogram shown here, which was obtained on a 2-m column, determine values for t r , w, t r ￠ , k￠ , N, and H. 6. Complete the following table. N B a k￠ B R 100,000 1.05 0.50 10,000 1.10 1.50 10,000 4.0 1.00 1.05 3.0 1.75 7. Moody 19 studied the efficiency of a GC separation of 2- butanone on a dinonyl phthalate column. Evaluating the plate height as a function of flow rate gave a van Deemter equation for which A is 1.65 mm, B is 25.8 mm mL min –1 , and C is 0.0236 mm min mL –1 . (a) Prepare a graph of H versus u for flow rates in the range of 5–120 mL/min. (b) For what range of flow rates does each term in the van Deemter equation have the greatest effect? (c) What are the optimum flow rate and the height of a theoretical plate at that flow rate? (d) For open tubular columns the A term is no longer needed. If the B and C terms remain unchanged, what are the optimum flow rate and the height of a theoretical plate at that flow rate? (e) How many more theoretical plates will there be in the open tubular column compared with the packed column? (f ) Equation 12.28 is written in terms of the linear velocity (centimeters per second), yet we have evaluated it in this problem using the flow rate (milliliters per minute). Why can we do this? 8. Hsieh and Jorgenson 20 prepared 12–33-m m HPLC columns packed with 5.44-m m spherical stationary phase particles. To evaluate these columns they measured reduced plate height, h, as a function of reduced flow rate, v, where d p is the particle diameter, and D m is the solute’s diffusion coefficient in the mobile phase. The data were analyzed using van Deemter plots, with a portion of their results summarized in the following table for the solute norepinephrine. Column Internal Diameter (lm) ABC 33 0.63 1.32 0.10 33 0.67 1.30 0.08 23 0.40 1.34 0.09 23 0.58 1.11 0.09 17 0.31 1.47 0.09 17 0.40 1.41 0.11 12 0.22 1.53 0.11 12 0.19 1.27 0.12 v ud D = p m h H d = p 12N PROBLEMS Signal Retention time (s) 0 500100 Nonretained solutes 200 300 400 4. Using the partial chromatogram shown here, determine the resolution between the two solute bands. Signal Retention time (s) 300 450325 350 375 400 425 5. The chromatogram in Problem 4 was obtained on a 2-m column with a column dead time of 50 s. How long a column is needed to achieve a resolution of 1.5? What height of a theoretical plate is needed to achieve a resolution of 1.5 without increasing the length of the column? 1400-CH12 9/8/99 4:29 PM Page 615 (a) Construct separate van Deemter plots using the data in the first and last rows for flow rates in the range 0.7–15. Determine the optimum flow rate and plate height for each case, given d p = 5.44 m m and D m = 6.23 · 10 –6 cm 2 s –1 . (b) The A term in the van Deemter equation appears to be strongly correlated with the column’s inner diameter, with smaller diameter columns providing smaller values of A. Explain why this effect is seen (Hint: Consider how many particles can fit across a capillary of each diameter). 9. Method 12.1 describes the analysis of the trihalomethanes CHCl 3 , CHBr 3 , CHCl 2 Br, and CHClBr 2 in drinking water using a packed column with a nonpolar stationary phase. Predict the order in which these four trihalomethanes will elute. 10. A mixture of n-heptane, tetrahydrofuran, 2-butanone, and n-propanol elutes in this order when using a polar stationary phase such as Carbowax. The elution order is exactly the opposite when using a nonpolar stationary phase such as polydimethyl siloxane. Explain the order of elution in each case. 11. The analysis of trihalomethanes in drinking water is described in Method 12.1. A single standard gives the following results when carried through the described procedure. Concentration Trihalomethane (ppb) Peak Area CHCl 3 1.30 1.35 · 10 4 CHCl 2 Br 0.90 6.12 · 10 4 CHClBr 2 4.00 1.71 · 10 4 CHBr 3 1.20 1.52 · 10 4 Analysis of water from a drinking fountain gives areas of 1.56 · 10 4 , 5.13 · 10 4 , 1.49 · 10 4 , and 1.76 · 10 4 for CHCl 3 , CHCl 2 Br, CHClBr 2 , and CHBr 3 , respectively. Determine the concentration of each of the trihalomethanes in the sample of water. 12. Zhou and colleagues determined the %w/w H 2 O in methanol by GC, using a capillary column coated with a nonpolar stationary phase and a thermal conductivity detector. 21 A series of calibration standards gave the following results. Peak Height % w/w H 2 O (arb. units) 0.00 1.15 0.0145 2.74 0.0472 6.33 0.0951 11.58 0.1757 20.43 0.2901 32.97 (a) What is the %w/w H 2 O in a sample giving a peak height of 8.63? (b) The %w/w H 2 O in a freeze-dried antibiotic is determined in the following manner. A 0.175-g sample is placed in a vial along with 4.489 g of methanol. Water in the vial extracts into the methanol. Analysis of the sample gave a peak height of 13.66. What is the %w/w H 2 O in the antibiotic? 13. Loconto and co-workers describe a method for determining trace levels of water in soil. 22 The method takes advantage of the reaction of water with calcium carbide, CaC 2 , to produce acetylene gas, C 2 H 2 . By carrying out the reaction in a sealed vial, the amount of acetylene produced may be determined by sampling the headspace. In a typical analysis a sample of soil is placed in a sealed vial with CaC 2 . Analysis of the headspace gave a blank-corrected signal of 2.70 · 10 5 . A second sample is prepared in the same manner except that a standard addition of 5.0 mg H 2 O/g solid is added, giving a blank-corrected signal of 1.06 · 10 6 . Determine the number of milligrams of H 2 O/g soil in the soil sample. 14. Van Atta and Van Atta used gas chromatography to determine the %v/v methyl salicylate in rubbing alcohol. 23 A set of standard additions was prepared by transferring 20.00 mL of rubbing alcohol to separate 25-mL volumetric flasks and pipeting 0.00 mL, 0.20 mL, and 0.50 mL of methyl salicylate to the flasks. All three flasks were then diluted to volume using isopropanol. Analysis of the three samples gave peak heights for methyl salicylate of 57.00 mm, 88.5 mm, and 132.5 mm, respectively. Determine the %v/v methyl salicylate in the rubbing alcohol. 15. The amount of camphor in an analgesic ointment can be determined by GC using the method of internal standards. 24 A standard sample was prepared by placing 45.2 mg of camphor and 2.00 mL of a 6.00 mg/mL internal standard solution of terpene hydrate in a 25-mL volumetric flask and diluting to volume with CCl 4 . When an approximately 2-m L sample of the standard was injected, the FID signals for the two components were measured (in arbitrary units) as 67.3 for camphor and 19.8 for terpene hydrate. A 53.6-mg sample of an analgesic ointment was prepared for analysis by placing it in a 50-mL Erlenmeyer flask along with 10 mL of CCl 4 . After heating to 50 °C in a water bath, the sample was cooled to below room temperature and filtered. The residue was washed with two 5-mL portions of CCl 4 , and the combined filtrates were collected in a 25-mL volumetric flask. After adding 2.00 mL of the internal standard solution, the contents of the flask were diluted to volume with CCl 4 . Analysis of an approximately 2-m L sample gave FID signals of 13.5 for the terpene hydrate and 24.9 for the camphor. Report the %w/w camphor in the analgesic ointment. 16. The concentration of pesticide residues on agricultural products, such as oranges, may be determined by GC-MS. 25 Pesticide residues are extracted from the sample using methylene chloride, and the concentrations of the extracted pesticides are concentrated by evaporating the methylene chloride to a smaller volume. Calibration is accomplished using anthracene-d 10 as an internal standard. In a study to determine the parts per billion of heptachlor epoxide on oranges, a 50.0-g sample of orange rinds was chopped and extracted with 50.00 mL of methylene chloride. After removing any insoluble material by filtration, the methylene chloride was reduced in volume, spiked with a known amount of the internal standard, and diluted to 10 mL in a volumetric flask. Analysis of the sample gives a peak–area ratio (A anal / A int stan ) of 0.108. A series of calibration standards, each 616 Modern Analytical Chemistry 1400-CH12 9/8/99 4:29 PM Page 616 containing the same amount of anthracene-d 10 as the sample, give the following results. ppb Heptachlor Epoxide A anal /A int stan 20.0 0.065 60.0 0.153 200.0 0.637 500.0 1.554 1000.0 3.198 Report the concentration of heptachlor epoxide residue (in nanograms per gram) on the oranges. 17. The adjusted retention times for octane, toluene, and nonane on a particular GC column are 15.98 min, 17.73 min, and 20.42 min, respectively. What is the retention index for all three compounds? 18. The following data were collected for a series of normal alkanes using a stationary phase of Carbowax 20M. t r ￠ Alkane (min) pentane 0.79 hexane 1.99 heptane 4.47 octane 14.12 nonane 33.11 What is the retention index for a compound whose adjusted retention time is 9.36 min? 19. The following data have been reported for the gas chromatographic analysis of p-xylene and methylisobutylketone (MIBK) on a capillary column. 8 Injection t r Peak Peak Width Mode Compound (min) Area (min) split MIBK 1.878 54285 0.028 p-xylene 5.234 123483 0.044 splitless MIBK 3.420 2493005 1.057 p-xylene 5.795 3396656 1.051 Explain the difference in the retention times, the peak areas, and the peak widths when switching from a split injection to a splitless injection. 20. Otto and Wegscheider report the following capacity factors for the reverse phase separation of 2-aminobenzoic acid on a C 18 column when using 10% v/v methanol as a mobile phase. 26 pH k￠ 2.0 10.5 3.0 16.7 4.0 15.8 5.0 8.0 6.0 2.2 7.0 1.8 Explain the changes in capacity factor. 21. Haddad and associates report the following capacity factors for the reverse-phase separation of salicylamide (k ￠ sal ) and caffeine (K ￠ caff ). 27 %v/v methanol 30% 35% 40% 45% 50% 55% k ￠ sal 2.4 1.6 1.6 1.0 0.7 0.7 k ￠ caff 4.3 2.8 2.3 1.4 1.1 0.9 Explain the changes in capacity factor. What is the advantage of using a mobile phase with a smaller %v/v methanol? Are there any disadvantages? 22. Suppose that you are to separate a mixture of benzoic acid, aspartame, and caffeine in a diet soda. The following information is available to you. t r in Aqueous Mobile Phase Buffered to a pH of Compound 3.0 3.5 4.0 4.5 benzoic acid 7.4 7.0 6.9 4.4 aspartame 5.9 6.0 7.1 8.1 caffeine 3.6 3.7 4.1 4.4 (a) Explain the change in retention time for each compound. (b) Plot retention time versus pH for each compound on the same graph, and identify a pH level that will yield an acceptable separation. 23. The composition of a multivitamin tablet is conveniently determined using an HPLC with a diode array UV/Vis detector. A 5-m L standard sample containing 170 ppm vitamin C, 130 ppm niacin, 120 ppm niacinamide, 150 ppm pyridoxine, 60 ppm thiamine, 15 ppm folic acid, and 10 ppm riboflavin is injected into the HPLC, giving signals (in arbitrary units) of, respectively, 0.22, 1.35, 0.90, 1.37, 0.82, 0.36, and 0.29. The multivitamin tablet is prepared for analysis by grinding into a powder and transferring to a 125-mL Erlenmeyer flask containing 10 mL of 1% v/v NH 3 in dimethyl sulfoxide. After sonicating in an ultrasonic bath for 2 min, 90 mL of 2% acetic acid is added, and the mixture is stirred for 1 min and sonicated at 40 °C for 5 min. The extract is then filtered through a 0.45-m m membrane filter. Injection of a 5-m L sample into the HPLC gives signals of 0.87 for vitamin C, 0.00 for niacin, 1.40 for niacinamide, 0.22 for pyridoxine, 0.19 for thiamine, 0.11 for folic acid, and 0.44 for riboflavin. Report the number of milligrams of each vitamin present in the tablet. 24. The amount of caffeine in an analgesic tablet was determined by HPLC using a normal calibration curve. Standard solutions of caffeine were prepared and analyzed using a 10-m L fixed-volume injection loop. Results for the standards are summarized in the following table. Concentration of Standards Signal (ppm) (arbitrary units) 50.0 8354 100.0 16925 150.0 25218 200.0 33584 250.0 42002 Chapter 12 Chromatographic and Electrophoretic Methods 617 1400-CH12 9/8/99 4:29 PM Page 617 The sample was prepared by placing a single analgesic tablet in a small beaker and adding 10 mL of methanol. After allowing the sample to dissolve, the contents of the beaker, including the insoluble binder, were quantitatively transferred to a 25-mL volumetric flask and diluted to volume with methanol. The sample was then filtered, and a 1.00-mL aliquot was transferred to a 10-mL volumetric flask and diluted to volume with methanol. When analyzed by HPLC, the signal for the caffeine was found to be 21469. Report the number of milligrams of caffeine in the analgesic tablet. 25. Kagel and Farwell report a reverse-phase HPLC method for determining the concentration of acetylsalicylic acid (ASA) and caffeine (CAF) in analgesic tablets using salicylic acid (SA) as an internal standard. 28 A series of standards was prepared by adding known amounts of acetylsalicylic acid and caffeine to 250-mL Erlenmeyer flasks and adding 100 mL of methanol. A 10.00-mL aliquot of a standard solution of salicylic acid was then added to each. The following results are obtained for a typical set of standard solutions. Peak Height Peak Height Milligrams Milligrams Ratio Ratio Standard ASA CAF ASA/SA CAF/SA 1 200.0 20.0 20.5 10.6 2 250.0 40.0 25.1 23.0 3 300.0 60.0 30.9 36.8 A sample of an analgesic tablet was placed in a 250-mL Erlenmeyer flask and dissolved in 100 mL of methanol. After adding a 10.00-mL portion of the internal standard, the solution was filtered. Analysis of the sample gave a peak height ratio of 23.2 for ASA and 17.9 for CAF. (a) Determine the number of milligrams ASA and CAF in the tablet. (b) Why was it necessary to filter the sample? (c) The directions indicate that approximately 100 mL of methanol is used to dissolve the standards and samples. Why is it not necessary to measure this volume more precisely? (d) In the presence of moisture, ASA decomposes to SA and acetic acid. What complication might this present for this analysis? How might you evaluate whether this is a problem? 26. Bohman and colleagues described a reverse-phase HPLC method for the quantitative analysis of vitamin A in food using the method of standard additions. 29 In a typical example, a 10.067-g sample of cereal is placed in a 250-mL Erlenmeyer flask along with 1 g of sodium ascorbate, 40 mL of ethanol, and 10 mL of 50% w/v KOH. After refluxing for 30 min, 60 mL of ethanol is added, and the solution is cooled to room temperature. Vitamin A is extracted using three 100-mL portions of hexane. The combined portions of hexane are evaporated, and the residue containing vitamin A is transferred to a 5-mL volumetric flask and diluted to volume with methanol. A standard addition is prepared in a similar manner using a 10.093-g sample of the cereal and spiking it with 0.0200 mg of vitamin A. Injecting the sample and standard addition into the HPLC gives peak areas of 6.77 · 10 3 and 1.32 · 10 4 , respectively. Report the vitamin A content of the sample in milligrams/100 g cereal. 27. Ohta and Tanaka reported a method for the simultaneous analysis of several inorganic anions and the cations Mg 2+ and Ca 2+ in water by ion-exchange chromatography. 30 The mobile phase includes 1,2,4-benzenetricarboxylate, which absorbs strongly at 270 nm. Indirect detection of the analytes is possible because their presence in the detector leads to a decrease in absorbance. Unfortunately, Ca 2+ and Mg 2+ , which are present at high concentrations in many environmental waters, form stable complexes with 1,2,4- benzenetricarboxylate that interfere with the analysis. (a) Adding EDTA to the mobile phase eliminates the interference caused by Ca 2+ and Mg 2+ ; explain why. (b) A standard solution containing 1.0 M NaHCO 3 , 0.20 mM NaNO 2 , 0.20 mM MgSO 4 , 0.10 mM CaCl 2 , and 0.10 mM Ca(NO 3 ) 2 gives the following typical peak areas (arbitrary units). Ion HCO 3 – Cl – NO 2 – NO 3 – Peak Area 373.5 322.5 264.8 262.7 Ion Ca 2+ Mg 2+ SO 4 2– Peak Area 458.9 352.0 341.3 Analysis of a river water sample (pH of 7.49) gives the following results. Ion HCO 3 – Cl – NO 2 – NO 3 – Peak Area 310.0 403.1 3.97 157.6 Ion Ca 2+ Mg 2+ SO 4 2– Peak Area 734.3 193.6 324.3 Determine the concentration of each ion in the sample of rain water. (c) The detection of HCO 3 – actually gives the total concentration of carbonate in solution ([CO 3 2– ] + [HCO 3 – ] + [H 2 CO 3 ]). Given that the pH of the water is 7.49, what is the actual concentration of HCO 3 – ? (d) An independent analysis gives the following additional concentrations. [Na + ] = 0.60 mM [NH 4 + ] = 0.014 mM [K + ] = 0.046 mM A solution’s ionic balance is defined as the ratio of the total cation charge to the total anion charge. Determine the ion balance for this sample of water, and comment on whether the result is reasonable. 28. The concentrations of Cl – , NO 3 – , and SO 4 2– may be determined by ion chromatography. A 50-m L standard sample of 10.0-ppm Cl – , 2.00-ppm NO 3 – , and 5.00-ppm SO 4 2– gave signals (in arbitrary units) of 59.3, 16.1, and 6.08, respectively. A sample of effluent from a wastewater treatment plant was diluted tenfold, and a 50-m L portion gave signals of 44.2 for 618 Modern Analytical Chemistry 1400-CH12 9/8/99 4:29 PM Page 618 Cl – , 2.73 for NO 3 – , and 5.04 for SO 4 2– . Report the parts per million of each anion in the effluent sample. 29. A series of polyvinylpyridine standards of different molecular weight were analyzed by size-exclusion chromatography, yielding the following results. Retention Volume Formula Weight (mL) 600,000 6.42 100,000 7.98 20,000 9.30 3,000 10.94 When a preparation of polyvinylpyridine of unknown formula weight was analyzed the retention volume was found to be 8.45. Report the average formula weight for the preparation. 30. Diet soft drinks contain appreciable quantities of aspartame, benzoic acid, and caffeine. What is the expected order of elution for these compounds in a capillary zone electrophoresis separation using a pH 9.4 buffer solution, given that aspartame has pK a values of 2.964 and 7.37, benzoic acid’s pK a is 4.2, and the pK a for caffeine is less than 0. 32. The analysis of NO 3 – in aquarium water was carried out by CZE using IO 4 – as an internal standard. A standard solution of 15.0-ppm NO 3 – and 10.0-ppm IO 4 – gives peak heights (arbitrary units) of 95.0 and 100.1, respectively. A sample of water from an aquarium is diluted 1:100, and sufficient internal standard added to make its concentration 10.0 ppm. Analysis gives signals of 29.2 and 105.8 for NO 3 – and IO 4 – , respectively. Report the parts per million of NO 3 – in the sample of aquarium water. 33. Suggest conditions for separating a mixture of 2-aminobenzoic acid (pK a1 = 2.08, pK a2 = 4.96), benzylamine (pK a = 9.35), and 4-methylphenol (pK a = 10.26) by capillary zone electrophoresis. 34. McKillop and associates have examined the electrophoretic separation of alkylpyridines by CZE. 32 Separations were carried out using either 50-m m or 75-m m inner diameter capillaries, with a total length of 57 cm and a length of 50 cm from the point of injection to the detector. The run buffer was a pH 2.5 lithium phosphate buffer. Separations were achieved using an applied voltage of 15 kV. The electroosmotic flow velocity, as measured using a neutral marker, was found to be 6.398 · 10 –5 cm 2 V –1 s –1 . The diffusion coefficient, D, for the alkylpyridines may be taken to be 1.0 · 10 –5 cm 2 s –1 .(a) Calculate the electrophoretic mobility for 2-ethylpyridine, given that its elution time is 8.20 min. (b) How many theoretical plates are there for 2-ethylpyridine? (c) The electrophoretic mobilities for 3-ethylpyridine and 4-ethylpyridine are 3.366 · 10 –4 cm 2 V –1 s –1 and 3.397 · 10 –4 cm 2 V –1 s –1 , respectively. What is the expected resolution between these two alkylpyridines? (d) Explain the trends in electrophoretic mobility shown in the following table. Alkylpyridine l ep (cm 2 V –1 s –1 ) 2-methylpyridine 3.581 · 10 –4 2-ethylpyridine 3.222 · 10 –4 2-propylpyridine 2.923 · 10 –4 2-pentylpyridine 2.534 · 10 –4 2-hexylpyridine 2.391 · 10 –4 (e) Explain the trends in electrophoretic mobility shown in the following table. Alkylpyridine l ep (cm 2 V –1 s –1 ) 2-ethylpyridine 3.222 · 10 –4 3-ethylpyridine 3.366 · 10 –4 4-ethylpyridine 3.397 · 10 –4 (f) The pK a for pyridine is 5.229. At a pH of 2.5 the electrophoretic mobility of pyridine is 4.176 · 10 –4 cm 2 V –1 s –1 . What is the expected electrophoretic mobility if the run buffer’s pH is 7.5? Chapter 12 Chromatographic and Electrophoretic Methods 619 + H 3 N N H O COOH OCH 3 aspartame O COOH benzoic acid H 3 C CH 3 CH 3 N N O O N N caffeine 31. Janusa and co-workers report the determination of chloride by CZE. 31 Analysis of a series of external standards gives the following calibration curve. Area = -883 + 5590(ppm Cl – ) A standard sample of 57.22% w/w Cl – was analyzed by placing 0.1011 g in a 100-mL volumetric flask and diluting to volume. Three unknowns were prepared by pipeting 0.250 mL, 0.500 mL, and 0.750 mL of the bulk unknown into separate 50-mL volumetric flasks and diluting to volume. Analysis of the three unknowns gave areas of 15310, 31546, and 47582, respectively. Evaluate the accuracy of this analysis. 1400-CH12 9/8/99 4:29 PM Page 619 620 Modern Analytical Chemistry 1. Craig, L. C. J. Biol. Chem. 1944, 155, 519. 2. Martin, A. J. P.; Synge, R. L. M. Biochem. J. 1941, 35, 1358. 3. Giddings, J. C. Unified Separation Science. Wiley-Interscience: New York, 1991. 4. Hawkes, S. J. J. Chem. Educ. 1983, 60, 393–398. 5. Kennedy, R. T.; Jorgenson, J. W. Anal. Chem. 1989, 61, 1128–1135. 6. Hinshaw, J. V. LC?GC, 1993, 11, 644–648. 7. Grob, K. Anal. Chem. 1994, 66, 1009A–1019A. 8. Marriott, P. J.; Carpenter, P. D. J. Chem. Educ. 1996, 73, 96–99. 9. Method 6232B as published in Standard Methods for the Examination of Water and Wastewater, 18th ed. American Public Health Association: Washington, D.C., 1992. 10. Novotny, M. Science 1989, 246, 51–57. 11. (a) Jorgenson, J. W.; Guthrie, E. J. J. Chromatog. 1983, 255, 335; (b) Kennedy, R. T.; Oates, M. D.; Cooper, B. R.; et al. Science 1989, 246, 57–63. 12P REFERENCES The following texts provide a good introduction to the broad field of separations, including chromatography and electrophoresis. Giddings, J. C. Unified Separation Science. Wiley-Interscience: New York, 1991. Karger, B. L.; Snyder, L. R.; Harvath, C. An Introduction to Separation Science. Wiley-Interscience: New York, 1973. Miller, J. M. Separation Methods in Chemical Analysis. Wiley- Interscience: New York, 1975. A more recent discussion of peak capacity is presented in the following paper. Shen, Y.; Lee, M. “General Equation for Peak Capacity in Column Chromatography,” Anal. Chem. 1998, 70, 3853–3856. The following references may be consulted for more information on gas chromatography. Grob, R. L., ed. Modern Practice of Gas Chromatography. Wiley- Interscience: New York, 1972. Hinshaw, J. V. “A Compendium of GC Terms and Techniques,” LC?GC 1992, 10, 516–522. Ioffe, B. V.; Vitenberg, A. G. Head-Space Analysis and Related Methods in Gas Chromatography. Wiley-Interscience: New York, 1982. Kitson, F. G.; Larsen, B. S.; McEwen, C. N. Gas Chromatography and Mass Spectrometry: A Practical Guide. Academic Press: San Diego, 1996. The following references may be consulted for more information on high-performance liquid chromatography. Dorschel, C. A.; Ekmanis, J. L.; Oberholtzer, J. E.; et al. “LC Detectors,” Anal. Chem. 1989, 61, 951A–968A. Simpson, C. F., ed. Techniques in Liquid Chromatography. Wiley- Hayden: Chichester, England, 1982. Snyder, L. R.; Glajch, J. L.; Kirkland, J. J. Practical HPLC Method Development. Wiley-Interscience: New York, 1988. The following references may be consulted for more information on ion chromatography. Shpigun, O. A.; Zolotov, Y. A. Ion Chromatography in Water Analysis. Ellis Horwood: Chichester, England, 1988. Smith, F. C. Jr.; Chang, R. C. The Practice of Ion Chromatography. Wiley-Interscience: New York, 1983. The following references may be consulted for more information on supercritical fluid chromatography. Palmieri, M. D. “An Introduction to Supercritical Fluid Chromatography. Part I: Principles and Applications,” J. Chem. Educ. 1988, 65, A254–A259. Palmieri, M. D. “An Introduction to Supercritical Fluid Chromatography. Part II: Applications and Future Trends,” J. Chem. Educ. 1989, 66, A141–A147. The following references may be consulted for more information on capillary electrophoresis. Baker, D. R. Capillary Electrophoresis. Wiley-Interscience: New York, 1995. Copper, C. L. “Capillary Electrophoresis: Part I. Theoretical and Experimental Background,” J. Chem. Educ. 1998, 75, 343–347. Copper, C. L.; Whitaker, K. W. “Capillary Electrophoresis: Part II. Applications,” J. Chem. Educ. 1998, 75, 347–351. The application of spreadsheets and computer programs for modeling chromatography is described in the following papers. Abbay, G. N.; Barry, E. F.; Leepipatpiboon, S.; et al. “Practical Applications of Computer Simulation for Gas Chromatography Method Development,” LC?GC 1991, 9, 100–114. Drouen, A.; Dolan, J. W.; Snyder, L. R.; et al. “Software for Chromatographic Method Development,” LC?GC 1991, 9, 714–724. Kevra, S. A.; Bergman, D. L.; Maloy, J. T. “A Computational Introduction to Chromatographic Bandshape Analysis,” J. Chem. Educ. 1994, 71, 1023–1028. Sundheim, B. R. “Column Operations: A Spreadsheet Model.” J. Chem. Educ. 1992, 69, 1003–1005. 12O SUGGESTED READINGS 1400-CH12 9/8/99 4:29 PM Page 620 12. Snyder, L. R.; Glajch, J. L.; Kirkland, J. J. Practical HPLC Method Development. Wiley-Interscience: New York, 1988. 13. Foley, J. P. Chromatography 1987, 2(6), 43–51. 14. Yeung, E. S. LC?GC 1989, 7, 118–128. 15. Smyth, W. F. Analytical Chemistry of Complex Matrices. Wiley Teubner: Chichester, England, 1996, pp. 187–189. 16. Baker, D. R. Capillary Electrophoresis. Wiley-Interscience: New York, 1995. 17. Jones, W. R.; Jandik, P. J. Chromatog. 1992, 608, 385–393. 18. Smyth, W. F. Analytical Chemistry of Complex Matrices. Wiley Teubner: Chichester, England, 1996, pp. 154–156. 19. Moody, H. W. J. Chem. Educ. 1982, 59, 218–219. 20. Hsieh, S.; Jorgenson, J. W. Anal. Chem. 1996, 68, 1212–1217. 21. Zhou, X.; Hines, P. A.; White, K. C.; et al. Anal. Chem. 1998, 70, 390–394. 22. Loconto, P. R.; Pan, Y. L.; Voice, T. C. LC?GC 1996, 14, 128–132. 23. Van Atta, R. E.; Van Atta, R. L. J. Chem. Educ. 1980, 57, 230–231. 24. Pant, S. K.; Gupta, P. N.; Thomas, K. M.; et al. LC?GC 1990, 8, 322–325. 25. Feigel, C. Varian GC/MS Application Note, Number 22. 26. Otto, M.; Wegscheider, W. J. Chromatogr. 1983, 258, 11–22. 27. Haddad, P.; Hutchins, S.; Tuffy, M. J. Chem. Educ. 1983, 60, 166–168. 28. Kagel, R. A.; Farwell, S. O. J. Chem. Educ. 1983, 60, 163–166. 29. Bohman, O.; Engdahl, K. A.; Johnsson, H. J. Chem. Educ. 1982, 59, 251–252. 30. Ohta, K.; Tanaka, K. Anal. Chim. Acta 1998, 373, 189–195. 31. Janusa, M. A.; Andermann, L. J.; Kliebert, N. M.; et al. J. Chem. Educ. 1998, 75, 1463–1465. 32. McKillop, A. G.; Smith, R. M.; Rowe, R. C.; et al. Anal. Chem. 1999, 71, 497–503. Chapter 12 Chromatographic and Electrophoretic Methods 621 1400-CH12 9/8/99 4:29 PM Page 621 622 \Chapter 1 3 Kinetic Methods of Analysis Asystem under thermodynamic control is in a state of equilibrium, and its signal has a constant, or steady-state value (Figure 13.1a). When a system is under kinetic control, however, its signal changes with time (Figure 13.1b) until equilibrium is established. Thus far, the techniques we have considered have involved measurements made when the system is at equilibrium. By changing the time at which measurements are made, an analysis can be carried out under either thermodynamic control or kinetic control. For example, one method for determining the concentration of NO 2 — in groundwater involves the diazotization reaction shown in Figure 13.2. 1 The final product, which is a reddish-purple azo dye, absorbs visible light at a wavelength of 543 nm. Since the concentration of dye is determined by the amount of NO 2 — in the original sample, the solution s absorbance can be used to determine the concentration of NO 2 — . The reaction in the second step, however, is not instantaneous. To achieve a steady-state signal, such as that in Figure 13.1a, the absorbance is measured following a 10-min delay. By measuring the signal during the 10-min development period, information about the rate of the reaction is obtained. If the reaction s rate is a function of the concentration of NO 2 — , then the rate also can be used to determine its concentration in the sample. 2 There are many potential advantages to kinetic methods of analysis, perhaps the most important of which is the ability to use chemical reactions that are slow to reach equilibrium. In this chapter we examine three techniques that rely on measurements made while the analytical system is under kinetic rather than thermodynamic control: chemical kinetic techniques, in which the rate of a chemical reaction is measured; radiochemical techniques, in which a radioactive element s rate of nuclear decay is measured; and flow injection analysis, in which the analyte is injected into a continuously flowing carrier stream, where its mixing and reaction with reagents in the stream are controlled by the kinetic processes of convection and diffusion. 1400-CH13 9/8/99 4:32 PM Page 622 Figure 13.1 Plot of signal versus time for an analytical system that is under (a) thermodynamic control; and (b) under kinetic control. Chapter 13 Kinetic Methods of Analysis 623 Signal Time Signal Time H 2 NO 3 S Step 1 sulfanilamide diazonium ion NH 2 + NO 2 – + 2H + H 2 NO 3 SNN + N + 2H 2 O Step 2 diazonium ion H 2 NO 3 S C 2 H 4 NH 3 N N-(1-naphthyl)-ethylenediamine dihydrochloride N + + NNH 2 + + azo dye H 2 NO 3 S C 2 H 4 NH 3 N N + H + NNH 2 + + (a) (b) Figure 13.2 Analytical scheme for the analysis of NO 2 – in groundwater. 1 3 A Methods Based on Chemical Kinetics The earliest examples of analytical methods based on chemical kinetics, which date from the late nineteenth century, took advantage of the catalytic activity of en- zymes. Typically, the enzyme was added to a solution containing a suitable sub- strate, and the reaction between the two was monitored for a fixed time. The en- zyme’s activity was determined by measuring the amount of substrate that had reacted. Enzymes also were used in procedures for the quantitative analysis of hy- drogen peroxide and carbohydrates. The application of catalytic reactions contin- ued in the first half of the twentieth century, and developments included the use of nonenzymatic catalysts, noncatalytic reactions, and differences in reaction rates when analyzing samples with several analytes. 1400-CH13 9/8/99 4:32 PM Page 623 Despite the variety of methods that had been developed, by 1960 kinetic meth- ods were no longer in common use. The principal limitation to a broader accep- tance of chemical kinetic methods was their greater susceptibility to errors from un- controlled or poorly controlled variables, such as temperature and pH, and the presence of interferents that activate or inhibit catalytic reactions. Many of these limitations, however, were overcome during the 1960s, 1970s, and 1980s with the development of improved instrumentation and data analysis methods compensat- ing for these errors. 3 1 3 A.1 Theory and Practice Every chemical reaction occurs at a finite rate and, therefore, can potentially serve as the basis for a chemical kinetic method of analysis. To be effective, however, the chemical reaction must meet three conditions. First, the rate of the chemical reac- tion must be fast enough that the analysis can be conducted in a reasonable time, but slow enough that the reaction does not approach its equilibrium position while the reagents are mixing. As a practical limit, reactions reaching equilibrium within 1 s are not easily studied without the aid of specialized equipment allowing for the rapid mixing of reactants. A second requirement is that the rate law for the chemical reaction must be known for the period in which measurements are made. In addition, the rate law should allow the kinetic parameters of interest, such as rate constants and concen- trations, to be easily estimated. For example, the rate law for a reaction that is first order in the concentration of the analyte, A, is expressed as 13.1 where k is the reaction’s rate constant. As shown in Appendix 5,* the integrated form of this rate law ln [A] t =ln[A] 0 – kt or [A] t = [A] 0 e –kt 13.2 provides a simple mathematical relationship between the rate constant, the reac- tion’s elapsed time, t, the initial concentration of analyte, [A] 0 , and the analyte’s concentration at time t, [A] t . Unfortunately, most reactions of analytical interest do not follow the simple rate laws shown in equations 13.1 and 13.2. Consider, for example, the following re- action between an analyte, A, and a reagent, R, to form a product, P where k f is the rate constant for the forward reaction, and k b is the rate constant for the reverse reaction. If the forward and reverse reactions occur in single steps, then the rate law is Rate = k f [A][R] – k b [P] 13.3 Although the rate law for the reaction is known, there is no simple integrated form. We can simplify the rate law for the reaction by restricting measurements to the A+R P b f t k k rate = A] A]?= d dt k [ [ 624 Modern Analytical Chemistry rate constant In a rate law, the proportionality constant between a reaction’s rate and the concentrations of species affecting the rate (k). rate The change in a property’s value per unit change in time; the rate of a reaction is a change in concentration per unit change in time. rate law An equation relating a reaction’s rate at a given time to the concentrations of species affecting the rate. *Appendix 5 provides a general review of kinetics. 1400-CH13 9/8/99 4:32 PM Page 624 beginning of the reaction when the product’s concentration is negligible. Under these conditions, the second term in equation 13.3 can be ignored; thus Rate = k f [A][R] 13.4 The integrated form of the rate law for equation 13.4, however, is still too compli- cated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. 4 For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R] 0 >> [A] 0 ), such that its concentration remains essentially constant. Under these conditions 13.5 ln [A] t =ln[A] 0 – k￠ t or [A] t = [A] 0 e –k￠ t 13.6 It may even be possible to adjust conditions such that measurements are made under pseudo-zero-order conditions where 13.7 [A] t = [A] 0 – k? t 13.8 A final requirement for a chemical kinetic method of analysis is that it must be possible to monitor the reaction’s progress by following the change in concentra- tion for one of the reactants or products as a function of time. Which species is used is not important; thus, in a quantitative analysis the rate can be measured by moni- toring the analyte, a reagent reacting with the analyte, or a product. For example, the concentration of phosphate can be determined by monitoring its reaction with Mo(VI) to form 12-molybdophosphoric acid (12-MPA). H 3 PO 4 + 6Mo(VI) + 9H 2 O fi 12-MPA + 9H 3 O + 13.9 We can monitor the progress of this reaction by coupling it to a second reaction in which 12-MPA is reduced to form heteropolyphosphomolybdenum blue, PMB, 12-MPA + nRed fi PMB + nOx where Red is a suitable reducing agent, and Ox is its conjugate form. 5,6 The rate of formation of PMB is measured spectrophotometrically and is proportional to the concentration of 12-MPA. The concentration of 12-MPA, in turn, is proportional to the concentration of phosphate. Reaction 13.9 also can be followed spectropho- tometrically by monitoring the formation of 12-MPA. 6,7 Classifying Chemical Kinetic Methods A useful scheme for classifying chemical ki- netic methods of analysis is shown in Figure 13.3. 3 Methods are divided into two main categories. For those methods identified as direct-computation methods, the concentration of analyte, [A] 0 , is calculated using the appropriate rate law. Thus, for a first-order reaction in A, equation 13.2 is used to determine [A] 0 , provided that values for k, t, and [A] t are known. With a curve-fitting method, regression is used to find the best fit between the data (e.g., [A] t as a function of time) and the known mathematical model for the rate law. In this case, kinetic parameters, such as k and [A] 0 , are adjusted to find the best fit. Both categories are further subdivided into rate methods and integral methods. Rate A] A] R] 00 =? = = ′′ d dt kk [ [[ Rate A] RA] A]=? = = ′ d dt kk [ [][ [ 0 Chapter 13 Kinetic Methods of Analysis 625 1400-CH13 9/8/99 4:32 PM Page 625 Figure 13.3 Classification of chemical kinetic methods of analysis. 626 Modern Analytical Chemistry Chemical kinetic methods Direct-computation methods Curve-fitting methods Integral methods Rate methods Fixed-time One-point Two-point Variable-time One-point Two-point Initial rate Intermediate rate Linear response Nonlinear response Integral methods Rate methods Direct-Computation Integral Methods Integral methods for analyzing kinetic data make use of the integrated form of the rate law. In the one-point fixed-time integral method, the concentration of analyte is determined at a single time. The initial con- centration of analyte, [A] 0 , is calculated using equation 13.2, 13.6, or 13.8, depend- ing on whether the reaction follows first-order, pseudo-first-order, or pseudo-zero- order kinetics. The rate constant for the reaction is determined in a separate experiment using a standard solution of analyte. Alternatively, the analyte’s initial concentration can be determined using a calibration curve consisting of a plot of [A] t for several standard solutions of known [A] 0 . EXAMPLE 1 3 .1 The concentration of nitromethane, CH 3 NO 2 , can be determined from the kinetics of its decomposition in basic solution. In the presence of excess base the reaction is pseudo-first-order in nitromethane. For a standard solution of 0.0100 M nitromethane, the concentration of nitromethane after 2.00 s was found to be 4.24 · 10 –4 M. When a sample containing an unknown amount of nitromethane was analyzed, the concentration remaining after 2.00 s was found to be 5.35 · 10 –4 M. What is the initial concentration of nitromethane in the sample? SOLUTION The value for the pseudo-first-order rate constant is determined by solving equation 13.6 for k￠ and making appropriate substitutions; thus Equation 13.6 can then be solved for the initial concentration of nitromethane. This is easiest to do using the exponential form of equation 13.6. [[ [ [. . ( . )( . A] A] A] A] M M 0 0 s s) t kt t kt e e e = == × = ? ′ ? ′ ? ? ? 535 10 0 0126 4 158 200 1 ′ = ? = ?× = ? ? k t t ln[ ln[ ln( . ) ln( . ) . . A] A] s 0 0 0100 4 24 10 200 158 4 1 s 1400-CH13 9/8/99 4:32 PM Page 626 Chapter 13 Kinetic Methods of Analysis 627 In Example 13.1 the initial concentration of analyte is determined by measur- ing the amount of unreacted analyte at a fixed time. Sometimes it is more conven- ient to measure the concentration of a reagent reacting with the analyte or the con- centration of one of the reaction’s products. The one-point fixed-time integral method can still be applied if the stoichiometry is known between the analyte and the species being monitored. For example, if the concentration of the product in the reaction A+Rfi P is monitored, then the concentration of the analyte at time t is [A] t = [A] 0 – [P] t 13.10 since the stoichiometry between the analyte and product is 1:1. Substituting equa- tion 13.10 into equation 13.6 gives ln([A] 0 – [P] t )=ln[A] 0 – k￠ t 13.11 which is simplified by writing in exponential form [A] 0 – [P] t = [A] 0 e –k￠ t and solving for [A] 0 . 13.12 EXAMPLE 1 3 .2 The concentration of thiocyanate, SCN – , can be determined from the pseudo- first-order kinetics of its reaction with excess Fe 3+ to form a reddish colored complex of Fe(SCN) 2+ . The reaction’s progress is monitored by measuring the absorbance of Fe(SCN) 2+ at a wavelength of 480 nm. When a standard solution of 0.100 M SCN – is used, the concentration of Fe(SCN) 2+ after 10.0 s is found to 0.0516 M. The analysis of a sample containing an unknown amount of SCN – results in a concentration of Fe(SCN) 2+ of 0.0420 M after 10.0 s. What is the initial concentration of SCN – in the sample? SOLUTION The pseudo-first-order rate constant is determined by solving equation 13.11 for k￠ and making appropriate substitutions Equation 13.12 then can be used to determine the initial concentration of SCN – . The one-point fixed-time integral method has the advantage of simplicity since only a single measurement is needed to determine the analyte’s initial con- centration. As with any method relying on a single determination, however, a [ [. . ( . )( . A] P] M M 0 s s) = ? = ? = ? ′ ? ? t kt e e1 0 0420 1 0 0814 0 0726 10 0 1 ′ = ?? = ?? = ? k t t ln[ ln([ [ ) ln( . ) ln( . . ) . . A] A] P] s 00 0 1 0 1 0 0516 10 0 0 0726 1 s [ [ A] P] 0 = ? ? ′ t kt e1 1400-CH13 9/8/99 4:32 PM Page 627 one-point fixed-time integral method cannot compensate for constant sources of determinate error. Such corrections can be made by making measurements at two points in time and using the difference between the measurements to determine the analyte’s initial concentration. Constant sources of error affect both measurements equally, thus, the difference between the measurements is independent of these er- rors. For a two-point fixed-time integral method, in which the concentration of an- alyte for a pseudo-first-order reaction is measured at times t 1 and t 2 , we can write [A] t 1 = [A] 0 e –k￠ t l 13.13 [A] t 2 = [A] 0 e –k￠ t 2 Subtracting the second equation from the first equation and solving for [A] 0 gives 13.14 The rate constant for the reaction can be calculated from equation 13.14 by measur- ing [A] t 1 and [A] t 2 for a standard solution of analyte. The analyte’s initial concentra- tion also can be found using a calibration curve consisting of a plot of ([A] t 1 – [A] t 2 ) versus [A] 0 . Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times t 1 or t 2 , because the relevant con- centration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer’s law (Abs) t = e b[A] t is valid, equations 13.6 and 13.14 can be rewritten as (Abs) t = [A] 0 (e –k￠ t )e b = c[A] 0 where (Abs) t is the absorbance at time t, and c is a constant. An alternative to a fixed-time method is a variable-time method, in which we measure the time required for a reaction to proceed by a fixed amount. In this case the analyte’s initial concentration is determined by the elapsed time, D t, with a higher concentration of analyte producing a smaller D t. For this reason variable- time integral methods are appropriate when the relationship between the detector’s response and the concentration of analyte is not linear or is unknown. In the one- point variable-time integral method, the time needed to cause a desired change in concentration is measured from the start of the reaction. With the two-point vari- able-time integral method, the time required to effect a change in concentration is measured. One important application of the variable-time integral method is the quantita- tive analysis of catalysts, which is based on the catalyst’s ability to increase the rate of a reaction. As the initial concentration of catalyst is increased, the time needed to reach the desired extent of reaction decreases. For many catalytic systems the rela- tionship between the elapsed time, D t, and the initial concentration of analyte is 1 ?t FF=+ cat 0 uncat A][ [ (( [( ( ]A] Abs) Abs) Abs) Abs) 0 = ? ? ×= ? ? ′ ? ′ tt kt kt tt ee b c 12 12 1 ε [ [[ A] A] A] 0 = ? ? ? ′ ? ′ tt kt kt ee 12 628 Modern Analytical Chemistry 1400-CH13 9/8/99 4:32 PM Page 628 Figure 13.5 Determination of reaction rate from a tangent line at time t. Chapter 13 Kinetic Methods of Analysis 629 where F cat and F uncat are constants that are functions of the rate constants for the catalyzed and uncatalyzed reactions, and the extent of the reaction during the time span D t. 8 EXAMPLE 1 3 . 3 Sandell and Kolthoff 9 developed a quantitative method for iodide based on its catalytic effect on the following redox reaction. As 3+ + 2Ce 4+ fi As 5+ + 2Ce 3+ Standards were prepared by adding a known amount of KI to fixed amounts of As 3+ and Ce 4+ and measuring the time for all the Ce 4+ to be reduced. The following results were obtained: D t Micrograms I – (min) 5.0 0.9 2.5 1.8 1.0 4.5 How many micrograms of I – are in a sample for which D t is found to be 3.2 min? SOLUTION The relationship between the concentration of I – and D t is shown by the calibration curve in Figure 13.4, for which Substituting 3.2 min for D t in the preceding equation gives 1.4 m g as the amount of I – originally present in the sample. 1 8 67 10 0 222 9 ?t =? × + ?? ..()μg I 012345 0.0 1.2 1.0 0.8 0.6 0.4 0.2 μg I – 6 1 Dt Figure 13.4 Calibration curve for the variable-time integral determination of I – . Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time t, (rate) t , is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first- order, or pseudo-first-order in analyte, the rate at time t is given as (rate) t = k[A] t Time [P] t D [P] (rate) t = D [P] D t D t 1400-CH13 9/8/99 4:32 PM Page 629 630 Modern Analytical Chemistry Substituting an equation similar to 13.13 into the preceding equation gives the fol- lowing relationship between the rate at time t and the analyte’s initial concentration. (rate) t = k[A] 0 e –kt If the rate is measured at a fixed time, then both k and e –kt are constant, and a cali- bration curve of (rate) t versus [A] 0 can be used for the quantitative analysis of the analyte. The use of the initial rate (t = 0) has the advantage that the rate is at its maxi- mum, providing an improvement in sensitivity. Furthermore, the initial rate is measured under pseudo-zero-order conditions, in which the change in concentra- tion with time is effectively linear, making the determination of slope easier. Finally, when using the initial rate, complications due to competing reactions are avoided. One disadvantage of the initial rate method is that there may be insufficient time for a complete mixing of the reactants. This problem is avoided by using a rate mea- sured at an intermediate time (t > 0). EXAMPLE 1 3 . 4 The concentration of aluminum in serum can be determined by adding 2-hydroxy-1-naphthaldehyde p-methoxybenzoyl-hydrazone and measuring the initial rate of the resulting complexation reaction under pseudo-first-order conditions. 10 The rate of reaction is monitored by the fluorescence of the metal–ligand complex. Initial rates, with units of emission intensity per second, were measured for a set of standard solutions, yielding the following results [Al 3+ ] (m M) 0.300 0.500 1.00 3.00 (rate) t = 0 0.261 0.599 1.44 4.82 A serum sample treated in the same way as the standards has an initial rate of 0.313 emission intensity/s. What is the concentration of aluminum in the serum sample? SOLUTION A calibration curve of emission intensity per second versus the concentration of Al 3+ (Figure 13.6) is a straight line, where (rate) t =0 = 1.69 · [Al 3+ (m M)] – 0.246 Substituting the sample’s initial rate into the calibration equation gives an aluminum concentration of 0.331 m M. 0123 0 5 4 3 2 1 [Al 3+ ] 4 (rate) t = 0 Figure 13.6 Result of curve-fitting for the kinetic data in Example 13.4. 1400-CH13 9/8/99 4:32 PM Page 630 Chapter 13 Kinetic Methods of Analysis 631 Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte’s concentration is determined by solving the appropriate rate equa- tion at one or two discrete times. The relationship between the analyte’s concen- tration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). In a curve-fitting method the concentration of a reactant or product is moni- tored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. For example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentra- tion of a product is followed as a function of time, can be determined by fitting a re- arranged form of equation 13.12 [P] t = [A] 0 (1 – e –k￠ t ) to the kinetic data using both [A] 0 and k￠ as adjustable parameters. By using data from more than one or two discrete times, curve-fitting methods are cap- able of producing more reliable results. Although curve-fitting methods are computationally more demanding, the calculations are easily handled by computer. EXAMPLE 1 3 . 5 The data shown in the following table were collected for a reaction known to follow pseudo-zero-order kinetics during the time in which the reaction was monitored. Time [A] t (s) (mM) 3 0.0731 4 0.0728 5 0.0681 6 0.0582 7 0.0511 8 0.0448 9 0.0404 10 0.0339 11 0.0217 12 0.0143 What are the rate constant and the initial concentration of analyte in the sample? SOLUTION For a pseudo-zero-order reaction a plot of [A] t versus time should be linear with a slope of –k, and a y-intercept of [A] 0 (equation 13.8). A plot of the kinetic data is shown in Figure 13.7. Linear regression gives an equa- tion of [A] t = 0.0986 – 0.00677t 1400-CH13 9/8/99 4:32 PM Page 631 632 Modern Analytical Chemistry 024681012 0.00 0.10 0.08 0.06 0.04 0.02 Time [A] t 14 Figure 13.7 Result of curve-fitting for the kinetic data in Example 13.5. Representative Methods —Continued Thus, the rate constant is 0.00677 s –1 , and the initial concentration of analyte is 0.0986 mM. Miscellaneous Methods At the beginning of this section we noted that kinetic methods are susceptible to significant errors when experimental variables affecting the reaction’s rate are difficult to control. Many variables, such as temperature, can be controlled with proper instrumentation. Other variables, such as interferents in the sample matrix, are more difficult to control and may lead to significant errors. Although not discussed in this text, direct-computation and curve-fitting methods have been developed that compensate for these sources of error. 3 Representative Method Although each chemical kinetic method has its own unique considerations, the determination of creatinine in urine based on the ki- netics of its reaction with picrate provides an instructive example of a typical procedure. Method 1 3 .1 Determination of Creatinine in Urine 11 Description of Method. Creatine is an organic acid found in muscle tissue that supplies energy for muscle contractions. One of its metabolic products is creatinine, which is excreted in urine. Because the concentration of creatinine in urine and serum is an important indication of renal function, rapid methods for its analysis are clinically important. In this method the rate of reaction between creatinine and picrate in an alkaline medium is used to determine the concentration of creatinine in urine. Under the conditions of the analysis, the reaction is first-order in picrate, creatinine, and hydroxide. Rate = k[picrate][creatinine][OH – ] The rate of reaction is monitored using a picrate ion-selective electrode. Procedure. Prepare a set of external standards containing 0.5 g/L to 3.0 g/L creatinine (in 5 mM H 2 SO 4 ) using a stock solution of 10.00 g/L creatinine in 5 mM H 2 SO 4 . In addition, prepare a solution of 1.00 · 10 –2 M sodium picrate. Pipet 25.00 mL of 0.20 M NaOH, adjusted to an ionic strength of 1.00 M using Na 2 SO 4 , into a thermostated reaction cell at 25 °C. Add 0.500 mL of the 1.00 · 10 –2 M picrate solution to the reaction cell. Suspend a picrate ion-selective electrode in the solution, and monitor the potential until it stabilizes. When the potential is stable, add 2.00 mL of a 1400-CH13 9/8/99 4:32 PM Page 632 Chapter 13 Kinetic Methods of Analysis 633 creatinine external standard, and record the potential as a function of time. Repeat the procedure using the remaining external standards. Construct a calibration curve of rate (D E/D t) versus [creatinine]. Samples of urine (2.00 mL) are analyzed in a similar manner, with the concentration of creatinine determined from the calibration curve. Questions 1. The analysis is carried out under conditions in which the reaction’s kinetics are pseudo-first-order in picrate. Show that under these conditions, a plot of potential as a function of time will be linear. The response of the picrate ion-selective electrode is We know from equation 13.6 that for a pseudo-first-order reaction, the concentration of picrate at time t is ln [picrate] t = ln [picrate] 0 – k￠ t where k￠ is the pseudo-first-order rate constant. Substituting the kinetic expression into the equation for the ion-selective electrode’s potential leaves us with Since both K and (RT/F) ln [picrate] 0 are constants, a plot of E t versus t will be a straight line whose slope, RTk￠ /F, is the reaction’s rate (D E/D t). 2. As carried out the rate of the reaction is pseudo-first-order in picrate and pseudo-zero-order in creatinine and OH – . Explain why it is possible to prepare a calibration curve of rate versus [creatinine]. Since the reaction is carried out under conditions in which it is pseudo-zero- order in creatinine and OH – , the rate constant, k￠ , is k￠ = k[creatinine][OH – ] where k is the reaction’s true rate constant. The rate, therefore, is where c is a constant. 3. Why is it necessary to use a thermostat in the reaction cell? The rate of a reaction is temperature-dependent. To avoid a determinate error resulting from a systematic change in temperature or to minimize indeterminate errors due to fluctuations in temperature, the reaction cell must have a thermostat to maintain a constant temperature. 4. Why is it necessary to prepare the NaOH solution so that it has an ionic strength of 1.00 M? The potential of the ion-selective electrode actually responds to the activity of picrate in solution. By adjusting the NaOH solution to a high ionic strength, we maintain a constant ionic strength in all standards and samples. Because the relationship between activity and concentration is a function of ionic strength (see Chapter 6), the use of a constant ionic strength allows us to treat the potential as though it were a function of the concentration of picrate. Rate creatinine][OH [creatinine]== ? RTk F c [] EK RT F kt EK RT F RTk F t t t =? ?′ =? + ′ (ln[ ) ln[ picrate] picrate] 0 0 EK RT F =? ln[picrate] 1400-CH13 9/8/99 4:32 PM Page 633 1 3 A.2 Instrumentation Quantitative information about a chemical reaction can be made using any of the techniques described in the preceding chapters. For reactions that are kineti- cally slow, an analysis may be performed without worrying about the possibility that significant changes in concentration occur while measuring the signal. When the reaction’s rate is too fast, which is usually the case, significant errors may be introduced if changes in concentration are ignored. One solution to this problem is to stop, or quench, the reaction by suitably adjusting experimental conditions. For example, many reactions involving enzymes show a strong pH dependency and may be quenched by adding a strong acid or strong base. Once the reaction is stopped, the concentration of the desired species can be deter- mined at the analyst’s convenience. Another approach is to use a visual indicator that changes color after the reaction occurs to a fixed extent. You may recall that this variable-time method is the basis of the so-called “clock reactions” com- monly used to demonstrate kinetics in the general chemistry classroom and labo- ratory. Finally, reactions with fast kinetics may be monitored continuously using the same types of spectroscopic and electrochemical detectors found in chro- matographic instrumentation. Two additional problems for chemical kinetic methods of analysis are the need to control the mixing of the sample and reagents in a rapid and repro- ducible fashion and the need to control the acquisition and analysis of the signal. Many kinetic determinations are made early in the reaction when pseudo-zero- order or pseudo-first-order conditions are in effect. Depending on the rate of re- action, measurements are typically made within a span of a few milliseconds or seconds. This is both an advantage and a disadvantage. The disadvantage is that transferring the sample and reagent to a reaction vessel and their subsequent mixing must be automated if a reaction with rapid kinetics is to be practical. This usually requires a dedicated instrument, thereby adding an additional ex- pense to the analysis. The advantage is that a rapid, automated analysis allows for a high throughput of samples. For example, an instrument for the automated ki- netic analysis of phosphate, based on reaction 13.9, has achieved sampling rates of 3000 determinations per hour. 6 A variety of designs have been developed to automate kinetic analyses. 6 The stopped-flow apparatus, which is shown schematically in Figure 13.8, has found use in kinetic determinations involving very fast reactions. Sample and reagents are loaded into separate syringes, and precisely measured volumes are dispensed by the action of a syringe drive. The two solutions are rapidly mixed in the mix- ing chamber before flowing through an observation cell. The flow of sample and reagents is stopped by applying back pressure with the stopping syringe. The back pressure completes the mixing, after which the reaction’s progress is moni- tored spectrophotometrically. With a stopped-flow apparatus, it is possible to complete the mixing of sample and reagent and initiate the kinetic measure- ments within approximately 0.5 ms. The stopped-flow apparatus shown in Fig- ure 13.8 can be modified by attaching an automatic sampler to the sample sy- ringe, thereby allowing the sequential analysis of multiple samples. In this way the stopped-flow apparatus can be used for the routine analysis of several hun- dred samples per hour. Another automated approach to kinetic analyses is the centrifugal analyzer, a partial cross section of which is shown in Figure 13.9. In this technique the sample and reagents are placed in separate wells oriented radially around a circu- lar transfer disk attached to the rotor of a centrifuge. As the centrifuge spins, the 634 Modern Analytical Chemistry quench To stop a reaction by suddenly changing the reaction conditions. stopped flow A kinetic method of analysis designed to rapidly mix samples and reagents when using reactions with very fast kinetics. 1400-CH13 9/8/99 4:32 PM Page 634 Figure 13.8 Schematic diagram of a stopped-flow analyzer. 635 Light source Monochromator Detector Observation cell Mixing chamber Syringe drive Syringe drive Sample syr inge Reagent syr inge Stopping syr inge Cross-section Cuvette Rotor Transfer disk Sample Reagents Optically transparent windows Detector Light source Monochromator (a) (b) (c) Figure 13.9 Schematic diagram of a centrifugal analyzer showing (a) the wells for holding the sample and reagent; (b) mixing of the sample and reagent; and (c) the configuration of the spectrophotometric detector. 1400-CH13 9/8/99 4:32 PM Page 635 sample and reagents are pulled by the centrifugal force to the cuvette, where mixing occurs. A single optical source and detector, located above and below the transfer disk’s outer edge, allows the absorbance of the reaction mixture to be measured as it passes through the optical beam. The centrifugal analyzer allows a number of samples to be analyzed simultaneously. For example, if a transfer plate contains 30 cuvettes and rotates with a speed of 600 rpm, it is possible to collect 10 data points per sample for each second of rotation. The ability to collect kinetic data for several hundred samples per hour is of little consequence if the analysis of the data must be accomplished manually. Be- sides time, the manual analysis of kinetic data is limited by noise in the detector’s signal and the accuracy with which the analyst can determine reaction rates from tangents drawn to differential rate curves. Not surprisingly, the development of automated kinetic analyzers was paralleled by the development of analog and digital circuitry, as well as computer software for the smoothing, on-line integra- tion and differentiation, and analysis of kinetic signals. 12 1 3 A. 3 Quantitative Applications Chemical kinetic methods of analysis continue to find use for the analysis of a vari- ety of analytes, most notably in clinical laboratories, where automated methods aid in handling a large volume of samples. In this section several general quantitative applications are considered. Enzyme-Catalyzed Reactions Enzymes are highly specific catalysts for biochemi- cal reactions, with each enzyme showing a selectivity for a single reactant, or sub- strate. For example, acetylcholinesterase is an enzyme that catalyzes the decomposi- tion of the neurotransmitter acetylcholine to choline and acetic acid. Many enzyme–substrate reactions follow a simple mechanism consisting of the initial for- mation of an enzyme–substrate complex, ES, which subsequently decomposes to form product, releasing the enzyme to react again. 13.15 If measurements are made early in the reaction, the product’s concentration is neg- ligible, and the step described by the rate constant k –2 can be ignored. Under these conditions the rate of the reaction is 13.16 To be analytically useful equation 13.16 needs to be written in terms of the concentrations of enzyme and substrate. This is accomplished by applying the “steady-state” approximation, in which we assume that the concentration of ES is essentially constant. After an initial period in which the enzyme–substrate complex first forms, the rate of formation of ES and its rate of disappearance ?? ? ? ? ? ? = ? d dt kk d [ [[ ES] ES] + ES] 12 d dt kk f [ [([[ ES] E][S] E] ES])[S] 0 ? ? ? ? ? ? ==? 11 d dt k [ [ P] ES]= 2 E+S ES E+P 12 tt k k k k ??12 636 Modern Analytical Chemistry enzyme A protein that catalyzes biochemical reactions. substrate The specific molecule for which an enzyme serves as a catalyst. steady-state approximation In a kinetic process, the assumption that a compound formed during the reaction reaches a concentration that remains constant until the reaction is nearly complete. 1400-CH13 9/8/99 4:32 PM Page 636 Figure 13.10 Plot of equation 13.18 showing limits for which a chemical kinetic method of analysis can be used to determine the concentration of a catalyst or a substrate. are equal. Combining these equations gives k 1 ([E] 0 – [ES])[S] = k –1 [ES] + k 2 [ES] which is solved for the concentration of the enzyme–substrate complex 13.17 where K m is called the Michaelis constant. Substituting equation 13.17 into equa- tion 13.16 leads to the final rate equation 13.18 A plot of equation 13.18, shown in Figure 13.10, is instructive for defining con- ditions under which the rate of an enzymatic reaction can be used for the quantita- tive analysis of enzymes and substrates. For high substrate concentrations, where [S] >> K m , equation 13.18 simplifies to 13.19 where V max is the maximum rate for the catalyzed reaction. Under these conditions the rate of the reaction is pseudo-zero-order in substrate, and the maximum rate can be used to calculate the enzyme’s concentration. Typically, this determination is made by a variable-time method. At lower substrate concentrations, where [S] << K m , equation 13.18 becomes 13.20 The reaction is now first-order in substrate, and the rate of the reaction can be used to determine the substrate’s concentration by a fixed-time method. Chemical kinetic methods have been applied to the quantitative analysis of a number of enzymes and substrates. 13 One example, is the determination of glucose based on its oxidation by the enzyme glucose oxidase. 6 glucose oxidase Glucose + H 2 O+O 2 ——————fi gluconolactone + H 2 O 2 Conditions are controlled, such that equation 13.20 is valid. The reaction is moni- tored by following the rate of change in the concentration of dissolved O 2 using an appropriate voltammetric technique. Nonenzyme-Catalyzed Reactions The variable-time method has also been used to determine the concentration of nonenzymatic catalysts. Because a trace amount of catalyst can substantially enhance a reaction’s rate, a kinetic determination of a cat- alyst’s concentration is capable of providing an excellent detection limit. One of the most commonly used reactions is the reduction of H 2 O 2 by reducing agents, such as thiosulfate, iodide, and hydroquinone. These reactions are catalyzed by trace levels of selected metal ions. For example the reduction of H 2 O 2 by I – 2I – +H 2 O 2 +2H 3 O + t 2H 2 O+I 2 is catalyzed by Mo(VI), W(VI), and Zr(IV). A variable-time analysis is conducted by adding a small, fixed amount of ascorbic acid to each solution. As I 2 is produced, d dt k K V K [[[ [P] E] S] S] 0 m max m == 2 d dt kV [ [ P] E] 0 max == 2 d dt k K [[[ [ P] E] S] S] 0 m = + 2 [ [ )/ } [ [[ [ ES] = [E] S] {( S] E] S] S] 00 m kkk K ? ++ = + 121 Chapter 13 Kinetic Methods of Analysis 637 Michaelis constant A combination of several rate constants affecting the rate of an enzyme-substrate reaction. Concentration of substrate Analytical region for the analysis of enzymes Maximum rate = V max Analytical region for the analysis of substrates d[P] dt 1400-CH13 9/8/99 4:32 PM Page 637 it rapidly oxidizes the ascorbic acid and is, itself, reduced back to I – . Once all the ascorbic acid is consumed, the presence of excess I 2 provides a visual end point. Noncatalytic Reactions Chemical kinetic methods are not as common for the quantitative analysis of analytes in noncatalytic reactions. Because they lack the en- hancement of reaction rate obtained when using a catalyst, noncatalytic methods generally are not used for the determination of analytes at low concentrations. 4 Noncatalytic methods for analyzing inorganic analytes are usually based on a com- plexation reaction. One example was outlined in Example 13.4, in which the con- centration of aluminum in serum was determined by the initial rate of formation of its complex with 2-hydroxy-1-naphthaldehyde p-methoxybenzoyl-hydrazone. 10 The greatest number of noncatalytic methods, however, are for the quantitative analysis of organic analytes. For example, the insecticide methyl parathion has been determined by measuring its rate of hydrolysis in alkaline solutions. 14 1 3 A. 4 Characterization Applications Chemical kinetic methods also find use in determining rate constants and elucidat- ing reaction mechanisms. These applications are illustrated by two examples from the chemical kinetic analysis of enzymes. Determining V max and K m for Enzyme-Catalyzed Reactions The value of V max and K m for an enzymatic reaction are of significant interest in the study of cellular chemistry. 15 From equation 13.19 we see that V max provides a means for determin- ing the rate constant k 2 . For enzymes that follow the mechanism shown in reaction 13.15, k 2 is equivalent to the enzyme’s turnover number, k cat . The turnover number is the maximum number of substrate molecules converted to product by a single ac- tive site on the enzyme, per unit time. Thus, the turnover number provides a direct indication of the catalytic efficiency of an enzyme’s active site. The Michaelis con- stant, K m , is significant because it provides an estimate of the substrate’s intracellu- lar concentration. 15b Values of V max and K m for reactions obeying the mechanism shown in reaction 13.15 can be determined using equation 13.18 by measuring the rate of reaction as a function of the substrate’s concentration. The curved nature of the relationship be- tween rate and the concentration of substrate (see Figure 13.10), however, is incon- venient for this purpose. Equation 13.18 can be rewritten in a linear form by taking its reciprocal A plot of 1/v versus 1/[S], which is called a double reciprocal, or Lineweaver–Burk plot, is a straight line with a slope of K m /V max , a y-intercept of 1/V max , and an x-intercept of –1/K m (Figure 13.11). Elucidating Mechanisms for the Inhibition of Enzyme Catalysis An inhibitor inter- acts with an enzyme in a manner that decreases the enzyme’s catalytic efficiency. Examples of inhibitors include some drugs and poisons. Irreversible inhibitors co- valently bind to the enzyme’s active site, producing a permanent loss in catalytic ef- ficiency even when the inhibitor’s concentration is decreased. Reversible inhibitors form noncovalent complexes with the enzyme, thereby causing a temporary de- 11 11 ddtv K VV[[P]/ S] m max max == ? ? ? ? ? ? + 638 Modern Analytical Chemistry Lineweaver–Burk plot A graphical means for evaluating enzyme kinetics. 1400-CH13 9/8/99 4:32 PM Page 638 Figure 13.11 Lineweaver–Burk plot of equation 13.18. crease in catalytic efficiency. If the inhibitor is removed, the enzyme’s catalytic effi- ciency returns to its normal level. The reversible binding of an inhibitor to an enzyme can occur through several pathways, as shown in Figure 13.12. In competitive inhibition (Figure 13.12a), the substrate and the inhibitor compete for the same active site on the enzyme. Because the substrate cannot bind to an enzyme that is already bound to an inhibitor, the enzyme’s catalytic efficiency for the substrate is decreased. With noncompetitive in- hibition (Figure 13.12b) the substrate and inhibitor bind to separate active sites on the enzyme, forming an enzyme–substrate–inhibitor, or ESI complex. The presence of the ESI complex, however, decreases catalytic efficiency since only the enzyme–substrate complex can react to form product. Finally, in uncompetitive in- hibition (Figure 13.12c) the inhibitor cannot bind to the free enzyme, but can bind to the enzyme–substrate complex. Again, the result is the formation of an inactive ESI complex. The three reversible mechanisms for enzyme inhibition are distinguished by ob- serving how changing the inhibitor’s concentration affects the relationship between the rate of reaction and the concentration of substrate. As shown in Figure 13.13, when kinetic data are displayed as a Lineweaver–Burk plot, it is possible to determine which mechanism is in effect. 1 3 A. 5 Evaluation of Chemical Kinetic Methods Scale of Operation The detection limit for chemical kinetic methods ranges from minor components to ultratrace components (see Figure 3.6 in Chapter 3) and is principally determined by two factors: the rate of the reaction and the instrumental method used for monitoring the rate. Because the signal is directly proportional to the reaction rate, faster reactions generally result in lower detection limits. All other considerations being equal, the detection limit is smaller for catalytic reactions than for noncatalytic reactions. Not surprisingly, chemical kinetic methods based on catalysis were among the earliest techniques for trace-level analysis. Ultratrace analysis is also possible when using catalytic reactions. For example, ultratrace levels of Cu (<1 ppb) can be determined by measuring its catalytic effect on the redox re- action between hydroquinone and H 2 O 2 . Without a catalyst, most chemical kinetic methods for organic compounds involve reactions with relatively slow rates, limit- ing the method to minor and higher concentration trace analytes. Noncatalytic chemical kinetic methods for inorganic compounds involving metal–ligand com- plexation may be fast or slow, with detection limits ranging from trace to minor lev- els of analyte. Chapter 13 Kinetic Methods of Analysis 639 1 V max y-intercept = 1 V 1 [S] K m V max Slope = –1 K m x-intercept = Competitive inhibition E + S ES E + P + I EI Noncompetitive inhibition E + S ES E + P + I EI + S ESI + I Uncompetitive inhibition E + S ES E + P ESI + I (a) (b) (c) Figure 13.12 Mechanisms for the inhibition of enzyme catalysis. 1400-CH13 9/8/99 4:32 PM Page 639 The second factor influencing detection limits is the instru- mental method used to monitor the reaction’s progress. Most reactions are monitored spectrophotometrically or electro- chemically. The scale of operation for these methods was dis- cussed in Chapters 10 and 11 and, therefore, is not discussed here. Accuracy As noted earlier, chemical kinetic methods are po- tentially subject to larger errors than equilibrium methods due to the effect of uncontrolled or poorly controlled variables, such as temperature and solution pH. Although the direct- computation chemical kinetic methods described in this chapter can yield results with moderate accuracy (1–5%), re- action systems are encountered in which accuracy is quite poor. An improvement in accuracy may be realized by using curve-fitting methods. In one study, for example, accuracy was improved by two orders of magnitude (500–5%) by re- placing a direct-computation analysis with a curve-fitting analysis. 16 Although not discussed in this chapter, data analy- sis methods that include the ability to compensate for experi- mental error 3,17–19 can lead to a significant improvement in accuracy. Precision The precision of a chemical kinetic method is lim- ited by the signal-to-noise ratio of the instrumental method used to monitor the reaction’s progress. With integral methods, precisions of 1–2% are routinely possible. The precision for dif- ferential methods may be somewhat poorer, particularly for noisy signals, due to the difficulty in measuring the slope of a noisy rate curve. 19 It may be possible to improve the precision in this case by using a combination of signal averaging and smoothing of the data before its analysis. Sensitivity The sensitivity for a one-point fixed-time integral method of analysis is improved by making measurements under conditions in which the concentration of the monitored species is larger rather than smaller. When the analyte’s concentra- tion, or the concentration of any other reactant, is monitored, measurements are best made early in the reaction before its concentration has substantially decreased. On the other hand, when a product is used to monitor the reaction, measurements are more appropriately made at longer times. For a two-point fixed-time integral method, sensitivity is improved by increasing the difference between times t 1 and t 2 . As discussed earlier, the sensitivity of a rate method improves when using the initial rate. Selectivity The analysis of closely related compounds, as we have seen in ear- lier chapters, is often complicated by their tendency to interfere with one an- other. To overcome this problem, the analyte and interferent must first be sepa- rated. An advantage of chemical kinetic methods is that conditions can often be adjusted so that the analyte and interferent have different reaction rates. If the difference in rates is large enough, one species may react completely before the other species has a chance to react. For example, many enzymes selectively cat- 640 Modern Analytical Chemistry 1 V 1 [S] [I] [I] = 0 1 V 1 [S] [I] [I] = 0 1 V 1 [S] [I] [I] = 0 (a) (b) (c) Figure 13.13 Effect of the concentration of inhibitor on the Lineweaver–Burk plots for (a) competitive inhibition, (b) noncompetitive inhibition, and (c) uncompetitive inhibition. The inhibitor’s concentration increases in the direction shown by the arrows. 1400-CH13 9/8/99 4:32 PM Page 640 alyze a single substrate, allowing the quantitative analysis of that substrate in the presence of similar substrates. The conditions necessary to ensure that a faster-reacting species can be kineti- cally separated from a more slowly reacting species can be determined from the ap- propriate integrated rate laws. As an example, let’s consider a system consisting of an analyte, A, and an interferent, B, both of which show first-order kinetics with a common reagent. To avoid an interference, the relative magnitudes of their rate constants must be sufficiently different. The fractions, f, of A and B remaining at any point in time, t, are given as 13.21 13.22 For a first-order reaction we can write, from equations 13.2, 13.21, and 13.22 ln(f A ) t = –k A t ln(f B ) t = –k B t Taking the ratio of these two equations gives Thus, for example, if 99% of A is to react before 1% of B has reacted then the rate constant for A must be 458 times larger than that for B. Under these conditions the analyte’s concentration can be determined before the interferent be- gins to react. If the analyte has the slower reaction, then it can be determined after the interferent’s reaction is complete. The method described here is impractical when the simultaneous analysis of both A and B is desired. The difficulty in this case is that conditions favoring the analysis of A generally do not favor the analysis of B. For example, if condi- tions are adjusted such that 99% of A reacts in 5 s, then B must reach 99% com- pletion in either 0.01 s if it has the faster kinetics or 2300 s if it has the slower kinetics. Several additional approaches for analyzing mixtures have been developed that do not require such a large difference in rate constants. 3,4 Because both A and B react at the same time, the integrated form of the first-order rate law becomes C t = [A] t + [B] t = [A] 0 e –k A t + [B] 0 e –k B t 13.23 where C t is the total concentration of A and B. If C t is measured at times t 1 and t 2 , the resulting pair of simultaneous equations can be solved to give [A] 0 and [B] 0 . The rate constants k A and k B must be determined in separate experiments using standard solutions of A and B. Alternatively, if A and B react to form a common product, P, equation 13.23 can be written as P t = [A] 0 (1 – e –k A t ) + [B] 0 (1 – e –k B t ) k k A B == ln( . ) ln( . ) 001 099 458 k k f f t t A B A B = ln( ) ln( ) () [ [ f t t B 0 B] B] = () [ [ f t t A 0 A] A] = Chapter 13 Kinetic Methods of Analysis 641 1400-CH13 9/8/99 4:32 PM Page 641 Figure 13.14 Determination of the concentration of a slowly reacting analyte, B, in the presence of a faster reacting analyte, A. 642 Modern Analytical Chemistry 0 –4 0 Time ln( C t ) 205 ln[B] 0 10 15 –3.5 –3 –2.5 –2 –1.5 –1 –0.5 isotopes Atoms with the same number of protons but different numbers of neutrons are called isotopes. alpha particle A positively charged subatomic particle equivalent to a helium nucleus (a ). beta particle A charged subatomic particle produced when a neutron converts to a proton, or a proton converts to a neutron (b ). negatron The beta particle formed when a neutron converts to a proton; equivalent to an electron ( 0 –1 b ). Again, a pair of simultaneous equations at times t 1 and t 2 can be solved for [A] 0 and [B] 0 . Equation 13.23 can also be used as the basis for a curve-fitting method. As shown in Figure 13.14, a plot of ln(C t ) as a function of time consists of two regions. At short times the plot is curved since A and B are reacting simultaneously. At later times, however, the concentration of the faster-reacting component, A, decreases to 0, and equation 13.23 simplifies to C t ? [B] t = [B] 0 e –k B t Under these conditions a plot of ln(C t ) versus time is linear. Extrapolating the linear portion to time 0 gives [B] 0 , and [A] 0 is determined by difference. Time, Cost, and Equipment Automated chemical kinetic methods of analysis pro- vide a rapid means for analyzing samples, with throughputs ranging from several hundred to several thousand determinations per hour. The initial start-up costs, however, may be fairly high because an automated analysis requires a dedicated in- strument designed to meet the specific needs of the analysis. When handled manu- ally, chemical kinetic methods can be accomplished using equipment and instru- mentation routinely available in most laboratories. Sample throughput, however, is much lower than with automated methods. 1 3 B Radiochemical Methods of Analysis Atoms with the same number of protons but a different number of neutrons are called isotopes. To identify an isotope we use the symbol A Z E, where E is the ele- ment’s atomic symbol, Z is the element’s atomic number (which is the number of protons), and A is the element’s atomic mass number (which is the sum of the number of protons and neutrons). Although isotopes of a given element have the same chemical properties, their nuclear properties are different. The most impor- tant difference between isotopes is their stability. The nuclear configuration of a sta- ble isotope remains constant with time. Unstable isotopes, however, spontaneously disintegrate, emitting radioactive particles as they transform into a more stable form. The most important types of radioactive particles are alpha particles, beta parti- cles, gamma rays, and X-rays. An alpha particle, which is symbolized as a , is equiv- alent to a helium nucleus, 4 2 He. Thus, emission of an alpha particle results in a new isotope whose atomic number and atomic mass number are, respectively, 2 and 4 less than that for the unstable parent isotope. 238 92 U fi 234 90 Th + a Beta particles, b , come in two forms. A negatron, 0 –1 b , is equivalent to an electron, and is produced when a neutron is converted to a proton, increasing the atomic number by 1. 214 82 Pb fi 214 83 Bi + 0 –1 b Converting a proton to a neutron results in the emission of a positron, 0 1 b . 30 15 P fi 30 14 Si + 0 1 b Emission of an alpha or beta particle often produces an isotope in an unstable, high-energy state. This excess energy is released as a gamma ray, g , or an X-ray. Gamma ray and X-ray emission may also occur without the release of alpha or beta particles. positron The beta particle formed when a proton converts to a neutron ( 0 1 b ). gamma ray High-energy radiation emitted by nuclei (g ). 1400-CH13 9/8/99 4:32 PM Page 642 Although similar to chemical kinetic methods of analysis, radiochemical meth- ods are best classified as nuclear kinetic methods. In this section we review the ki- netics of radioactive decay and examine several quantitative and characterization applications. 1 3 B.1 Theory and Practice The rate of decay, or activity, for a radioactive isotope follows first-order kinetics 13.24 where A is the activity, N is the number of radioactive atoms present in the sample at time t, and l is the radioisotope’s decay constant. Activity is given in units of dis- integrations per unit time, which is equivalent to the number of atoms undergoing radioactive decay per unit time. As with any first-order process, equation 13.24 can be expressed in an inte- grated form. N = N 0 e –l t 13.25 Substituting equation 13.25 into equation 13.24 gives A = l N 0 e –l t = A 0 e –l t 13.26 By measuring the activity at time t, therefore, we can determine the initial activity, A 0 , or the number of radioactive atoms originally present in the sample, N 0 . An important characteristic property of a radioactive isotope is its half-life, t 1/2 , which is the amount of time required for half of the radioactive atoms to disinte- grate. For first-order kinetics the half-life is independent of concentration and is given as 13.27 Since the half-life is independent of the number of radioactive atoms, it remains constant throughout the decay process. Thus, 50% of the radioactive atoms disinte- grate in one half-life, 75% in two half-lives, and 87.5% in three half-lives. Kinetic information about radioactive isotopes is usually given in terms of the half-life because it provides a more intuitive sense of the isotope’s stability. Know- ing, for example, that the decay constant for 90 38 Sr is 0.0247 yr –1 does not give an im- mediate sense of how fast it disintegrates. On the other hand, knowing that the half- life for 90 38 Sr is 28.1 years makes it clear that the concentration of 90 38 Sr in a sample remains essentially constant over a short period of time. 1 3 B.2 Instrumentation Alpha particles, beta particles, gamma rays, and X-rays are measured using the par- ticle’s energy to produce an amplified pulse of electric current in a detector. These pulses are counted to give the rate of disintegration. Three types of detectors com- monly are encountered: gas-filled detectors, scintillation counters, and semiconduc- tor detectors. The gas-filled detector consists of a tube filled with an inert gas, such as Ar. When radioactive particles enter the tube, they ionize the inert gas, producing a large number of Ar + /e – ion pairs. Movement of the electrons toward an anode and the Ar + toward a cathode generates a measurable electric current. A Geiger counter is one example of a gas-filled detector. A scintillation counter uses a fluorescent t 12 0 693 / . = λ A dN dt N=? =λ Chapter 13 Kinetic Methods of Analysis 643 half-life The time required for half of the initial number of a radioactive isotope’s atoms to disintegrate (t 1/2 ). Geiger counter An instrument for counting radioactive particles based on their ability to ionize an inert gas such as Ar. scintillation counter An instrument for counting radioactive particles based on their ability to initiate fluorescence in another substance. 1400-CH13 9/8/99 4:32 PM Page 643 644 Modern Analytical Chemistry material to convert radioactive particles into easily measurable photons. For exam- ple, one solid-state scintillation counter, consisting of a crystal of NaI with 0.2% TlI, produces several thousand photons for each radioactive particle. Finally, in a semi- conductor detector the adsorption of a radioactive particle promotes thousands of electrons to the semiconductor’s conduction band, leading to an increase in its con- ductivity. Further details about radiation detectors, and the signal processors used to count particles, can be found in the suggested readings listed at the end of this chapter. 1 3 B. 3 Quantitative Applications Three common quantitative applications of radiochemical methods of analysis are considered in this section: the direct analysis of radioactive isotopes by measuring their rate of disintegration, neutron activation, and the use of radioactive isotopes as tracers in isotope dilution. Direct Analysis of Radioactive Analytes The concentration of a long-lived ra- dioactive isotope is essentially constant during the period of analysis. As shown in Example 13.6, the sample’s activity can be used to calculate the number of radioac- tive particles that are present. EXAMPLE 1 3 .6 The activity in a 10.00-mL sample of radioactive wastewater containing 90 38 Sr was found to be 9.07 · 10 6 disintegrations/s. What is the molar concentration of 90 38 Sr in the sample? The half-life for 90 38 Sr is 28.1 years. SOLUTION Substituting equation 13.27 into equation 13.24 and solving for N gives Before the number of atoms of 90 38 Sr can be determined, it is necessary to express the activity and the half-life in the same units. Converting the half-life for 90 38 Sr to seconds gives t 1/2 as 8.86 · 10 8 s. Substituting known values gives the number of atoms of 90 38 Sr as Thus, the concentration of 90 38 Sr in the sample is The direct analysis of short-lived radioactive isotopes using the method out- lined in Example 13.6 is less useful since it provides only a transient measure of the isotope’s concentration. The concentration of the isotope at a particular moment 116 10 193 10 16 6 90 38 . . × × =× ? atoms (6.022 10 atoms/mol)(0.01000 L) M Sr 23 (. . 907 10 116 10 6 16 90 38 ×× =× disintegrations/s)(8.86 10 s) 0.693 atoms of Sr 8 N At = × 12 0 693 / . 1400-CH13 9/8/99 4:32 PM Page 644 Figure 13.15 Plot of gamma-ray emission as a function of time showing how the analyte’s initial activity is determined experimentally. can be determined by measuring its activity after an elapsed time, t, and using equa- tion 13.26 to calculate N 0 . Neutron Activation Analysis Few samples of interest are naturally radioactive. For many elements, however, radioactivity may be induced by irradiating the sample with neutrons in a process called neutron activation analysis (NAA). The radioac- tive element formed by neutron activation decays to a stable isotope by emitting gamma rays and, if necessary, other nuclear particles. The rate of gamma-ray emis- sion is proportional to the analyte’s initial concentration in the sample. For exam- ple, when a sample containing nonradioactive 27 13 Al is placed in a nuclear reactor and irradiated with neutrons, the following nuclear reaction results. 27 13 Al + 1 0 n fi 28 13 Al The radioactive isotope of 28 13 Al has a characteristic decay process that includes the release of a beta particle and a gamma ray. 28 13 Al fi 28 14 Si + 0 -1 b + g When irradiation is complete, the sample is removed from the nuclear reactor, al- lowed to cool while any short-lived interferences that might be present decay to the background, and the rate of gamma-ray emission is measured. The initial activity at the end of irradiation depends on the number of 28 13 Al atoms that are present. This, in turn, is equal to the difference between the rate of formation for 28 13 Al and its rate of disintegration, 13.28 where F is the neutron flux, and s is the reaction cross-section, or probability for the capture of a neutron by the 27 13 Al nucleus. Integrating equation 13.28 over the time of irradiation, t i , and multiplying by l gives the initial activity, A 0 , at the end of irradiation as A 0 = l( N28 13 A1 ) = Fs( N28 13 A1 )(1 – e –l t i ) Thus, the number of atoms of 27 13 Al initially present in the sample can be calculated from A 0 if values for F , s , l , and t i are known. A simpler approach for analyzing neutron activation data is to use one or more external standards. Letting (A 0 ) x and (A 0 ) s represent the initial activity for the ana- lyte in an unknown and a single external standard, and letting w x and w s represent the weight of analyte in the unknown and the external standard, gives a pair of equations (A 0 ) x = kw x 13.29 (A 0 ) s = kw s 13.30 that can be solved to determine the mass of analyte in the sample. As noted earlier, gamma-ray emission is measured following a “cooling” period in which short-lived interferents are allowed to decay away. The initial activity therefore, is determined by extrapolating a curve of activity versus time back to t = 0 (Figure 13.15). Alternatively, if the samples and standards are irradiated simul- taneously, and the activities are measured at the same time, then these activities may be used in place of (A 0 ) x and (A 0 ) s in the preceding equations. dN dt NN () () 28 13 27 13 28 13 Al Al Al =?Φσ λ Chapter 13 Kinetic Methods of Analysis 645 Activity Elapsed time after irradiation Gamma-ray emission from analyte Short-lived gamma-ray Emission from impurities Initial rate of gamma-ray emission for analyte neutron activation A means of inducing radioactivity in a nonradioactive sample by bombarding the sample with neutrons. 1400-CH13 9/8/99 4:32 PM Page 645 646 Modern Analytical Chemistry EXAMPLE 1 3 . 7 The concentration of Mn in steel can be determined by a neutron activation analysis using the method of external standards. A 1.000-g sample of an unknown steel sample and a 0.950-g sample of a standard steel known to contain 0.463% w/w Mn, are irradiated with neutrons in a nuclear reactor for 10 h. After a 40-min “cooling” period, the activities for gamma-ray emission were found to be 2542 cpm (counts per minute) for the unknown and 1984 cpm for the standard. What is the %w/w Mn in the unknown steel sample? SOLUTION Combining equations 13.29 and 13.30 gives The weight of Mn in the standard sample is Substituting into the previous equation gives Since the original mass of steel is 1.000 g, the %w/w Mn is 0.564%. One of the important advantages of NAA is its applicability to almost all ele- ments in the periodic table. Another advantage of neutron activation is that it is nondestructive. Consequently, NAA is an important technique for analyzing ar- chaeological and forensic samples, as well as works of art. Isotope Dilution Another important quantitative radiochemical method is isotope dilution. In this method of analysis a sample of analyte, called a tracer, is prepared in a radioactive form with a known activity, A T , for its radioactive decay. A mea- sured mass of the tracer, w T , is added to a sample containing an unknown mass, w x , of a nonradioactive analyte, and the material is homogenized. The sample is then processed to isolate w A grams of purified analyte, containing both radioactive and nonradioactive materials. The activity of the isolated sample, A A , is measured. If all the analyte, both radioactive and nonradioactive, is recovered, then A A and A T will be equal. Normally, some of the analyte is lost during isolation and purification. In this case A A is less than A T , and 13.31 The ratio of weights in equation 13.31 accounts for the “dilution” of the activity due to a failure to recover all the analyte. Solving equation 13.31 for w x gives 13.32 w A A ww x =? T A AT AA w ww x AT A T = + ? ? ? ? ? ? w x =× = 2542 0 00440 0 00564 cpm 1984 cpm g Mn g Mn.. w s g Mn g steel g steel g Mn=×= 0 00463 0 950 0 00440 . .. w A A w x x =× s s isotope dilution A form of internal standardization in which a radioactive form of the analyte serves as the internal standard. tracer A radioactive species used as an internal standard. 1400-CH13 9/8/99 4:32 PM Page 646 Chapter 13 Kinetic Methods of Analysis 647 EXAMPLE 1 3 .8 The concentration of insulin in a production vat is determined by the method of isotope dilution. A 1.00-mg sample of insulin labeled with 14 C, with an activity of 549 cpm, is added to a 10.0-mL sample taken from the production vat. After homogenizing the sample, a portion of the insulin is separated and purified, yielding 18.3 mg of pure insulin. The activity for the isolated insulin is measured at 148 cpm. How many milligrams of insulin are in the original sample? SOLUTION Substituting known values into equation 13.32 gives Equations 13.31 and 13.32 are only valid if the radioactive element in the tracer has a half-life that is considerably longer than the time needed to conduct the analysis. If this is not the case, then the decrease in activity is due both to the effect of dilu- tion and the natural decrease in the isotope’s activity. Some common radioactive isotopes for use in isotope dilution are listed in Table 13.1. An important feature of isotope dilution is that it is not necessary to recover all the analyte to determine the amount of analyte present in the original sample. Iso- tope dilution, therefore, is useful for the analysis of samples with complex matrices, when a complete recovery of the analyte is difficult. 1 3 B. 4 Characterization Applications One example of a characterization application is the determination of a sample’s age based on the kinetics for the decay of a radioactive isotope present in the sam- ple. The most common example is carbon-14 dating, which is used to determine the age of natural organic materials. As cosmic rays pass through the upper atmosphere, some of the 14 7 N present is converted to 14 6 C by the capture of high-energy neutrons. The 14 6 C then migrates into w x =×?= 549 148 18 3 1 00 66 9 cpm cpm mg mg mg insulin.. . Table 1 3 .1 Common Isotopes for Use as Tracers Isotope Half-Life 3 H 12.5 years 14 C 5730 years 32 P 14.3 days 35 S 87.1 days 45 Ca 152 days 55 Fe 2.91 years 60 Co 5.3 years 131 I 8 days 1400-CH13 9/8/99 4:32 PM Page 647 648 Modern Analytical Chemistry the lower atmosphere, where it is oxidized to produce radioactive CO 2 , which is subsequently incorporated into living organisms. As a result, all living plants and animals have approximately the same ratio of 14 6 C to 12 6 C in their tissues. When an organism dies, however, the radioactive decay of 14 6 C to 14 7 N by b – emission (t 1/2 = 5730 years) results in a gradual reduction in the ratio of 14 6 C to 12 6 C. The change in this ratio can be used to date samples that are as much as 30,000 years old, although the precision of the analysis is best when the sample’s age is less than 7000 years. The accuracy of carbon-14 data is limited by the assumption that the ratio of 14 6 C to 12 6 C has remained constant over time. Some variation in the ratio has occurred as the result of the increased consumption of fossil fuel, and the produc- tion of 14 6 C during the testing of nuclear weapons. Correction factors have been de- veloped that increase the accuracy of carbon-14 dating. EXAMPLE 1 3 . 9 A sample of fabric was dated by measuring the relative ratio of 14 6 C to 12 6 C, with the result being 80.9% of that found in modern fibers. How old is the fabric? SOLUTION Equations 13.26 and 13.27 provide us with a way to convert a change in the ratio of 14 6 C to 12 6 C to the fabric’s age. Letting A 0 be the ratio of 14 6 C to 12 6 C in modern fibers, we assign it a value of 1.00. The ratio of 14 6 C to 12 6 C in the sample, A, is 0.809. Solving gives Other isotopes can be used to determine the age of samples. The age of rocks, for example, has been determined from the ratio of the number of radioactive 238 92 U atoms to the number of stable 206 82 Pb atoms produced by radioactive decay. For rocks that do not contain uranium, dating is accomplished by comparing the ratio of radioactive 40 19 K to the stable 40 18 Ar. Another example is the dating of sediments col- lected from lakes by measuring the amount of 210 82 Pb present. 1 3 B. 5 Evaluation Radiochemical methods are routinely used for the analysis of trace analytes in macro- and mesosamples. The accuracy and precision of radiochemical analyses are generally within the range of 1–5%. Precision is limited by the random nature of ra- dioactive decay and is improved by counting the emission of radioactive particles over as long a time as is practical. If the number of counts, M, is reasonably large (M ? 100), and the counting period is significantly less than the isotopes half-life, then the percent relative standard deviation for the activity, (s A ) rel , is estimated as For example, when the activity is determined by counting 10,000 radioactive parti- cles, the relative standard deviation is 1%. The analytical sensitivity of a radiochemi- cal method is inversely proportional to the standard deviation of the measured ac- ()σ A M rel =× 1 100 t A A t = ? ? ? ? ? ? ×= ? ? ? ? ? ? × ? ? ? ? ? ? =ln . ln . . /012 0 693 100 0 809 5730 1750 years 0.693 years 1400-CH13 9/8/99 4:32 PM Page 648 Figure 13.16 Schematic diagram of a simple flow injection analyzer showing the principal units. Chapter 13 Kinetic Methods of Analysis 649 Reagent reservoir Detector Mixing/reaction zone Sample injection Propelling unit Direction of flow tivity and, therefore, is improved by increasing the number of particles that are counted. Selectivity rarely is of concern with radiochemical methods because most sam- ples contain only a single radioactive isotope. When several radioactive isotopes are present, differences in the energies of their respective radioactive particles can be used to determine each isotope’s activity. In comparison with most other analytical techniques, radiochemical methods are usually more expensive and require more time to complete an analysis. Radio- chemical methods also are subject to significant safety concerns due to the analyst’s potential exposure to high-energy radiation and the need to safely dispose of ra- dioactive waste. 1 3 C Flow Injection Analysis The focus of this chapter is on methods in which the signal is time-dependent. Methods based on the kinetics of chemical and nuclear reactions were presented in the previous two sections. In this section we consider the technique of flow injec- tion analysis. In this technique the sample is injected into a flowing carrier stream and swept through the detector, giving rise to a transient signal. The shape of this signal depends on the physical and chemical kinetic processes occurring in the car- rier stream during the time between injection and detection. 1 3 C.1 Theory and Practice Flow injection analysis (FIA) was developed in the mid-1970s as a highly efficient technique for the automated analyses of samples. 20,21 Unlike the centrifugal analyzer described earlier in this chapter, in which samples are simultaneously analyzed in batches of limited size, FIA allows for the rapid, sequential analysis of an unlimited number of samples. FIA is one member of a class of techniques called continuous- flow analyzers, in which samples are introduced sequentially at regular intervals into a liquid carrier stream that transports the samples to the detector. 22 Figure 13.16 is a schematic diagram detailing the basic components of a flow injection analyzer. The reagent serving as the carrier is stored in a reservoir, and a propelling unit maintains a constant flow of the carrier through the system of tub- ing comprising the instrument. The sample is injected directly into the flowing car- rier stream, where it travels through a mixing and reaction zone before passing through the detector’s flow-cell. Figure 13.16 is the simplest design for a flow injec- tion analyzer, consisting of a single channel with one reagent reservoir. Multiple- channel instruments, in which reagents contained in separate reservoirs are combined flow injection analysis An analytical technique in which samples are injected into a carrier stream of reagents, or in which the sample merges with other streams carrying reagents before passing through a detector. 1400-CH13 9/8/99 4:32 PM Page 649 Figure 13.17 Effect of dispersion on a sample’s flow profile at different times during a flow injection analysis: (a) at injection; and when the dispersion is due to (b) convection; (c) convection and diffusion; and (d) diffusion. The widths of the flow profiles also are shown. by merging channels, also are possible. A more detailed discussion of FIA instru- mentation is found in the next section. When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample dis- perses into the carrier stream. Dispersion results from two processes: convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube’s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample’s injection. The second contribution to the sample’s dispersion is diffusion due to the con- centration gradient between the sample and the carrier stream. Diffusion occurs parallel (axial) and perpendicular (radial) to the flow of the carrier stream, with only the latter contribution being important. Radial diffusion decreases the linear velocity of the sample at the center of the tubing, but the sample at the edge of the tubing experiences an increase in its linear velocity. Diffusion helps to maintain the integrity of the sample’s flow profile (Figure 13.17c), preventing samples in the car- rier stream from dispersing into one another. Both convection and diffusion make significant contributions to dispersion from approximately 3–20 s after the sample’s injection. This is the normal time scale for a flow injection analysis. After approxi- mately 25 s, diffusion becomes the only significant contributor to dispersion, result- ing in a flow profile similar to that shown in Figure 13.17d. An FIA curve, or “fiagram,” is a plot of the detector’s signal as a function of time. Figure 13.18 shows a typical fiagram for conditions in which both convection and diffusion contribute to the sample’s dispersion. Also shown on the figure are several parameters used to characterize the fiagram. Two parameters are used to de- fine the time required for the sample to move from the injector to the detector. The travel time, t a , is the elapsed time from the sample’s injection to the arrival of the leading edge of its flow profile at the detector. Residence time, T, on the other hand, is the time required to obtain the maximum signal. The difference between the resi- dence time and travel time is given as t￠ . The value for t￠ approaches 0 when convec- tion is the primary means of dispersion and increases in value as the contribution from diffusion becomes more important. The time required for the sample to pass through the detector’s flow cell, and for the signal to return to the baseline, is also described by two parame- ters. The baseline-to-baseline time, D t, is the elapsed time between the arrival of the leading edge of the sample’s flow profile to the departure of its trailing edge. The elapsed time between the maximum signal and its return to the baseline is called the return time, T￠ . The final characteristic parameter of a fi- agram is the peak height, h, which is equivalent to the difference between the maximum signal and the signal at the baseline. Of the six parameters shown in Figure 13.18, the most important are peak height and return time. The peak height is related, directly or indirectly, to the analyte’s concentration and is used for quantitative work. The sensitiv- ity of the method, therefore, is also determined by the peak height. The return time determines the frequency with which samples may be injected. 22 Figure 13.19 shows that when a second sample is injected at a time T￠ after injecting the first sample, 650 Modern Analytical Chemistry w w w w (a) (b) (c) (d) Detector signal Time h t a D t t' TT' Figure 13.18 Typical FIA curve showing characteristic parameters. fiagram A graph showing the detector’s response as a function of time in a flow injection analysis. 1400-CH13 9/8/99 4:33 PM Page 650 Figure 13.19 Effect of return time on sampling frequency. the overlap of the two FIA curves is minimal. By injecting samples at intervals of T ￠ , the maximum sampling rate is realized. Peak heights and return times are influenced by the dispersion of the sample’s flow profile and are influenced by the physical and chemical properties of the flow injection system. Physical parameters affecting the peak height and return time in- clude the volume of sample injected; the flow rate; the length, diameter, and geom- etry of the mixing and reaction zone; and the presence of mixing points where sep- arate channels merge together. The kinetics of any chemical reactions involving the sample and reagents in the carrier stream also influences the peak height and re- turn time. Unfortunately, there is no good theory that can be used to consistently predict the peak height and return time for a given set of physical and chemical parameters. The design of a flow injection analyzer for a particular analytical problem still oc- curs largely by a process of experimentation. Nevertheless, some general observa- tions about the effects of physical and chemical parameters can be made. In the ab- sence of chemical effects, sensitivity (larger peak height) is improved by injecting larger samples, increasing the flow rate, decreasing the length and diameter of the tubing in the mixing and reaction zone, and merging separate channels before the point where the sample is injected. Except for sample volume, an improvement in the sampling rate (smaller return time) is achieved by the same combination of physical parameters. Larger sample volumes, however, lead to longer return times and a decrease in sample throughput. The effect of chemical reactivity depends on whether the species monitored by the detector is a reactant or a product. For exam- ple, when the monitored species is a reactant, sensitivity is improved by selecting a combination of physical parameters that enables the sample to reach the detector more quickly. Adjusting the chemical composition of the carrier stream in a man- ner that decreases the rate of the reaction also improves sensitivity in this case. 1 3 C.2 Instrumentation The basic components of a flow injection analyzer are shown in Figure 13.16 and include a unit for propelling the carrier stream, a means for injecting the sample into the carrier stream, and a detector for monitoring the composition of the carrier stream. These units are connected by a transport system that provides a means for bringing together separate channels and that enables an appropriate mixing of the sample with the carrier stream. Separations modules also may be incorporated in the flow injection analyzer. Each of these components is considered in greater detail in this section. Chapter 13 Kinetic Methods of Analysis 651 Detector signal Time Second injection T' First injection 1400-CH13 9/8/99 4:33 PM Page 651 Figure 13.20 Schematic diagram of a peristaltic pump. Propelling Unit The propelling unit is used to move the carrier stream in the main channel, as well as any additional reagent streams in secondary channels, through the flow injection analyzer. Although several different propelling units have been used, the most common is a peristaltic pump. A peristaltic pump consists of a set of rollers attached to the outside of a rotating drum (Figure 13.20). Tubing from the reagent reservoirs is placed in between the rollers and a fixed plate. As the drum ro- tates the rollers squeeze the tubing, forcing the contents of the tubing to move in the direction of the rotation. Peristaltic pumps are capable of providing a constant flow rate, which is controlled by the drum’s speed of rotation and the inner diame- ter of the tubing. Flow rates from 0.0005–40 mL/min are possible, which is more than adequate to meet the needs of FIA, for which flow rates of 0.5–2.5 mL/min are common. One limitation to peristaltic pumps is that they produce a pulsed flow, particularly at higher flow rates, which may lead to oscillations in the signal. Injector The sample, typically 5–200 m L, is placed in the carrier stream by injec- tion. Although syringe injections through a rubber septum are used, a more com- mon means of injection is the rotary, or loop, injector used in HPLC and shown in Figure 12.28 of Chapter 12. This type of injector provides reproducible injection volumes and is easily adaptable to automation, a feature that is particularly impor- tant when high sampling rates are desired. Detector Detection in FIA may be accomplished using many of the electrochemi- cal and optical detectors used in HPLC. These detectors were discussed in Chapter 12 and are not considered further in this section. In addition, FIA detectors also have been designed around the use of ion-selective electrodes and atomic ab- sorption spectroscopy. Transport System The heart of a flow injection analyzer is the transport system used to bring together the carrier stream, the sample, and any reagents that must react with the sample to generate the desired signal. Each reagent reservoir con- nected to the flow injection analyzer is considered a separate channel. All channels must merge before the carrier stream reaches the detector, with the merging points determined by the chemistry involved in the method. The completed assembly of channels is called a manifold. The simplest manifold includes only a single channel, the basic outline of which is shown in Figure 13.21. This type of manifold is commonly used for direct analyses that do not require a chemical reaction. In this case the carrier stream only serves as a means for transporting the sample to the detector rapidly and repro- ducibly. For example, this manifold design has been used as a means of sample in- troduction in atomic absorption spectroscopy, achieving sampling rates as high as 700 samples/h. This manifold also is used for determining a sample’s pH or the concentration of metal ions using ion-selective electrodes. A single-channel manifold also can be used for systems in which a chemical re- action generates the species responsible for the analytical signal. In this case the car- rier stream both transports the sample to the detector and reacts with the sample. Because the sample must mix with the carrier stream, flow rates are lower than when no chemical reaction is involved. One example is the determination of chlo- ride in water, which is based on the following sequence of reactions. 23 Hg(SCN) 2 (aq) + 2Cl – (aq) t HgCl 2 (aq) + 2SCN – (aq) Fe 3+ (aq) + SCN – (aq) t Fe(SCN) 2+ (aq) 652 Modern Analytical Chemistry Drum Tubing Roller peristaltic pump A device for propelling liquids through flexible tubing. manifold The complete system of tubing for merging together samples and reagents in a flow injection analysis. W W B I C S P R1 D Figure 13.21 Example of a single-channel manifold for use in flow injection analysis where R1 is a reagent reservoir; P is the pump; S is the sample; I is the injector; B is a bypass loop; W is waste; C is the mixing and reaction coil; and D is the detector. 1400-CH13 9/8/99 4:33 PM Page 652 Figure 13.22 Two examples of dual-channel manifolds for use in flow injection analysis where R1 and R2 are reagent reservoirs; P is the pump; S is the sample; I is the injector; B is a bypass loop; W is waste; C is the mixing and reaction coil; and D is the detector. The carrier stream consists of an acidic solution of Hg(SCN) 2 and Fe 3+ . When a sample containing chloride is injected into the carrier stream, the chloride displaces the thiocyanate from Hg(SCN) 2 . The displaced thiocyanate then reacts with Fe 3+ to form the reddish colored Fe(SCN) 2+ complex, the absorbance of which is moni- tored at a wavelength of 480 nm. Sampling rates of approximately 120 samples/h have been achieved with this system. Most flow injection analyses requiring a chemical reaction use a manifold containing more than one channel. This provides more control over the mixing of reagents and the interaction of the reagents with the sample. Two configura- tions are possible for dual-channel systems. The dual-channel manifold shown in Figure 13.22a is used when a mixture of the reagents is unstable. For example, in acidic solutions phosphate reacts with molybdate to form the heteropoly acid H 3 P(Mo 12 O 40 ). In the presence of ascorbic acid the molybdenum in the het- eropoly acid is reduced from Mo(VI) to Mo(V), forming a blue-colored complex that is monitored spectrophotometrically at 660 nm. 23 Solutions of molybdate and ascorbic acid are maintained in separate reservoirs because they are not sta- ble when mixed together. The two reagent channels are merged and mixed just before the point where the sample is injected. A dual-channel manifold can also be used to mix a reagent in a secondary chan- nel with a sample that has been injected into the primary channel (Figure 13.22b). This style of manifold has been used for the quantitative analysis of many analytes, including the determination of chemical oxygen demand (COD) in wastewater. 24 The COD is a measure of the amount of oxygen required to completely oxidize the organic matter in the water sample and is an indicator of the level of organic pollu- tion. In the conventional method of analysis, COD is determined by refluxing the sample for 2 h in the presence of acid and a strong oxidizing agent, such as K 2 Cr 2 O 7 or KMnO 4 . When refluxing is complete, the amount of oxidant consumed in the re- action is determined by a redox titration. In the flow injection version of this analy- sis, the sample is injected into a carrier stream of aqueous H 2 SO 4 , which merges with a solution of the oxidant from a secondary channel. The oxidation reaction is kineti- cally slow. As a result, the mixing and reaction coils are very long, typically 40 m, and are submerged in a thermostated bath. The sampling rate is lower than that for most flow injection analyses, but at 10–30 samples/h it is substantially greater than that for the conventional method. More complex manifolds involving three or more channels are common, 22 but the possible combination of designs is too numerous to discuss in this text. One ex- ample of a four-channel manifold is shown in Figure 13.23. Separation Modules Incorporating a separation module in the flow injection manifold allows separations, such as dialysis, gaseous diffusion, and liquid–liquid extraction, to be included in a flow injection analysis. Such separations are never complete, but are reproducible if the operating conditions are carefully controlled. Chapter 13 Kinetic Methods of Analysis 653 W W B I CC S P R1 R2 D R1 R2 P W W B I C S D (a) (b) 1400-CH13 9/8/99 4:33 PM Page 653 Figure 13.23 Example of a four-channel manifold for a flow injection analysis where R1 through R4 are reagent reservoirs; P is the pump; S is the sample; I is the injector; B is a bypass loop; W is waste; C is a mixing and reaction coil; and D is the detector. Figure 13.24 Separation module for a flow injection analysis using a semipermeable membrane for dialysis and gaseous diffusion. Dialysis and gaseous diffusion are accomplished by placing a semipermeable membrane between the carrier stream containing the sample and an acceptor stream (Figure 13.24). As the sample stream passes through the separation mod- ule, a portion of those species capable of crossing the semipermeable membrane do so, entering the acceptor stream. This type of separation module is common in the analysis of clinical samples, such as serum and urine, for which dialysis separates the analyte from its complex matrix. Semipermeable gaseous diffusion membranes have been used for the determination of ammonia and carbon diox- ide in blood. For example, ammonia is determined by injecting the sample into a carrier stream of aqueous NaOH. Ammonia diffuses across the semipermeable membrane into an acceptor stream containing an acid–base indicator. The re- sulting acid–base reaction between ammonia and the indicator is monitored spectrophotometrically. Liquid–liquid extractions are accomplished by merging together two immis- cible fluids, each carried in a separate channel. The result is a segmented flow through the separation module, consisting of alternating portions of the two phases. At the outlet of the separation module, the two fluids are separated by taking advantage of the difference in their densities. Figure 13.25 shows a typical configuration for a separation module in which the sample is injected into an aqueous phase and extracted into a less dense organic phase that passes through the detector. 654 Modern Analytical Chemistry W B I C C S P R1 R2 R3 R4 W C D Semipermeable membrane Acceptor stream Detector WasteSample stream A org Organic phase Detector WasteAqueous phase (with sample) A aq Figure 13.25 Separation module for a flow injection analysis using a liquid–liquid extraction (inset shows the equilibrium reaction). 1400-CH13 9/8/99 4:33 PM Page 654 Figure 13.26 Use of FIA for titration experiments showing the determination of D t. The concentration of the analyte decreases from (a) to (d). 1 3 C. 3 Quantitative Applications In a quantitative flow injection analysis a calibration curve is determined by injecting standard samples containing known concentrations of analyte. The format of the cali- bration curve, such as absorbance versus concentration, is determined by the method of detection. Calibration curves for standard spectroscopic and electrochemical methods were discussed in Chapters 10 and 11 and are not considered further in this chapter. Flow injection analysis has been applied to a wide variety of samples, including environmental, clinical, agricultural, industrial, and pharmaceutical samples. The majority of analyses to date involve environmental and clinical samples, which is the focus of this section. Quantitative analytical methods using FIA have been developed for cationic, anionic, and molecular pollutants in wastewater, fresh waters, groundwaters, and marine waters, several examples of which were described in the previous section. Table 13.2 provides a partial listing of other analytes that have been determined using FIA, many of which are modifications of conventional standard spectropho- tometric and potentiometric methods. An additional advantage of FIA for environ- mental analysis is its ability to provide for the continuous, in situ monitoring of pollutants in the field. 25 As noted in Chapter 9, several standard methods for the analysis of water in- volve acid–base, complexation, or redox titrations. Flow injection analysis also can be used for rapidly conducting such titrations using a single-channel manifold simi- lar to that shown in Figure 13.21. 26 A solution, consisting of a titrant (the concen- tration of which is stoichiometrically less than that of the analyte) and an indicator, is placed in the reservoir and continuously pumped through the system. When a sample is injected, it thoroughly mixes with the titrant in the carrier stream. The re- action between the analyte, which is in excess, and the titrant produces a relatively broad flow profile for the sample. As the sample moves toward the detector, addi- tional mixing occurs. Due to the continued reaction between titrant and analyte, the analyte’s concentration decreases, and the width of its flow profile also de- creases. As the sample passes through the detector, the width of its flow profile, D t, is determined by monitoring the indicator’s absorbance (Figure 13.26). Solutions with higher initial concentrations of analyte have a greater D t. Calibration curves of D t versus log [analyte] are prepared using standard solutions of analyte. Chapter 13 Kinetic Methods of Analysis 655 Table 1 3 .2 Selected Examples of the Application of FIA to the Analysis of Environmental Samples Volume Sample Concentration Sampling Frequency Analtye Sample (m L) Range (h –1 ) Ca 2+ fresh water 20 0.8–7.2 ppm 80 Cu 2+ groundwater 70–700 100–400 ppb 20 Pb 2+ groundwater 70–700 0–40 ppb 20 Zn 2+ sea water 1000 1–100 ppb 30–60 NH 4 + sea water 60 0.18–18.1 ppm 288 NO 3 – rain water 1000 1–10 ppm 40 SO 4 2– fresh water 400 4–140 ppm 180 CN – industrial 10 0.3–100 ppm 40 Source: Adapted from Valcárcel, M.; Luque de Castro, M. D. Flow-Injection Analysis: Principles and Applications. Ellis Horwood: Chichester, England, 1987. Detector signal Time (a) D t (b) (c) (d) 1400-CH13 9/8/99 4:33 PM Page 655 656 Modern Analytical Chemistry Table 1 3 . 3 Selected Examples of the Application of FIA to the Analysis of Clinical Samples Volume Sample Sampling Frequency Analyte Sample (m L) Concentration Range (h –1 ) Nonenzymatic Methods Cu 2+ serum 20 0.7–1.5 ppm 70 Cl – serum 60 50–150 meq/L 125 PO 4 3– serum 200 10–60 ppm 130 total CO 2 blood plasma 50 10–50 mM 70 chloropromazine urine 200 1.5–9 m M24 Enzymatic Methods glucose blood serum 26.5 0.5–15 mM 60 urea blood serum 30 4–20 mM 60 ethanol blood 30 5–30 ppm 50 Source: Adapted from Valcárcel, M.; Luque de Castro, M. D. Flow-Injection Analysis: Principles and Applications. Ellis Horwood: Chichester, England, 1987. Representative Methods —Continued Flow injection analysis has also found numerous applications in the analysis of clinical samples, using both enzymatic and nonenzymatic methods. A list of selected examples is given in Table 13.3. Representative Method Although each FIA method has its own unique considera- tions, the determination of phosphate described in the following method provides an instructive example of a typical procedure. Method 1 3 .2 Determination of Phosphate by FIA 27 Description of Method. The FIA determination of phosphate is an adaptation of a standard spectrophotometric analysis for phosphate. In the presence of acid, phosphate reacts with molybdate to form a yellow-colored complex in which molybdenum is present as Mo(VI). H 3 PO 4 (aq) + 12H 2 MoO 4 (aq) t H 3 P(Mo 12 O 40 )(aq) + 12H 2 O(l) In the presence of a reducing agent, such as ascorbic acid, the yellow-colored complex is reduced to a blue-colored complex of Mo(V). Procedure. Prepare reagent solutions of 0.005 M ammonium molybdate in 0.40 M HNO 3 , and 0.7% w/v ascorbic acid in 1% v/v glycerin. Using a stock solution of 100.0-ppm phosphate, prepare a set of external standards with phosphate concentrations of 10, 20, 30, 40, 50, and 60 ppm. Use a manifold similar to that shown in Figure 13.22a, with a 50-cm mixing coil and a 50-cm reaction coil. Set the flow rate to 0.5 mL/min. Prepare a calibration curve by injecting 50 m L of each standard, measuring the absorbance at 650 nm. Samples are analyzed in the same manner. 1400-CH13 9/8/99 4:33 PM Page 656 Chapter 13 Kinetic Methods of Analysis 657 Questions 1. As noted, this procedure is adapted from a standard spectrophotometric method. The instructions for the spectrophotometric method indicate that the absorbance should be measured 5–10 min after adding the ascorbic acid. Why is this waiting period necessary in the spectrophotometric method, but not necessary in the FIA method? The reduction of the yellow-colored Mo(VI) complex to the blue-colored Mo(V) complex is a slow reaction. In the standard spectrophotometric method, it is difficult to reproducibly control the amount of time that reagents are allowed to react before measuring the absorbance. To achieve good precision, therefore, the reaction is allowed sufficient time to proceed to completion before measuring the absorbance. In the FIA method, the flow rate and the dimensions of the reaction coil determine the elapsed time between sample introduction and the measurement of absorbance (about 30 s in this configuration). Since this time is precisely controlled, the reaction time is the same for all standards and samples. 2. The spectrophotometric method recommends using standard solutions of phosphate in the range of 2–10 ppm, whereas the FIA method recommends standards in the range of 10–60 ppm. Explain why the methods use a different range of standards. In the FIA method we measure the absorbance before the reduction of the yellow-colored Mo(VI) complex is complete. For this reason, the absorbance for any standard solution of phosphate will always be smaller when using the FIA method. This means that the FIA method is less sensitive, and higher concentrations of phosphate are necessary. 3. How would you incorporate a reagent blank into the FIA analysis? A reagent blank can be obtained by injecting a sample of distilled water in place of the external standard or sample. The reagent blank’s absorbance is subtracted from the absorbances obtained for the standards and samples. 4. The following data were obtained for a set of external phosphate standards. All absorbances have been corrected for a reagent blank. Parts per Million PO 4 3– Absorbance 10.00 0.079 20.00 0.160 30.00 0.233 40.00 0.316 60.00 0.482 What is the concentration of phosphate in the sample if it gives an absorbance of 0.287? A calibration curve of absorbance versus parts per million PO 4 3– gives the following calibration relationship. Absorbance = –0.0030 + 8.04 · 10 –3 (ppm PO 4 3– ) Substituting the sample’s absorbance gives its concentration of phosphate as 36.07 ppm. 1400-CH13 9/8/99 4:33 PM Page 657 1 3 C. 4 Evaluation The majority of FIA applications are modifications of conventional titrimetric, spectrophotometric, and electrochemical methods of analysis. For this reason it is appropriate to evaluate FIA in relation to these conventional methods. The scale of operations for FIA allows for the routine analysis of minor and trace analytes and for macro-, meso-, and microsamples. The ability to work with microliter injection volumes is useful when the sample is scarce. Conventional methods of analysis, however, may allow the determination of smaller concentrations of analyte. The accuracy and precision of FIA are comparable to that obtained by conven- tional methods of analysis. The precision of a flow injection analysis is influenced by variables that are not encountered in conventional methods, including the stabil- ity of the flow rate and the reproducibility of the sample’s injection. In addition, re- sults from FIA may be more susceptible to temperature variations. These variables, therefore, must be carefully controlled. In general, the sensitivity of FIA is less than that for conventional methods of analysis for two principal reasons. First, as with chemical kinetic methods, measure- ments in FIA are made under nonequilibrium conditions when the signal has yet to reach its maximum value. Second, dispersion of the sample as it progresses through the system results in its dilution. As discussed earlier, however, the variables that in- fluence sensitivity are known. As a result the FIA manifold can be designed to opti- mize the sensitivity of the analysis. Selectivity in FIA is often better than that for conventional methods of analysis. In many cases this is due to the kinetic nature of the measurement process, in which potential interferents may react more slowly than the analyte. Contamination from external sources also is less of a problem since reagents are stored in closed reser- voirs and are pumped through a system of transport tubing that, except for waste lines, is closed to the environment. Finally, FIA is an attractive technique with respect to demands on time, cost, and equipment. When employed for automated analyses, FIA provides for very high sam- pling rates. Most analyses can be operated with sampling rates of 20–120 samples/h, but rates as high as 1700 samples/h have been realized. 22 Because the volume of the flow injection manifold is small, typically less than 2 mL, consumption of reagents is substantially less than with conventional methods. This can lead to a significant de- crease in the cost per analysis. Flow injection analysis requires additional equipment, beyond that used for similar conventional methods of analysis, which adds to the ex- pense of the analysis. On the other hand, flow injection analyzers can be assembled from equipment already available in many laboratories. 658 Modern Analytical Chemistry 1 3 D KEY TERMS alpha particle (p. 642) beta particle (p. 642) enzyme (p. 636) fiagram (p. 650) flow injection analysis (p. 649) gamma ray (p. 642) Geiger counter (p. 643) half-life (p. 643) isotope dilution (p. 646) isotopes (p. 642) Lineweaver–Burk plot (p. 638) manifold (p. 652) Michaelis constant (p. 637) negatron (p. 642) neutron activation (p. 645) peristaltic pump (p. 652) positron (p. 642) quench (p. 634) rate (p. 624) rate constant (p. 624) rate law (p. 624) scintillation counter (p. 643) steady-state approximation (p. 636) stopped flow (p. 634) substrate (p. 636) tracer (p. 646) 1400-CH13 9/8/99 4:33 PM Page 658 Chapter 13 Kinetic Methods of Analysis 659 Kinetic methods of analysis are based on the rate at which a chem- ical or physical process involving the analyte occurs. Three types of kinetic methods are discussed in this chapter: chemical kinetic methods, radiochemical methods, and flow injection analysis. Chemical kinetic methods are based on the rate at which a chemical reaction involving the analyte proceeds. Either the inte- grated or differential form of the rate law may be used. When using an integral method, the concentration of analyte, or a reac- tant or product stoichiometrically related to the analyte, is deter- mined at one or more points in time following the reactions initia- tion. The initial concentration of analyte is then determined using the integral form of the reaction’s rate law. Alternatively, the time required to effect a given change in concentration may be mea- sured. In a differential kinetic method the rate of the reaction is measured at a time, t, and the differential form of the rate law used to determine the analyte’s concentration. Chemical kinetic methods are particularly useful for reactions that are too slow for a convenient analysis by other analytical methods. In addition, chemical kinetic methods are often easily adapted to an automated analysis. For reactions with fast kinetics, automation allows hundreds (or more) of samples to be analyzed per hour. Another important application of chemical kinetic methods is for the quantitative analysis of enzymes and their sub- strates and for the characterization of enzyme catalysis. Radiochemical methods of analysis take advantage of the decay of radioactive isotopes. A direct measurement of the rate at which a radioactive isotope decays may be used to determine its concen- tration in a sample. For analytes that are not naturally radioactive, neutron activation often can be used to induce radioactivity. Iso- tope dilution, in which a radioactively labeled form of an analyte is spiked into the sample, can be used as an internal standard for quantitative work. In a flow injection analysis the sample is injected into a flowing carrier stream that often is merged with additional streams carry- ing reagents. As the sample moves with the carrier stream, it both reacts with the contents of the carrier stream and any additional reagent streams, and undergoes dispersion. The resulting fiagram of signal versus time bears some resemblance to a chromatogram. Unlike chromatography, however, flow injection analysis is not a separation technique. Because all components in a sample move with the carrier stream’s flow rate, it is possible to introduce a sec- ond sample before the first sample reaches the detector. As a re- sult, flow injection analysis is ideally suited for the rapid through- put of samples. 1 3 E SUMMARY The following experiments introduce the application of chemical kinetics, including enzyme kinetics. Bateman, Jr. R. C.; Evans, J. A. “Using the Glucose Oxidase/Peroxidase Systems in Enzyme Kinetics,” J. Chem. Educ. 1995, 72, A240–A241. Glucose oxidase catalyzes the oxidation of glucose to gluconolactone and hydrogen peroxide. In the presence of horseradish peroxidase, the hydrogen peroxide is reduced by a water-soluble dye that forms a blue-green color when oxidized. The kinetics of the enzymatic reactions are followed by monitoring absorbance at 725 nm or 414 nm. Procedures are given for determining the activity of glucose oxidase and for studying the effect of glucose concentration on the reaction rate. The adaptation of the method for other sugars is also discussed. Diamandis, E. P.; Koupparis, M. A.; Hadjiionnou, T. P. “Kinetic Studies with Ion-Selective Electrodes: Determination of Creatinine in Urine with a Picrate Ion- Selective Electrode,” J. Chem. Educ. 1983, 60, 74–76. This experiment includes instructions for preparing a picrate ion-selective electrode. The application of the electrode in determining the concentration of creatinine in urine (which is further described in Method 13.1) also is outlined. Pandey, S.; McHale, M. E. R.; Horton, A. M.; et al. “Kinetics- Based Indirect Spectrophotometric Method for the Simultaneous Determination of MnO 4 – and Cr 2 O 7 2– , J. Chem. Educ. 1998, 75, 450–452. This experiment describes a kinetic method for determining the concentrations of MnO 4 – and Cr 2 O 7 2– in a mixture. The analysis is based on a difference in the rate at 1 3 F Suggested EXPERIMENTS The following experiments may be used to illustrate the application of kinetic methods of analysis. Experiments are divided into two groups: those based on chemical kinetics and those using flow injection analysis. Each suggested experiment includes a brief description. Experiments —Continued 1400-CH13 9/8/99 4:33 PM Page 659 660 Modern Analytical Chemistry Continued from page 659 Experiments which the analytes oxidize a solution of pyrogallol red; oxidation of pyrogallol red by MnO 4 – is faster than that for Cr 2 O 7 2– . The reaction is followed by measuring the absorbance of pyrogallol at a wavelength of 488 nm at two reaction times. Calibration curves prepared using external standards provide the necessary information to determine the concentrations of MnO 4 – and Cr 2 O 7 2– in an unknown. Results can be compared with more traditional spectrophotometric methods for analyzing mixtures. The following set of experiments provide examples of the application of flow injection analysis or the characterization of the behavior of a flow injection analysis system. Carroll, M. K.; Tyson, J. F. “An Experiment Using Time- Based Detection in Flow Injection Analysis,” J. Chem. Educ. 1993, 70, A210–A216. Most flow injection analyses use peak height as the analytical signal. When there is insufficient time for reagents to merge with the sample, the result is a split-peak, or “doublet,” due to reaction at the sample’s leading and trailing edges. This experiment describes how the difference between the peak times can be used for quantitative work. Hansen, E. H.; Ruzicka, J. “The Principles of Flow Injection Analysis as Demonstrated by Three Lab Exercises,” J. Chem. Educ. 1979, 56, 677–680. Directions are provided for the quantitative analyses of Cl – and PO 4 3– , and for conducting titrations of strong acids and strong bases. McKelvie, I. D.; Cardwell, T. J.; Cattrall, R. W. “A Microconduit Flow Injection Analysis Demonstration Using a 35-mm Slide Projector,” J. Chem. Educ. 1990, 67, 262–263. Directions are provided for constructing a small-scale FIA system that can be used to demonstrate the features of flow injection analysis. For another example see Grudpan, K.; Thanasarn, T. “Overhead Projector Injection Analysis,” Anal. Proc. 1993, 30, 10–12. Meyerhoff, M. E.; Kovach, P. M. “An Ion-Selective Electrode/Flow Injection Analysis Experiment: Determination of Potassium in Serum,” J. Chem. Educ. 1983, 60, 766–768. This experiment describes the preparation of a flow-through potentiometric electrode assembly incorporating a valinomycin–PVC membrane in the transport tubing. The application of this detector to the analysis of K + in serum is outlined. Ríos, A.; Luque de Castro, M.; Valcárcel, M. “Determination of Reaction Stoichiometries by Flow Injection Analysis,” J. Chem. Educ. 1986, 63, 552–553. This experiment describes the use of FIA for determining the stoichiometry of the Fe 2+ –o-phenanthroline complex using the method of continuous variations and the mole-ratio method. Directions are also provided for determining the stoichiometry of the oxidation of ascorbic acid by dichromate and for determining the rate constant for the reaction at different pH levels and different concentration ratios of the reactants. Stults, C. L. M.; Wade, A. P.; Crouch, S. R. “Investigation of Temperature Effects on Dispersion in a Flow Injection Analyzer,” J. Chem. Educ. 1988, 65, 645–647. Many factors affect a sample’s dispersion in an FIA analysis. In this experiment students study the effect of temperature on dispersion. Wolfe, C. A. C.; Oates, M. R.; Hage, D. S. “Automated Protein Assay Using Flow Injection Analysis,” J. Chem. Educ. 1998, 75, 1025–1028. This experiment describes the adaptation of the bicinchoninic acid (BCA) protein assay to a flow injection analysis. The assay is based on the reduction of Cu 2+ to Cu + by the protein, followed by the reaction of Cu + with bicinchoninic acid to form a purple complex that absorbs at 562 nm. Directions are provided for the analysis of bovine serum albumin and rabbit immunoglobulin G, and suggestions are provided for additional analyses. 1400-CH13 9/8/99 4:33 PM Page 660 Chapter 13 Kinetic Methods of Analysis 661 1 Equation 13.14 shows how [A] 0 is determined for a two-point fixed-time integral method in which the concentration of A for the pseudo-first-order reaction A+Rfi P is measured at times t 1 and t 2 . Derive a similar equation for the case when the product is monitored under pseudo-first- order conditions. 2. The concentration of phenylacetate can be determined from the kinetics of its pseudo-first-order hydrolysis reaction in an ethylamine buffer. When a standard solution of 0.55 mM phenylacetate is analyzed, the concentration of phenylacetate after 60 s is found to be 0.17 mM. When an unknown is analyzed, the concentration of phenylacetate remaining after 60 s is found to be 0.23 mM. What is the initial concentration of phenylacetate in the unknown? 3. In the presence of acid, solutions of iodide are oxidized by hydrogen peroxide 2I – +H 2 O 2 +2H 3 O + fi 4H 2 O+I 2 When I – and H 3 O + are present in excess, the kinetics of the reaction are pseudo-first-order in H 2 O 2 , and can be used to determine the concentration of H 2 O 2 by following the production of I 2 with time. In one analysis the absorbance of the solution was measured after 240 s at 348 nm (where Beer’s law holds for I 2 ). When a set of standard solutions of H 2 O 2 was analyzed, the following results were obtained [H 2 O 2 ] (m M) Absorbance 100.0 0.236 200.0 0.471 400.0 0.933 800.0 1.872 What is the concentration of H 2 O 2 in a sample that has an absorbance of 0.669 after 240 s? 4. The concentration of chromic acid can be determined from its reduction by alcohols under conditions when the kinetics are pseudo-first-order in analyte. One approach is to monitor the absorbance of the solution at a wavelength of 355 nm. A standard solution of 5.1 · 10 –4 M chromic acid yields absorbances of 0.855 and 0.709 at, 100 s and 300 s, respectively, after the reaction’s initiation. When a sample with an unknown amount of chromic acid is analyzed under identical conditions, absorbances of 0.883 and 0.706 are obtained. What is the concentration of chromic acid in this sample? 5. Malmstadt and Pardue developed a variable-time method for the determination of glucose based on its oxidation by the enzyme glucose oxidase. 28 To monitor the reaction’s progress, iodide is added to the samples and standards. The H 2 O 2 produced by the oxidation of glucose reacts with the I – , giving I 2 as a product. The time required to produce a fixed amount of I 2 is determined spectrophotometrically. The following data were reported for a set of calibration standards Glucose Time (ppm) (s) 5.0 146.5 150.0 149.6 10.0 69.2 67.1 66.0 20.0 34.8 35.0 34.0 30.0 22.3 22.7 22.6 40.0 16.7 16.5 17.0 50.0 13.3 13.3 13.8 To verify the method a 1.00-mL aliquot of a standard solution of 40.0-ppm glucose was added to 1.00 mL of the combined reagents, requiring 34.6 s to produce the same extent of reaction. Determine the calculated concentration of glucose in the standard and the percent error for the analysis. 6. Deming and Pardue studied the kinetics for the hydrolysis of p-nitrophenyl phosphate by the enzyme alkaline phosphatase. 29 The progress of the reaction was monitored by measuring the absorbance due to p-nitrophenol, which is one of the products of the reaction. A plot of the rate of the reaction (with units of m mol mL –1 s –1 ) versus the volume, V, (in milliliters) of a serum calibration standard containing the enzyme yielded a straight line with the following equation Rate (in m mol mL –1 s –1 ) = 2.7 · 10 –7 + 3.485 · 10 –5 · V A 10.00-mL sample of serum is analyzed, yielding a rate of 6.84 · 10 –5 m mol mL –1 s –1 . How much more dilute is the enzyme in the serum sample than in the serum calibration standard? 7. The following data were collected for a reaction known to be pseudo-first-order in analyte, A, during the time in which the reaction is monitored. 1 3 G PROBLEMS 1400-CH13 9/8/99 4:33 PM Page 661 Time [A] t (s) (mM) 2 1.36 4 1.24 6 1.12 8 1.02 10 0.924 12 0.838 14 0.760 16 0.690 18 0.626 20 0.568 What are the rate constant and the initial concentration of analyte in the sample? 8. The enzyme acetylcholinesterase catalyzes the decomposition of acetylcholine to choline and acetic acid. Under a given set of conditions the enzyme has a K m of 9 · 10 –5 M, and a k 2 of 1.4 · 10 4 s –1 . What is the concentration of acetylcholine in a sample for which the rate of reaction in the presence of 6.61 · 10 –7 M enzyme is 12.33 m M s –1 ? You may assume that the concentration of acetylcholine is significantly smaller than K m . 9. The enzyme fumarase catalyzes the stereospecific addition of water to fumarate to form L-malate. A standard solution of fumarase, with a concentration of 0.150 m M, gave a rate of reaction of 2.00 m M min –1 under conditions in which the concentration of the substrate was significantly greater than K m . The rate of reaction for a sample, under identical conditions, was found to be 1.15 m M min –1 . What is the concentration of fumarase in the sample? 10. The enzyme urease catalyzes the hydrolysis of urea. The rate of this reaction was determined for a series of solutions in which the concentration of urea was changed while maintaining a fixed urease concentration of 5.0 m M. The following data were obtained. [urea] Rate (M) (Ms –1 ) 1.00 · 10 –7 6.25 · 10 –6 2.00 · 10 –7 1.25 · 10 –5 3.00 · 10 –7 1.88 · 10 –5 4.00 · 10 –7 2.50 · 10 –5 5.00 · 10 –7 3.12 · 10 –5 6.00 · 10 –7 3.75 · 10 –5 7.00 · 10 –7 4.37 · 10 –5 8.00 · 10 –7 5.00 · 10 –5 9.00 · 10 –7 5.62 · 10 –5 1.00 · 10 –6 6.25 · 10 –5 Determine the values of V max , k 2 , and K m for urease. 11. In a study of the effect of an enzyme inhibitor, values for V max and K m were measured for several concentrations of inhibitor. As the concentration of the inhibitor was increased, V max was found to remain essentially constant while the value of K m increased. Which mechanism for enzyme inhibition is in effect? 12. In the case of competitive inhibition, the equilibrium between the enzyme, E, the inhibitor, I, and the enzyme–inhibitor complex, EI, is described by the equilibrium constant K I . Show that for competitive inhibition the equation for the rate of reaction is where K I is the dissociation constant for the EI complex E+I t EI You may assume that k 2 << k –1 . 13. Analytes A and B react with a common reagent R with first- order kinetics. If 99.9% of A must react before 0.1% of B has reacted, what is the minimum acceptable ratio for their respective rate constants? 14. A mixture of two analytes, A and B, is analyzed simultaneously by monitoring their combined concentration, C = [A] + [B], as a function of time when they react with a common reagent. Both A and B are known to follow first- order kinetics with the reagent, and A is known to react faster than B. Given the data in the following table, Time C (min) (mM) 1 0.313 6 0.200 11 0.136 16 0.098 21 0.074 26 0.058 31 0.047 36 0.038 41 0.032 46 0.027 51 0.023 56 0.019 61 0.016 66 0.014 71 0.012 determine the initial concentrations of A and B, and the first- order rate constants, k A and k B . 15. Table 13.1 provides a list of several isotopes commonly used as tracers. The half-lives for these isotopes also are listed. What is the rate constant for the radioactive decay of each isotope? 16. 60 Co is a long-lived isotope (t 1/2 = 5.3 years) that frequently is used as a radiotracer. The activity in a 5.00-mL sample of a solution of 60 Co was found to be 2.1 · 10 7 disintegrations/s. What is the molar concentration of 60 Co in the sample? d dt V KK [] [] {([]/)}[] max PS IS mI = ++1 662 Modern Analytical Chemistry 1400-CH13 9/8/99 4:33 PM Page 662 17. The concentration of Ni in a new alloy is determined by a neutron activation analysis using the method of external standards. A 0.500-g sample of the alloy and a 1.000-g sample of a standard alloy known to contain 5.93% w/w Ni are irradiated with neutrons in a nuclear reactor. When irradiation is complete, the sample and standard are allowed to cool, and the gamma-ray activities are measured. Given that the activity is 1020 cpm for the sample and 3540 cpm for the standard, determine the %w/w Ni in the alloy. 18. The vitamin B 12 content of a multivitamin tablet is determined by dissolving ten tablets in water. The dissolved tablets are transferred to a 100-mL volumetric flask and diluted to volume. A 50.00-mL portion is removed and treated with 0.500 mg of radioactive vitamin B 12 having an activity of 572 cpm. After homogenization, the vitamin B 12 in the sample is isolated and purified, producing 18.6 mg with an activity of 361 cpm. Calculate the average concentration of vitamin B 12 in the tablet (in milligrams per tablet). 19. The oldest sample that can be dated by carbon-14 is approximately 30,000 years. What percentage of the 14 C still remains after this time span? 20. Potassium–argon dating is based on the nuclear decay of 40 K (t 1/2 = 1.3 · 10 9 years) to 40 Ar. Assuming that there is no 40 Ar originally present in the rock and that the 40 Ar cannot escape to the atmosphere, then the relative amounts of 40 K and 40 Ar can be used to determine the age of the rock. When 100.0 mg of a rock sample was analyzed, it was found to contain 4.63 · 10 –6 mol of 40 K and 2.09 · 10 –6 mol 40 Ar. How old is the rock sample? 21. The steady-state activity for 14 C is 15 cpm/g of carbon. What mass of carbon is needed to give a percent relative standard deviation of 1.0% for the activity of a sample if counting is limited to 1 h? How long must the radioactive decay from a 0.50-g sample of carbon be monitored to give a percent relative standard deviation of 1% for the activity? 22. Shown here is a fiagram obtained for a solution of 100.0-ppm PO 4 3– . Determine h, t a , T, t￠ , D t, and T￠ . What is the sensitivity of this FIA method (assuming a linear relationship between absorbance and concentration)? How many samples can be analyzed per hour? 23. A sensitive method for the flow injection analysis of Cu 2+ is based on its ability to catalyze the oxidation of di-2-pyridyl ketone hydrazone (DPKH) by atmospheric oxygen. 30 The product of the reaction is fluorescent and can be used to generate a signal when using a fluorometer as a detector. The yield of the reaction is at a maximum when the solution is made basic with NaOH. The fluorescence, however, is greatest in the presence of HCl. Sketch an FIA manifold that will be appropriate for this analysis. 24. The concentration of chloride in sea water can be determined by a flow injection analysis. The analysis of a set of calibration standards gives the results in the following table. Cl – (ppm) Absorbance 5.00 0.057 10.00 0.099 20.00 0.230 30.00 0.354 40.00 0.478 50.00 0.594 75.00 0.840 A 1.00-mL sample of sea water is placed in a 500-mL volumetric flask and diluted to volume with distilled water. When injected into the flow injection analyzer, an absorbance of 0.317 is measured. What is the concentration of Cl – in the sample? 25. Ramsing and colleagues 26 developed an FIA method for acid– base titrations using a carrier stream mixture of 2.0 · 10 –3 M NaOH and the acid–base indicator bromthymol blue. Standard solutions of HCl were injected, and the following values of D t were measured from the resulting fiagrams. [HCl] D t (M) (s) 0.008 3.13 0.010 3.59 0.020 5.11 0.040 6.39 0.060 7.06 0.080 7.71 0.100 8.13 0.200 9.27 0.400 10.45 0.600 11.40 A sample with an unknown concentration of HCl was analyzed five times, giving values for D t of 7.43, 7.28, 7.41, 7.37, and 7.33 s. Determine the concentration of HCl in the sample. 26. Milardovíc and co-workers used a flow injection analysis method with an amperometric biosensor to determine the concentration of glucose in blood. 31 Given that a blood sample that is 6.93 mM in glucose has a signal of 7.13 nA, what is the concentration of glucose in a sample of blood giving a signal of 11.50 nA? Chapter 13 Kinetic Methods of Analysis 663 Absorbance 0 0 Time (s) 5 10 15 20 25 30 0.2 0.4 0.6 0.8 35 1400-CH13 9/8/99 4:33 PM Page 663 27. Fernández-Abedul and Costa-García developed an FIA method for detecting cocaine (C 17 H 21 NO 4 ) in samples. 32 The following data were collected for 12 replicate injections of a sample with a cocaine concentration of 6.2 · 10 –6 M. Signal Replicate (arbitrary units) 1 24.5 2 24.1 3 24.1 4 23.8 5 23.9 6 25.1 7 23.9 8 24.8 9 23.7 10 23.3 11 23.2 12 23.2 (a) What is the relative standard deviation for the analysis of this sample? (b) Given that these replicate analyses required 10.5 min, what is the expected sample throughput in samples per hour? (c) The following calibration data are available. [cocaine] Signal (M) (arbitrary units) 1.8 · 10 –7 0.8 3.6 · 10 –7 2.1 6.0 · 10 –7 2.4 8.1 · 10 –7 3.2 1.0 · 10 –6 4.5 2.0 · 10 –6 8.1 4.0 · 10 –6 14.4 6.0 · 10 –6 21.6 8.0 · 10 –6 27.1 1.0 · 10 –5 32.9 In a typical analysis a 10.0-mg sample is dissolved in water and diluted to volume in a 25-mL volumetric flask. A 125-m L aliquot is transferred to a 25-mL volumetric flask and diluted to volume with a pH 9 buffer. When injected into the flow injection apparatus a signal of 21.4 is obtained. What is the %w/w cocaine in the sample? 28. Holman and associates described an FIA method for determining the concentration of H 2 SO 4 in nonaqueous solvents. 33 Agarose beads (22 – 45-m m diameter) with a bonded acid–base indicator are soaked in NaOH and immobilized in the detector’s flow cell. Solutions of H 2 SO 4 in n-butanol are injected into the carrier stream. As the sample passes through the flow cell, the reaction of H 2 SO 4 and NaOH takes place. The end point of this neutralization reaction is signaled by a change in the bound indicator’s color and is detected spectrophotometrically. The elution volume needed to reach the titration’s end point is inversely proportional to the concentration of H 2 SO 4 ; thus, a plot of end point volume versus [H 2 SO 4 ] –1 should be linear. The following data are typical of that obtained using a set of external standards. [H 2 SO 4 ] End Point Volume (M) (mL) 3.58 · 10 –4 0.266 4.36 · 10 –4 0.227 5.60 · 10 –4 0.176 7.52 · 10 –4 0.136 1.38 · 10 –3 0.075 2.98 · 10 –3 0.037 5.62 · 10 –3 0.017 What is the concentration of H 2 SO 4 in a sample whose end point volume is 0.157 mL? 664 Modern Analytical Chemistry The following source provides a general review of the importance of kinetics in analytical chemistry. Mottola, H. A. “Some Kinetic Aspects Relevant to Contemporary Analytical Chemistry.” J. Chem. Ed. 1981, 58, 399–403. A brief history of chemical kinetic methods of analysis is found in the following text. Laitinen, H. A.; Ewing, G. W., eds. A History of Analytical Chemistry, The Division of Analytical Chemistry of the American Chemical Society: Washington, D.C., 1977, pp. 97–102. The following are useful resources for further information regarding chemical kinetic methods of analysis. Malmstadt, H. V.; Delaney, C. J.; Cordos, E. A. “Reaction-Rate Methods of Chemical Analysis,” Crit. Rev. Anal. Chem. 1972, 2, 559–619. Mark, H. B.; Rechnitz, G. A. Kinetics in Analytical Chemistry. Wiley: New York, 1968. Mottola, H. A. “Catalytic and Differential Reaction-Rate Methods of Chemical Analysis,” Crit. Rev. Anal. Chem. 1974, 4, 229–280. Mottola, H. A. Kinetic Aspects of Analytical Chemistry. Wiley: New York, 1988. 1 3 H SUGGESTED READINGS 1400-CH13 9/8/99 4:33 PM Page 664 Pardue, H. L. “Kinetic Aspects of Analytical Chemistry,” Anal. Chim. Acta 1989, 216, 69–107. Perez-Bendito, D.; Silva, M. Kinetic Methods in Analytical Chemistry. Ellis Horwood: Chichester, England, 1988. Additional information on the kinetics of enzyme catalyzed reactions may be found in the following texts. Bergmyer, H. U.; Grassl, M. Methods of Enzymatic Analysis, 3rd ed. Verlag Chemie: Deerfield Beach, FL, 1983. Pisakiewicz, D. Kinetics of Chemical and Enzyme-Catalyzed Reactions. Oxford University Press: New York, 1977. The following instrumental analysis textbooks may be consulted for further information on the detectors and signal analyzers used in radiochemical methods of analysis. Skoog, D. A.; Leary, J. J. Principles of Instrumental Analysis, 4th ed., Saunders College Publishing/Harcourt Brace and Co.: Philadelphia, 1998, Chapter 32. Strobel, H. A.; Heineman, W. R. Chemical Instrumentation: A Systematic Approach, 3rd ed., Wiley-Interscience: New York, 1989. The following resources provide additional information on the theory and application of flow injection analysis. Betteridge, D. “Flow Injection Analysis,” Anal. Chem. 1978, 50, 832A–846A. Kowalski, B. R.; Ruzicka, J. Christian, G. D. “Flow Chemography—The Future of Chemical Education,” Trends Anal. Chem. 1990, 9, 8–13. Mottola, H. A. “Continuous Flow Analysis Revisited,” Anal. Chem. 1981, 53, 1312A–1316A. Ruzicka, J. “Flow Injection Analysis: From Test Tube to Integrated Microconduits,” Anal. Chem. 1983, 55, 1040A–1053A. Ruzicka, J.; Hansen, E. H. “Flow-Injection Analysis,” Wiley- Interscience: New York, 1989. Stewart, K. K. “Flow Injection Analysis: New Tools for Old Assays, New Approaches to Analytical Measurements,” Anal. Chem. 1983, 55, 931A–940A. Valcárcel, M.; Luque de Castro, M. D. Flow-Injection Analysis: Principles and Applications, Ellis Horwood: Chichester, England, 1987. The use of flow injection as a means of introducing samples in atomic absorption spectroscopy is covered in the following paper. Tyson, J. F. “Atomic Spectrometry and Flow Injection Analysis: A Synergic Combination,” Anal. Chim. Acta 1988, 214, 57–75. Chapter 13 Kinetic Methods of Analysis 665 1. Method 4500-NO 2 – B in Standard Methods for the Analysis of Waters and Wastewaters, 20th ed. American Public Health Association: Washington, D.C., 1998, pp. 4-112–4-114. 2. Karayannis, M. I.; Piperaki, E. A.; Maniadake, M. M. Anal. Lett. 1986, 19, 13–23. 3. Pardue, H. L. Anal. Chim. Acta 1989, 216, 69–107. 4. Mottola, H. A. Anal. Chim. Acta 1993, 280, 279–287. 5. Crouch, S. R.; Malmstadt, H. V.; Anal. Chem. 1967, 39, 1084–1089, 1090–1093. 6. Malmstadt, H. V.; Cordos, E. A.; Delaney, C. J. Anal. Chem. 1972, 44(12), 26A–41A. 7. Javier, A. C.; Crouch, S. R.; Malmstadt, H. V. Anal. Chem. 1969, 41, 239–243. 8. Mark, H. B.; Rechnitz, G. A. Kinetics in Analytical Chemistry. Interscience: New York, 1968. 9. Sandell, E. B.; Kolthoff, I. M. J. Am. Chem. Soc. 1934, 56, 1426. 10. Ioannou, P. C.; Piperaki, E. A. Clin. Chem. (Winston-Salem, NC) 1986, 32, 1481–1483. 11. Diamandis, E. P.; Koupparis, M. A.; Hadjiioannou, T. P. J. Chem. Educ. 1983, 60, 74–76. 12. Malmstadt, H. V.; Delaney, C. J.; Cordos, E. A. Anal. Chem. 1972, 44(12), 79A–89A. 13. Guilbault, G. G. Enzymatic Methods of Analysis, Pergamon: New York, 1970. 14. Cruces Blanco, C.; Garcia Sanchez, F. Int. J. Environ. Anal. Chem. 1990, 38, 513. 15. (a) Northrop, D. B. J. Chem. Educ. 1998, 75, 1153–1157; (b) Zubzy, G. Biochemistry, 2nd ed. Macmillan Publishing Co.: New York, 1988, p. 269. 16. Pausch, J. B.; Margerum, D. W. Anal. Chem. 1969, 41, 226–232. 17. Holler, F. J.; Calhoun, R. K.; McClanahan, S. F. Anal. Chem. 1982, 54, 755–761. 18. Wentzell, P. D.; Crouch, S. R. Anal. Chem. 1986, 58, 2851–2855. 19. Wentzell, P. D.; Crouch, S. R. Anal. Chem. 1986, 58, 2855–2858. 20. Ruzicka, J.; Hansen, E. H. Anal. Chim. Acta 1975, 78, 145–157. 21. Stewart, K. K.; Beecher, G. R.; Hare, P. E. Anal. Biochem. 1976, 70, 167–173. 22. Valcárcel, M.; Luque de Castro, M. D. Flow-Injection Analysis: Principles and Applications, Ellis Horwood: Chichester, England, 1987. 23. Hansen, E. H.; Ruzicka, J. J. Chem. Educ. 1979, 56, 677–680. 24. Korenaga, T.; Ikatsu, H. Anal. Chim. Acta 1982, 141, 301–309. 25. Andrew, K. N.; Blundell, N. J.; Price, D.; et al. Anal. Chem. 1994, 66, 916A–922A. 26. Ramsing, A. U.; Ruzicka, J.; Hansen, E. H. Anal. Chim. Acta 1981, 129, 1–17. 27. Guy, R. D.; Ramaley, L.; Wentzell, P. D. J. Chem. Educ. 1998, 75, 1028–1033. 28. Malmstadt, H. V.; Pardue, H. L. Anal. Chem. 1961, 33, 1040–1047. 29. Deming, S. N.; Pardue, H. L. Anal. Chem. 1971, 43, 192–200. 30. Lazaro, F.; Luque de Castro, M. D.; Valcárcel, M. Analyst 1984, 109, 333. 31. Milardovíc, S.; Kruhak, I.; Ivekovíc, D.; et al. Anal. Chim. Acta 1997, 350, 91–96. 32. Fernández-Abedul, M; Costa-García, A. Anal. Chim. Acta 1996, 328, 67–71. 33. Holman, D. A.; Christian, G. D.; Ruzicka, J. Anal. Chem. 1997, 69, 1763–1765. 1 3 I REFERENCES 1400-CH13 9/8/99 4:33 PM Page 665 666 Chapter 1 4 Developing a Standard Method In Chapter 1 we made a distinction between analytical chemistry and chemical analysis. The goals of analytical chemistry are to improve established methods of analysis, to extend existing methods of analysis to new types of samples, and to develop new analytical methods. Once a method has been developed and tested, its application is best described as chemical analysis. We recognize the status of such methods by calling them standard methods. A standard method may be unique to a particular laboratory, which developed the method for their specific purpose, or it may be a widely accepted method used by many laboratories. Numerous examples of standard methods have been presented and discussed in the preceding six chapters. What we have yet to consider, however, is what constitutes a standard method. In this chapter we consider how a standard method is developed, including optimizing the experimental procedure, verifying that the method produces acceptable precision and accuracy in the hands of a single analyst, and validating the method for general use. 1400-CH14 9/8/99 4:35 PM Page 666 Figure 14.1 Example of a one-factor response surface. Chapter 14 Developing a Standard Method 667 Absorbance 01234 0.000 0.500 0.450 0.400 0.350 0.300 0.250 0.200 0.150 0.100 0.050 Analyte (ppm) 5 response The property of a system that is measured (R). factor A property of a system that is experimentally varied and that may affect the response. factor level A factor’s value. response surface A graph showing how a system’s response changes as a function of its factors. 1 4 A Optimizing the Experimental Procedure In the presence of H 2 O 2 and H 2 SO 4 , solutions of vanadium form a reddish brown color that is believed to be a compound with the general formula (VO) 2 (SO 4 ) 3 . Since the intensity of the color depends on the concentration of vanadium, the ab- sorbance of the solution at a wavelength of 450 nm can be used for the quantitative analysis of vanadium. The intensity of the color, however, also depends on the amount of H 2 O 2 and H 2 SO 4 present. In particular, a large excess of H 2 O 2 is known to decrease the solution’s absorbance as it changes from a reddish brown to a yel- lowish color. 1 Developing a standard method for vanadium based on its reaction with H 2 O 2 and H 2 SO 4 requires that their respective concentrations be optimized to give a max- imum absorbance. Using terminology adapted by statisticians, the absorbance of the solution is called the response. Hydrogen peroxide and sulfuric acid are factors whose concentrations, or factor levels, determine the system’s response. Optimiza- tion involves finding the best combination of factor levels. Usually we desire a max- imum response, such as maximum absorbance in the quantitative analysis for vana- dium as (VO) 2 (SO 4 ) 3 . In other situations, such as minimizing percent error, we seek a minimum response. 1 4 A.1 Response Surfaces One of the most effective ways to think about optimization is to visualize how a sys- tem’s response changes when we increase or decrease the levels of one or more of its factors. A plot of the system’s response as a function of the factor levels is called a response surface. The simplest response surface is for a system with only one fac- tor. In this case the response surface is a straight or curved line in two dimensions. A calibration curve, such as that shown in Figure 14.1, is an example of a one-factor response surface in which the response (absorbance) is plotted on the y-axis versus the factor level (concentration of analyte) on the x-axis. Response surfaces can also be expressed mathematically. The response surface in Figure 14.1, for example, is A = 0.008 + 0.0896C A where A is the absorbance, and C A is the analyte’s concentration in parts per million. For a two-factor system, such as the quantitative analysis for vanadium de- scribed earlier, the response surface is a flat or curved plane plotted in three dimen- sions. For example, Figure 14.2a shows the response surface for a system obeying the equation R = 3.0 – 0.30A + 0.020AB where R is the response, and A and B are the factor levels. Alternatively, we may represent a two-factor response surface as a contour plot, in which contour lines in- dicate the magnitude of the response (Figure 14.2b). The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 ≤ A ≤ 10, 0 ≤ B ≤ 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most re- sponse surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural con- straint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. standard method A method that has been identified as providing acceptable results. 1400-CH14 9/8/99 4:35 PM Page 667 Figure 14.3 Mountain-climbing analogy to using a searching algorithm to find the optimum response for a response surface. The path on the left leads to the global optimum, and the path on the right leads to a local optimum. If the equation for the response surface is known, then the optimum response is easy to locate. Unfortunately, the response surface is usually unknown; instead, its shape, and the location of the optimum response must be determined experimen- tally. The focus of this section is on useful experimental designs for optimizing ana- lytical methods. These experimental designs are divided into two broad categories: searching methods, in which an algorithm guides a systematic search for the opti- mum response; and modeling methods, in which we use a theoretical or empirical model of the response surface to predict the optimum response. 1 4 A.2 Searching Algorithms for Response Surfaces Imagine that you wish to climb to the top of a mountain. Because the mountain is covered with trees that obscure its shape, the shortest path to the summit is un- known. Nevertheless, you can reach the summit by always walking in a direction that moves you to a higher elevation. The route followed (Figure 14.3) is the result of a systematic search for the summit. Of course, many routes are possible leading from the initial starting point to the summit. The route taken, therefore, is deter- mined by the set of rules (the algorithm) used to determine the direction of each step. For example, one algorithm for climbing a mountain is to always move in the direction that has the steepest slope. A systematic searching algorithm can also be used to locate the optimum re- sponse for an analytical method. To find the optimum response, we select an initial set of factor levels and measure the response. We then apply the rules of the search- ing algorithm to determine the next set of factor levels. This process is repeated until the algorithm indicates that we have reached the optimum response. Two common searching algorithms are described in this section. First, however, we must consider how to evaluate a searching algorithm. Effectiveness and Efficiency A searching algorithm is characterized by its effec- tiveness and its efficiency. To be effective, the experimentally determined optimum should closely coincide with the system’s true optimum. A searching algorithm may fail to find the true optimum for several reasons, including a poorly designed algo- rithm, uncertainty in measuring the response, and the presence of local optima. A poorly designed algorithm may prematurely end the search. For example, an algo- rithm for climbing a mountain that allows movement to the north, south, east, or west will fail to find a summit that can only be reached by moving to the northwest. When measuring the response is subject to relatively large random errors, or noise, a spuriously high response may produce a false optimum from which the 668 Modern Analytical Chemistry Start 0 10 8 6 4 2 3 2.5 2 1.5 1 0.5 0 0 1.5 3 4.5 6 7.5 9 Factor A Factor B Response 0 0 1 10 9 8 7 6 5 4 3 2 Factor A F actor B 10 987654321 Figure 14.2 Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. (a) (b) 1400-CH14 9/8/99 4:35 PM Page 668 Figure 14.4 Factor effect plot for two independent factors. searching algorithm cannot move. When climbing a mountain, boulders encoun- tered along the way are examples of “noise” that must be avoided if the true opti- mum is to be found. The effect of noise can be minimized by increasing the size of the individual steps such that the change in response is larger than the noise. Finally, a response surface may contain several local optima, only one of which is the system’s true, or global, optimum. This is a problem because a set of initial conditions near a local optimum may be unable to move toward the global opti- mum. The mountain shown in Figure 14.3, for example, contains two peaks, with the peak on the left being the true summit. A search for the summit beginning at the position identified by the dot may find the local peak instead of the true sum- mit. Ideally, a searching algorithm should reach the global optimum regardless of the initial set of factor levels. One way to evaluate a searching algorithm’s effective- ness, therefore, is to use several sets of initial factor levels, finding the optimum re- sponse for each, and comparing the results. A second desirable characteristic for a searching algorithm is efficiency. An effi- cient algorithm moves from the initial set of factor levels to the optimum response in as few steps as possible. The rate at which the optimum is approached can be in- creased by taking larger steps. If the step size is too large, however, the difference between the experimental optimum and the true optimum may be unacceptably large. One solution is to adjust the step size during the search, using larger steps at the beginning, and smaller steps as the optimum response is approached. One-Factor-at-a-Time Optimization One approach to optimizing the quantitative method for vanadium described earlier is to select initial concentrations for H 2 O 2 and H 2 SO 4 and measure the absorbance. We then increase or decrease the concentration of one reagent in steps, while the second reagent’s concentration remains constant, until the absorbance decreases in value. The concentration of the second reagent is then adjusted until a decrease in ab- sorbance is again observed. This process can be stopped after one cycle or repeated until the absorbance reaches a maximum value or exceeds an acceptable threshold value. A one-factor-at-a-time optimization is consistent with a commonly held belief that to determine the influ- ence of one factor it is necessary to hold constant all other factors. This is an effective, although not necessarily an ef- ficient, experimental design when the factors are indepen- dent. 2 Two factors are considered independent when changing the level of one factor does not influence the ef- fect of changing the other factor’s level. Table 14.1 provides an example of two in- dependent factors. When factor B is held at level B 1 , changing factor A from level A 1 to level A 2 increases the response from 40 to 80; thus, the change in response, D R, is D R =80–40=40 In the same manner, when factor B is at level B 2 , we find that D R =100–60=40 when the level of factor A changes from A 1 to A 2 . We can see this independence graphically by plotting the response versus the factor levels for factor A (Figure 14.4). The parallel lines show that the level of factor B does not influence the effect on the response of changing factor A. In the same manner, the effect of changing factor B’s level is independent of the level of factor A. Chapter 14 Developing a Standard Method 669 Response Level for factor A Factor B 2 constant Factor B 1 constant Table 1 4 .1 Example of Two Independent Factors Factor A Factor B Response A 1 B 1 40 A 2 B 1 80 A 1 B 2 60 A 2 B 2 100 1400-CH14 9/8/99 4:35 PM Page 669 Figure 14.5 Two views of a two-factor response surface for which the factors are independent. The optimum response in (b) is indicated by the ? at (2, 8). Contour lines are shown for intervals of 0.5 response units. Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as R = 2.0 + 0.12A + 0.48B – 0.03A 2 – 0.03B 2 14.1 The resulting response surface for equation 14.1 is shown in Figure 14.5. The progress of a searching algorithm is best followed by mapping its path on a contour plot of the response surface. Positions on the response surface are identified as (a, b) where a and b are the levels for factors A and B. Four examples of a one- factor-at-a-time optimization of the response surface for equation 14.1 are shown in Figure 14.5b. For those paths indicated by a solid line, factor A is optimized first, followed by factor B. The order of optimization is reversed for paths marked by a dashed line. The effectiveness of this algorithm for optimizing independent factors is shown by the fact that the optimum response at (2, 8) is reached in a single cycle from any set of initial factor levels. Further- more, it does not matter which factor is optimized first. Although this algorithm is effective at locating the opti- mum response, its efficiency is limited by requiring that only a single factor can be changed at a time. Unfortunately, it is more common to find that two factors are not independent. In Table 14.2, for instance, changing the level of factor B from level B 1 to level B 2 has a significant effect on the response when factor A is at level A 1 D R =60–20=40 but has no effect when factor A is at level A 2 . D R =80–80=0 This effect is seen graphically in Figure 14.6. Factors that are not independent are said to interact. In this case the equation for the response includes an interaction term in which both factors A and B are present. Equation 14.2, for example, con- tains a final term accounting for the interaction between the factors A and B. R = 5.5 + 1.5A + 0.6B – 0.15A 2 – 0.0245B 2 – 0.0857AB 14.2 The resulting response surface for equation 14.2 is shown in Figure 14.7a. 670 Modern Analytical Chemistry 0 10 8 6 4 2 4.5 4 3.5 3 2 2.5 1.5 0.5 1 0 0 1.5 3 4.5 6 7.5 9 Factor A Factor B Response 0 0 1 10 9 8 7 6 5 4 3 2 Factor A F actor B 10 987654321 (a) (b) Table 1 4 .2 Example of Two Dependent Factors Factor A Factor B Response A 1 B 1 20 A 2 B 1 80 A 1 B 2 60 A 2 B 2 80 Response Level for factor B Factor A 2 constant Factor A 1 constant Figure 14.6 Factor effect plot for two interacting factors. 1400-CH14 9/8/99 4:35 PM Page 670 Figure 14.7 Two views of a two-factor response surface for which the factors interact. The optimum response in (b) is indicated by the ? at (3, 7). The response at the end of the first cycle is shown in (b) by the currency1 . Contour lines are shown for intervals of 1.0 response units. The progress of a one-factor-at-a-time optimization for the response surface given by equation 14.2 is shown in Figure 14.7b. In this case the optimum response of (3, 7) is not reached in a single cycle. If we start at (0, 0), for example, optimizing factor A follows the solid line to the point (5, 0). Optimizing factor B completes the first cycle at the point (5, 3.5). If our algorithm allows for only a single cycle, then the optimum response is not found. The optimum response usually can be reached by continuing the search through additional cycles, as shown in Figure 14.7b. The efficiency of a one-factor-at-a-time searching algorithm is significantly less when the factors interact. An additional complication with interacting factors is the possi- bility that the search will end prematurely on a ridge of the response surface, where a change in level for any single factor results in a smaller response (Figure 14.8). Simplex Optimization The efficiency of a searching algorithm is improved by al- lowing more than one factor to be changed at a time. A convenient way to accom- plish this with two factors is to select three sets of initial factor levels, positioned as the vertices of a triangle (Figure 14.9), and to measure the response for each. The set of factor levels giving the smallest response is rejected and replaced with a new set of factor levels using a set of rules. This process is continued until no further optimiza- tion is possible. The set of factor levels is called a simplex. In general, for k factors a simplex is a (k + 1)-dimensional geometric figure. 3,4 The initial simplex is determined by choosing a starting point on the response surface and selecting step sizes for each factor. Ideally the step sizes for each factor should produce an approximately equal change in the response. For two factors a convenient set of factor levels is (a, b), (a + s A , b), and (a + 0.5s A , b + 0.87s B ), where s A and s B are the step sizes for factors A and B. 5 Optimization is achieved using the following set of rules: Rule 1. Rank the response for each vertex of the simplex from best to worst. Rule 2. Reject the worst vertex, and replace it with a new vertex generated by reflecting the worst vertex through the midpoint of the remaining vertices. The factor levels for the new vertex are twice the average factor levels for the retained vertices minus the factor levels for worst vertex. Rule 3. If the new vertex has the worst response, then reject the vertex with the second-worst response, and calculate the new vertex using rule 2. This rule ensures that the simplex does not return to the previous simplex. Rule 4. Boundary conditions are a useful way to limit the range of possible factor levels. For example, it may be necessary to limit the concentration of a factor Chapter 14 Developing a Standard Method 671 (a) (b) 0 10 8 6 4 2 10 9 8 7 5 6 4 1 2 3 0 0 1.5 3 4.5 6 7.5 9 Factor A Factor B Response 0 0 1 10 9 8 7 6 5 4 3 2 Factor A F actor B 10 987654321 True optimum False optimum X Figure 14.8 Example of a false optimum for a one- factor-at-a-time searching algorithm. F actor B Factor A Figure 14.9 Simplex for two factors. 1400-CH14 9/8/99 4:35 PM Page 671 672 Modern Analytical Chemistry for solubility reasons or to limit temperature due to a reagent’s thermal stability. If the new vertex exceeds a boundary condition, then assign it a response lower than all other responses, and follow rule 3. Because the size of the simplex remains constant during the search, this algorithm is called a fixed-sized simplex optimization. Example 14.1 illustrates the application of these rules. EXAMPLE 1 4 .1 Find the optimum response for the response surface in Figure 14.7 using the fixed-sized simplex searching algorithm. Use (0, 0) for the initial factor levels, and set the step size for each factor to 1.0. SOLUTION Letting a = 0, b = 0, s a = 1 and s b = 1 gives the vertices for the initial simplex as Vertex 1: (a, b) = (0, 0) Vertex 2: (a + s A , b) = (1, 0) Vertex 3: (a + 0.5s A , b + 0.87s B ) = (0.5, 0.87) The responses (calculated using equation 14.2) for the three vertices are shown in the following table Vertex Factor A Factor B Response V 1 0 0 5.50 V 2 1.00 0 6.85 V 3 0.50 0.87 6.68 with V 1 giving the worst response and V 3 the best response (rule 1). We reject V 1 and replace it with a new vertex whose factor levels are calculated using rule 2; thus The new simplex, therefore, is Vertex Factor A Factor B Response V 2 1.00 0 6.85 V 3 0.50 0.87 6.68 V 4 1.50 0.87 7.80 The worst response is for vertex 3, which we replace with the following new vertex ab=× + ?= =× + ?=2 100 150 2 05 200 2 0087 2 087 0 .. .. . . New for V + for V 2 for V ) = 2 0 + 0.87 2 1 b bb b=× ? ? ? ? ? ? ?×?=2087 23 (. New for V + for V 2 for V ) = 2 1.00 + 0.50 2 3 1 a aa a=× ? ? ? ? ? ? ?×?=150 2 simplex optimization An efficient optimization method that allows several factors to be optimized at the same time. 1400-CH14 9/8/99 4:35 PM Page 672 The resulting simplex now consists of the following vertices Vertex Factor A Factor B Response V 2 1.00 0 6.85 V 4 1.50 0.87 7.80 V 5 2.00 0 7.90 The calculation of the remaining vertices is left as an exercise. The progress of the completed optimization is shown in Table 14.3 and in Figure 14.10. The optimum response of (3, 7) first appears in the twenty-fourth simplex, but a total of 29 steps is needed to verify that the optimum has been found. Chapter 14 Developing a Standard Method 673 Table 1 4 . 3 Progress of Fixed-Sized Simplex Optimization for Response Surface in Figure 14.10 Simplex Vertices Notes 1 1, 2, 3 2 2, 3, 4 3 2, 4, 5 4 4, 5, 6 5 5, 6, 7 6 6, 7, 8 7 7, 8, 9 8 8, 9, 10 9 8, 10, 11 10 10, 11, 12 11 11, 12, 13 12 12, 13, 14 follow rule 3 13 13, 14, 15 14 13, 15, 16 15 13, 16, 17 follow rule 3 16 16, 17, 18 17 16, 18, 19 18 16, 19, 20 follow rule 3 19 19, 20, 21 20 19, 21, 22 follow rule 3 21 21, 22, 23 22 21, 23, 24 follow rule 3 23 23, 24, 25 24 23, 25, 26 25 23, 26, 27 follow rule 3 26 26, 27, 28 follow rule 3 27 26, 28, 29 28 26, 29, 30 follow rule 3 29 26, 30, 31 vertex 31 same as vertex 25 1400-CH14 9/8/99 4:35 PM Page 673 The fixed-size simplex searching algorithm is effective at locating the optimum response for both independent and interacting factors. Its efficiency, however, is limited by the simplex’s size. We can increase its efficiency by allowing the size of the simplex to expand or contract in response to the rate at which the optimum is being approached. 3,6 Although the algorithm for a variable-sized simplex is not pre- sented here, an example of its increased efficiency is shown Figure 14.11. The refer- ences and suggested readings may be consulted for further details. 1 4 A. 3 Mathematical Models of Response Surfaces Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then lin- ear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treat- ment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for 674 Modern Analytical Chemistry 6 0 8 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 7.6 7.2 6.8 6.4 6 5.6 5.2 4.8 4.4 4 3.6 3.2 2.8 2.4 2 1.6 1.2 0.8 0.4 1 34 6 8 25 79 10 11 12 13 16 14 17 1518 19 20 21 22 27 28 25, 31 29 30 26 23 24 Figure 14.10 Progress of a fixed-sized simplex optimization for the response surface of Example 14.1. The optimum response at (3, 7) corresponds to vertex 26. 1400-CH14 9/8/99 4:35 PM Page 674 Figure 14.11 Progress of a variable-sized simplex optimization for the response surface of Example 14.1. The optimum response is at (3, 7). a few special cases are straightforward and are considered in this section. A more comprehensive treatment of linear regression can be found in several of the sug- gested readings listed at the end of this chapter. Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories: those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical rela- tionships between the response and the factors. In spectrophotometry, for example, Beer’s law is a theoretical model relating a substance’s absorbance, A, to its concen- tration, C A A = e bC A where e is the molar absorptivity, and b is the pathlength of the electromagnetic ra- diation through the sample. A Beer’s law calibration curve, therefore, is a theoretical model of a response surface. Empirical Models of the Response Surface In many cases the underlying theoreti- cal relationship between the response and its factors is unknown, making impossi- ble a theoretical model of the response surface. A model can still be developed if we make some reasonable assumptions about the equation describing the response sur- face. For example, a response surface for two factors, A and B, might be represented by an equation that is first-order in both factors Chapter 14 Developing a Standard Method 675 6 0 8 0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 5.1 5.4 5.7 7.6 7.2 6.8 6.4 6 5.6 5.2 4.8 4.4 4 3.6 3.2 2.8 2.4 2 1.6 1.2 0.8 0.4 1 3 4 6 8 2 5 7 9 10 11 13 16 14 17 15 18 19 20 21 12 theoretical model A model describing a system’s response that has a theoretical basis and can be derived from theoretical principles. 1400-CH14 9/8/99 4:35 PM Page 675 Figure 14.12 2 k factorial designs for (a) k = 2 and (b) k =3. R = b 0 + b a A + b b B first-order in both factors with an additional interaction term R = b 0 + b a A + b b B + b ab AB or a second-order polynomial equation R = b 0 + b a A + b b B + b aa A 2 + b bb B 2 The terms b 0 , b a , b b , b ab , b aa , and b bb are adjustable parameters whose values are de- termined by using linear regression to fit the data to the equation. Such equations are empirical models of the response surface because they have no basis in a theo- retical understanding of the relationship between the response and its factors. An empirical model may provide an excellent description of the response surface over a wide range of factor levels. It is more common, however, to find that an empirical model only applies to the range of factor levels for which data have been collected. To develop an empirical model for a response surface, it is necessary to collect the right data using an appropriate experimental design. Two popular experimental designs are considered in the following sections. Factorial Designs To determine a factor’s effect on the response, it is necessary to measure the response for at least two factor levels. For convenience these levels are labeled high, H f , and low, L f , where f is the factor; thus H A is the high level for factor A, and L B is the low level for factor B. When more than one factor is included in the empirical model, then each factor’s high level should be paired with both the high and low levels for all other factors. In the same way, the low level for each factor should be paired with the high and low levels for all other factors (Figure 14.12). All together, a minimum of 2 k experiments is necessary, where k is the number of fac- tors. This experimental design is known as a 2 k factorial design. 676 Modern Analytical Chemistry F actor A Factor B 34 12 Runs 1 2 3 4 A H A H A L A L A Factor levels B L B H B L B H B F actor A Factor B Factor C 3 4 5 6 7 8 1 2 Runs 1 2 3 4 5 6 7 8 A H A H A H A H A L A L A L A L A Factor levels B L B L B H B H B L B L B H B H B C L C H C H C L C L C H C H C L C (a) (b) empirical model A model describing a system’s response that is not derived from theoretical principles. 1400-CH14 9/8/99 4:35 PM Page 676 Chapter 14 Developing a Standard Method 677 The linear regression calculations for a 2 k factorial design are straightforward and can be done without the aid of a sophisticated statistical software package. To simplify the computations, factor levels are coded as +1 for the high level, and –1 for the low level. The relationship between a factor’s coded level, x f * , and its actual value, x f , is given as x f = c f + d f x f * 14.3 where c f is the factor’s average level, and d f is the absolute difference between the factor’s average level and its high and low values. Equation 14.3 is used to transform coded results back to their actual values. EXAMPLE 1 4 .2 The factor A has coded levels of +1 and –1 with an average factor level of 100, and d A equal to 5. What are the actual factor levels? SOLUTION The actual factor levels are H A = 100 + (1)(5) = 105 L A = 100 + (–1)(5) = 95 Let’s start by considering a simple example involving two factors, A and B, to which we wish to fit the following empirical model. R = b 0 + b a A + b b B + b ab AB 14.4 A 2 k factorial design with two factors requires four runs, or sets of experimental conditions, for which the uncoded levels, coded levels, and responses are shown in Table 14.4. The terms b 0 , b a , b b , and b ab in equation 14.4 account for, respectively, the mean effect (which is the average response), first-order effects due to factors A and B, and the interaction between the two factors. Estimates for these parameters are given by the following equations 14.5 14.6 14.7 β bb ii b n BR≈=∑ 1 * β aa ii b n AR≈=∑ 1 * β 00 1 ≈=∑b n R i Table 1 4 . 4 Example of Uncoded and Coded Factor Levels and Responses for a 2 2 Factorial Design Run ABA* B* Response 1 15 30 +1 +1 22.5 2 15 10 +1 –1 11.5 3 5 30 –1 +1 17.5 4 5 10 –1 –1 8.5 1400-CH14 9/8/99 4:35 PM Page 677 678 Modern Analytical Chemistry 14.8 where n is the number of runs, and A i * and B i * are the coded factor levels for the ith run. Solving for the estimated parameters using the data in Table 14.4 leaves us with the following empirical model for the response surface R = 15.0 + 2.0A* + 5.0B* + 0.5A*B* 14.9 The suitability of this model can be evaluated by substituting values for A* and B* from Table 14.4 and comparing the calculated response to the known response. Using the values for the first run as an example gives R = 15.0 + (2.0)(+1) + (5.0)(+1) + (0.5)(+1)(+1) = 22.5 which agrees with the known response. EXAMPLE 1 4 . 3 Equation 14.9 gives the empirical model of the response surface for the data in Table 14.4 when the factors are in coded form. Convert the equation to its uncoded form. SOLUTION To convert the equation to its uncoded form, it is necessary to solve equation 14.3 for each factor. Values for c f and d f are determined from the high and low levels for each factor; thus Substituting known values into equation 14.3, A =10+5A* B =20+10B* and rearranging, gives A* = 0.2A –2 B* = 0.1B –2 Substituting these equations into equation 14.9 gives, after simplifying, the uncoded equation for the response surface. c HL dHc B BB BBB = + = + ==?=?= 2 30 10 2 20 30 20 10 c HL dHc A AA AAA = + = + = = ?=?= 2 15 5 2 10 15 10 5 b ab = ??+= 1 4 22 5 11 5 17 5 8 5 0 5(....). b b =?+?= 1 4 22 5 11 5 17 5 8 5 5 0(. . . .) . b a = +??= 1 4 22 5 11 5 17 5 8 5 2 0(. . . .) . b 0 1 4 22 5 11 5 17 5 8 5 15 0= +++=(....). β ab ab i i i b n ABR≈=∑ 1 ** 1400-CH14 9/8/99 4:35 PM Page 678 Chapter 14 Developing a Standard Method 679 R = 15.0 + 2(0.2A – 2) + 5(0.1B – 2) + 0.5(0.2A – 2)(0.1B –2) = 15.0 + 0.4A –4+0.5B – 10 + 0.01AB – 0.2A – 0.1B +2 = 3.0 + 0.2A + 0.4B + 0.01AB We can verify this equation by substituting values for A and B from Table 14.4 and solving for the response. Using values for the first run, for example, gives R = 3.0 + (0.2)(15) + (0.4)(30) + (0.01)(15)(30) = 22.5 which agrees with the expected value. The computation just outlined is easily extended to any number of factors. For a system with three factors, for example, a 2 3 factorial design can be used to deter- mine the parameters for the empirical model described by the following equation R = b 0 + b a A + b b B + b c C + b ab AB + b ac AC + b bc BC + b abc ABC 14.10 where A, B, and C are the factors. The terms b 0 , b a , b b , and b ab are estimated using equations 14.6–14.9. The remaining parameters are estimated using the following equations. 14.11 14.12 14.13 14.14 EXAMPLE 1 4 . 4 Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 3 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. SOLUTION We begin by calculating the estimated parameters using equations 14.6–14.9 and 14.11–14.14. b b = +??++?? = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 15 0(........). b a = +++???? = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 18 0(........). b 0 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 56 0= +++++++ =(........). β abc abc i i i i b n ABCR≈=∑ 1 *** β bc bc i i i b n BCR≈=∑ 1 ** β ac ac i i i b n ACR≈=∑ 1 ** β cc ii b n CR≈=∑ 1 * 1400-CH14 9/8/99 4:35 PM Page 679 The coded empirical model, therefore, is R =56+18A*+15B* + 22.5C*+7A*B*+9A*C*+6B*C* + 3.75A*B*C* To check the result we substitute the coded factor levels for the first run into the coded empirical model, giving R = 56 + (18)(+1) + (15)(+1) + (22.5)(+1) + (7)(+1)(+1) + (9)(+1)(+1) + (6)(+1)(+1) + (3.75)(+1)(+1)(+1) = 137.25 which agrees with the measured response. To transform the coded empirical model into its uncoded form, it is necessary to replace A*, B*, and C* with the following relationships the derivations of which are left as an exercise. Substituting these relationships into the coded empirical model and simplifying (which also is left as an exercise) gives the following result for the uncoded empirical model R =3+0.2A + 0.4B + 0.5C – 0.01AB + 0.02AC – 0.01BC + 0.005ABC AA BB C C *** ..=? =? =?02 2 01 2 15 2 b abc = ??+ ++? = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 3 75(....–....). b bc = ??++??+ = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 6 0(........). b ac = ?+??+?+ = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 9 0(........). b ab = +????++ = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 7 0(........). b c = ?+?+?+? = 1 8 137 25 54 75 73 75 30 25 61 75 30 25 41 25 18 75 22 5(........). 680 Modern Analytical Chemistry Table 1 4 . 5 Uncoded and Coded Factor Levels and Responses for the 2 3 Factorial Design of Example 14.4 Run ABCA* B* C* Response 1 15 30 45 +1 +1 +1 137.25 2 15 30 15 +1 +1 –1 54.75 3 15 10 45 +1 –1 +1 73.75 4 15 10 15 +1 –1 –1 30.25 5 5 30 45 –1 +1 +1 61.75 6 5 30 15 –1 +1 –1 30.25 7 5 10 45 –1 –1 +1 41.25 8 5 10 15 –1 –1 –1 18.75 1400-CH14 9/8/99 4:35 PM Page 680 Figure 14.13 Curved one-factor response surface showing (a) the limitation of a 2 k factorial design for modeling second-order effects; and (b) the application of a 3 k factorial design for modeling second-order effects. Chapter 14 Developing a Standard Method 681 Response Level for factor A Response Level for factor A (a) (b) A 2 k factorial design is limited to models that include only a factor’s first-order effects on the response. Thus, for a 2 2 factorial design, it is possible to determine the first-order effect for each factor (b a and b b ), as well as the interaction between the factors (b ab ). There is insufficient information in the factorial design, however, to determine any higher order effects or interactions. This limitation is a consequence of having only two levels for each factor. Consider, for example, a system in which the response is a function of a single factor. Figure 14.13a shows the experimentally measured response for a 2 1 factorial design in which only two levels of the factor are used. The only empirical model that can be fit to the data is that for a straight line. R = b 0 + b a A If the actual response is that represented by the dashed curve, then the empirical model is in error. To fit an empirical model that includes curvature, a minimum of three levels must be included for each factor. The 3 1 factorial design shown in Fig- ure 14.13b, for example, can be fit to an empirical model that includes second-order effects for the factor. R = b 0 + b a A + b aa A 2 In general, an n-level factorial design can include single-factor and interaction terms up to the (n – 1)th order. The effectiveness of a first-order empirical model can be judged by measuring the response at the center of the factorial design. If there are no higher order effects, the average response of the runs in a 2 k factorial design should be equal to the mea- sured response at the center of the factorial design. The influence of random error can be accounted for by making several determinations of the response at the center of the factorial design and establishing a suitable confidence interval. If the differ- ence between the two responses is significant, then a first-order empirical model is probably not appropriate. EXAMPLE 1 4 . 5 At the beginning of this section we noted that the concentration of vanadium can be determined spectrophotometrically by making the solution acidic with H 2 SO 4 and reacting with H 2 O 2 to form a reddish brown compound with the general formula (VO) 2 (SO 4 ) 3 . Palasota and Deming 7 studied the effect on the absorbance of the relative amounts of H 2 SO 4 and H 2 O 2 , reporting the following results for a 2 2 factorial design. 1400-CH14 9/8/99 4:35 PM Page 681 Figure 14.14 3 k factorial design for k =2. H 2 SO 4 H 2 O 2 Response +1 +1 0.330 +1 –1 0.359 –1 +1 0.293 –1 –1 0.420 Four replicate measurements were made at the center of the factorial design, giving responses of 0.334, 0.336, 0.346, and 0.323. Determine if a first-order empirical model is appropriate for this system. Use a 90% confidence interval when accounting for the effect of random error. SOLUTION We begin by determining the confidence interval for the response at the center of the factorial design. The mean response is 0.335, with a standard deviation of 0.0094. The 90% confidence interval, therefore, is The average response, – R, from the factorial design is Because – R exceeds the confidence interval’s upper limit of 0.346, there is reason to believe that a 2 2 factorial design and a first-order empirical model are inappropriate for this system. A complete empirical model for this system is presented in problem 10 in the end-of-chapter problem set. Many systems that cannot be represented by a first-order empirical model can be described by a full second-order polynomial equation, such as that for two factors. R = b 0 + b a A + b b B + b aa A 2 + b bb B 2 + b ab AB Because each factor must be measured for at least three levels, a convenient experi- mental design is a 3 k factorial design. A 3 2 factorial design for two factors, for exam- ple, is shown in Figure 14.14. The computations for 3 k factorial designs are not as easily generalized as those for a 2 k factorial design and are not considered in this text. Further details about the calculations are found in the suggested readings at the end of the chapter. Central Composite Designs One limitation to a 3 k factorial design is the num- ber of trials that must be run. For two factors, as shown in Figure 14.14, a total of nine trials is needed. This number increases to 27 for three factors and 81 for four factors. A more efficient experimental design for systems containing more than two factors is the central composite design, two examples of which are shown in Figure 14.15. The central composite design consists of a 2 k factorial de- sign, which provides data for estimating the first-order effects for each factor and interactions between the factors, and a “star” design consisting of 2k + 1 points, which provides data for estimating second-order effect. Although a central com- posite design for two factors requires the same number of trials, 9, as a 3 2 facto- rial design, it requires only 15 trials and 25 trials, respectively, for systems in- R n R i = = +++ = ∑ 11 4 0 330 0 359 0 293 0 420 0 350(....). μ= ± = ± = ±X ts n 0 335 2 35 0 0094 4 0 335 0 011. (. )(. ) .. 682 Modern Analytical Chemistry F actor A Factor B 1400-CH14 9/8/99 4:35 PM Page 682 Figure 14.15 Central composite designs for (a) k = 2 and (b) k =3. Chapter 14 Developing a Standard Method 683 F actor A Factor B F actor A Factor B (a) (b) volving three or four factors. A discussion of central composite designs, includ- ing computational considerations, can be found in the suggested readings at the end of the chapter. 1 4 B Verifying the Method After developing and optimizing a method, it is necessary to determine the quality of results that can reasonably be expected when the method is used by a single ana- lyst. Generally, three steps are included in the process of verifying a method: deter- mining single-operator characteristics, the blind analysis of standards, and deter- mining the method’s ruggedness. In addition, if an alternative standard method exists, both the standard method and the new method can be used to analyze the same sample, and the results compared. If the quality of the results is unacceptable, the method is not suitable for consideration as a standard method. 1 4 B.1 Single-Operator Characteristics The first step in verifying a method is to determine the precision, accuracy, and de- tection limit when a single analyst uses the method to analyze a standard sample of known composition. The detection limit, which was discussed in Chapter 4, is de- termined by analyzing a reagent blank for each type of sample matrix for which the method will be used. Precision is determined by analyzing replicate portions, preferably more than ten, of a standard sample. Finding the method’s accuracy is evaluated by a t-test, as described in Chapter 4. Precision and accuracy should be evaluated for several different concentration levels of analyte, including at least one concentration near the detection limit, and for each type of sample matrix that will be encountered. The analysis of several concentrations allows for the detection of constant sources of determinate error and establishes the range of concentrations for which the method is applicable. 1 4 B.2 Blind Analysis of Standard Samples Single-operator characteristics are determined by analyzing a sample whose concen- tration of analyte is known to the analyst. The second step in verifying a method is the blind analysis of standard samples where the analyte’s concentration remains unknown to the analyst. The standard sample is analyzed several times, and the av- erage concentration of the analyte is determined. This value should be within three, and preferably two standard deviations (as determined from the single-operator characteristics) of the analyte’s known concentration. blind analysis The analysis of a standard sample whose composition is unknown to the analyst. 1400-CH14 9/8/99 4:35 PM Page 683 1 4 B. 3 Ruggedness Testing In many cases an optimized method may produce excellent results in the laboratory developing the method, but poor results in other laboratories. This is not surprising since a method is often optimized by a single analyst under an ideal set of condi- tions, in which the sources of reagents, equipment, and instrumentation remain the same for each trial. The procedure might also be influenced by environmental fac- tors, such as the temperature or relative humidity in the laboratory, whose levels are not specified in the procedure and which may differ between laboratories. Finally, when optimizing a method the analyst usually takes particular care to perform the analysis in exactly the same way during every trial. An important step in developing a standard method is to determine which fac- tors have a pronounced effect on the quality of the analytical method’s result. The procedure can then be written to specify the degree to which these factors must be controlled. A procedure that, when carefully followed, produces high-quality results in different laboratories is considered rugged. The method by which the critical fac- tors are discovered is called ruggedness testing. 8 Ruggedness testing is often performed by the laboratory developing the stan- dard method. Potential factors are identified and their effects evaluated by perform- ing the analysis while the factors are held at two levels. Normally one level for each factor is that given in the procedure, and the other is a level likely to be encountered when the procedure is used by other laboratories. This approach to ruggedness testing can be time-consuming. If seven poten- tial factors are identified, for example, ruggedness testing can be accomplished with a 2 7 factorial design. This requires a total of 128 trials, which is a prohibitively large amount of work. A simpler experimental design is shown in Table 14.6, in which the two factor levels are identified by upper case and lower case letters. This design, which is similar to that for the 2 3 factorial design discussed in the previous section, is called a fractional factorial design and provides information about the first-order effect of each factor. It does not, however, provide suffi- cient information to evaluate higher order effects or potential interactions be- tween factors, both of which are assumed to be of less importance than first- order effects. The experimental design for ruggedness testing is balanced in that each factor level is paired an equal number of times with the upper case and lower case levels 684 Modern Analytical Chemistry ruggedness testing The process of evaluating a method to determine those factors for which a small change in value has a significant effect on the method’s results. Table 1 4 .6 Experimental Design for a Ruggedness Test Involving Seven Factors Factor Run ABCDEFGResponse 1ABCDEFG R 1 2A B c D e f g R 2 3AbCdEfg R 3 4 bcdeFG R 4 5aBCdeFg R 5 6 BcdEfG R 6 7a b C D e f G R 7 8a b c D E F g R 8 1400-CH14 9/8/99 4:35 PM Page 684 Chapter 14 Developing a Standard Method 685 for every other factor. The effect of changing the level for any one factor, E f , is de- termined by subtracting the average response when the factor is at its upper case level from the average value when it is at its lower case level 14.15 Because the design is balanced, the levels for the remaining factors appear an equal number of times in both summation terms, and, consequently, their effect on the response is canceled. For example, the effect of changing the level for fac- tor A is determined by averaging the responses from runs 1 through 4 and sub- tracting the average response from runs 5 through 8. Factor B does not affect E f because its upper case levels in runs 1 and 2 are canceled by the upper case levels in runs 5 and 6, and its lower case levels in runs 3 and 4 are canceled by the lower case levels in runs 7 and 8. Similar reasoning shows that the remaining factors are also canceled. After each of the effects is calculated, they are ranked from largest to smallest, without regard to sign, and those factors whose effects are substantially larger than the other factors are identified. The estimated standard deviation for the analysis is given by 4.16 This provides the best estimate of the expected standard deviation for results ob- tained by several laboratories, due to the effects of small changes in uncontrolled or poorly controlled factors. If this standard deviation is unacceptably large, then the procedure may be modified to bring under greater control those factors whose lev- els have the greatest effect on the response. EXAMPLE 1 4 .6 The concentration of trace metals in sediment samples collected from rivers and lakes can be determined by extracting with acid and analyzing the extract by atomic absorption spectrophotometry. One procedure calls for an overnight extraction with dilute HCl or HNO 3 . The samples are placed in plastic bottles with 25 mL of acid and extracted on a shaking table at a moderate speed and ambient temperature. To determine the ruggedness of the method, the effect of a change in level for the following factors was studied using the experimental design shown in Table 14.6. Factor A – extraction time A = 24 h a = 12 h Factor B – shaking speed B = medium b = high Factor C – acid type C = HCl c = HNO 3 Factor D – acid concentration D = 0.1 M d = 0.05 M Factor E – volume of acid E = 25 mL e = 35 mL Factor F – type of container F = plastic f = glass Factor G – temperature G = ambient g =25°C sE i = ∑ 2 7 2 E R Ri i f upper case lower case = ∑ () ? ∑ () 44 1400-CH14 9/8/99 4:35 PM Page 685 A standard sample containing a known amount of analyte was carried through the procedure. The percentage of analyte actually found in the eight trials were found to be R 1 = 98.9 R 2 = 99.0 R 3 = 97.5 R 4 = 97.7 R 5 = 97.4 R 6 = 97.3 R 7 = 98.6 R 8 = 98.6 Determine which factors, if any, appear to have a significant effect on the response, and estimate the method’s expected standard deviation. SOLUTION The effect of a change in level for each factor is calculated using equation 14.15 Ordering the factors by their absolute values Factor D 1.30 Factor A 0.35 Factor E –0.10 Factor B 0.05 Factor C –0.05 Factor F 0.05 Factor G 0.00 shows that the concentration of acid (factor D) has a substantial effect on the response, with a concentration of 0.05 M providing a much lower percent recovery. The extraction time (factor A) also appears to be significant, but its effect is not as important as that for the acid’s concentration. All other factors appear to be insignificant. The method’s estimated standard deviation is s =++?++?++= 2 7 030 005 005 130 010 005 000 072 22 22 222 {. .(.).(.). .}. E RRRR RRRR G = +++ ? +++ = 1467 2358 44 000. E RRRR RRRR F = +++ ? +++ = 1458 2367 44 005. E RRRR RRRR E = +++ ? +++ =? 1368 2457 44 010. E RRRR RRRR D = +++ ? +++ = 1278 3456 44 130. E RRRR RRRR C = +++ ? +++ =? 1357 2468 44 005. E RRRR RRRR B = +++ ? +++ = 1256 3478 44 005. E RRRR RRRR A = +++ ? +++ = 1234 5678 44 030. 686 Modern Analytical Chemistry 1400-CH14 9/8/99 4:36 PM Page 686 Chapter 14 Developing a Standard Method 687 collaborative testing A validation method used to evaluate the sources of random and systematic errors affecting an analytical method. which, for an average recovery of 98.1% gives a relative standard deviation of approximately 0.7%. If the acid’s concentration is controlled such that its effect approaches that for factors B, C, and F, then the relative standard deviation becomes 0.18, or approximately 0.2%. 1 4 B. 4 Equivalency Testing If a standard method is available, the performance of a new method can be evalu- ated by comparing results with those obtained with an approved standard method. The comparison should be done at a minimum of three concentrations to evaluate the applicability of the new method for different amounts of analyte. Alternatively, we can plot the results obtained by the new method against those obtained by the approved standard method. A linear regression analysis should give a slope of 1 and a y-intercept of 0 if the results of the two methods are equivalent. 1 4 C Validating the Method as a Standard Method For an analytical method to be of use, it must be capable of producing results with acceptable accuracy and precision. The process of verifying a method as de- scribed in the previous section determines whether the method meets this goal for a single analyst. A further requirement for a standard method is that an analysis should not be affected by a change in the analyst performing the work, the laboratory in which the work is performed, or the time when the analysis is conducted. The process by which a method is approved for general use is known as validation and involves a collaborative test of the method by analysts in sev- eral laboratories. Collaborative testing is used routinely by regulatory agencies and professional organizations, such as the U.S. Environmental Protection Agency, the American Society for Testing and Materials, the Association of Offi- cial Analytical Chemists, and the American Public Health Association, in estab- lishing their standard methods of analysis. When an analyst performs a single analysis on a sample, the difference be- tween the experimentally determined value and the expected value is influenced by three sources of error: random error, systematic errors inherent to the method, and systematic errors unique to the analyst. If enough replicate analyses are performed, a distribution of results can be plotted (Figure 14.16a). The width of this distribution is described by the standard deviation and can be used to de- termine the effect of random error on the analysis. The position of the distribu- tion relative to the sample’s true value, m , is determined both by systematic er- rors inherent to the method and those systematic errors unique to the analyst. For a single analyst there is no way to separate the total systematic error into its component parts. The goal of a collaborative test is to determine the expected magnitude of all three sources of error when a method is placed into general practice. When several analysts each analyze the same sample one time, the variation in their col- lective results (Figure 14.16b) includes contributions from random errors and those systematic errors (biases) unique to the analysts. Without additional infor- mation, the standard deviation for the pooled data cannot be used to separate the precision of the analysis from the systematic errors of the analysts. The posi- tion of the distribution, however, can be used to detect the presence of a system- atic error in the method. 1400-CH14 9/8/99 4:36 PM Page 687 Figure 14.16 Partitioning of random error, systematic errors due to the analyst, and systematic error due to the method for (a) replicate analyses performed by a single analyst and (b) single determinations performed by several analysts. 1 4 C.1 Two-Sample Collaborative Testing The design of a collaborative test must provide the additional information needed to separate the effect of random error from that due to systematic errors intro- duced by the analysts. One simple approach, which is accepted by the Association of Official Analytical Chemists, is to have each analyst analyze two samples, X and Y, that are similar in both matrix and concentration of analyte. The results ob- tained by each analyst are plotted as a single point on a two-sample chart, using the result for one sample as the x-coordinate and the value for the other sample as the y-coordinate. 8,9 A two-sample chart is divided into four quadrants, identified as (+, +), (–, +), (–, –), and (+, –), depending on whether the points in the quadrant have values for the two samples that are larger or smaller than the mean values for samples X and Y. Thus, the quadrant (+, –) contains all points for which the result for sample X is larger than the mean for sample X, and for which the result for sample Y is less than the mean for sample Y. If the variation in results is dominated by random errors, 688 Modern Analytical Chemistry Frequenc y Effect of random error Effect of systematic error due to method and the analyst μ X Frequenc y Effect of random error and systematic error due to the analysts Effect of systematic error due to the method μ X (a) (b) 1400-CH14 9/8/99 4:36 PM Page 688 Figure 14.17 Typical two-sample plot when (a) random errors are larger than systematic errors due to the analysts and (b) systematic errors due to the analysts are larger than the random errors. then the points should be distributed randomly in all four quadrants, with an equal number of points in each quadrant. Furthermore, the points will be clustered in a circular pattern whose center is the mean values for the two samples (Figure 14.17a). When systematic errors are significantly larger than random errors, then the points will be found primarily in the (+, +) and (–, –) quadrants and will be clus- tered in an elliptical pattern around a line bisecting these quadrants at a 45° angle (Figure 14.17b). A visual inspection of a two-sample chart provides an effective means for qualitatively evaluating the results obtained by each analyst and of the capabilities of a proposed standard method. If no random errors are present, then all points will be found on the 45° line. The length of a perpendicular line from any point to the 45° line, there- fore, is proportional to the effect of random error on that analyst’s re- sults (Figure 14.18). The distance from the intersection of the lines for the mean values of samples X and Y, to the perpendicular projection of a point on the 45° line, is proportional to the analyst’s systematic error (Figure 14.18). An ideal standard method is characterized by small random errors and small systematic errors due to the analysts and should show a compact clustering of points that is more circular than elliptical. The data used to construct a two-sample chart can also be used to separate the total variation of the data, s tot , into contributions from random error, s rand , and systematic errors due to the analysts, s sys . 9 Since an analyst’s systematic errors should be present at the same level in the analysis of samples X and Y, the differ- ence, D, between the results for the two samples D i = X i – Y i is influenced only by random error. The contribution from random error, there- fore, is estimated by the standard deviation of the differences for each analyst 14.17 where n is the number of analysts. The inclusion of a factor of 2 in the denominator of equation 14.17 is the result of using two values to determine D i . The total, T, of each analyst’s results for the two samples T i = X i + Y i s DD n s i D rand rand = ?∑ ? =≈ () () 2 21 σ Chapter 14 Developing a Standard Method 689 Result f or sample Y Result for sample X X Y (–,+) (+,+) (–,–) (+,–) Result f or sample Y Result for sample X X Y (–,+) (+,+) (–,–) (+,–) (a) (b) Result f or sample Y Result for sample X X Proportional to systematic error due to analyst Proportional to random error Y (–,+) (+,+) (X i ,Y i ) (–,–) (+,–) Figure 14.18 Relationship between point in a two-sample plot and the random error and systematic error due to the analyst. 1400-CH14 9/8/99 4:36 PM Page 689 690 Modern Analytical Chemistry contains contributions from both random error and twice the analyst’s systematic error; thus s 2 tot = s 2 rand + 2s 2 sys 14.18 The standard deviation of the totals provides an estimate for s tot . 14.19 Again, the factor of 2 in the denominator is the result of using two values to deter- mine T i . If systematic errors due to the analysts are significantly larger than random er- rors, then s T should be larger than s D . This can be tested statistically using a one- tailed F-test where the degrees of freedom for both the numerator and the denominator are n – 1. If s T is significantly larger than s D , then we can use equation 14.18 to separate the data’s total variation into that due to random error and that due to the system- atic errors of the analysts. EXAMPLE 1 4 . 7 As part of a collaborative study of a new method for determining the amount of total cholesterol in blood, two samples were sent to ten analysts with instructions to analyze each sample one time. The following results, in milligrams of total cholesterol per 100 mL of serum, were obtained Analyst Tablet 1 Tablet 2 1 245.0 229.4 2 247.4 249.7 3 246.0 240.4 4 244.9 235.5 5 255.7 261.7 6 248.0 239.4 7 249.2 255.5 8 225.1 224.3 9 255.0 246.3 10 243.1 253.1 Using this data estimate the values for s rand and s sys for the method assuming a = 0.05. SOLUTION A two-sample plot of the data is shown in Figure 14.19, with the average value for sample 1 shown by the vertical line at 245.9, and the average value for sample 2 shown by the horizontal line at 243.5. To estimate s rand and s sys , it first is necessary to calculate values for D i and T i ; thus F s s = T 2 D 2 s TT n s i T tot tot = ?∑ ? =≈ () () 2 21 σ 1400-CH14 9/8/99 4:36 PM Page 690 Analyst D i T i 1 15.6 474.4 2 –2.3 497.1 3 5.6 486.4 4 9.4 480.4 5 –6.0 517.4 6 8.6 487.4 7 –6.3 504.7 8 0.8 449.4 9 8.7 501.3 10 –10.0 496.2 The standard deviations for the differences, s D , and the totals, s T , are calculated* using equations 14.17 and 14.19, giving s D = 5.95 s T = 13.3 To determine if the systematic errors of the analysts are significant, an F-test is performed using s T and s D Because the F-ratio is larger than F(0.05, 9, 9), which is 3.179, we conclude that the systematic errors of the analysts are significant at the 95% confidence levels. The estimated precision for a single analyst is s rand ? s D = 5.95 The estimated standard deviation due to systematic errors between analysts is calculated from equation 14.18. σ σσ sys tot 2 rand 2 T 2 D 2 = ? ≈ ? = ? = 22 13 3 5 95 2 841 22 ss (.) (.) . F s s == = T 2 D 2 (.) (. ) . 13 3 595 500 2 2 Chapter 14 Developing a Standard Method 691 Sample 2 220.0 220 265 Sample 1 260.0230.0 8 10 1 4 6 3 9 2 7 5 240.0 250.0 260 255 250 245 240 235 230 225 Figure 14.19 Two-sample plot for data in Example 14.7. The analyst responsible for each data point is indicated by the associated number. The true values for the two samples are indicated by the *. *Here is a short-cut that simplifies the calculation of s D and s T . Enter the values for D i into your calculator, and use its built-in functions to find the standard deviation. Divide this result by 2 to obtain s D . You can use the same approach to calculate s T . 1400-CH14 9/8/99 4:36 PM Page 691 692 Modern Analytical Chemistry When the true values for the two samples are known, it is possible to test for the presence of systematic errors in the method. If no systematic method errors occur, then the sum of the true values for the samples, m tot m tot = m samp X + m samp Y will fall within the confidence interval around the average value, T – . A two-tailed t-test, as described in Chapter 4, with the following null and alternative hypotheses H 0 : T – = m tot H A : T – ? m tot is used to determine whether there is any evidence for systematic method errors. The test statistic, t exp , is given as 14.20 and has n – 1 degrees of freedom. The 2 in the denominator is included because s T , as given in equation 14.19, underestimates the standard deviation when comparing T – to m tot . EXAMPLE 1 4 .8 The two samples analyzed in Example 14.7 are known to contain the following concentrations of cholesterol. m samp 1 = 248.3 mg/100 mL m samp 2 = 247.6 mg/100 mL Determine if there is any evidence for a systematic error in the method at the 95% confidence level. SOLUTION Using the data from Example 14.7 and the true values for the samples, we know that s T is 13.3, and T – = X – samp 1 + X – samp 2 = 245.9 + 243.5 = 489.4 m tot = m samp 1 + m samp 2 = 248.3 + 247.6 = 495.9 Substituting these values into equation 14.20 gives This value for t exp is smaller than the critical value of 2.26 for t(0.05, 9). Thus, there is no evidence for a systematic error in the method at the 95% confidence level. In Examples 14.7 and 14.8 we have seen how a collaborative test using a pair of closely related samples can be used to evaluate a method. Ideally, a collaborative test should involve several pairs of samples, whose concentrations of analyte span the t exp = ? = 489 4 495 9 10 13 3 2 109 .. . . t Tn s exp tot T = ?μ 2 1400-CH14 9/8/99 4:36 PM Page 692 method’s anticipated useful range. In this way the method can be evaluated for con- stant sources of error, and the expected relative standard deviation and bias for dif- ferent levels of analyte can be determined. 1 4 C.2 Collaborative Testing and Analysis of Variance In the two-sample collaborative test, each analyst performs a single determina- tion on two separate samples. The resulting data are reduced to a set of differ- ences, D, and a set of totals, T, each characterized by a mean value and a stan- dard deviation. Extracting values for random errors affecting precision and systematic differences between analysts is relatively straightforward for this ex- perimental design. An alternative approach for collaborative testing is to have each analyst per- form several replicate determinations on a single, common sample. This approach generates a separate data set for each analyst, requiring a different statistical treat- ment to arrive at estimates for s rand and s sys . A variety of statistical methods may be used to compare three or more sets of data. The most commonly used method is an analysis of variance (ANOVA). In its simplest form, a one-way ANOVA allows the importance of a single variable, such as the identity of the analyst, to be determined. The importance of this variable is evaluated by comparing its variance with the variance explained by indeterminate sources of error inherent to the analytical method. Variance was introduced in Chapter 4 as one measure of a data set’s spread around its central tendency. In the context of an analysis of variance, it is useful to see that variance is simply a ratio of the sum of squares for the differences between individual values and their mean, to the degrees of freedom. For example, the vari- ance, s 2 , of a data set consisting of n measurements is given as where X i is the value of a single measurement, and X – is the mean. As we will see, the ability to partition the variance into separate terms for the sum of squares and the degrees of freedom greatly simplifies the calculations in a one-way ANOVA. Let’s use a simple example to develop the rationale behind a one-way ANOVA calculation. The data in Table 14.7 show the results obtained by several analysts in de- termining the purity of a single pharmaceutical preparation of sulfanilamide. Each column in this table lists the results obtained by an individual analyst. For conve- nience, entries in the table are represented by the symbol X ij , where i identifies the an- alyst and j indicates the replicate number; thus X 3,5 is the fifth replicate for the third analyst (and is equal to 94.24%). The variability in the results shown in Table 14.7 arises from two sources: indeterminate errors associated with the analytical procedure that are experienced equally by all analysts, and systematic or determinate errors in- troduced by the analysts. One way to view the data in Table 14.7 is to treat it as a single system, charac- terized by a global mean, = X, and a global variance, = s 2 . These parameters are calcu- lated using the following equations. X X N ij j n i h i = == ∑∑ 11 s XX n i i n 2 2 1 1 == ? ? = ∑ sum of squares degrees of freedom () Chapter 14 Developing a Standard Method 693 analysis of variance A statistical method for comparing three or more sets of data. 1400-CH14 9/8/99 4:36 PM Page 693 694 Modern Analytical Chemistry Table 1 4 . 7 Results of Four Analysts for the %Purity of a Preparation of Sulfanilamide Replicate Analyst A Analyst B Analyst C Analyst D 1 94.09 99.55 95.14 93.88 2 94.64 98.24 94.62 94.23 3 95.08 101.1 95.28 96.05 4 94.54 100.4 94.59 93.89 5 95.38 100.1 94.24 94.95 6 93.62 95.49 14.21 where h is the total number of samples (in this case the number of analysts), n i is the number of replicates for the ith sample (in this case the ith analyst), and N is the total number of data points in the system. The global variance provides a measure of the combined influence of indeterminate and systematic errors. A second way to work with the data in Table 14.7 is to treat the results for each analyst separately. Because the repeatability for any analyst is influenced by indeter- minate errors, the variance, s 2 , of the data in each column provides an estimate of s 2 rand . A better estimate is obtained by pooling the individual variances. The result, which is called the within-sample variance (s 2 w ), is calculated by summing the squares of the differences between the replicates for each sample and that sample’s mean, and dividing by the degrees of freedom. 14.22 Finally, the data for each analyst can be reduced to separate mean values, X – i . The variance of the individual means about the global mean is called the between- sample variance, s 2 b , and is calculated as 14.23 where n i is the number of replicates for the ith sample. The between-sample vari- ance includes contributions from both random and systematic errors and, there- fore, provides an estimate for both s 2 rand and s 2 sys s 2 b = s 2 rand + – ns 2 sys 14.24 where – n is approximated as the average number of replicates per analyst. n n h i i h = = ∑ 1 s nX X h ii i h b 2 2 1 1 = ? ? ? ? ? ? = ∑ s XX Nh ij i j n i h w 2 rand 2 == ? ? == ∑∑ σ () 2 11 s XX N ij j n i h i 2 2 11 1 = ? ? ? ? ? ? == ∑∑ 1400-CH14 9/8/99 4:36 PM Page 694 Chapter 14 Developing a Standard Method 695 Table 1 4 .8 Summary of Calculations for a One-Way Analysis of Variance Source Sum of Squares Degrees of Freedom Variance Expected Variance F-ratio Between sample h – 1 Within sample SS w = SS t – SS b N – h Total SS X X sN ij j n i h t =?∑∑ =? == () () 2 11 2 1 s w rand 22 =σ s SS Nh w w 2 = ? F s s = b w 2 2 sn b rand sys 22 2 =+σσ s SS h b b 2 1 = ? SS n X X ii i h b =?∑ = () 2 1 In a one-way ANOVA of the data in Table 14.7, the null hypothesis is that no significant differences exist between the mean values for each analyst. The alterna- tive hypothesis is that at least one of the means is significantly different. If the null hypothesis is true, then s 2 sys must be zero. Thus, from equations 14.22 and 14.24, both s 2 w and s 2 b are predictors of s 2 rand and should have similar values. If s 2 b is signifi- cantly greater than s 2 w , then s 2 sys is greater than zero. In this case the alternative hy- pothesis must be accepted, and a significant difference between the means for the analysts has been demonstrated. The test statistic is the F-ratio which is compared with the critical value F(a , h –1,N – h). This is a one-tailed signifi- cance test because we are only interested in whether s 2 b is significantly greater than s 2 w . Both s 2 b and s 2 w are easy to determine for small data sets. For larger data sets, how- ever, calculating s 2 w becomes tedious.* Its calculation is simplified by taking advantage of the relationship between the sum-of-squares terms for the global variance, the within-sample variance, and the between-sample variance. The numerator of equation 14.21, which also is known as the total sum of squares, SS t , can be split into two terms SS t = SS w + SS b where the sum of squares for the variation within the sample, SS w , is the numerator of equation 14.22, and the sum of squares between the sample, SS b , is the numera- tor of equation 14.23. Calculating SS t and SS b gives SS w by difference. Dividing SS w and SS b by their respective degrees of freedom gives s 2 w and s 2 b . Table 14.8 summa- rizes all the necessary equations for a one-way ANOVA calculation. The application of a one-way ANOVA is outlined in Example 14.9. EXAMPLE 1 4 . 9 Table 14.7 shows the results obtained by four analysts determining the purity of a pharmaceutical preparation of sulfanilamide. Determine if the difference in their results is significant at a = 0.05. If such a difference exists, estimate values for s 2 rand and s 2 sys . F s s = b 2 w 2 *A number of statistics and spreadsheet software packages are available to perform ANOVA calculations. 1400-CH14 9/8/99 4:36 PM Page 695 SOLUTION To begin, we calculate the global mean and variance and the means for each analyst. These results are X = = 95.87 = s 2 = 5.506 X – A = 94.56 X – B = 99.88 X – C = 94.77 X – D = 94.75 Using these values, we then calculate the total sum of squares SS t = = s 2 (N – 1) = (5.506)(22 – 1) = 115.63 the between-sample sum of squares and the within-sample sum of squares SS w = SS t – SS b = 115.63 – 104.27 = 11.36 The remainder of the necessary calculations are summarized in the following table. Source Sum of Squares Degrees of Freedom Variance between 104.27 h –1=4–1=3 34.76 within 11.36 N – h =22–4=18 0.631 The value of the test statistic is for which the critical value of F(0.05, 3, 18) is 3.16. Because F exp is greater than F(0.05, 3, 18), the null hypothesis is rejected, and the results for at least one analyst are significantly different from those for the other analysts. The value for s 2 rand is estimated by the within-sample variance s 2 rand ≈ s 2 w = 0.631 whereas, from equation 14.24, s 2 sys is In this example the variance due to systematic differences between the analysts is almost an order of magnitude greater than the variance due to the method’s precision. Once a significant difference has been demonstrated by an analysis of variance, a modified version of the t-test, known as Fisher’s least significant difference, can be used to determine which analyst or analysts are responsible for the difference. The test statistic for comparing the mean values X – 1 and X – 2 is the t-test described in Chapter 4, except that s pool is replaced by the square root of the within-sample vari- ance obtained from an analysis of variance. σ sys 2 b 2 w 2 ≈ ? = ? = ss n 34 76 0 631 22 4 6 205 .. / . F s s exp . . .== = b 2 w 2 34 76 0 631 55 08 SS n X X ii i b =? ? ? ? ? =?+? +?+?= = ∑ 2 1 4 22 6 94 56 95 87 5 99 88 95 87 5 94 77 95 87 6 94 75 95 87 104 27 (. .) (. .) (. .) (. .) . 696 Modern Analytical Chemistry Fisher’s least significant difference A modified form of the t-test for comparing several sets of data. 1400-CH14 9/8/99 4:36 PM Page 696 Chapter 14 Developing a Standard Method 697 14.25 This value of t exp is compared with the critical value for t(a , n ), where the signifi- cance level is the same as that used in the ANOVA calculation, and the degrees of freedom is the same as that for the within-sample variance. Because we are inter- ested in whether the larger of the two means is significantly greater than the other mean, the value of t(a , n ) is that for a one-tail significance test. EXAMPLE 1 4 .10 Determine the source of the significant difference for the data in Example 14.9. SOLUTION Individual comparisons using Fisher’s least significant difference test are based on the following null hypothesis and one-tailed alternative hypothesis H 0 : X – i = X – j H A : X – i > X – j or X – i < X – j Using equation 14.25, we can calculate values of t exp for each possible comparison. These values can then be compared with the one-tailed critical value of 1.73 for t(0.05, 18), as found in Appendix 1B. For example, t exp when comparing the results for analysts A and B is Because (t exp ) AB is greater than t(0.05, 18), we reject the null hypothesis and accept the alternative hypothesis that the results for analyst B are significantly greater than those for analyst A. Working in the same fashion, it is easy to show that (t exp ) AC = 0.437 H 0 is retained (t exp ) AD = 0.414 H 0 is retained (t exp ) BC = 10.17 H 0 is rejected and H A is accepted (t exp ) BD = 10.67 H 0 is rejected and H A is accepted (t exp ) CD = 0.04 H 0 is accepted Taken together, these results suggest that there is a significant systematic difference between the work of analyst B and that of the other analysts. There is no way to decide, however, whether any of the four analysts has done accurate work. An analysis of variance can be extended to systems involving more than a single variable. For example, a two-way ANOVA can be used in a collaborative study to determine the importance to an analytical method of both the analyst and the in- strumentation used. The treatment of multivariable ANOVA is beyond the scope of this text, but is covered in several of the texts listed as suggested readings at the end of the chapter. () [( / ) ( / )] .. ( . )[( / ) ( / )] .t XX sn n exp AB AB w 2 AB = ? + = ? + = 11 94 56 99 88 0631 16 15 11 06 t XX sn n exp w 2 = ? + 12 12 11[( / ) ( / )] 1400-CH14 9/8/99 4:36 PM Page 697 698 Modern Analytical Chemistry 1 4 C. 3 What Is a Reasonable Result for a Collaborative Study? Collaborative testing provides a means for estimating the variability (or repro- ducibility) among analysts in different labs. If the variability is significant, we can determine that portion due to random errors traceable to the method (s rand ) and that due to systematic differences between the analysts (s sys ). In the previous two sections we saw how a two-sample collaborative test, or an analysis of variance can be used to estimate s rand and s sys (or s 2 rand and s 2 sys ). We have not considered, however, what is a reasonable value for a method’s reproducibility. An analysis of nearly 10,000 collaborative studies suggests that a reasonable es- timate for a method’s reproducibility is R =2 (1–0.5 log C) 14.26 where R is the percent relative standard deviation for the results included in the col- laborative study, and C is the fractional amount of analyte in the sample on a weight-to-weight basis. 10 For example, when the sample used in a collaborative study is 1 ppm (m g/g) in analyte, C has a value of 10 –6 . The estimated percent rela- tive standard deviation, therefore, is R =2 [1–0.5 log(10 –6 )] = 16% Equation 14.26 appears to be independent of the type of analyte, the type of matrix, and the method of analysis. EXAMPLE 1 4 .11 What is the estimated relative standard deviation for the results of a collaborative study in which the sample is pure analyte (100% w/w analyte)? Repeat for the case in which the analyte’s concentration is 0.1% w/w. SOLUTION When the sample is 100% w/w analyte (C = 1), the estimated percent relative standard deviation is R =2 [1–0.5 log(1)] =2% Thus, we expect that approximately 67% of the participants in the collaborative study (– 1s ) will report the analyte’s concentration within the range of 98% w/w to 102% w/w. When the analyte’s concentration is 0.1% w/w (C = 0.001), the estimated percent relative standard deviation is R=2 [1 – 0.5 log(0.001)] = 5.7% and we expect that 67% of the analysts will report the analyte’s concentration within the range of 0.094% w/w to 0.106% w/w. Of course, equation 14.26 only provides an estimate of the variability in results submitted by those participating in a collaborative study. A relative standard devia- tion that is within one-half to twice the estimated value suggests that the method performs acceptably when used by analysts in different laboratories. The percent relative standard deviation for a single analyst should be one-half to two-thirds that for the variability between analysts. 1400-CH14 9/8/99 4:36 PM Page 698 Chapter 14 Developing a Standard Method 699 1 4 D KEY TERMS analysis of variance (p. 693) blind analysis (p. 683) collaborative testing (p. 687) empirical model (p. 676) factor (p. 667) factor level (p. 667) Fisher’s least significant difference (p. 696) response (p. 667) response surface (p. 667) ruggedness testing (p. 684) simplex optimization (p.672) standard method (p. 667) theoretical model (p. 675) One of the goals of analytical chemistry is to develop new analyti- cal methods that are recognized or accepted as being standard methods. In this chapter we considered how a standard method is developed, including finding the optimum experimental condi- tions, verifying that the method produces acceptable precision and accuracy, and validating the method for general use. In optimizing a method, we seek to find the combination of experimental parameters producing the best result or response. We can visualize this process as being similar to finding the high- est point on a mountain, in which the mountain’s topography, called a response surface, is a plot of the system’s response as a function of the factors under our control. One approach to finding the optimum response is to use a searching algorithm. In a one-factor-at-a-time optimization, one factor is varied while holding constant all other factors, until there is no further improvement in the response. The process then con- tinues with the next factor until no further improvement is found when changing any of the factors. This approach to finding the optimum response is often effective, but not efficient. Another searching algorithm, that is both effective and efficient, is a sim- plex optimization, the rules of which allow us to change the values of all factors simultaneously. Another approach to optimizing a method is to develop a mathematical model of the response surface. Such models can be theoretical, in that they are derived from a known chemical and physical relationship between the response and its factors. Alterna- tively, an empirical model, which does not have a firm theoretical basis, can be developed by fitting an empirical equation to experi- mental data. One approach is to use a 2 k factorial design in which each factor is tested at both a high and a low level, and against every other factor at each of their levels. After optimizing a method, it is necessary to demonstrate that it can produce acceptable results. Verifying a method usu- ally includes establishing single-operator characteristics, per- forming a blind analysis of standard samples, and determining the method’s ruggedness. Single-operator characteristics include the method’s precision, accuracy, and detection limit when used by a single analyst. To test against possible bias on the part of the analyst, the analyst then analyzes a set of blind samples in which the analyst does not know the concentration of analyte in the samples. Finally, ruggedness testing is used to determine which experimental factors must be carefully controlled to avoid unexpectedly large determinate or indeterminate sources of error. The last step in establishing a standard method is to validate its transferability to other laboratories. An important step in the process of validating a method is collaborative testing, in which a common set of samples is analyzed by different laboratories. In a well-designed collaborative test, it is possible to establish limits for the method’s precision and accuracy. 1 4 E SUMMARY Harvey, D. T.; Byerly, S.; Bowman, A.; et al. “Optimization of HPLC and GC Separations Using Response Surfaces,” J. Chem. Educ. 1991, 68, 162–168. In this experiment a theoretical model is used to optimize the HPLC separation of substituted benzoic acids by adjusting the pH of the mobile phase. An empirical model is then used to find the optimum combination of organic modifiers (methanol, acetonitrile, and tetrahydrofuran) to shorten the analysis time. A fixed-size simplex optimization of a GC separation using column temperature and carrier gas flow rate is also presented. 1 4 F Suggested EXPERIMENTS The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. Experiments —Continued 1400-CH14 9/8/99 4:36 PM Page 699 700 Modern Analytical Chemistry Continued from page 699 Experiments Leggett, D. L. “Instrumental Simplex Optimization,” J. Chem. Educ. 1983, 60, 707–710. A variable-size simplex optimization of a gas chromatographic separation using oven temperature and carrier gas flow rate as factors is described in this experiment. Oles, P. J. “Fractional Factorial Experimental Design as a Teaching Tool for Quantitative Analysis,” J. Chem. Educ. 1998, 75, 357–359. This experiment describes the use of a fractional factorial design to examine the effects of volume of HNO 3 , molarity of AgNO 3 , volume of AgNO 3 , digestion temperature, and composition of wash water on the gravimetric analysis for chloride. Sangsila, S.; Labinaz, G.; Poland, J. S.; et al. “An Experiment on Sequential Simplex Optimization of an Atomic Absorption Analysis Procedure,” J. Chem. Educ. 1989, 66, 351–353. This experiment describes a fixed-size simplex optimization of a system involving four factors. The goal of the optimization is to maximize the absorbance of As by hydride generation atomic absorption spectroscopy using the concentration of HCl, the N 2 flow rate, the mass of NaBH 4 , and reaction time as factors. Shavers, C. L.; Parsons, M. L.; Deming, S. N. “Simplex Optimization of Chemical Systems,” J. Chem Educ. 1979, 56, 307–309. This experiment describes a variable-size simplex optimization of the quantitative analysis of vanadium as (VO) 2 (SO 4 ) 3 using the amount of H 2 O 2 and H 2 SO 4 as factors. A related experiment [Palasota, J. A.; Deming, S. N. “Central Composite Experimental Design,” J. Chem. Educ. 1992, 69, 560–561] examines this same analytical method using a central composite experimental design. Stieg, S. “A Low-Noise Simplex Optimization Experiment,” J. Chem. Educ. 1986, 63, 547–548. In this experiment the goal is to mix solutions of 1 M HCl and 20-ppm methyl violet to give the maximum absorbance at a wavelength of 425 nm (corresponding to a maximum concentration for the acid form of methyl violet). A variable- size simplex optimization is used to find the optimum mixture. Stolzberg, R. J. “Screening and Sequential Experimentation: Simulations and Flame Atomic Absorption Spectrometry Experiments,” J. Chem. Educ. 1997, 74, 216–220. This experiment describes a fractional factorial design used to examine the effects of flame height, flame stoichiometry, acetic acid, lamp current, wavelength, and slit width on the flame atomic absorbance obtained using a solution of 2.00-ppm Ag + . Van Ryswyk, H.; Van Hecke, G. R. “Attaining Optimal Conditions,” J. Chem. Educ. 1991, 66, 878–882. This experiment examines the effect of reaction time, temperature, and mole ratio of reactants on the synthetic yield of acetylferrocene by a Friedel–Crafts acylation of ferrocene. A central composite experimental design is used to find the optimum conditions, but the experiment could be modified to use a factorial design. 1. For each of the following equations, determine the optimum response, using the one-factor-at-a-time searching algorithm. Begin the search at (0, 0) with factor A, and use a step size of 1 for both factors. The boundary conditions for each response surface are 0 ≤ A ≤ 10 and 0 ≤ B ≤ 10. Continue the search through as many cycles as necessary until the optimum response is found. Compare your optimum response for each equation with the true optimum. (a) R = 1.68 + 0.24A + 0.56B – 0.04A 2 – 0.04B 2 m opt = (3, 7) (b) R = 4.0 – 0.4A + 0.08AB m opt = (10, 10) (c) R = 3.264 + 1.537A + 0.5664B – 0.1505A 2 – 0.02734B 2 – 0.05785AB m opt = (3.91, 6.22) Note: These equations are from Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach. Elsevier: Amsterdam, 1987, and pseudo-three-dimensional plots of the response surfaces can be found in their Figures 11.4, 11.5, and 11.14. The response surface for problem (a) also is shown in Color Plate 13. 2. Determine the optimum response for the equation in problem 1c, using the fixed-sized simplex searching algorithm. Compare your optimum response with the true optimum. 3. A 2 k factorial design was used to determine the equation for the response surface in problem 1b. The uncoded levels, coded levels, and the responses are shown in the following table. 1 4 G PROBLEMS 1400-CH14 9/8/99 4:36 PM Page 700 ABA* B* Response 8 8 +1 +1 5.92 8 2 +1 –1 2.08 2 8 –1 +1 4.48 2 2 –1 –1 3.52 Determine the coded and uncoded equation for the response surface. 4. Koscielniak and Parczewski investigated the influence of Al on the determination of Ca by atomic absorption spectrophotometry using the 2 k factorial design shown in the following table. 11 Ca 2+ Al 3+ (ppm) (ppm) Ca* Al* Response 10 160 +1 +1 54.29 10 0 +1 –1 98.44 4 160 –1 +1 19.18 4 0 –1 –1 38.53 (a) Determine the coded equation for the response surface. (b) If you wish to analyze a sample for which the concentration of Ca 2+ is 6.0 ppm, what is the maximum concentration of Al 3+ that can be present if the error in the response must be less than 5.0%? 5. Strange reports the following information for a 2 3 factorial design used to investigate the yield of a chemical process. 12 Factor High (+1) Level Low (–1) Level X: Temperature 140 °C 120 °C Y: Catalyst Type B Type A Z: [Reactant] 0.50 M 0.25 M Run X* Y* Z* % Yield 1–1–1–128 2 1 –1 –1 17 3 –1 1 –1 41 41 –134 5 –1–11 56 6 1 –1 1 51 7– 1 42 81 136 (a) Determine the coded equation for this data. (b) If b terms of less than – 1 are insignificant, what main effect and interaction terms in the coded equation are important? Write down this simpler form for the coded equation. (c) Explain why the coded equation for these data cannot be transformed into an uncoded form. (d) Which is the better catalyst, A or B? (e) What is the yield using this catalyst if the temperature is set to 125 °C and the concentration of the reactant is 0.45 M? 6. Pharmaceutical tablets coated with lactose often develop a brown discoloration. The factors that primarily affect the discoloration are temperature, relative humidity, and the presence of a base acting as a catalyst. The following data for a 2 3 factorial design have been reported. 13 Factor High (+1) Level Low (–1) Level X: benzocaine present absent Y: temperature 40 °C 25 °C Z: relative humidity 75% 50% Color Run X* Y* Z* (arb. units) 1 –1 –1 –1 1.55 2 1 –1 –1 5.40 3 –1 1 –1 3.50 4 1 1 –1 6.75 5 –1 –1 1 2.45 6 1 –1 1 3.60 7 –1 1 1 3.05 8 1 1 1 7.10 (a) Determine the coded equation for these data. (b) If b terms of less than 0.5 are insignificant, what main effect and interaction terms in the coded equation are important? Write down this simpler form for the coded equation. 7. The following data for a 2 3 factorial design were collected during a study of the effect of temperature, pressure, and residence time on the %yield of a reaction. 14 Factor High (+1) Level Low (–1) Level X: temperature 200 °C 100 °C Y: pressure 0.6 MPa 0.2 MPa Z: residence time 20 min 10 min Run X* Y* Z* Yield 1–1–1–12 2 1 –1 –1 6 3 –1 1 –1 4 41 –18 5 –1–1110 6 1 –1 1 18 7– 1 81 112 (a) Determine the coded equation for these data. (b) If b terms of less than ±1 are insignificant, what main effect and interaction terms in the coded equation are important? Write down this simpler form for the coded equation. (c) Three runs were made at the center of the factorial design (temperature = 150 °C, pressure = 0.4 MPa, residence time = 15 min), giving yields of 8, 9, and 8.8. Determine if a first-order empirical model is appropriate for this system at a = 0.05. Chapter 14 Developing a Standard Method 701 1400-CH14 9/8/99 4:36 PM Page 701 8. Duarte and colleagues used a factorial design to optimize a flow injection analysis method for determining penicillin potentiometrically. 15 Three factors were studied—reactor length, carrier flow rate, and sample volume, with the high and low values summarized in the following table. Factor Low (–1) Level High (+1) Level X: reactor length (cm) 1.5 2.0 Y: carrier flow rate (mL/min) 1.6 2.2 Z: sample volume (m L) 100 150 An optimum response was defined as the greatest sensitivity, as determined by the measured potential for a standard solution of penicillin, and the largest sampling rate. The results of the optimization studies are shown in the following table. D E Run X* Y* Z* (mV) Samples/h 1 –1 –1 –1 37.45 21.5 2 1 –1 –1 31.70 26.0 3 –1 1 –1 32.10 30.0 4 1 1 –1 27.20 33.0 5 –1 –1 1 39.85 21.0 6 1 –1 1 32.85 19.5 7 –1 1 1 35.00 30.0 8 1 1 1 32.15 34.0 (a) Determine the coded equation for the response surface where ?E is the response. (b) Determine the coded equation for the response surface where samples/h is the response. (c) Based on the coded equations, do the conditions favoring sensitivity also improve the sampling rate? (d) What conditions would you choose if your goal is to optimize both sensitivity and sampling rate? 9. Here is a challenge! McMinn and co-workers investigated the effect of five factors for optimizing an H 2 -atmosphere flame ionization detector using a 2 5 factorial design. 16 The factors and their levels were Factor High (+1) Level Low (–1) Level A: H 2 flow rate (mL/min) 1460 1382 B: SiH 4 (ppm) 20.0 12.2 C: O 2 +N 2 flow rate (mL/min) 255 210 D: O 2 /N 2 1.36 1.19 E: electrode height 75 55 The coded (“+” = +1, “–” = –1) factor levels and responses, R, for the 32 experiments are shown in the following table Run A* B* C* D* E* R Run A* B* C* D* E* R 1–––––0.36 17 – – – – + 0.39 2 + – – – – 0.51 18 + – – – + 0.45 3 – + – – – 0.15 19 – + – – + 0.32 4 + + – – – 0.39 20 + + – – + 0.25 5 – – + – – 0.79 21 – – + – + 0.18 6+–+––0.83 22 + – + – + 0.29 7 – + + – – 0.74 23 – + + – + 0.07 8 + + + – – 0.69 24 + + + – + 0.19 9 – – – + – 0.60 25 – – – + + 0.53 10 + – – + – 0.82 26 + – – + + 0.60 1 –+–+–0.42 27 – + – + + 0.36 12 + + – + – 0.59 28 + + – + + 0.43 13 – – + + – 0.96 29 – – + + + 0.23 14 + – + + – 0.87 30 + – + + + 0.51 15 – + + + – 0.76 31 – + + + + 0.13 16++++–0.74 32 + + + + + 0.43 (a) Determine the coded equation for this response surface, ignoring b terms less than ±0.03. (b) A simplex optimization of this system finds optimal values for the factors of A = 2278 mL/min, B = 9.90 ppm, C = 260.6 mL/min, and D = 1.71. The value for E was maintained at its high level. Are these values consistent with your analysis of the factorial design? 10. A good empirical model provides an accurate picture of the response surface for factor levels included within the experimental design. The same model, however, may yield an inaccurate prediction for the response at other factor levels. An empirical model, therefore, should be tested before extrapolating to conditions other than those used to determine the model. For example, Palasota and Deming 7 studied the effect of the relative amounts of H 2 SO 4 and H 2 O 2 on the absorbance of solutions of vanadium using the following central composite design. Run Drops 1% H 2 O 2 Drops 20% H 2 SO 4 115 2 210 0 32 2 48 15 515 61 15 715 81 15 922 10 10 10 11 20 10 12 15 8 702 Modern Analytical Chemistry 1400-CH14 9/8/99 4:36 PM Page 702 The reaction of H 2 O 2 and H 2 SO 4 generates a reddish brown solution whose absorbance is measured at a wavelength of 450 nm. A regression analysis on their data yielded the following uncoded equation for the response (Absorbance · 1000). R = 835.90 – 36.82X 1 – 21.34X 2 + 0.52(X 1 ) 2 + 0.15(X 2 ) 2 + 0.98X 1 X 2 where X 1 is the drops of H 2 O 2 , and X 2 is the drops of H 2 SO 4 . Calculate the predicted absorbances for 10 drops of H 2 O 2 and 0 drops of H 2 SO 4 , 0 drops of H 2 O 2 and 10 drops of H 2 SO 4 , and for 0 drops of each reagent. Are these results reasonable? Explain. What does your answer tell you about this empirical model? 11. A newly proposed method is to be tested for its single- operator characteristics. To be competitive with the standard method, the new method must have a relative standard deviation of less than 10%, with a bias of less than 10%. To test the method, an analyst performs ten replicate analyses on a standard sample known to contain 1.30 ppm of the analyte. The results for the ten trials are 1.25 1.26 1.29 1.56 1.46 1.23 1.49 1.27 1.31 1.43 Are the single-operator characteristics for this method acceptable? 12. A proposed gravimetric method was evaluated for its ruggedness by varying the following factors Factor A—sample size A =1 g a = 1.1 g Factor B—pH B = 6.5 b = 6.0 Factor C—digestion time C =3 h c =1 h Factor D—number of rinses D = 3 rinses d = 5 rinses Factor E—precipitant E = reagent 1 e = reagent 2 Factor F—digestion temperature F =50°C f =60°C Factor G—drying temperature G = 110 °C g = 140 °C A standard sample containing a known amount of analyte was carried through the procedure using the experimental design in Table 14.6. The percentage of the known amount of analyte actually found in the eight trials were found to be R 1 = 98.9 R 2 = 98.5 R 3 = 97.7 R 4 = 97.0 R 5 = 98.8 R 6 = 98.5 R 7 = 97.7 R 8 = 97.3 Determine which factors, if any, appear to have a significant effect on the response, and estimate the expected standard deviation for the method. 13. The two-sample plot for the data in Example 14.7 is shown in Figure 14.19. Identify the analyst whose work is (a) the most accurate; (b) the most precise; (c) the least accurate; and (d) the least precise. 14. Chichilo reports the following data for the determination of the %w/w Al in two samples of limestone. 17 Analyst Sample 1 Sample 2 1 1.35 1.57 2 1.35 1.33 3 1.34 1.47 4 1.50 1.60 5 1.52 1.62 6 1.39 1.52 7 1.30 1.36 8 1.32 1.53 Construct a two-sample plot for these data, and estimate values for s rand and s sys assuming a = 0.05. 15. The importance of between-laboratory variability on the results of an analytical method can be determined by having several laboratories analyze the same sample. In one such study seven laboratories analyzed a sample of homogenized milk for a selected alfatoxin. 18 The results, in parts per billion, are summarized in the following table. Replicate Lab A Lab B Lab C Lab D Lab E Lab F Lab G 1 1.6 4.6 1.2 1.5 6.0 6.2 3.3 2 2.9 2.8 1.9 2.7 3.9 3.8 3.8 3 3.5 3.0 2.9 3.4 4.3 5.5 5.5 4 1.8 4.5 1.1 2.0 5.8 4.2 4.9 5 2.2 3.1 2.9 3.4 4.0 5.3 4.5 (a) Determine if the between-laboratory variability is significantly greater than the within-laboratory variability at a = 0.05. If the between-laboratory variability is significant, then determine the source of that variability. (b) Estimate values for s 2 rand and s 2 sys . 16. A class of analytical students is asked to analyze a steel sample to determine the %w/w Mn. (a) Given that the steel sample is 0.26% w/w Mn, estimate the expected relative standard deviation for the class’ results. (b) The actual results obtained by the students are 0.26% 0.28% 0.27% 0.24% 0.26% 0.25% 0.26% 0.28% 0.25% 0.24% 0.26% 0.25% 0.29% 0.24% 0.27% 0.23% 0.26% 0.24% Are these results consistent with the estimated relative standard deviation? Chapter 14 Developing a Standard Method 703 1400-CH14 9/8/99 4:36 PM Page 703 704 Modern Analytical Chemistry The following texts and articles provide an excellent discussion of optimization methods based on searching algorithms and mathematical modeling, including a discussion of the relevant calculations. Bayne, C. K.; Rubin, I. B. Practical Experimental Designs and Optimization Methods for Chemists. VCH Publishers: Deerfield Beach, FL, 1986. Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach. Elsevier: Amsterdam, 1987. Hendrix, C. D. “What Every Technologist Should Know About Experimental Design,” Chemtech 1979, 9, 167–174. Hendrix, C. D. “Through the Response Surface with Test Tube and Pipe Wrench,” Chemtech 1980, 10, 488–497. Morgan, E. Chemometrics: Experimental Design. John Wiley and Sons: Chichester, England, 1991. The following texts provide additional information about ANOVA calculations, including discussions of two-way analysis of variance. Graham, R. C. Data Analysis for the Chemical Sciences. VCH Publishers: New York, 1993. Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed. Ellis Horwood Limited: Chichester, England, 1993. 1 4 H SUGGESTED READINGS 1. Vogel’s Textbook of Quantitative Inorganic Analysis. Longman: London, 1978; p. 752. 2. Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Chemometrics. Wiley- Interscience: New York, 1986. 3. Spendley, W.; Hext, G. R.; Himsworth, F. R. Technometrics 1962, 4, 441–461. 4. Deming, S. N.; Parker, L. R. CRC Crit. Rev. Anal. Chem. 1978, 7(3), 187–202. 5. Long, D. E. Anal. Chim. Acta 1969, 46, 193–206. 6. Nelder, J. A.; Mead, R. Computer J. 1965, 7, 308–313. 7. Palasota, J. A.; Deming, S. N. J. Chem. Educ. 1992, 62, 560–563. 8. Youden, W. J. Anal. Chem. 1960, 32(13), 23A–37A. 9. Youden, W. J. “Statistical Techniques for Collaborative Tests” in Statistical Manual of the Association of Official Analytical Chemists. Association of Official Analytical Chemists: Washington, D.C., 1975. 10. (a) Horwitz, W. Anal. Chem. 1982, 54, 67A–76A; (b) Hall, P.; Selinger, B. Anal. Chem. 1989, 61, 1465–1466; (c) Albert, R.; Horwitz, W. Anal. Chem. 1997, 69, 789–790. 11. Koscielniak, P.; Parczewski, A. Anal. Chim. Acta 1983, 153, 111–119. 12. Strange, R. S. J. Chem. Educ. 1990, 67, 113–115. 13. Armstrong, N. A.; James, K. C. Pharmaceutical Experimental Design and Interpretation. Taylor and Francis: London, 1996 as cited in González, A. G. Anal. Chim. Acta 1998, 360, 227–241. 14. Akhnazarova, S.; Kafarov, V. Experimental Optimization in Chemistry and Chemical Engineering. MIR Publishers: Moscow, 1982 as cited in González, A. G. Anal. Chim. Acta 1998, 360, 227–241. 15. Duarte, M. M. M. B.; de O. Netro, G.; Kubota, L. T.; et al. Anal. Chim. Acta 1997, 350, 353–357. 16. McMinn, D. G.; Eatherton, R. L.; Hill, H. H. Anal. Chem. 1984, 56, 1293–1298. 17. Chichilo, P. J. Assoc. Offc. Agr. Chemists 1964, 47, 1019 as reported in Youden, W. J. “Statistical Techniques for Collaborative Tests,” in Statistical Manual of the Association of Official Analytical Chemists. Association of Official Analytical Chemists: Washington, D.C., 1975. 18. Massart, D. L.; Vandeginste, B. G. M.; Deming, S. N.; et al. Chemometrics: A Textbook. Elsevier: Amsterdam, 1988. 1 4 I REFERENCES 1400-CH14 9/8/99 4:36 PM Page 704 Chapter 1 5 705 Quality Assurance In Chapter 1 we noted that each field of chemistry brings a unique perspective to the broader discipline of chemistry. For analytical chemistry this perspective was identified as an approach to solving problems, which was presented as a five-step process: (1) Identify and define the problem; (2) Design the experimental procedure; (3) Conduct an experiment and gather data; (4) Analyze the experimental data; and (5) Propose a solution to the problem. The analytical approach, as presented thus far, appears to be a straightforward process of moving from problem-to-solution. Unfortunately (or perhaps fortunately for those who consider themselves to be analytical chemists!), an analysis is seldom routine. Even a well-established procedure, carefully followed, can yield poor data of little use. An important feature of the analytical approach, which we have neglected thus far, is the presence of a feedback loop involving steps 2, 3, and 4. As a result, the outcome of one step may lead to a reevaluation of the other two steps. For example, after standardizing a spectrophotometric method for the analysis of iron we may find that its sensitivity does not meet the original design criteria. Considering this information we might choose to select a different method, to change the original design criteria, or to improve the sensitivity. The feedback loop in the analytical approach is maintained by a quality assurance program (Figure 15.1), whose objective is to control systematic and random sources of error. 1 — 5 The underlying assumption of a quality assurance program is that results obtained when an analytical system is in statistical control are free of bias and are characterized by well-defined confidence intervals. When used properly, a quality assurance program identifies the practices necessary to bring a system into statistical control, allows us to determine if the system remains in statistical control, and suggests a course of corrective action when the system has fallen out of statistical control. The focus of this chapter is on the two principal components of a quality assurance program: quality control and quality assessment. In addition, considerable attention is given to the use of control charts for routinely monitoring the quality of analytical data. 1400-CH15 9/8/99 4:38 PM Page 705 706 Modern Analytical Chemistry 1 5 A Quality Control Quality control encompasses all activities used to bring a system into statistical control. The most important facet of quality control is a set of written directives de- scribing all relevant laboratory-specific, technique-specific, sample-specific, method-specific, and protocol-specific operations. 1,3,6 Good laboratory practices (GLPs) describe the general laboratory operations that need to be followed in any analysis. These practices include properly recording data and maintaining records, using chain-of-custody forms for samples that are submitted for analysis, specifying and purifying chemical reagents, preparing commonly used reagents, cleaning and calibrating glassware, training laboratory personnel, and maintaining the laboratory facilities and general laboratory equipment. Good measurement practices (GMPs) describe operations specific to a tech- nique. In general, GMPs provide instructions for maintaining, calibrating, and using the equipment and instrumentation that form the basis for a specific tech- nique. For example, a GMP for a titration describes how to calibrate a buret (if nec- 1. Identify the problem Determine type of information needed (qualitative, quantitative, characterization, or fundamental) Establish context of the problem 2. Design the experimental procedure Establish design criteria (accuracy, precision, scale of operation, sensitivity, selectivity, cost, speed) Identify interferents Select method Establish validation criteria Establish sampling strategy Quality assurance program 3. Conduct an experiment Calibrate instruments and equipment Standardize reagents Gather data 4. Analyze the experimental data Reduce or transform data Analyze statistics Verify results Interpret results 5. Propose a solution Conduct external evaluation Q u a l i t y C o n t r o l A s s e s s m e n t Q u a l i t y Figure 15.1 Schematic diagram of the analytical approach to problem solving, showing the role of the quality assurance program. quality control Those steps taken to ensure that an analysis is under statistical control. good laboratory practices Those general laboratory procedures that, when followed, help ensure the quality of analytical work. good measurement practices Those instructions outlining how to properly use equipment and instrumentation to ensure the quality of measurements. quality assurance The steps taken during an analysis to ensure that the analysis is under control and that it is properly monitored. 1400-CH15 9/8/99 4:38 PM Page 706 Chapter 15 Quality Assurance 707 standard operations procedure The procedure followed in collecting and analyzing samples and in interpreting the results of an analysis. essary), how to fill a buret with the titrant, the correct way to read the volume of titrant in the buret, and the correct way to dispense the titrant. The operations that need to be performed when analyzing a specific analyte in a specific matrix are defined by a standard operations procedure (SOP). The SOP describes all steps taken during the analysis, including: how the sample is processed in the laboratory, the analyte’s separation from potential interferents, how the method is standardized, how the analytical signal is measured, how the data are transformed into the desired result, and the quality assessment tools that will be used to maintain quality control. If the laboratory is responsible for sampling, then the SOP will also state how the sample is to be collected and preserved and the na- ture of any prelaboratory processing. A SOP may be developed and used by a single laboratory, or it may be a standard procedure approved by an organization such as the American Society for Testing and Materials or the Federal Food and Drug Ad- ministration. A typical SOP is provided in the following example. EXAMPLE 1 5 .1 Provide an SOP for the determination of cadmium in lake sediments by atomic absorption spectrophotometry using a normal calibration curve. SOLUTION Sediment samples should be collected using a bottom grab sampler and stored at 4 °C in acid-washed polyethylene bottles during transportation to the laboratory. Samples should be dried to constant weight at 105 °C and ground to a uniform particle size. The cadmium in a 1-g sample of the sediment is extracted by adding the sediment and 25 mL of 0.5 M HCl to an acid-washed 100-mL polyethylene bottle and shaking for 24 h. After filtering, the sample is analyzed by atomic absorption spectrophotometry using an air–acetylene flame, a wavelength of 228.8 nm, and a slit width of 0.5 nm. A normal calibration curve is prepared using five standards with nominal concentrations of 0.20, 0.50, 1.00, 2.00, and 3.00 ppm. The accuracy of the calibration curve is checked periodically by analyzing the 1.00-ppm standard. An accuracy of – 10% is considered acceptable. Although an SOP provides a written procedure, it is not necessary to follow the procedure exactly as long as any modifications are identified. On the other hand, a protocol for a specific purpose (PSP), which is the most detailed of the written quality control directives, must be followed exactly if the results of the analysis are to be accepted. In many cases the required elements of a PSP are established by the agency sponsoring the analysis. For example, labs working under contract with the Environmental Protection Agency must develop a PSP that addresses such items as sampling and sample custody, frequency of calibration, schedules for the preventive maintenance of equipment and instrumentation, and management of the quality assurance program. Two additional aspects of a quality control program deserve mention. The first is the physical inspection of samples, measurements and results by the individuals responsible for collecting and analyzing the samples. 1 For example, sediment sam- ples might be screened during collection, and samples containing “foreign objects,” such as pieces of metal, be discarded without being analyzed. Samples that are dis- carded can then be replaced with additional samples. When a sudden change in the protocol for a specific purpose A precisely written protocol for an analysis that must be followed exactly. 1400-CH15 9/8/99 4:38 PM Page 707 708 Modern Analytical Chemistry performance of an instrument is observed, the analyst may choose to repeat those measurements that might be adversely influenced. The analyst may also decide to reject a result and reanalyze the sample when the result is clearly unreasonable. By identifying samples, measurements, and results that may be subject to gross errors, inspection helps control the quality of an analysis. A final component of a quality control program is the certification of an ana- lyst’s competence to perform the analysis for which he or she is responsible. 7 Before an analyst is allowed to perform a new analytical method, he or she may be required to successfully analyze an independent check sample with acceptable accuracy and precision. The check sample should be similar in composition to samples that the analyst will routinely encounter, with a concentration that is 5 to 50 times that of the method’s detection limit. 1 5 B Quality Assessment The written directives of a quality control program are a necessary, but not a suffi- cient, condition for obtaining and maintaining an analysis in a state of statistical control. Although quality control directives explain how an analysis should be properly conducted, they do not indicate whether the system is under statistical control. This is the role of quality assessment, which is the second component of a quality assurance program. The goals of quality assessment are to determine when a system has reached a state of statistical control; to detect when the system has moved out of statistical control; and, if possible, to suggest why a loss of statistical control has occurred so that corrective ac- tions can be taken. For convenience, the methods of quality assessment are divided into two categories: internal methods that are coordinated within the laboratory and exter- nal methods for which an outside agency or individual is responsible. The incorpora- tion of these methods into a quality assurance program is covered in Section 15C. 1 5 B.1 Internal Methods of Quality Assessment The most useful methods for quality assessment are those that are coordinated by the laboratory and that provide the analyst with immediate feedback about the sys- tem’s state of statistical control. Internal methods of quality assessment included in this section are the analysis of duplicate samples, the analysis of blanks, the analysis of standard samples, and spike recoveries. Analysis of Duplicate Samples An effective method for determining the precision of an analysis is to analyze duplicate samples. In most cases the duplicate samples are taken from a single gross sample (also called a split sample), although in some cases the duplicates must be independently collected gross samples. The results from the duplicate samples, X 1 and X 2 , are evaluated by determining the difference, d, or the relative difference, (d) r , between the samples d = X 1 – X 2 and comparing the results with accepted values, such as those shown in Table 15.1 for the analysis of waters and wastewaters. 7 Alternatively, the results for a set of n duplicates are combined to estimate the standard deviation for the analysis () ()/ d d XX r = + × 12 2 100 quality assessment The steps taken to evaluate whether an analysis is under statistical control. duplicate samples Two samples taken from a single gross sample and used to evaluate an analytical method’s precision. 1400-CH15 9/8/99 4:38 PM Page 708 Chapter 15 Quality Assurance 709 Table 1 5 .1 Selected Quality Assessment Limits for the Analysis of Waters and Wastewaters Limits for Spike Recovery (d) r When [Analyte] < 20 · MDL (d) r When [Analyte] > 20 · MDL Analyte (%) (– %) (– %) acids 60–140 40 20 anions 80–120 25 10 bases or neutrals 70–130 40 20 carbamate pesticides 50–150 40 20 herbicides 40–160 40 20 metals 80–120 25 10 other inorganics 80–120 25 10 volatile organics 70–130 40 20 Abbreviation: MDL = method’s detection limit. where d i is the difference between the ith pair of duplicates. The degrees of freedom for the standard deviation is the same as the number of duplicate samples. If dupli- cate samples from several sources are combined, then the precision of the measure- ment process must be approximately the same for each. The precision obtained is then compared with the precision needed to accept the results of the analysis. EXAMPLE 1 5 .2 To evaluate the precision for the determination of potassium in blood serum, duplicate analyses were performed on six samples, yielding the following results. Duplicate X 1 X 2 1 160 147 2 196 202 3 207 196 4 185 193 5 172 188 6 133 119 Calculate the standard deviation for the analysis. SOLUTION The standard deviation is determined as follows. Duplicate d = X 1 – X 2 d 2 1 13 169 2–636 3 11 121 4–864 5 –16 256 6 14 196 s d n i = ∑ 2 2 1400-CH15 9/8/99 4:38 PM Page 709 The Analysis of Blanks The use of a blank was introduced in Chapter 3 as a means of correcting the measured signal for contributions from sources other than the analyte. The most common blank is a method, or reagent blank, in which an analyte-free sample, usually distilled water, is carried through the analysis using the same reagents, glassware, and instrumentation. Method blanks are used to identify and correct systematic errors due to impurities in the reagents and con- tamination in the glassware and instrumentation. At a minimum, method blanks should be analyzed whenever new reagents are used, although a more frequent analysis provides an ongoing monitoring of the purity of the reagents. A new method blank should also be run whenever a sample with a high concentration of the analyte is analyzed, because any residual carryover of the analyte may contami- nate the glassware or instrumentation. When samples are collected in the field, the method blank may be augmented with field and trip blanks. 8 A field blank is an analyte-free sample carried from the laboratory to the sampling site. At the sampling site the blank is transferred to a clean sample container, exposing it to the local environment, preserved, and trans- ported back to the laboratory for analysis. Field blanks are used to identify and correct systematic errors due to sampling, transport, and analysis. Trip blanks are analyte-free samples carried from the laboratory to the sampling site and returned to the laboratory without being opened. A trip blank is used to identify and correct systematic errors due to cross-contamination of volatile organic compounds during transport, handling, storage, and analysis. Analysis of Standards The analysis of a standard containing a known concentra- tion of analyte also can be used to monitor a system’s state of statistical control. Ide- ally, a standard reference material (SRM) should be used, provided that the matrix of the SRM is similar to that of the samples being analyzed. A variety of appropriate SRMs are available from the National Institute of Standards and Technology (NIST). If a suitable SRM is not available, then an independently prepared synthetic sample can be used if it is prepared from reagents of known purity. At a minimum, a standardization of the method is verified by periodically analyzing one of the cali- bration standards. In all cases, the analyte’s experimentally determined concentra- tion in the standard must fall within predetermined limits if the system is to be con- sidered under statistical control. Spike Recoveries One of the most important quality assessment tools is the recov- ery of a known addition, or spike, of analyte to a method blank, field blank, or sam- ple. To determine a spike recovery, the blank or sample is split into two portions, and a known amount of a standard solution of the analyte is added to one portion. The concentration of the analyte is determined for both the spiked, F, and unspiked portions, I, and the percent recovery, %R, is calculated as where A is the concentration of the analyte added to the spiked portion. %R FI A = ? × 100 s = ++ ++ + == 169 36 121 64 256 196 26 842 12 84 ()() . 710 Modern Analytical Chemistry field blank A blank sample collected in the field. trip blank A blank prepared in the laboratory that accompanies a set of sample containers in the field and laboratory. spike recovery An analysis of a sample after spiking with a known amount of analyte. 1400-CH15 9/8/99 4:38 PM Page 710 Chapter 15 Quality Assurance 711 EXAMPLE 1 5 . 3 A spike recovery for the analysis of chloride in well water was performed by adding 5.00 mL of a 25,000-ppm solution of Cl – to a 500-mL volumetric flask and diluting to volume with the sample. Analysis of the sample and the spiked sample resulted in chloride concentrations of 183 ppm and 409 ppm, respectively. Determine the percent recovery of the spike. SOLUTION The concentration of the added spike is calculated by taking into account the effect of dilution. Thus, the spike recovery is Spike recoveries on method blanks and field blanks are used to evaluate the general performance of an analytical procedure. The concentration of analyte added to the blank should be between 5 and 50 times the method’s detection limit. Sys- tematic errors occurring during sampling and transport will result in an unaccept- able recovery for the field blank, but not for the method blank. Systematic errors occurring in the laboratory, however, will affect the recoveries for both the field and method blanks. Spike recoveries for samples are used to detect systematic errors due to the sample matrix or the stability of the sample after its collection. Ideally, samples should be spiked in the field at a concentration between 1 and 10 times the expected concentration of the analyte or 5 to 50 times the method’s detection limit, whichever is larger. If the recovery for a field spike is unacceptable, then a sample is spiked in the laboratory and analyzed immediately. If the recovery for the labora- tory spike is acceptable, then the poor recovery for the field spike may be due to the sample’s deterioration during storage. When the recovery for the laboratory spike also is unacceptable, the most probable cause is a matrix-dependent relationship be- tween the analytical signal and the concentration of the analyte. In this case the samples should be analyzed by the method of standard additions. Typical limits for acceptable spike recoveries for the analysis of waters and wastewaters are shown in Table 15.1. 7 1 5 B.2 External Methods of Quality Assessment Internal methods of quality assessment should always be viewed with some level of skepticism because of the potential for bias in their execution and interpretation. For this reason, external methods of quality assessment also play an important role in quality assurance programs. One external method of quality assessment is the certification of a laboratory by a sponsoring agency. Certification is based on the successful analysis of a set of proficiency standards prepared by the sponsoring agency. For example, laboratories involved in environmental analyses may be re- quired to analyze standard samples prepared by the Environmental Protection %.%R = ? ×= 409 183 250 100 90 4 A =×=25 000 250, ppm 5.00 mL 500.0 mL ppm proficiency standard A standard sample provided by an external agency as part of certifying the quality of a laboratory’s work. 1400-CH15 9/8/99 4:38 PM Page 711 712 Modern Analytical Chemistry Agency. A second example of an external method of quality assessment is the volun- tary participation of the laboratory in a collaborative test (Chapter 14) sponsored by a professional organization such as the Association of Official Analytical Chemists. Finally, individuals contracting with a laboratory can perform their own external quality assessment by submitting blind duplicate samples and blind standard sam- ples to the laboratory for analysis. If the results for the quality assessment samples are unacceptable, then there is good reason to consider the results suspect for other samples provided by the laboratory. 1 5 C Evaluating Quality Assurance Data In the previous section we described several internal methods of quality assessment that provide quantitative estimates of the systematic and random errors present in an analytical system. Now we turn our attention to how this numerical information is incorporated into the written directives of a complete quality assurance program. Two approaches to developing quality assurance programs have been described 9 : a prescriptive approach, in which an exact method of quality assessment is prescribed; and a performance-based approach, in which any form of quality assessment is ac- ceptable, provided that an acceptable level of statistical control can be demonstrated. 1 5 C.1 Prescriptive Approach With a prescriptive approach to quality assessment, duplicate samples, blanks, stan- dards, and spike recoveries are measured following a specific protocol. The result for each analysis is then compared with a single predetermined limit. If this limit is exceeded, an appropriate corrective action is taken. Prescriptive approaches to qual- ity assurance are common for programs and laboratories subject to federal regula- tion. For example, the Food and Drug Administration (FDA) specifies quality as- surance practices that must be followed by laboratories analyzing products regulated by the FDA. A good example of a prescriptive approach to quality assessment is the protocol outlined in Figure 15.2, published by the Environmental Protection Agency (EPA) for laboratories involved in monitoring studies of water and wastewater. 10 Indepen- dent samples A and B are collected simultaneously at the sample site. Sample A is split into two equal-volume samples, and labeled A 1 and A 2 . Sample B is also split into two equal-volume samples, one of which, B SF , is spiked with a known amount of analyte. A field blank, D F , also is spiked with the same amount of analyte. All five samples (A 1 , A 2 , B, B SF , and D F ) are preserved if necessary and transported to the laboratory for analysis. The first sample to be analyzed is the field blank. If its spike recovery is unac- ceptable, indicating that a systematic error is present, then a laboratory method blank, D L , is prepared and analyzed. If the spike recovery for the method blank is also unsatisfactory, then the systematic error originated in the laboratory. An ac- ceptable spike recovery for the method blank, however, indicates that the systematic error occurred in the field or during transport to the laboratory. Systematic errors in the laboratory can be corrected, and the analysis continued. Any systematic er- rors occurring in the field, however, cast uncertainty on the quality of the samples, making it necessary to collect new samples. If the field blank is satisfactory, then sample B is analyzed. If the result for B is above the method’s detection limit, or if it is within the range of 0.1 to 10 times the amount of analyte spiked into B SF , then a spike recovery for B SF is determined. An 1400-CH15 9/8/99 4:38 PM Page 712 unacceptable spike recovery for B SF indicates the presence of a systematic error in- volving the sample. To determine the source of the systematic error, a laboratory spike, B SL , is prepared using sample B and analyzed. If the spike recovery for B SL is acceptable, then the systematic error requires a long time to have a noticeable effect on the spike recovery. One possible explanation is that the analyte has not been properly preserved or has been held beyond the acceptable holding time. An unac- ceptable spike recovery for B SL suggests an immediate systematic error, such as that due to the influence of the sample’s matrix. In either case, the systematic errors are fatal and must be corrected before the sample is reanalyzed. If the spike recovery for B SF is acceptable, or if the result for sample B is below the method’s detection limit or outside the range of 0.1 to 10 times the amount of analyte spiked in B SF , then the duplicate samples A 1 and A 2 are analyzed. The results for A 1 and A 2 are discarded if the difference between their values is excessive. If the difference between the results for A 1 and A 2 is within the accepted limits, then the results for samples A 1 and B are compared. Since samples collected from the same sampling site at the same time should be identical in composition, the results are discarded if the difference between their values is unsatisfactory, and accepted if the difference is satisfactory. Chapter 15 Quality Assurance 713 D F recovery within limits D L recovery within limits B > MDL, or B > 0.1 · [spike], and B < 10 · [spike] Systematic error in laboratory Systematic error in field Ye sNo Ye sNo Immediate systematic errors Time- dependent systematic errors Ye sNo No ‰ A 1 – B‰ within limits Poor Replication NoYe s ‰ A 1 – A 2 ‰ within limits B SF recovery within limits B SL recovery within limits Poor duplicate samples Ye s Ye s Ye s No No Data valid Figure 15.2 Example of a prescriptive approach to quality assurance. Adapted from Environmental Monitoring and Support Laboratory, U.S. Environmental Protection Agency, “Handbook for Analytical Quality Control in Water and Wastewater Laboratories,” March 1979. 1400-CH15 9/8/99 4:38 PM Page 713 This protocol requires four to five evaluations of quality assessment data before the result for a single sample can be accepted; a process that must be repeated for each analyte and for each sample. Other prescriptive protocols are equally demand- ing. For example, Figure 3.7 in Chapter 3 shows a portion of the quality assurance protocol used for the graphite furnace atomic absorption analysis of trace metals in aqueous solutions. This protocol involves the analysis of an initial calibration verifi- cation standard and an initial calibration blank, followed by the analysis of samples in groups of ten. Each group of samples is preceded and followed by continuing cal- ibration verification (CCV) and continuing calibration blank (CCB) quality assess- ment samples. Results for each group of ten samples can be accepted only if both sets of CCV and CCB quality assessment samples are acceptable. The advantage to a prescriptive approach to quality assurance is that a single con- sistent set of guidelines is used by all laboratories to control the quality of analytical results. A significant disadvantage, however, is that the ability of a laboratory to pro- duce quality results is not taken into account when determining the frequency of col- lecting and analyzing quality assessment data. Laboratories with a record of producing high-quality results are forced to spend more time and money on quality assessment than is perhaps necessary. At the same time, the frequency of quality assessment may be insufficient for laboratories with a history of producing results of poor quality. 1 5 C.2 Performance-Based Approach In a performance-based approach to quality assurance, a laboratory is free to use its experience to determine the best way to gather and monitor quality assessment data. The quality assessment methods remain the same (duplicate samples, blanks, standards, and spike recoveries) since they provide the necessary information about precision and bias. What the laboratory can control, however, is the fre- quency with which quality assessment samples are analyzed, and the conditions in- dicating when an analytical system is no longer in a state of statistical control. Fur- thermore, a performance-based approach to quality assessment allows a laboratory to determine if an analytical system is in danger of drifting out of statistical con- trol. Corrective measures are then taken before further problems develop. The principal tool for performance-based quality assessment is the control chart. In a control chart the results from the analysis of quality assessment samples are plotted in the order in which they are collected, providing a continuous record of the statistical state of the analytical system. Quality assessment data collected over time can be summarized by a mean value and a standard deviation. The fundamen- tal assumption behind the use of a control chart is that quality assessment data will show only random variations around the mean value when the analytical system is in statistical control. When an analytical system moves out of statistical control, the quality assessment data is influenced by additional sources of error, increasing the standard deviation or changing the mean value. Control charts were originally developed in the 1920s as a quality assurance tool for the control of manufactured products. 11 Two types of control charts are commonly used in quality assurance: a property control chart in which results for single measurements, or the means for several replicate measurements, are plotted sequentially; and a precision control chart in which ranges or standard deviations are plotted sequentially. In either case, the control chart consists of a line represent- ing the mean value for the measured property or the precision, and two or more boundary lines whose positions are determined by the precision of the measure- ment process. The position of the data points about the boundary lines determines whether the system is in statistical control. 714 Modern Analytical Chemistry control chart A graph showing the time-dependent change in the results of an analysis that is used to monitor whether an analysis is in a state of statistical control. 1400-CH15 9/8/99 4:38 PM Page 714 Chapter 15 Quality Assurance 715 Construction of Property Control Charts The simplest form for a property control chart is a sequence of points, each of which represents a single determination of the property being monitored. To construct the control chart, it is first necessary to de- termine the mean value of the property and the standard deviation for its measure- ment. These statistical values are determined using a minimum of 7 to 15 samples (although 30 or more samples are desirable), obtained while the system is known to be under statistical control. The center line (CL) of the control chart is determined by the average of these n points The positions of the boundary lines are determined by the standard deviation, S, of the points used to determine the central line with the upper and lower warning limits (UWL and LWL), and the upper and lower control limits (UCL and LCL) given by UWL = CL +2S LWL = CL –2S UCL = CL +3S LCL = CL –3S EXAMPLE 1 5 . 4 Construct a property control chart for the following spike recovery data (all values are for percentage of spike recovered). Sample: 1 2 3 4 5 Result: 97.3 98.1 100.3 99.4 100.9 Sample: 6 7 8 9 10 Result: 98.6 96.9 99.6 101.1 100.4 Sample: 11 12 13 14 15 Result: 100.0 95.9 98.3 99.2 102.1 Sample: 16 17 18 19 20 Result: 98.5 101.7 100.4 99.1 100.3 SOLUTION The mean and the standard deviation for the 20 data points are 99.4 and 1.6, respectively, giving the UCL as 104.2, the UWL as 102.6, the LWL as 96.2 and the LCL as 94.6. The resulting property control chart is shown in Figure 15.3. S XX n i = ? ? ∑ () 2 1 CL X X n i == ∑ 1400-CH15 9/8/99 4:38 PM Page 715 Property control charts can also be constructed using points that are the mean value, – X i , for a set of r replicate determinations on a single sample. The mean for the ith sample is given by where X ij is the jth replicate. The center line for the control chart, therefore, is To determine the standard deviation for the warning and control limits, it is neces- sary to calculate the variance for each sample, s i 2 . The overall standard deviation, S, is the square root of the average variance for the samples used to establish the control plot. Finally, the resulting warning and control limits are UCL CL S r =+ 3 LWL CL S r =? 2 UWL CL S r =+ 2 S s n i = ∑ 2 s XX r i ij j r i 2 1 2 1 = ? ? = ∑ () CL X X n i == ∑ X X j i ij j r = = ∑ 1 716 Modern Analytical Chemistry P e r cent reco ver y 1 94 105 Sequence 20 LCL LWL CL = X UWL UCL 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 104 103 102 101 100 99 98 97 96 95 Figure 15.3 Property control chart for Example 15.4. 1400-CH15 9/8/99 4:38 PM Page 716 Chapter 15 Quality Assurance 717 Table 1 5 .2 Statistical Factors for the Upper Warning Limit and Upper Control Limit Replicates f UWL f UCL 2 2.512 3.267 3 2.050 2.575 4 1.855 2.282 5 1.743 2.115 6 1.669 2.004 Constructing a Precision Control Chart The most common measure of precision used in constructing a precision control chart is the range, R, between the largest and smallest results for a set of j replicate analyses on a sample. R = X large – X small To construct the control chart, ranges for a minimum of 15–20 samples (prefer- ably 30 or more samples) are obtained while the system is known to be in statistical control. The line for the average range, R – , is determined by the mean of these n samples The upper control line and the upper warning line are given by UCL = f UCL · R – UWL = f UWL · R – where f UCL and f UWL (Table 15.2) are statistical factors determined by the number of replicates used to determine the range. Because the range always is greater than or equal to zero, there is no lower control limit or lower warning limit. EXAMPLE 1 5 . 5 Construct a precision control chart using the following 20 ranges, each determined from a duplicate analysis of a 10-ppm calibration standard Sample: 1 2 3 4 5 Result: 0.36 0.09 0.11 0.06 0.25 Sample: 6 7 8 9 10 Result: 0.15 0.28 0.27 0.03 0.28 Sample: 11 12 13 14 15 Result: 0.21 0.19 0.06 0.13 0.37 Sample: 16 17 18 19 20 Result: 0.01 0.19 0.39 0.05 0.05 R R n i = ∑ LCL CL S r =? 3 1400-CH15 9/8/99 4:38 PM Page 717 SOLUTION The average range for the 20 duplicate samples is 0.177. Because two replicates were used for each point, the UWL and UCL are UWL = (2.512)(0.177) = 0.44 UCL = (3.267)(0.177) = 0.58 The complete control chart is shown in Figure 15.4. The precision control chart is strictly valid only for the replicate analysis of identical samples, such as a calibration standard or a standard reference material. Its use for the analysis of nonidentical samples, such as a series of clinical or environ- mental samples, is complicated by the fact that the range usually is not independent of the magnitude of X large and X small . For example, Table 15.3 shows the relationship between – R and the concentration of chromium in water. 10 Clearly the significant difference in the average range for these concentrations of Cr makes a single preci- sion control chart impossible. One solution to this problem is to prepare separate precision control charts, each of which covers a range of concentrations for which – R is approximately constant (Figure 15.5). Interpreting Control Charts The purpose of a control chart is to determine if a sys- tem is in statistical control. This determination is made by examining the location of individual points in relation to the warning limits and the control limits, and the distribution of the points around the central line. If we assume that the data are normally distributed, then the probability of finding a point at any distance from the mean value can be determined from the normal distribution curve. The upper and lower control limits for a property control chart, for example, are set to – 3S, which, if S is a good approximation for s , includes 99.74% of the data. The proba- bility that a point will fall outside the UCL or LCL, therefore, is only 0.26%. The 718 Modern Analytical Chemistry 1 0.00 0.60 R Sequence 20 R UWL UCL 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0.50 0.40 0.30 0.20 0.10 Figure 15.4 Precision control chart for Example 15.5. 1400-CH15 9/8/99 4:38 PM Page 718 Figure 15.5 Example of the use of subrange precision control charts for samples that span a range of analyte concentrations. The precision control charts are used for (a) low concentrations of analyte; (b) intermediate concentrations of analyte; and (c) high concentrations of analyte. most likely explanation when a point exceeds a control limit is that a systematic error has occurred or that the precision of the measurement process has deterio- rated. In either case the system is assumed to be out of statistical control. Rule 1. A system is considered to be out of statistical control if any single point exceeds either the UCL or the LCL. The upper and lower warning limits, which are located at – 2S, should only be ex- ceeded by 5% of the data; thus Rule 2. A system is considered to be out of statistical control if two out of three consecutive points are between the UWL and UCL or between the LWL and LCL. When a system is in statistical control, the data points should be randomly dis- tributed about the center line. The presence of an unlikely pattern in the data is an- other indication that a system is no longer in statistical control. 4,12 Thus, Rule 3. A system is considered to be out of statistical control if a run of seven consecutive points is completely above or completely below the center line (Figure 15.6a). Chapter 15 Quality Assurance 719 Table 1 5 . 3 Average Range for Duplicate Samples for Different Concentrations of Chromium in Water Cr Number of (ppb) Duplicate Samples – R 5 to <10 32 0.32 10 to <25 15 0.57 25 to <50 16 1.12 50 to <150 15 3.80 150 to <500 8 5.25 >500 5 76.0 Rang e Sequence UCL (a) (b) (c) UCL UCL UWL UWL UWL Sequence Sequence 1400-CH15 9/8/99 4:38 PM Page 719 Figure 15.6 Examples of property control charts that show a run of data (highlighted in box) indicating that the system is out of statistical control. 720 1 Sequence 20 UCL UWL CL LWL LCL 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 Sequence 202 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 UCL UWL CL LWL LCL 1 Sequence 202 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 UCL UWL CL LWL LCL (a) (b) (c) 1400-CH15 9/8/99 4:38 PM Page 720 Rule 4. A system is considered to be out of statistical control if six consecutive points are all increasing in value or all decreasing in value (Figure 15.6b). The points may be on either side of the center line. Rule 5. A system is considered to be out of statistical control if 14 consecutive points alternate up and down in value (Figure 15.6c). The points may be on either side of the center line. Rule 6. A system is considered to be out of statistical control if any obvious “nonrandom” pattern is observed. The same rules apply to precision control charts with the exception that there are no lower warning and lower control limits. Using Control Charts for Quality Assurance Control charts play an important role in a performance-based program of quality assurance because they provide an easily interpreted picture of the statistical state of an analytical system. Quality assessment samples such as blanks, standards, and spike recoveries can be monitored with prop- erty control charts. A precision control chart can be used to monitor duplicate samples. The first step in using a control chart for quality assurance is to determine the mean value and the standard deviation (except when using the range) for the quality assessment data while the system is under statistical control. These values must be es- tablished under the same conditions that will be present during the normal use of the control chart. Thus, preliminary data should be randomly collected throughout the day, as well as over several days, to account for short-term and long-term variability. The preliminary data are used to construct an initial control chart, and discrepant points are determined using the rules discussed in the previous section. Questionable points are dropped, and the control chart is replotted. As the control chart is used, it may become apparent that the original limits need adjusting. Control limits can be re- calculated if the number of new data points is at least equivalent to the amount of data used to construct the original control chart. For example, if 15 points were initially used, the limits can be reevaluated after 15 additional points are collected. The 30 points are pooled together to calculate the new limits. A second modification can be made after a further 30 points have been collected. Another indication that a control chart needs to be modified is when points rarely exceed the warning limits. In this case the new limits can be recalculated using the last 20 points. Once a control chart is in use, new quality assessment data should be added at a rate sufficient to ensure that the system remains in statistical control. As with prescriptive approaches to quality assurance, when a quality assessment sample is found to be out of statistical control, all samples analyzed since the last successful verification of statistical control must be reanalyzed. The advantage of a performance-based approach to quality assurance is that a laboratory may use its experience, guided by control charts, to determine the frequency for collect- ing quality assessment samples. When the system is stable, quality assessment samples can be acquired less frequently. Chapter 15 Quality Assurance 721 1 5 D KEY TERMS control chart (p. 714) duplicate samples (p. 708) field blank (p. 710) good laboratory practices (p. 706) good measurement practices (p. 706) proficiency standard (p. 711) protocol for a specific purpose (p. 707) quality assessment (p. 708) quality assurance (p. 706) quality control (p. 706) spike recovery (p. 710) standard operations procedure (p. 707) trip blank (p. 710) 1400-CH15 9/8/99 4:38 PM Page 721 722 Modern Analytical Chemistry Few analyses are so straightforward that high-quality results are easily obtained. Good analytical work requires careful planning and an attention to detail. Creating and maintaining a quality as- surance program is one way to help ensure the quality of analytical results. Quality assurance programs usually include elements of quality control and quality assessment. Quality control encompasses all activities used to bring a sys- tem into statistical control. The most important facet of quality control is written documentation, including statements of good laboratory practices, good measurement practices, standard oper- ating procedures, and protocols for a specific purpose. Quality assessment includes the statistical tools used to deter- mine whether an analysis is in a state of statistical control and, if possible, to suggest why an analysis has drifted out of statistical control. Among the tools included in quality assessment are the analysis of duplicate samples, the analysis of blanks, the analysis of standards, and the analysis of spike recoveries. Another important quality assessment tool, which provides an ongoing evaluation of an analysis, is a control chart. A control chart plots a property, such as a spike recovery, as a function of time. Re- sults exceeding warning and control limits, or unusual patterns of data indicate that an analysis is no longer under statistical control. 1 5 E SUMMARY Bell, S. C.; Moore, J. “Integration of Quality Assurance/ Quality Control into Quantitative Analysis,” J. Chem. Educ. 1998, 75, 874–877. The use of several QA/QC methods is described in this article, including control charts for monitoring the concentration of solutions of thiosulfate that have been prepared and stored with and without proper preservation; the use of method blanks and standard samples to determine the presence of determinate error and to establish single-operator characteristics; and the use of spiked samples and recoveries to identify the presence of determinate errors associated with collecting and analyzing samples. Laquer, F. C. “Quality Control Charts in the Quantitative Analysis Laboratory Using Conductance Measurement,” J. Chem. Educ. 1990, 67, 900–902. The conductivities of a standard solution of KCl, laboratory distilled water, and synthetic-process samples are monitored weekly and evaluated using a control chart. Marcos, J.; Ríos, A.; Valcárcel, M. “Practicing Quality Control in a Bioanalytical Experiment,” J. Chem. Educ. 1995, 72, 947–949. This experiment demonstrates how control charts and an analysis of variance can be used to evaluate the quality of results in a quantitative analysis for chlorophyll a and b in plant material. 1 5 F Suggested EXPERIMENTS The following three experiments introduce aspects of quality assurance and quality control. Experiments 1. Make a list of good laboratory practices for the lab accompanying this course (or another lab if this course does not have an associated laboratory). Explain the rationale for each item on your list. 2. Write directives outlining good measurement practices for (a) a buret, (b) a pH meter, and (c) a spectrophotometer. 3. A method for the analysis of lead in industrial wastewater has a method detection limit of 10 ppb. The relationship between the analytical signal and the concentration of lead, as determined from a calibration curve is S meas = 0.349 · (ppm Pb) Analysis of a sample in duplicate gives S meas as 0.554 and 0.516. Is the precision between these two duplicates acceptable based on the limits shown in Table 15.1? 4. The following data were obtained for the duplicate analysis of a 5.00-ppm NO 3 – standard 1 5 G PROBLEMS 1400-CH15 9/8/99 4:38 PM Page 722 X 1 X 2 Sample (ppm) (ppm) 1 5.02 4.90 2 5.10 5.18 3 5.07 4.95 4 4.96 5.01 5 4.88 4.98 6 5.04 4.97 Calculate the standard deviation for the analysis of these duplicate samples. If the maximum limit for the relative standard deviation is 1.5%, are these results acceptable? 5. Gonzalez and colleagues developed a voltammetric procedure for the determination of tert-butylhydroxyanisole (BHA) in chewing gum. 13 Analysis of a commercial chewing gum gave results of 0.20 mg/g. To evaluate the accuracy of their results, they performed five spike recoveries, adding an amount of BHA equivalent to 0.135 mg/g to each sample. The experimentally determined concentrations of BHA in these samples were reported as 0.342, 0.340, 0.340, 0.324, and 0.322 mg/g. Determine the % recovery for each sample and the average % recovery. 6. A sample is to be analyzed following the protocol shown in Figure 15.2, using a method with a detection limit of 0.05 ppm. The relationship between the analytical signal and the concentration of the analyte, as determined from a calibration curve is S meas = 0.273 · (ppm analyte) Answer the following questions if the limits for a successful spike recovery are – 10%. (a) A field blank is spiked with the analyte to a concentration of 2.00 ppm and returned to the lab. Analysis of the spiked field blank gives a signal of 0.573. Is the spike recovery for the field blank acceptable? (b) The analysis of a spiked field blank is found to be unacceptable. To determine the source of the problem, a spiked method blank is prepared by spiking distilled water with the analyte to a concentration of 2.00 ppm. Analysis of the spiked method blank gives a signal of 0.464. Is the source of the problem in the laboratory or in the field? (c) The analysis for a spiked field sample, B SF , was found to be unacceptable. To determine the source of the problem, the sample was spiked in the laboratory by adding sufficient analyte to increase the concentration by 2.00 ppm. Analysis of the sample before and after the spike gave signals of 0.456 for B and 1.03 for sample B SL . Considering these data, what is the most likely source of the systematic error? 7. The following data were obtained for the repetitive analysis of a stable standard. 14 X i X i X i Sample (ppm) Sample (ppm) Sample (ppm) 1 35.1 10 35.0 18 36.4 2 33.2 11 31.4 19 32.1 3 33.7 12 35.6 20 38.2 4 35.9 13 30.2 21 33.1 5 33.5 14 32.7 22 34.9 6 34.5 15 31.1 23 36.2 7 34.4 16 34.8 24 34.0 8 34.3 17 34.3 25 33.8 9 31.8 Construct a property control chart for these data, and evaluate the state of statistical control. 8. The following data were obtained for the repetitive spike recoveries of field samples. 15 Sample % Recovery Sample % Recovery Sample % Recovery 1 94.6 10 104.6 18 104.6 2 93.1 11 123.8 19 91.5 3 100.0 12 93.8 20 83.1 4 122.3 13 80.0 21 100.8 5 120.8 14 99.2 22 123.1 6 93.1 15 101.5 23 96.2 7 117.7 16 74.6 24 96.9 8 96.2 17 108.5 25 102.3 9 73.8 Construct a property control chart for these data, and evaluate the state of statistical control. 9. The following data were obtained for the duplicate analysis of a stable standard. 14 X 1 X 2 X 1 X 2 Sample (ppm) (ppm) Sample (ppm) (ppm) 15046143636 23736154745 32219161620 41720171821 5323418262 64646193536 72628202625 82630214951 961582 3 32 10 44 45 23 40 38 11 40 44 24 16 13 12 36 35 25 39 42 13 29 31 Construct a precision control chart for these data, and evaluate the state of statistical control. Chapter 15 Quality Assurance 723 1400-CH15 9/8/99 4:38 PM Page 723 724 Modern Analytical Chemistry The following texts and articles may be consulted for an additional discussion of the various aspects of quality assurance and quality control. Amore, F. “Good Analytical Practices,” Anal. Chem. 1979, 51, 1105A–1110A. Barnard, Jr. A. J.; Mitchell, R. M.; Wolf, G. E. “Good Analytical Practices in Quality Control,” Anal. Chem. 1978, 50, 1079A–1086A. Cairns, T.; Rogers, W. M. “Acceptable Analytical Data for Trace Analysis,” Anal. Chem. 1993, 55, 54A–57A. Taylor, J. K. Quality Assurance of Chemical Measurements. Lewis Publishers: Chelsea, MI, 1987. Additional information about the construction and use of control charts may be found in the following sources. Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed. Ellis Horwood Limited: Chichester, England, 1993. Ouchi, G. I. “Creating Control Charts with a Spreadsheet Program,” LC?GC 1993, 11, 416–423 and LC?GC 1997, 15, 336–344. Simpson, J. M. “Spreadsheet Statistics,” J. Chem. Educ. 1994, 71, A88–A89. 1 5 H SUGGESTED READINGS 1. Taylor, J. K. Anal. Chem. 1981, 53, 1588A–1596A 2. Taylor, J. K. Anal. Chem. 1983, 55, 600A–608A. 3. Taylor, J. K. Am. Lab. October 1985, 67–75. 4. Nadkarni, R. A. Anal. Chem. 1991, 63, 675A–682A. 5. Valcárcel, M.; Ríos, A. Trends Anal. Chem. 1994, 13, 17–23. 6. ACS Committee for Environmental Improvement, “Principles of Environmental Analysis,” Anal. Chem. 1983, 55, 2210–2218. 7. American Public Health Association, Standard Methods for the Analysis of Water and Wastewater, 18th ed. Washington, D.C., 1992. 8. Keith, L. H. Environmental Sampling and Analysis: A Practical Guide. Lewis Publishers, Chelsea, MI, 1991. 9. Poppiti, J. Environ. Sci. Technol. 1994, 28, 151A–152A. 10. Environmental Monitoring and Support Laboratory, U.S. Environmental Protection Agency, “Handbook for Analytical Quality Control in Water and Wastewater Laboratories,” March 1979. 11. Shewhart, W. A. Economic Control of the Quality of Manufactured Products. Macmillan: London, 1931. 12. Mullins, E. Analyst 1994, 119, 369–375. 13. Gonzalez, A.; Ruiz, M. A.; Yanez-Sedeno, P.; et al. Anal. Chim. Acta 1994, 285, 63–71. 14. American Public Health Association, Standard Methods for the Analysis of Water and Wastewater, 18th ed. Washington, D.C., 1992. Data from Table 1030:I on page 1–10. 15. American Public Health Association, Standard Methods for the Analysis of Water and Wastewater, 18th ed. Washington, D.C., 1992. Data adapted from Table 1030:II on page 1–10. 1 5 I REFERENCES 1400-CH15 9/8/99 4:38 PM Page 724 725 Appendix 1A Single-Sided Normal Distribution a u 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4365 0.4325 0.4286 0.4247 0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0253 2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 2.3 0.0107 0.0104 0.0102 0.00964 0.00914 0.00866 2.4 0.00820 0.00776 0.00734 0.00695 0.00657 2.5 0.00621 0.00587 0.00554 0.00523 0.00494 2.6 0.00466 0.00440 0.00415 0.00391 0.00368 2.7 0.00347 0.00326 0.00307 0.00289 0.00272 2.8 0.00256 0.00240 0.00226 0.00212 0.00199 2.9 0.00187 0.00175 0.00164 0.00154 0.00144 3.0 0.00135 3.1 0.000968 3.2 0.000687 AppendixAppendix 1 continued 1400-AP1 9/8/99 4:39 PM Page 725 726 Modern Analytical Chemistry Single-Sided Normal Distribution a —continued u 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 3.3 0.000483 3.4 0.000337 3.5 0.000233 3.6 0.000159 3.7 0.000108 3.8 0.0000723 3.9 0.0000481 4.0 0.0000317 4.1 0.0000207 4.2 0.0000133 4.3 0.00000854 4.4 0.00000541 4.5 0.00000340 4.6 0.00000211 4.7 0.00000130 4.8 0.000000793 4.9 0.000000479 5.0 0.000000287 a This table gives the proportion, P, of the area under a normal distribution curve that lies to the right of the deviation z, where z is defined as z =(X – m )/s For example, the proportion of the area under a normal distribution curve that lies to the right of a deviation of 0.04 is 0.4840, or 48.40%. The area to the left of the deviation is given as 1 – P. Thus, 51.60% of the area under the normal distribution curve lies to the left of a deviation of 0.04. When the deviation is negative, the values in the table give the proportion of the area under the normal distribution curve that lies to the left of z; therefore, 48.40% of the area lies to the left, and 51.60% of the area lies to the right of a deviation of –0.04. Appendix 1B t-Table a Value of t for confidence interval of: 90% 95% 98% 99% Critical value of t for a values of: 0.10 0.05 0.02 0.01 Degrees of Freedom 1 6.31 12.71 31.82 63.66 2 2.92 4.30 6.96 9.92 3 2.35 3.18 4.54 5.84 4 2.13 2.78 3.75 4.60 5 2.02 2.57 3.36 4.03 6 1.94 2.45 3.14 3.71 7 1.89 2.36 3.00 3.50 8 1.86 2.31 2.90 3.36 9 1.83 2.26 2.82 3.25 10 1.81 2.23 2.76 3.17 12 1.78 2.18 2.68 3.05 14 1.76 2.14 2.62 2.98 16 1.75 2.12 2.58 2.92 18 1.73 2.10 2.55 2.88 20 1.72 2.09 2.53 2.85 30 1.70 2.04 2.46 2.75 50 1.68 2.01 2.40 2.68 ￥ 1.64 1.96 2.33 2.58 a The t-values in this table are for a two-tailed test. For a one-tailed test, the a values for each column are half of the stated value. For example, the first column for a one-tailed test is for the 95% confidence level, a = 0.05. 1400-AP1 9/8/99 4:39 PM Page 726 Appendix 1 727 Appendix 1C.1 F-Table for One-Tailed Test at a = 0.05 (95% Confidence Level) n 2 /n 1 a 123456789101520 1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 245.9 248.0 254.3 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.43 19.45 19.50 3 10.13 9.552 9.277 9.117 9.013 8.941 8.887 8.845 8.812 8.786 8.703 8.660 8.526 4 7.709 6.944 6.591 6.388 6.256 6.163 6.094 6.041 5.999 5.964 5.858 5.803 5.628 5 6.608 5.786 5.409 5.192 5.050 4.950 4.876 4.818 4.772 4.735 4.619 4.558 4.365 6 5.987 5.143 4.757 4.534 4.387 4.284 4.207 4.147 4.099 4.060 3.938 3.874 3.669 7 5.591 4.737 4.347 4.120 3.972 3.866 3.787 3.726 3.677 3.637 3.511 3.445 3.230 8 5.318 4.459 4.066 3.838 3.687 3.581 3.500 3.438 3.388 3.347 3.218 3.150 2.928 9 5.117 4.256 3.863 3.633 3.482 3.374 3.293 3.230 3.179 3.137 3.006 2.936 2.707 10 4.965 4.103 3.708 3.478 3.326 3.217 3.135 3.072 3.020 2.978 2.845 2.774 2.538 11 4.844 3.982 3.587 3.357 3.204 3.095 3.012 2.948 2.896 2.854 2.719 2.646 2.404 12 4.747 3.885 3.490 3.259 3.106 2.996 2.913 2.849 2.796 2.753 2.617 2.544 2.296 13 4.667 3.806 3.411 3.179 3.025 2.915 2.832 2.767 2.714 2.671 2.533 2.459 2.206 14 4.600 3.739 3.344 3.112 2.958 2.848 2.764 2.699 2.646 2.602 2.463 2.388 2.131 15 4.534 3.682 3.287 3.056 2.901 2.790 2.707 2.641 2.588 2.544 2.403 2.328 2.066 16 4.494 3.634 3.239 3.007 2.852 2.741 2.657 2.591 2.538 2.494 2.352 2.276 2.010 17 4.451 3.592 3.197 2.965 2.810 2.699 3.614 2.548 2.494 2.450 2.308 2.230 1.960 18 4.414 3.555 3.160 2.928 2.773 2.661 2.577 2.510 2.456 2.412 2.269 2.191 1.917 19 4.381 3.552 3.127 2.895 2.740 2.628 2.544 2.477 2.423 2.378 2.234 2.155 1.878 20 4.351 3.493 3.098 2.866 2.711 2.599 2.514 2.447 2.393 2.348 2.203 2.124 1.843 ￥ 3.842 2.996 2.605 2.372 2.214 2.099 2.010 1.938 1.880 1.831 1.666 1.570 1.000 a n 1 = degrees of freedom in numerator; n 2 = degrees of freedom in denominator. Appendix 1C.2 F-Table for Two-Tailed Test at a = 0.05 (95% Confidence Level) n 2 /n 1 a 123456789101520 1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 984.9 993.1 1018 2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.43 39.45 39.498 3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.25 14.17 13.902 4 12.22 10.65 9.979 9.605 9.364 9.197 9.074 8.980 8.905 8.844 8.657 8.560 8.257 5 10.01 8.434 7.764 7.388 7.146 6.978 6.853 6.757 6.681 6.619 6.428 6.329 6.015 6 8.813 7.260 6.599 6.227 5.988 5.820 5.695 5.600 5.523 5.461 5.269 5.168 4.849 7 8.073 6.542 5.890 5.523 5.285 5.119 4.995 4.899 4.823 4.761 4.568 4.467 4.142 8 7.571 6.059 5.416 5.053 4.817 4.652 4.529 4.433 4.357 4.295 4.101 3.999 3.670 9 7.209 5.715 5.078 4.718 4.484 4.320 4.197 4.102 4.026 3.964 3.769 3.667 3.333 10 6.937 5.456 4.826 4.468 4.236 4.072 3.950 3.855 3.779 3.717 3.522 3.419 3.080 11 6.724 5.256 4.630 4.275 4.044 3.881 3.759 3.664 3.588 3.526 3.330 3.226 2.883 12 6.544 5.096 4.474 4.121 3.891 3.728 3.607 3.512 3.436 3.374 3.177 3.073 2.725 13 6.414 4.965 4.347 3.996 3.767 3.604 3.483 3.388 3.312 3.250 3.053 2.948 2.596 14 6.298 4.857 4.242 3.892 3.663 3.501 3.380 3.285 3.209 3.147 2.949 2.844 2.487 15 6.200 4.765 4.153 3.804 3.576 3.415 3.293 3.199 3.123 3.060 2.862 2.756 2.395 continued 1400-AP1 9/8/99 4:39 PM Page 727 728 Modern Analytical Chemistry F-Table for Two-Tailed Test at a = 0.05 (95% Confidence Level)—continued n 2 /n 1 a 123456789101520 16 6.115 4.687 4.077 3.729 3.502 3.341 3.219 3.125 3.049 2.986 2.788 2.681 2.316 17 6.042 4.619 4.011 3.665 3.438 3.277 3.156 3.061 2.985 2.922 2.723 2.616 2.247 18 5.978 4.560 3.954 3.608 3.382 3.221 3.100 3.005 2.929 2.866 2.667 2.559 2.187 19 5.922 4.508 3.903 3.559 3.333 3.172 3.051 2.956 2.880 2.817 2.617 2.509 2.133 20 5.871 4.461 3.859 3.515 3.289 3.128 3.007 2.913 2.837 2.774 2.573 2.464 2.085 ￥ 5.024 3.689 3.116 2.786 2.567 2.408 2.288 2.192 2.114 2.048 1.833 1.708 1.000 a n 1 = degrees of freedom in numerator; n 2 = degrees of freedom in denominator. Appendix 1D Critical Values for Q-Test N/a 0.1 0.05 0.04 0.02 0.01 3 0.941 0.970 0.976 0.988 0.994 4 0.765 0.829 0.846 0.889 0.926 5 0.642 0.710 0.729 0.780 0.821 6 0.560 0.625 0.644 0.698 0.740 7 0.507 0.568 0.586 0.637 0.680 8 0.468 0.526 0.543 0.590 0.634 9 0.437 0.493 0.510 0.555 0.598 10 0.412 0.466 0.483 0.527 0.568 Appendix 1E Random Number Table 23733 43499 92848 39382 65423 52443 41812 90777 42345 39906 57880 74874 94181 90599 55012 18321 18766 31656 22117 31932 16446 72008 81701 90740 56193 15918 11383 19009 12891 79288 29317 89360 22245 19021 13613 48029 18412 13461 76692 22631 33897 18660 58252 10512 64982 11378 20874 42643 79780 57597 42119 71919 34890 28512 46025 26436 90134 80172 88083 24811 99173 19823 55086 63276 26295 96233 89540 87300 37651 72632 26509 83658 57363 20297 43957 99727 24619 88851 46397 75952 99527 34791 35965 39549 31292 76368 64897 10940 31625 37920 51477 86644 97853 27286 52256 61036 13485 39191 86568 36054 77050 97664 52009 79924 46681 87091 91046 58698 96379 64325 36167 85529 43159 29083 10326 99195 25138 73863 62097 26824 58567 27030 52337 18779 79381 22082 50045 93486 70336 87715 86573 94804 42825 54559 97987 41256 60800 70200 31662 40669 31893 90234 41062 62755 33545 59069 17986 99255 98675 92860 88094 84469 50846 88467 50297 31536 36189 84433 48281 52258 45251 13773 60075 82857 31205 66074 68602 37893 33277 57724 66147 61078 91044 25890 85664 66375 21245 67183 69835 19025 82454 79002 50734 75464 26500 78228 77829 14984 42235 10653 12791 38550 18383 56867 77215 50166 97525 16877 67902 87906 1400-AP1 9/8/99 4:39 PM Page 728 729 FW Element Compound (g/mol) Comments aluminum Al metal 26.982 antimony Sb metal 121.760 KSbOC 4 H 4 O 6 324.92 compound prepared by drying KSbOC 4 H 4 O 6 · 1/2H 2 O at 110 °C and storing in desiccator arsenic As metal 74.922 As 2 O 3 197.84 toxic barium BaCO 3 197.35 dry at 200 °C for 4 h bismuth Bi metal 208.980 Bi 2 O 3 465.96 not considered a primary standard boron H 3 BO 3 61.83 do not dry bromine KBr 119.01 cadmium Cd metal 112.411 CdO 128.40 not considered a primary standard calcium CaCO 3 100.09 cerium Ce metal 140.116 (NH 4 ) 2 Ce(NO 3 ) 6 548.23 not considered a primary standard cesium Cs 2 CO 3 325.82 Cs 2 SO 4 361.87 chlorine NaCl 58.44 chromium Cr metal 51.996 K 2 Cr 2 O 7 294.19 cobalt Co metal 58.933 copper Cu metal 63.546 CuO 79.54 fluorine NaF 41.99 do not store in glass container iodine KI 166.00 KIO 3 214.00 iron Fe metal 55.845 lead Pb metal 207.2 Pb(NO 3 ) 2 331.20 approaches primary standard lithium Li 2 CO 3 73.89 AppendixAppendix 2 Recommended Reagents for Preparing Primary Standards continued constant weight at 110 °C. Most compounds can be dissolved in dilute acid (1:1 HCl or 1:1 HNO 3 ), with heating if necessary; some of the compounds are water- soluble. All compounds should be of the highest available purity. Metals should be cleaned with dilute acid to remove surface impurities and rinsed with distilled water. Unless otherwise indicated, compounds should be dried to 1400-AP2 9/8/99 4:39 PM Page 729 730 Modern Analytical Chemistry FW Element Compound (g/mol) Comments magnesium Mg metal 24.305 MgO 40.31 not considered a primary standard manganese Mn metal 54.938 MnSO 4 ?H 2 O 169.01 not considered a primary standard; may be dried at 110 °C without loss of hydrated water mercury Hg metal 200.59 molybdenum Mo metal 95.94 nickel Ni metal 58.693 phosphorus KH 2 PO 4 136.09 not considered a primary standard potassium KCl 74.56 K 2 CO 3 138.21 K 2 Cr 2 O 7 294.19 KHC 8 H 4 O 4 204.23 silicon Si metal 28.085 SiO 2 60.08 silver Ag metal 107.868 AgNO 3 169.87 approaches primary standard sodium NaCl 58.44 Na 2 CO 3 106.00 Na 2 C 2 O 4 134.00 strontium SrCO 3 147.63 sulfur elemental S 32.066 K 2 SO 4 174.27 Na 2 SO 4 142.04 tin Sn metal 118.710 titanium Ti metal 47.867 tungsten W metal 183.84 uranium U metal 238.029 U 3 O 8 842.09 vanadium V metal 50.942 V 2 O 5 181.88 not considered a primary standard zinc Zn metal 65.39 ZnO 81.37 not considered a primary standard Source: Information compiled from Moody, J. R.; Greenberg, R. R.; Pratt, K. W.; et al. Anal. Chem. 1988, 60, 1203A–1218A; and Smith, B. W.; Parsons, M. L. J. Chem. Educ. 1973, 50, 679–681. 1400-AP2 9/8/99 4:39 PM Page 730 731 Appendix 3 A Solubility Products Bromide (Br – )pK sp K sp CuBr 8.3 5 · 10 –9 AgBr 12.30 5.0 · 10 –13 Hg 2 Br 2 22.25 5.6 · 10 –23 HgBr 2 (m = 0.5 M) 18.9 1.3 · 10 –19 PbBr 2 (m = 4.0 M) 5.68 2.1 · 10 –6 Carbonate (CO 3 2– )pK sp K sp MgCO 3 7.46 3.5 · 10 –8 CaCO 3 (calcite) 8.35 4.5 · 10 –9 CaCO 3 (aragonite) 8.22 6.0 · 10 –9 SrCO 3 9.03 9.3 · 10 –10 BaCO 3 8.30 5.0 · 10 –9 MnCO 3 9.30 5.0 · 10 –10 FeCO 3 10.68 2.1 · 10 –11 CoCO 3 9.98 1.0 · 10 –10 NiCO 3 6.87 1.3 · 10 –7 Ag 2 CO 3 11.09 8.1 · 10 –12 Hg 2 CO 3 16.05 8.9 · 10 –17 ZnCO 3 10.00 1.0 · 10 –10 CdCO 3 13.74 1.8 · 10 –14 PbCO 3 13.13 7.4 · 10 –14 Chloride (Cl – )pK sp K sp CuCl 6.73 1.9 · 10 –7 AgCl 9.74 1.8 · 10 –10 Hg 2 Cl 2 17.91 1.2 · 10 –18 PbCl 2 4.78 1.7 · 10 –5 Chromate (CrO 4 2– )pK sp K sp BaCrO 4 9.67 2.1 · 10 –10 CuCrO 4 5.44 3.6 · 10 –6 Ag 2 CrO 4 11.92 1.2 · 10 –12 Hg 2 CrO 4 8.70 2.0 · 10 –9 AppendixAppendix 3 Cyanide (CN – )pK sp K sp AgCN 15.66 2.2 · 10 –16 Zn(CN) 2 (m = 3.0 M) 15.5 3 · 10 –16 Hg 2 (CN) 2 39.3 5 · 10 –40 Ferrocyanide ([Fe(CN) 6 4– ]) pK sp K sp Zn 2 [Fe(CN) 6 ] 15.68 2.1 · 10 –16 Cd 2 [Fe(CN) 6 ] 17.38 4.2 · 10 –18 Pb 2 [Fe(CN) 6 ] 18.02 9.5 · 10 –19 Fluoride (F – )pK sp K sp MgF 2 8.18 6.6 · 10 –9 CaF 2 10.41 3.9 · 10 –11 SrF 2 8.54 2.9 · 10 –9 BaF 2 5.76 1.7 · 10 –6 PbF 2 7.44 3.6 · 10 –8 Hydroxide (OH – )pK sp K sp Mg(OH) 2 11.15 7.1 · 10 –12 Ca(OH) 2 5.19 6.5 · 10 –6 Ba(OH) 2 8H 2 O 3.6 3 · 10 –4 La(OH) 3 20.7 2 · 10 –21 Mn(OH) 2 12.8 1.6 · 10 –13 Fe(OH) 2 15.1 8 · 10 –16 Co(OH) 2 14.9 1.3 · 10 –15 Ni(OH) 2 15.2 6 · 10 –16 Cu(OH) 2 19.32 4.8 · 10 –20 Fe(OH) 3 38.8 1.6 · 10 –39 Co(OH) 3 (T = 19 °C) 44.5 3 · 10 –45 Ag 2 O (+H 2 O t 2Ag + + 2OH – ) 15.42 3.8 · 10 –16 Cu 2 O (+ H 2 O t 2Cu + + 2OH – ) 29.4 4 · 10 –30 Zn(OH) 2 (amorphous) 15.52 3.0 · 10 –16 Cd(OH) 2 (b ) 14.35 4.5 · 10 –15 continued 1400-AP3 9/8/99 4:40 PM Page 731 Solubility Products—continued Hydroxide (OH – )pK sp K sp HgO (red) (+ H 2 O t Hg 2+ + 2OH – ) 25.44 3.6 · 10 –26 SnO (+ H 2 O t Sn 2+ + 2OH – ) 26.2 6 · 10 –27 PbO (yellow) (+ H 2 O t Pb 2+ + 2OH – ) 15.1 8 · 10 –16 Al(OH) 3 (a ) 33.5 3 · 10 –34 Iodate (IO 3 – )pK sp K sp Ca(IO 3 ) 2 6.15 7.1 · 10 –7 Ba(IO 3 ) 2 8.81 1.5 · 10 –9 AgIO 3 7.51 3.1 · 10 –8 Hg 2 (IO 3 ) 2 17.89 1.3 · 10 –18 Zn(IO 3 ) 2 5.41 3.9 · 10 –6 Cd(IO 3 ) 2 7.64 2.3 · 10 –8 Pb(IO 3 ) 2 12.61 2.5 · 10 –13 Iodide (I – )pK sp K sp AgI 16.08 8.3 · 10 –17 Hg 2 I 2 28.33 4.7 · 10 –29 HgI 2 (m = 0.5 M) 27.95 1.1 · 10 –28 PbI 2 8.10 7.9 · 10 –9 Oxalate (C 2 O 4 2– )pK sp K sp CaC 2 O 4 (m = 0.1 M, T = 20 °C) 7.9 1.3 · 10 –8 BaC 2 O 4 (m = 0.1 M, T = 20 °C) 6.0 1 · 10 –6 SrC 2 O 4 (m = 0.1 M, T = 20 °C) 6.4 4 · 10 –7 Phosphate (PO 4 3– )pK sp K sp Fe 3 (PO 4 ) 2 8H 2 O 36.0 1 · 10 –36 Zn 3 (PO 4 ) 2 4H 2 O 35.3 5 · 10 –36 Ag 3 PO 4 17.55 2.8 · 10 –18 Pb 3 (PO 4 ) 2 (T = 38 °C) 43.53 3.0 · 10 –44 Sulfate (SO 4 2– )pK sp K sp CaSO 4 4.62 2.4 · 10 –5 SrSO 4 6.50 3.2 · 10 –7 BaSO 4 9.96 1.1 · 10 –10 Ag 2 SO 4 4.83 1.5 · 10 –5 Hg 2 SO 4 6.13 7.4 · 10 –7 PbSO 4 7.79 1.6 · 10 –8 Sulfide (S 2– )pK sp K sp MnS (green) 13.5 3 · 10 –14 FeS 18.1 8 · 10 –19 CoS (b ) 25.6 3 · 10 –26 NiS (g ) 26.6 3 · 10 –27 CuS 36.1 8 · 10 –37 Cu 2 S 48.5 3 · 10 –49 Ag 2 S 50.1 8 · 10 –51 ZnS (a ) 24.7 2 · 10 –25 CdS 27.0 1 · 10 –27 Hg 2 S (red) 53.3 5 · 10 –54 PbS 27.5 3 · 10 –28 Thiocyanate (SCN – )pK sp K sp CuSCN (m = 5.0 M) 13.40 4.0 · 10 –14 AgSCN 11.97 1.1 · 10 –12 Hg 2 (SCN) 2 19.52 3.0 · 10 –20 Hg(SCN) 2 (m = 1.0 M) 19.56 2.8 · 10 –20 732 Modern Analytical Chemistry Source: All values are from Martell, A. E.; Smith, R. M. Critical Stability Constants, Vol. 4. Plenum Press: New York, 1976. Unless otherwise stated, values are for 25 °C and zero ionic strength. Appendix 3 B Acid Dissociation Constants Compound Conjugate Acid pK a K a acetic acid CH 3 COOH 4.757 1.75 · 10 –5 adipic acid HOOC(CH 2 ) 4 COOH 4.42 3.8 · 10 –5 5.42 3.8 · 10 –6 alanine 2.348 (COOH) 4.49 · 10 –3 9.867 (NH 3 ) 1.36 · 10 –10 aminobenzene 4.601 2.51 · 10 –5 NH 3 + NH 3 + CHCH 3 COOH 1400-AP3 9/8/99 4:40 PM Page 732 Compound Conjugate Acid pK a K a 4-aminobenzene sulfonic acid 3.232 5.86 · 10 –4 2-aminobenzoic acid 2.08 (COOH) 8.3 · 10 –3 4.96 (NH 3 ) 1.1 · 10 –5 2-aminophenol 4.78 (NH 3 ); (T = 20 °C) 1.7 · 10 –5 9.97 (OH); (T = 20 °C) 1.05 · 10 –10 ammonia NH 4 + 9.244 5.70 · 10 –10 arginine 1.823 (COOH) 1.50 · 10 –2 8.991 (NH 3 ) 1.02 · 10 –9 (12.48) (NH 2 ) 3.3 · 10 –13 arsenic acid H 3 AsO 4 2.24 5.8 · 10 –3 6.96 1.1 · 10 –7 11.50 3.2 · 10 –12 asparagine 2.14 (COOH); (m = 0.1 M) 7.2 · 10 –3 8.72 (NH 3 ); (m = 0.1 M) 1.9 · 10 –9 aspartic acid 1.990 (a -COOH) 1.02 · 10 –2 3.900 (b -COOH) 1.26 · 10 –4 10.002 (NH 3 ) 9.95 · 10 –11 benzoic acid 4.202 6.28 · 10 –5 benzylamine 9.35 4.5 · 10 –10 boric acid H 3 BO 3 9.236 5.81 · 10 –10 (12.74); (T = 20 °C) 1.82 · 10 –13 (13.80); (T = 20 °C) 1.58 · 10 –14 carbonic acid H 2 CO 3 6.352 4.45 · 10 –7 10.329 4.69 · 10 –11 catechol 9.40 4.0 · 10 –10 12.8 1.6 · 10 –13 Appendix 3 733 NH 3 +– O 3 S COOH NH 3 + NH 3 + NH 2 + NH 2 CHCH 2 CH 2 CH 2 NHC COOH OH NH 3 + NH 3 + O CHCH 2 CNH 2 COOH NH 3 + CHCH 2 COOH COOH COOH CH 2 NH 3 + OH OH continued 1400-AP3 9/8/99 4:40 PM Page 733 Acid Dissociation Constants—continued Compound Conjugate Acid pK a K a chloroacetic acid ClCH 2 COOH 2.865 1.36 · 10 –3 chromic acid H 2 CrO 4 –0.2; (T = 20 °C) 1.6 6.51 3.1 · 10 –7 citric acid 3.128 (COOH) 7.45 · 10 –4 4.761 (COOH) 1.73 · 10 –5 6.396 (COOH) 4.02 · 10 –7 cupferron 4.16; (m = 0.1 M) 6.9 · 10 –5 cysteine (1.71) (COOH) 1.9 · 10 –2 8.36 (SH) 4.4 · 10 –9 10.77 (NH 3 ) 1.7 · 10 –11 dichloroacetic acid Cl 2 CHCOOH 1.30 5.0 · 10 –2 diethylamine (CH 3 CH 2 ) 2 NH 2 + 10.933 1.17 · 10 –11 dimethylamine (CH 3 ) 2 NH 2 + 10.774 1.68 · 10 –11 dimethylgloxime 10.66 2.2 · 10 –11 12.0 1 · 10 –12 ethylamine CH 3 CH 2 NH 3 + 10.636 2.31 · 10 –11 ethylenediamine + H 3 NCH 2 CH 2 NH 3 + 6.848 1.42 · 10 –7 9.928 1.18 · 10 –10 ethylenediaminetetraacetic acid (EDTA) 0.0 (COOH); (m = 1.0 M) 1.0 1.5 (COOH); (m = 0.1 M) 3.2 · 10 –2 2.0 (COOH); (m = 0.1 M) 1.0 · 10 –2 2.68 (COOH); (m = 0.1 M) 2.1 · 10 –3 6.11 (NH); (m = 0.1 M) 7.8 · 10 –7 10.17 (NH); (m = 0.1 M) 6.8 · 10 –11 formic acid HCOOH 3.745 1.80 · 10 –4 fumaric acid 3.053 8.85 · 10 –4 4.494 3.21 · 10 –5 glutamic acid 2.23 (a -COOH) 5.9 · 10 –3 4.42 (l -COOH) 3.8 · 10 –5 9.95 (NH 3 ) 1.12 · 10 –10 734 Modern Analytical Chemistry OH CCH 2 COOHHOOCH 2 C COOH N NO OH NH 3 + CHCH 2 SH COOH NOHHON CH 3 H 3 C CH 2 COOHHOOCH 2 C HOOCH 2 CCH 2 COOH + HNCH 2 CH 2 NH + HOOC COOH NH 3 + CHCH 2 CH 2 COOH COOH 1400-AP3 9/8/99 4:40 PM Page 734 Compound Conjugate Acid pK a K a glutamine 2.17 (COOH); (m = 0.1 M) 6.8 · 10 –3 9.01 (NH 3 ); (m = 0.1 M) 9.8 · 10 –10 glycine + H 3 NCH 2 COOH 2.350 (COOH) 4.47 · 10 –3 9.778 (NH 3 ) 1.67 · 10 –10 glycolic acid HOCH 2 COOH 3.831 (COOH) 1.48 · 10 –4 histidine 1.7 (COOH); (m = 0.1 M) 2 · 10 –2 6.02 (NH); (m = 0.1 M) 9.5 · 10 –7 9.08 (NH 3 ); (m = 0.1 M) 8.3 · 10 –10 hydrogen cyanide HCN 9.21 6.2 · 10 –10 hydrogen fluoride HF 3.17 6.8 · 10 –4 hydrogen peroxide H 2 O 2 11.65 2.2 · 10 –12 hydrogen sulfide H 2 S 7.02 9.5 · 10 –8 13.9 1.3 · 10 –14 hydrogen thiocyanate HSCN 0.9 1.3 · 10 –1 8-hydroxyquinoline 4.91 (NH) 1.23 · 10 –5 9.81 (OH) 1.55 · 10 –10 hydroxylamine HONH 3 + 5.96 1.1 · 10 –6 hypobromous HOBr 8.63 2.3 · 10 –9 hypochlorous HOCl 7.53 3.0 · 10 –8 hypoiodous HOI 10.64 2.3 · 10 –11 iodic acid HIO 3 0.77 1.7 · 10 –1 isoleucine 2.319 (COOH) 4.80 · 10 –3 9.754 (NH 3 ) 1.76 · 10 –10 leucine 2.329 (COOH) 4.69 · 10 –3 9.747 (NH 3 ) 1.79 · 10 –10 lysine 2.04 (COOH); (m = 0.1 M) 9.1 · 10 –3 9.08 (a -NH 3 ); (m = 0.1 M) 8.3 · 10 –10 10.69 (e -NH 3 );(m = 0.1 M) 2.0 · 10 –11 Appendix 3 735 NH 3 + O CHCH 2 CH 2 CNH 2 COOH CHCH 2 COOH NH 3 + H N H + N OH H + N NH 3 + CHCH(CH 3 )CH 2 CH 3 COOH NH 3 + CHCH 2 CH(CH 3 ) 2 COOH NH 3 + CHCH 2 CH 2 CH 2 CH 2 NH 3 + COOH continued 1400-AP3 9/8/99 4:40 PM Page 735 Acid Dissociation Constants—continued Compound Conjugate Acid pK a K a maleic acid 1.910 1.23 · 10 –2 6.332 4.66 · 10 –7 malic acid 3.459 (COOH) 3.48 · 10 –4 5.097 (COOH) 8.00 · 10 –6 malonic acid HOOCCH 2 COOH 2.847 1.42 · 10 –3 5.696 2.01 · 10 –6 methionine 2.20 (COOH); (m = 0.1 M) 6.3 · 10 –3 9.05 (NH 3 ); (m = 0.1 M) 8.9 · 10 –10 methylamine CH 3 NH 3 + 10.64 2.3 · 10 –11 2-methylaniline 4.447 3.57 · 10 –5 4-methylaniline 5.084 8.24 · 10 –6 2-methylphenol 10.28 5.2 · 10 –11 4-methylphenol 10.26 5.5 · 10 –11 nitrilotriacetic acid 1.1 (COOH); (T = 20 °C , m = 1.0 M) 8 · 10 –2 1.650 (COOH); (T = 20 °C) 2.24 · 10 –2 2.940 (COOH); (T = 20 °C) 1.15 · 10 –3 10.334 (NH 3 ); (T = 20 °C) 4.63 · 10 –11 2-nitrobenzoic acid 2.179 6.62 · 10 –3 3-nitrobenzoic acid 3.449 3.56 · 10 –4 736 Modern Analytical Chemistry HOOC COOH CHCOOHHOOCH 2 C OH NH 3 + CHCH 2 CH 2 SCH 3 COOH CH 3 NH 3 + NH 3 + H 3 C CH 3 OH OHH 3 C CH 2 COOH + HN CH 2 COOH CH 2 COOH COOH NO 2 COOH NO 2 1400-AP3 9/8/99 4:40 PM Page 736 Compound Conjugate Acid pK a K a 4-nitrobenzoic acid 3.442 3.61 · 10 –4 2-nitrophenol 7.21 6.2 · 10 –8 3-nitrophenol 8.39 4.1 · 10 –9 4-nitrophenol 7.15 7.1 · 10 –8 nitrous acid HNO 2 3.15 7.1 · 10 –4 oxalic acid H 2 C 2 O 4 1.252 5.60 · 10 –2 4.266 5.42 · 10 –5 1,10-phenanthroline 4.86 1.38 · 10 –5 phenol 9.98 1.05 · 10 –10 phenylalanine 2.20 (COOH) 6.3 · 10 –3 9.31 (NH 3 ) 4.9 · 10 –10 phosphoric acid H 3 PO 4 2.148 7.11 · 10 –3 7.199 6.32 · 10 –8 12.35 4.5 · 10 –13 phthalic acid 2.950 1.12 · 10 –3 5.408 3.91 · 10 –6 piperidine 11.123 7.53 · 10 –12 proline 1.952 (COOH) 1.12 · 10 –2 10.640 (NH) 2.29 · 10 –11 Appendix 3 737 COOHO 2 N OH NO 2 OH NO 2 OHO 2 N NNH + OH NH 3 + CHCH 2 C 6 H 5 COOH COOH COOH NH 2 + COOH N H 2 + continued 1400-AP3 9/8/99 4:40 PM Page 737 Acid Dissociation Constants—continued Compound Conjugate Acid pK a K a propanoic acid CH 3 CH 2 COOH 4.874 1.34 · 10 –5 propylamine CH 3 CH 2 CH 2 NH 3 + 10.566 2.72 · 10 –11 pyridine 5.229 5.90 · 10 –6 resorcinol 9.30 5.0 · 10 –10 11.06 8.7 · 10 –12 salicylic acid 2.97 (COOH) 1.07 · 10 –3 13.74 (OH) 1.8 · 10 –14 serine 2.187 (COOH) 6.50 · 10 –3 9.209 (NH 3 ) 6.18 · 10 –10 succinic acid HOOCCH 2 CH 2 COOH 4.207 6.21 · 10 –5 5.636 2.31 · 10 –6 sulfuric acid H 2 SO 4 strong strong 1.99 1.0 · 10 –2 sulfurous acid H 2 SO 3 1.91 1.2 · 10 –2 7.18 6.6 · 10 –8 D-tartaric acid 3.036 (COOH) 9.20 · 10 –4 4.366 (COOH) 4.31 · 10 –5 threonine 2.088 (COOH) 8.17 · 10 –3 9.100 (NH 3 ) 7.94 · 10 –10 thiosulfuric acid H 2 S 2 O 3 0.6 3 · 10 –1 1.6 3 · 10 –2 trichloroacetic acid Cl 3 CCOOH 0.66; (m = 0.1 M) 2.2 · 10 –1 triethanolamine (HOCH 2 CH 2 ) 3 NH + 7.762 1.73 · 10 –8 triethylamine (CH 3 CH 2 ) 3 NH + 10.715 1.93 · 10 –11 738 Modern Analytical Chemistry NH + OH OH COOH OH NH 3 + CHCH 2 OH COOH CH CH COOHHOOC OH OH NH 3 + CHCHOHCH 3 COOH 1400-AP3 9/8/99 4:40 PM Page 738 Compound Conjugate Acid pK a K a trimethylamine (CH 3 ) 3 NH + 9.800 1.58 · 10 –10 tris(hydroxymethyl)- aminomethane (TRIS or THAM) (HOCH 2 ) 3 CNH 3 + 8.075 8.41 · 10 –9 tryptophan 2.35 (COOH); (m = 0.1 M) 4.5 · 10 –3 9.33 (NH 3 ); (m = 0.1 M) 4.7 · 10 –10 tyrosine 2.17 (COOH); (m = 0.1 M) 6.8 · 10 –3 9.19 (NH 3 ) 6.5 · 10 –10 10.47 (OH) 3.4 · 10 –11 valine 2.286 (COOH) 5.18 · 10 –3 9.718 (OH) 1.91 · 10 –10 Source: All values are from Martell, A. E.; Smith, R. M. Critical Stability Constants, Vol. 1–4. Plenum Press: New York, 1976. Unless otherwise stated, values are for 25 °C and zero ionic strength. Values in parentheses are considered less reliable. Appendix 3 C Metal–Ligand Formation Constants Acetate CH 3 COO – log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Mg 2+ 1.27 Ca 2+ 1.18 Ba 2+ 1.07 Mn 2+ 1.40 Fe 2+ 1.40 Co 2+ 1.46 Ni 2+ 1.43 Cu 2+ 2.22 1.41 Ag + 0.73 –0.09 Zn 2+ 1.57 Cd 2+ 1.93 1.22 –0.89 Pb 2+ 2.68 1.40 Appendix 3 739 CHCH 2 COOH NH 3 + H N OHCHCH 2 COOH NH 3 + NH 3 + CHCH(CH 3 ) 2 COOH continued 1400-AP3 9/8/99 4:40 PM Page 739 Metal–Ligand Formation Constants—continued Ammonia NH 3 log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Ag + 3.31 3.91 Co 2+ (T = 20 °C) 1.99 1.51 0.93 0.64 0.06 –0.74 Ni 2+ 2.72 2.17 1.66 1.12 0.67 –0.03 Cu 2+ 4.04 3.43 2.80 1.48 Zn 2+ 2.21 2.29 2.36 2.03 Cd 2+ 2.55 2.01 1.34 0.84 Chloride Cl – log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Cu 2+ 0.40 Fe 3+ 1.48 0.65 Ag + (m = 5.0 M) 3.70 1.92 0.78 –0.3 Zn 2+ 0.43 0.18 –0.11 –0.3 Cd 2+ 1.98 1.62 –0.2 –0.7 Pb 2+ 1.59 0.21 –0.1 –0.3 Cyanide CN – log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Fe 2+ 35.4 (b 6 ) Fe 3+ 43.6 (b 6 ) Ag + 20.48 (b 2 ) 0.92 Zn 2+ 11.07 (b 2 ) 4.98 3.57 Cd 2+ 6.01 5.11 4.53 2.27 Hg 2+ 17.00 15.75 3.56 2.66 Ni 2+ 30.22 (b 4 ) Ethylenediamine H 2 NCH 2 CH 2 NH 2 log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Ni 2+ 7.38 6.18 4.11 Cu 2+ 10.48 9.07 Ag + (T = 20 °C , m = 0.1 M) 4.70 3.00 Zn 2+ 5.66 4.98 3.25 Cd 2+ 5.41 4.50 2.78 EDTA log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Mg 2+ (T = 20 °C, m = 0.1 M) 8.79 Ca 2+ (T = 20 °C, m = 0.1 M) 10.69 Ba 2+ (T = 20 °C, m = 0.1 M) 7.86 Bi 3+ (T = 20 °C, m = 0.1 M) 27.8 Co 2+ (T = 20 °C, m = 0.1 M) 16.31 Ni 2+ (T = 20 °C, m = 0.1 M) 18.62 Cu 2+ (T = 20 °C, m = 0.1 M) 18.80 740 Modern Analytical Chemistry NCH 2 CH 2 N CH 2 COO – CH 2 COO – – OOCH 2 C – OOCH 2 C 1400-AP3 9/8/99 4:40 PM Page 740 EDTA log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Cr 3+ (T = 20 °C, m = 0.1 M) (23.4) Fe 3+ (T = 20 °C, m = 0.1 M) 25.1 Ag + (T = 20 °C, m = 0.1 M) 7.32 Zn 2+ (T = 20 °C, m = 0.1 M) 16.50 Cd 2+ (T = 20 °C, m = 0.1 M) 16.46 Hg 2+ (T = 20 °C, m = 0.1 M) 21.7 Pb 2+ (T = 20 °C, m = 0.1 M) 18.04 Al 3+ (T = 20 °C, m = 0.1 M) 16.3 Fluoride F – log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Al 3+ (m = 0.5 M) 6.11 5.01 3.88 3.0 1.4 0.4 Hydroxide OH – log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Al 3+ 9.01 (9.69) (8.3) 6.0 Co 2+ 4.3 4.1 1.3 0.5 Fe 2+ 4.5 (2.9) 2.6 –0.4 Fe 3+ 11.81 10.5 12.1 Ni 2+ 4.1 3.9 3 Pb 2+ 6.3 4.6 3.0 Zn 2+ 5.0 (6.1) 2.5 (1.2) Iodide I – log K 1 log K 2 log K 3 log K 4 log K 5 log K 6 Ag + (T = 18 °C) 6.58 (5.12) (1.4) Cd 2+ 2.28 1.64 1.08 1.0 Pb 2+ 1.92 1.28 0.7 0.6 Nitriloacetate log K 1 log K 2 log K 3 logK 4 log K 5 log K 6 Mg 2+ (T = 20 °C, m = 0.1 M) 5.41 Ca 2+ (T = 20 °C, m = 0.1 M) 6.41 Ba 2+ (T = 20 °C, m = 0.1 M) 4.82 Mn 2+ (T = 20 °C, m = 0.1 M) 7.44 Fe 2+ (T = 20 °C, m = 0.1 M) 8.33 Co 2+ (T = 20 °C, m = 0.1 M) 10.38 Ni 2+ (T = 20 °C, m = 0.1 M) 11.53 Cu 2+ (T = 20 °C, m = 0.1 M) 12.96 Fe 3+ (T = 20 °C, m = 0.1 M) 15.9 Zn 2+ (T = 20 °C, m = 0.1 M) 10.67 Cd 2+ (T = 20 °C, m = 0.1 M) 9.83 Pb 2+ (T = 20 °C, m = 0.1 M) 11.39 Appendix 3 741 NCH 2 CH 2 N CH 2 COO – CH 2 COO – – OOCH 2 C – OOCH 2 C NCH 2 COO – CH 2 COO – CH 2 COO – continued 1400-AP3 9/8/99 4:40 PM Page 741 Metal–Ligand Formation Constants—continued Oxalate C 2 O 4 2– log K 1 log K 2 log K 3 logK 4 log K 5 log K 6 Ca 2+ (m = 1 M) 1.66 1.03 Fe 2+ (m = 1 M) 3.05 2.10 Co 2+ 4.72 2.28 Ni 2+ 5.16 Cu 2+ 6.23 4.04 Fe 3+ (m = 0.5 M) 7.53 6.11 4.85 Zn 2+ 4.87 2.78 1,10-Phenanthroline log K 1 log K 2 log K 3 logK 4 log K 5 log K 6 Fe 2+ 20.7 (b 3 ) Mn 2+ (m = 0.1 M) 4.0 3.3 3.0 Co 2+ (m = 0.1 M) 7.08 6.64 6.08 Ni 2+ (m = 0.1 M) 8.6 8.1 7.6 Fe 3+ 13.8 (b 3 ) Ag + (m = 0.1 M) 5.02 7.04 Zn 2+ 6.2 (5.9) (5.2) Thiosulfate S 2 O 3 2– log K 1 log K 2 log K 3 logK 4 log K 5 log K 6 Ag + (T = 20 °C) 8.82 4.85 0.53 Thiocyanate SCN – log K 1 log K 2 log K 3 logK 4 log K 5 log K 6 Mn 2+ 1.23 Fe 2+ 1.31 Co 2+ 1.72 Ni 2+ 1.76 Cu 2+ 2.33 Fe 3+ 3.02 Ag + 4.8 3.43 1.27 0.2 Zn 2+ 1.33 0.58 0.09 –0.4 Cd 2+ 1.89 0.89 0.02 –0.5 Hg 2+ 17.26 (b 2 ) 2.71 1.83 Source: All values are from Martell, A. E.; Smith, R. M. Critical Stability Constants, Vol. 1–4, Plenum Press: New York, 1976. Unless otherwise stated, values are for 25 °C and zero ionic strength. Values in parentheses are considered less reliable. 742 Modern Analytical Chemistry NNH + 1400-AP3 9/8/99 4:40 PM Page 742 Appendix 3 D Standard Reduction Potentials a E° E°￠ Aluminum (V) (V) Al 3+ +3e – t Al(s) –1.676 Al(OH) 4 – +3e – t Al(s) + 4OH – –2.310 AlF 6 3– +3e – t Al(s)+6F – –2.07 E° E°￠ Antimony (V) (V) Sb(s)+3H + +3e – t SbH 3 (g) –0.510 Sb 2 O 5 (s)+6H + +4e – t 2SbO + +3H 2 O 0.605 SbO + +2H + +3e – t Sb(s)+H 2 O 0.212 E° E°￠ Arsenic (V) (V) As(s)+3H + +3e – t AsH 3 (g) –0.225 H 3 AsO 4 +2H + +2e – t HAsO 2 +2H 2 O 0.560 HAsO 2 +3H + +3e – t As(s)+2H 2 O 0.240 E° E°￠ Barium (V) (V) Ba 2+ +2e – t Ba(s) –2.92 BaO(s)+2H + +2e – t Ba(s)+H 2 O 2.365 E° E°￠ Beryllium (V) (V) Be 2+ +2e – t Be(s) –1.99 E° E°￠ Bismuth (V) (V) Bi 3+ +3e – t Bi(s) 0.317 BiCl 4 – +3e – t Bi(s) + 4Cl – 0.199 E° E°￠ Boron (V) (V) B(OH) 3 +3H + +3e – t B(s)+3H 2 O –0.890 B(OH) 4 – +3e – t B(s) + 4OH – –1.811 Appendix 3 743 E° E°￠ Bromine (V) (V) Br 2 +2e – t 2Br – 1.087 HOBr + H + +2e – t Br – +H 2 O 1.341 HOBr + H + + e – t 1 ?2Br 2 (l)+H 2 O 1.604 BrO – +H 2 O+2e – t Br – + 2OH – 0.76 1 M NaOH BrO 3 – +6H + +5e – t 1 ?2Br 2 (l)+3H 2 O 1.5 BrO 3 – +6H + +6e – t Br – +3H 2 O 1.478 E° E°￠ Cadmium (V) (V) Cd 2+ +2e – t Cd(s) –0.4030 Cd(CN) 4 2– +2e – t Cd(s) + 4CN – –0.943 Cd(NH 3 ) 4 2+ +2e – t Cd(s) + 4NH 3 –0.622 E° E°￠ Calcium (V) (V) Ca 2+ +2e – t Ca(s) –2.84 E° E°￠ Carbon (V) (V) CO 2 (g)+2H + +2e – t CO(g)+H 2 O –0.106 CO 2 (g)+2H + +2e – t HCO 2 H –0.20 2CO 2 (g)+2H + +2e – t H 2 C 2 O 4 –0.481 HCHO + 2H + +2e – t CH 3 OH 0.2323 E° E°￠ Cerium (V) (V) Ce 3+ +3e – t Ce(s) –2.336 Ce 4+ + e – t Ce 3+ 1.72 1.70 1 M HClO 4 1.44 1 M H 2 SO 4 1.61 1 M HNO 3 1.28 1 M HCl E° E°￠ Chlorine (V) (V) Cl 2 (g)+2e – t 2Cl – 1.396 ClO – +H 2 O+e – t 1 ?2Cl 2 (g) + 2OH – 0.421 1 M NaOH ClO – +H 2 O+2e – t Cl – + 2OH – 0.890 1 M NaOH HClO 2 +2H + +2e – t HOCl + H 2 O 1.64 ClO 3 – +2H + + e – t ClO 2 (g)+H 2 O 1.175 ClO 3 – +3H + +2e – t HClO 2 +H 2 O 1.181 ClO 4 – +2H + +2e – t ClO 3 – +H 2 O 1.201 continued 1400-AP3 9/8/99 4:40 PM Page 743 E° E°￠ Chromium (V) (V) Cr 3+ + e – t Cr 2+ –0.424 Cr 2+ +2e – t Cr(s) –0.90 Cr 2 O 7 2– + 14H + +6e – t 2Cr 3+ +7H 2 O 1.36 CrO 4 2– +4H 2 O+3e – t Cr(OH) 4 – + 4OH – –0.13 1 M NaOH E° E°￠ Cobalt (V) (V) Co 2+ +2e – t Co(s) –0.277 Co 3+ + e – t Co 2+ 1.92 Co(NH 3 ) 6 3+ + e – t Co(NH 3 ) 6 2+ 0.1 Co(OH) 3 (s)+e – t Co(OH) 2 (s)+OH – 0.17 Co(OH) 2 (s)+2e – t Co(s) + 2OH – –0.746 E° E°￠ Copper (V) (V) Cu + + e – t Cu(s) 0.520 Cu 2+ + e – t Cu + 0.159 Cu 2+ +2e – t Cu(s) 0.3419 Cu 2+ +I – + e – t CuI(s) 0.86 Cu 2+ +Cl – + e – t CuCl(s) 0.559 E° E°￠ Fluorine (V) (V) F 2 (g)+2H + +2e – t 2HF 3.053 F 2 (g)+2e – t 2F – 2.87 E° E°￠ Gallium (V) (V) Ga 3+ +3e – t Ga(s) –0.529 E° E°￠ Gold (V) (V) Au + + e – t Au(s) 1.83 Au 3+ +2e – t Au + 1.36 Au 3+ +3e – t Au(s) 1.52 AuCl 4 – +3e – t Au(s) + 4Cl – 1.002 E° E°￠ Hydrogen (V) (V) 2H + +2e – t H 2 (g) 0.00000 H 2 O+e – t 1 ?2H 2 (g)+OH – –0.828 E° E°￠ Iodine (V) (V) I 2 (s)+2e – t 2I – 0.5355 I 3 – +2e – t 3I – 0.536 HIO+H + +2e – t I – +H 2 O 0.985 IO 3 – +6H + +5e – t 1 ?2I 2 (s)+3H 2 O 1.195 IO 3 – +3H 2 O+6e – t I – + 6OH – 0.257 E° E°￠ Iron (V) (V) Fe 2+ +2e – t Fe(s) –0.44 Fe 3+ +3e – t Fe(s) –0.037 Fe 3+ + e – t Fe 2+ 0.771 0.70 1 M HCl 0.767 1 M HClO 4 0.746 1 M HNO 3 0.68 1 M H 2 SO 4 0.44 0.3 M H 3 PO 4 Fe(CN) 6 3– + e – t Fe(CN) 6 4– 0.356 0.71 1 M HCl Fe(phen) 3 3+ + e – t Fe(phen) 3 2+ 1.147 Fe(CN) 6 3– + e – t Fe(CN) 6 4– 0.356 E° E°￠ Ianthanum (V) (V) La 3+ +3e – t La(s) –2.38 E° E°￠ Lead (V) (V) Pb 2+ +2e – t Pb(s) –0.126 PbO 2 (s)+4H + + 2e – t Pb 2+ +2H 2 O 1.46 PbO 2 (s)+SO 4 2– +4H + + 2e – t PbSO 4 (s)+2H 2 O 1.690 PbSO 4 (s)+ 2e – t Pb(s)+SO 4 2– –0.356 E° E°￠ Lithium (V) (V) Li + + e – t Li(s) –3.040 E° E°￠ Magnesium (V) (V) Mg 2+ +2e – t Mg(s) –2.356 Mg(OH) 2 (s)+2e – t Mg(s) + 2OH – –2.687 E° E°￠ Manganese (V) (V) Mn 2+ +2e – t Mn(s) –1.17 M 3+ + e – t Mn 2+ 1.5 MnO 2 (s)+4H + +2e – t Mn 2+ +2H 2 O 1.23 MnO 4 – +4H + +3e – t MnO 2 (s)+2H 2 O 1.70 MnO 4 – +8H + +5e – t Mn 2+ +4H 2 O 1.51 MnO 4 – +2H 2 O+3e – t MnO 2 (s) + 4OH – 0.60 E° E°￠ Mercury (V) (V) Hg 2+ +2e – t Hg(l) 0.8535 2Hg 2+ +2e – t Hg 2 2+ 0.911 Hg 2 2+ +2e – t 2Hg(l) 0.7960 Hg 2 Cl 2 (s)+2e – t 2Hg(l) + 2Cl – 0.2682 HgO(s)+2H + +2e – t Hg(l)+H 2 O 0.926 Hg 2 Br 2 (s)+2e – t 2Hg(l) + 2Br – 0.1392 Hg 2 I 2 (s)+2e – t 2Hg(l)+2I – –0.0405 744 Modern Analytical Chemistry Standard Reduction Potentials a —continued 1400-AP3 9/8/99 4:40 PM Page 744 E° E°￠ Molybdenum (V) (V) Mo 3+ +3e – t Mo(s) –0.2 MoO 2 (s)+4H + +4e – t Mo(s)+2H 2 O –0.152 MoO 4 2– + 4H 2 O + 6e – t Mo(s) + 8OH – –0.913 E° E°￠ Nickel (V) (V) Ni 2+ +2e – t Ni(s) –0.257 Ni(OH) 2 (s) + 2e – t Ni(s) + 2OH – –0.72 Ni(NH 3 ) 6 2+ +2e – t Ni(s) + 6NH 3 –0.49 E° E°￠ Nitrogen (V) (V) N 2 (g) + 5H + + 4e – t N 2 H 5 + –0.23 N 2 O(g) + 2H + + 2e – t N 2 (g) + H 2 O1.77 2NO(g) + 2H + + 2e – t N 2 O(g)+H 2 O 1.59 HNO 2 + H + + e – t NO(g)+H 2 O 0.996 2HNO 2 + 4H + + 4e – t N 2 O(g)+3H 2 O 1.297 NO 3 – + 3H + + 2e – t HNO 2 +H 2 O 0.94 E° E°￠ Oxygen (V) (V) O 2 (g)+2H + + 2e – t H 2 O 2 0.695 O 2 (g)+4H + + 4e – t 2H 2 O 1.229 H 2 O 2 +2H + +2e – t 2H 2 O 1.763 O 2 (g)+2H 2 O + 4e – t 4OH – 0.401 O 3 (g)+2H + + 2e – t O 2 (g) + H 2 O 2.07 E° E°￠ Phosphorus (V) (V) P(s, white) +3H + +3e – t PH 3 (g) –0.063 H 3 PO 3 +2H + +2e – t H 3 PO 2 +H 2 O –0.499 H 3 PO 4 +2H + +2e – t H 3 PO 3 +H 2 O –0.276 E° E°￠ Platinum (V) (V) Pt 2+ +2e – t Pt(s) 1.188 PtCl 4 2– +2e – t Pt(s) + 4Cl – 0.758 E° E°￠ Potassium (V) (V) K + + e – t K(s) –2.924 E° E°￠ Ruthenium (V) (V) Ru 3+ + e – t Ru 2+ 0.249 RuO 2 (s)+4H + +4e – t Ru(s)+2H 2 O 0.68 Ru(NH 3 ) 6 3+ + e – t Ru(NH 3 ) 6 2+ 0.10 Ru(CN) 6 3– + e – t Ru(CN) 6 4– 0.86 E° E°￠ Selenium (V) (V) Se(s)+2e – t Se 2– –0.670 1 M NaOH Se(s)+2H + +2e – t H 2 Se(g) –0.115 H 2 SeO 3 +4H + +4e – t Se(s)+3H 2 O 0.74 SeO 4 3– +4H + + e – t H 2 SeO 3 +H 2 O 1.151 E° E°￠ Silicon (V) (V) SiF 6 2– +4e – t Si(s)+6F – –1.37 SiO 2 (s)+4H + +4e – t Si(s)+2H 2 O –0.909 SiO 2 (s)+8H + +8e – t SiH 4 (g)+2H 2 O –0.516 E° E°￠ Silver (V) (V) Ag + + e – t Ag(s) 0.7996 AgBr(s)+e – t Ag(s)+Br – 0.071 Ag 2 C 2 O 4 (s)+2e – t 2Ag(s)+C 2 O 4 2– 0.47 AgCl(s)+e – t Ag(s)+Cl – 0.2223 AgI(s)+e – t Ag(s)+I – –0.152 Ag 2 S(s)+2e – t 2Ag(s)+S 2– –0.71 Ag(NH 3 ) 2 + + e – t Ag(s) + 2NH 3 0.373 E° E°￠ Sodium (V) (V) Na + + e – t Na(s) –2.713 E° E°￠ Strontium (V) (V) Sr 2+ +2e – t Sr(s) –2.89 E° E°￠ Sulfur (V) (V) S(s)+2e – t S 2– –0.407 S(s)+2H + +2e – t H 2 S 0.144 S 2 O 6 2– +4H + +2e – t 2H 2 SO 3 0.569 S 2 O 8 2– +2e – t 2SO 4 2– 1.96 S 4 O 6 2– +2e – t 2S 2 O 3 2– 0.080 2SO 3 2– +2H 2 O+2e – t S 2 O 4 2– + 4OH – –1.13 2SO 3 2– +3H 2 O+4e – t S 2 O 3 2– + 6OH – –0.576 1 M NaOH 2SO 4 2– +4H + +2e – t S 2 O 6 2– +2H 2 O –0.25 SO 4 2– +H 2 O+2e – t SO 3 2– + 2OH – –0.936 SO 4 2– +4H + +2e – t H 2 SO 3 +H 2 O +0.172 E° E°￠ Thallium (V) (V) Tl 3+ +2e – t Tl + 1.25 1 M HClO 4 0.77 1 M HCL Tl 3 +3e – t Tl(s) 0.742 Appendix 3 745 continued 1400-AP3 9/8/99 4:40 PM Page 745 E° E°￠ Tin (V) (V) Sn 2+ +2e – t Sn(s) –0.19 1 M HCl Sn 4+ +2e – t Sn 2+ 0.154 0.139 1 M HCl E° E°￠ Titanium (V) (V) Ti 2+ +2e – t Ti(s) –1.63 Ti 3+ + e – t Ti 2+ –0.37 E° E°￠ Tungsten (V) (V) WO 2 (s)+4H + +4e – t W(s)+2H 2 O –0.119 WO 3 (s)+6H + +6e – t W(s)+3H 2 O –0.090 E° E°￠ Uranium (V) (V) U 3+ +3e – t U(s) –1.66 U 4+ +e – t U 3+ –0.52 UO 2 + +4H + + e – t U 4+ +2H 2 O 0.27 UO 2 2+ + e – t UO 2 + 0.16 UO 2 2+ +4H + +2e – t U 4+ +2H 2 O 0.327 E° E°￠ Vanadium (V) (V) V 2+ +2e – t V(s) –1.13 V 3+ + e – t V 2+ –0.255 VO 2+ +2H + + e – t V 3+ +H 2 O 0.337 VO 2 + +2H + + e – t VO 2+ +H 2 O 1.000 E° E°￠ Zinc (V) (V) Zn 2+ +2e – t Zn(s) –0.7618 Zn(OH) 4 2– +2e – t Zn(s) + 4OH – –1.285 Zn(NH 3 ) 4 2+ +2e – t Zn(s) + 4NH 3 –1.04 Zn(CN) 4 2– +2e – t Zn(s) + 4CN – –1.34 746 Modern Analytical Chemistry Source: Values are compiled from the following sources: Bard, A. J.; Parsons, R.; Jordon, J., eds. Standard Potentials in Aqueous Solutions. Dekker: New York, 1985; Milazzo, G.; Caroli, S.; Sharma, V. K. Tables of Standard Electrode Potentials. Wiley: London, 1978; Swift, E. H.; Butler, E. A. Quantitative Measurements and Chemical Equilibria. Freeman: New York, 1972. a Solids, gases, and liquids are identified; all other species are aqueous. Reduction reactions in acidic solution are written using H + instead of H 3 O + . Reactions may be rewritten by replacing H + with H 3 O + and adding one molecule of H 2 O to the opposite side of the reaction for each H + ; thus H 3 AsO 4 +2H + +2e – t HAsO 2 +2H 2 O becomes H 3 AsO 4 +2H 3 O + +2e – t HAsO 2 +4H 2 O Conditions for formal potentials (E°￠ ) are listed next to the potential. Standard Reduction Potentials a —continued 1400-AP3 9/8/99 4:40 PM Page 746 Appendix 3 E Selected Polarographic Half-Wave Potentials a E 1/2 Element (volts vs. SCE) Matrix Al 3+ +3e – t Al(s) –0.5 0.2 M acetate (pH 4.5–4.7) Cd 2+ +2e – t Cd(s) –0.60 0.1 M KCl 0.05 M H 2 SO 4 1 M HNO 3 –0.81 1 M NH 4 Cl plus 1 M NH 3 Cr +3 +3e – t Cr(s) –0.35 (+3 fi +2) 1 M NH 4 + /NH 3 buffer (pH 8–9) –1.70 (+2 fi 0) Co 3+ +3e – t Co(s) –0.5 (+3 fi +2) 1 M NH 4 Cl plus 1 M NH 3 –1.3 (+2 fi 0) Co 2+ +2e – t Co(s) –1.03 1 M KSCN Cu 2+ +2e – t Cu(s) 0.04 0.1 M KNO 3 0.1 M NH 4 ClO 4 1 M Na 2 SO 4 –0.22 0.5 M potassium citrate (pH 7.5) Fe 3+ +3e – t Fe(s) –0.17 (+3 fi +2) 0.5 M sodium tartrate (pH 5.8) –1.52 (+2 fi 0) Fe 3+ + e – t Fe 2+ –0.27 0.2 M Na 2 C 2 O 4 (pH < 7.9) Pb 2+ +2e – t Pb(s) –0.405 1 M HNO 3 –0.435 1 M KCl Mn 2+ +2e – t Mn(s) –1.65 1 M NH 4 Cl plus 1 M NH 3 Ni 2+ +2e – t Ni(s) –0.70 1 M KSCN –1.09 1 M NH 4 Cl plus 1 M NH 3 Zn 2+ +2e – t Zn(s) –0.995 0.1 M KCl –1.33 1 M NH 4 Cl plus 1 M NH 3 Source: All values from Dean, J. A. Analytical Chemistry Handbook. McGraw-Hill: New York, 1995. a In some cases two polarographic waves are seen as the redox reaction occurs in steps. Half-wave potentials are given for each wave, with the corresponding change in oxidation states shown in parentheses. Appendix 3 747 1400-AP3 9/8/99 4:40 PM Page 747 748 AppendixAppendix 4 Balancing Redox Reactions Balancing a redox reaction is often more challenging than balancing other types of reactions, because we must balance both electrons and elements. Perhaps the sim- plest way to balance a redox reaction is by the half-reaction method, which consists of the following steps. 1. Identify the oxidizing and reducing agents and the appropriate conjugate reducing and oxidizing agents. 2. Balance all atoms in each half-reaction except for O and H. 3. Balance the oxygen in each half-reaction by adding appropriate amounts of H 2 O. 4. For acidic solutions, balance the hydrogen in each half-reaction by adding H 3 O + and H 2 O to opposite sides of the reaction; for basic solutions, add OH – and H 2 O to opposite sides of the reaction. 5. Balance the net charge in each half-reaction by adding electrons; the electrons should be a reactant for the reduction half-reaction and a product for the oxidation half-reaction. 6. Adjust the coefficients of each half-reaction so that both half-reactions involve the same number of electrons. 7. Combine the two half-reactions, and simplify the stoichiometry. The application of the half-reaction method is shown in Example A4.1. EXAMPLE A 4 .1 Balance the following redox reactions. (a) Cr 2 O 7 2– (aq) + Zn(s) t Cr 3+ (aq)+Zn 2+ (aq) (acidic solution) (b) MnO 4 – (aq)+S 2– (aq) t MnO 2 (s)+S(s) (basic solution) SOLUTION We begin by writing unbalanced equations for the oxidation and reduction half-reactions in part (a). Cr 2 O 7 2– (aq) t Cr 3+ (aq) Zn(s) t Zn 2+ (aq) The atoms in each half-reaction are then balanced. The zinc half-reaction is already balanced in this respect, so we begin by balancing chromium in the dichromate half-reaction. Cr 2 O 7 2– (aq) t 2Cr 3+ (aq) 1400-AP4 9/8/99 4:41 PM Page 748 Oxygen is then balanced by adding H 2 O. Cr 2 O 7 2– (aq) t 2Cr 3+ (aq)+7H 2 O(l) and hydrogen is balanced by adding H 3 O + and sufficient additional water. 14H 3 O + (aq)+Cr 2 O 7 2– (aq) t 2Cr 3+ (aq) + 21H 2 O(l) Both half-reactions are balanced with respect to charge by adding electrons as needed. 14H 3 O + (aq)+Cr 2 O 7 2– (aq)+6e – t 2Cr 3+ (aq) + 21H 2 O(l) Zn(s) t Zn 2+ (aq)+2e – The stoichiometric coefficients for the zinc half-reaction are multiplied by 3 so that both half-reactions involve 6 electrons. 14H 3 O + (aq)+Cr 2 O 7 2– (aq)+6e – t 2Cr 3+ (aq) + 21H 2 O(l) 3Zn(s) t 3Zn 2+ (aq)+6e – Adding together the two half-reactions and simplifying gives us the final balanced redox reaction. 14H 3 O + (aq)+Cr 2 O 7 2– (aq) + 3Zn(s) t 2Cr 3+ (aq) + 21H 2 O(l) + 3Zn 2+ (aq) Balancing the reaction in part (b) is accomplished in the same manner, except that H 2 O and OH – are used to balance hydrogen. Balancing the reaction is left as an exercise; the resulting balanced reaction is 2MnO 4 – (aq)+4H 2 O(l)+3S 2– (aq) t 2MnO 2 (s) + 8OH – (aq) + 3S(s) Appendix 4 749 1400-AP4 9/8/99 4:41 PM Page 749 750 AppendixAppendix 5 Review of Chemical Kinetics Areaction’s equilibrium position defines the extent to which the reaction can occur. For example, we expect a reaction with a large equilibrium constant, such as the dissociation of HCl in water HCl(aq)+H 2 O(l) t H 3 O + (aq)+Cl – (aq) to proceed nearly to completion. The magnitude of an equilibrium constant, however, does not guarantee that a reaction will reach its equilibrium position. Many reactions with large equilibrium constants, such as the reduction of MnO 4 – by H 2 O 4MnO 4 – (aq)+2H 2 O(l) t 4MnO 2 (s)+3O 2 (g) + 4OH – (aq) do not occur to an appreciable extent. The study of the rate at which a chemical re- action approaches its equilibrium position is called kinetics. A 5 .1 Chemical Reaction Rates A study of the kinetics of a chemical reaction begins with the measurement of its re- action rate. Consider, for example, the general reaction shown in the following equation, involving the aqueous solutes A, B, C, and D, with stoichiometries of a, b, c, and d. aA+bB t cC+dD A5.1 The rate, or velocity, at which this reaction approaches its equilibrium position can be determined by following the change in concentration of a reactant or a product as a function of time. For example, if we monitor the concentration of reactant A, we express the rate as A5.2 where R is the measured rate expressed as a change in concentration as a function of time. Because a reactant’s concentration decreases with time, we include a negative sign so that the rate has a positive value. The rate also can be determined by following the change in concentration of a product as a function of time. A5.3 ′R d dt = [C] R d dt =– [A] 1400-AP5 9/8/99 4:41 PM Page 750 Appendix 5 751 Rates determined by monitoring different species in a chemical reaction need not have the same value. The rate R in equation A5.2 and the rate R￠ in equation A5.3 will have the same value only if the stoichiometric coefficients of A and C in reac- tion A5.1 are the same. In general, the relationship between the rates R and R￠ is A 5 .2 The Rate Law A rate law describes how the rate of a reaction is affected by the concentration of each species present in the reaction mixture. The rate law for reaction A5.1 takes the general form of R = k[A] a [B] b [C] g [D] d [E] e ... A5.4 where k is the rate constant, and a , b , g , d and e are the orders of the reaction with respect to the species present in the reaction. Several important points about the rate law are shown in equation A5.4. First, the rate of a reaction may depend on the concentrations of both reactants and prod- ucts, as well as the concentrations of species that do not appear in the reaction’s overall stoichiometry. Species E in equation A5.4, for example, may represent a cat- alyst. Second, the reaction order for a given species is not necessarily the same as its stoichiometry in the chemical reaction. Reaction orders may be positive, negative, or zero and may take integer or noninteger values. Finally, the overall reaction order is the sum of the individual reaction orders. Thus, the overall reaction order for equation A5.4 is a + b + g + d + e . A 5 . 3 Kinetic Analysis of Selected Reactions In this section we review the application of kinetics to several simple chemical reac- tions, focusing on how the integrated form of the rate law can be used to determine reaction orders. In addition, we consider how rate laws for more complex systems can be determined. First-Order Reactions The simplest case is a first-order reaction in which the rate depends on the concentration of only one species. The best example of a first-order reaction is an irreversible thermal decomposition, which we can represent as A fi Products A5.5 with a rate law of A5.6 The simplest way to demonstrate that a reaction is first-order in A, is to double the concentration of A and note the effect on the reaction’s rate. If the observed rate doubles, then the reaction must be first-order in A. Alternatively, we can derive a relationship between the [A] and time by rearranging equation A5.6 d kdt [] [ – A A] = R d dt k==– [] [] A A R a c R=×′ 1400-AP5 9/8/99 4:41 PM Page 751 and integrating. A5.7 Evaluating the integrals in equation A5.7, A5.8 and rearranging, ln [A] t =–kt +ln[A] 0 A5.9 shows that for a first-order reaction, a plot of ln [A] t versus time is linear with a slope of –k and an intercept of ln [A] 0 . Equations A5.8 and A5.9 are known as inte- grated forms of the rate law. Reaction A5.5 is not the only possible form of a first-order reaction. For exam- ple, the reaction A+B fi Products A5.10 shows first-order kinetics if the concentration of B does not affect the reaction rate. This may happen if the reaction’s mechanism involves at least two steps. Imagine that in the first step, A is slowly converted to an intermediate species C. The inter- mediate and the remaining reactant, B, then react rapidly, in one or more steps, to form the products. A fi C (slow) B+C fi Products (fast) Since the rate of a chemical reaction only depends on the slowest, or rate-determining step, and any preceding steps, species B will not show up in the rate law. Second-Order Reactions The simplest overall reaction demonstrating second- order behavior is 2A fi Products for which the rate law is Proceeding in the same manner as for a first-order reaction, the integrated form of the rate law is derived as follows d kdt d kdt kt t t t [] [ – [] [ – [[ [] [] A A] A A] A] A] 2 2 A A 0 = = =+ ∫∫ 0 0 11 R d dt k==– [] [] A A 2 ln [] [ – A A] 0 t kt= d kdt tt [] [ – [] [] A A] A A = ∫∫ 0 0 752 Modern Analytical Chemistry 1400-AP5 9/8/99 4:41 PM Page 752 Appendix 5 753 ln(% p -Metho xyphen ylacetylene) 0 3 5 Time (min) First-order plot 700100 200 300 400 500 600 4 (% p -Metho xyphen ylacetylene) –1 0 0.01 0.05 Time (min) Second-order plot 700100 200 300 400 500 600 0.04 0.03 0.02 Figure A5.1 Graphs used to determine the reaction order for the data in Example A5.1. (a) (b) Thus, for a second-order reaction, a plot of [A] t –1 versus t is linear, with a slope of k and an intercept of [A] 0 –1 . Alternatively, a reaction can be shown to be second-order in A by observing the effect on the rate of changing the concentration of A. In this case, doubling the concentration of A produces a fourfold increase in the reaction’s rate. EXAMPLE A 5 .1 The following data were obtained during a kinetic study of the hydration of p-methoxyphenylacetylene by measuring the relative amounts of reactants and products by nuclear magnetic resonance (NMR). 1 Time p-Methoxyphenylacetylene (min) (%) 67 85.9 161 70.0 241 57.6 381 40.7 479 32.4 545 27.7 604 24 Using appropriate graphs, determine whether this reaction is first- or second- order in p-methoxyphenylacetylene. SOLUTION To determine the reaction order we plot ln(%p-methoxyphenylacetylene) versus time for a first-order reaction, and (%p-methoxyphenylacetylene) –1 versus time for a second-order reaction (Figure A5.1). Because the straight-line for the first- order plot fits the data nicely, we conclude that the reaction is first-order in p-methoxyphenylacetylene. Note that when plotted using the equation for a second-order reaction, the data show curvature that does not fit the straight- line model. Pseudo-Order Reactions and the Method of Initial Rates Unfortunately, most re- actions of importance in analytical chemistry do not follow these simple first-order and second-order rate laws. We are more likely to encounter the second-order rate law given in equation A5.11 than that in equation A5.10. R = k[A][B] A5.11 Demonstrating that a reaction obeys the rate law in equation A5.11 is compli- cated by the lack of a simple integrated form of the rate law. The kinetics can be simplified, however, by carrying out the analysis under conditions in which the concentrations of all species but one are so large that their concentrations are effec- tively constant during the reaction. For example, if the concentration of B is se- lected such that [B] >> [A], then equation A5.11 simplifies to R = k￠ [A] where the rate constant k￠ is equal to k[B]. Under these conditions, the reaction appears to follow first-order kinetics in A and is termed a pseudo-first-order reaction in A. Veri- fying the reaction order for A can then be carried out by either using the integrated rate 1400-AP5 9/8/99 4:41 PM Page 753 754 Modern Analytical Chemistry law or by noting the effect on the reaction rate of changing the concentration of A. The process can be repeated to determine the reaction order for B by making [A] >> [B]. A variation on the use of pseudo-ordered reactions is the initial rate method. In this approach to determining a reaction’s rate law, a series of experiments is con- ducted in which the concentration of those species expected to affect the reaction’s rate are changed one at a time. The initial rate of the reaction is determined for each set of conditions. Comparing the reaction’s initial rate for two experiments in which the concentration of only a single species has been changed allows the reac- tion order for that species to be determined. The application of this method is out- lined in the following example. EXAMPLE A 5 .2 The following data were collected during a kinetic study of the iodination of acetone by measuring the concentration of unreacted I 2 in solution. 2 Experiment [C 3 H 6 O] [H + ][I 2 ] Rate Number (M) (M) (M) (M s –1 ) 1 1.33 0.0404 6.65 · 10 –3 1.78 · 10 –6 2 1.33 0.0809 6.65 · 10 –3 3.89 · 10 –6 3 1.33 0.162 6.65 · 10 –3 8.11 · 10 –6 4 1.33 0.323 6.65 · 10 –3 16.6 · 10 –6 5 0.167 0.323 6.65 · 10 –3 1.64 · 10 –6 6 0.333 0.323 6.65 · 10 –3 3.76 · 10 –6 7 0.667 0.323 6.65 · 10 –3 7.55 · 10 –6 8 0.333 0.323 3.32 · 10 –3 3.57 · 10 –6 Determine the rate law for this reaction and the value of the rate constant. SOLUTION The order of the rate law with respect to the three reactants can be determined by comparing the rates of two experiments in which the concentration of only one of the reactants is changed. For example, in experiment 2 the [H + ] and the rate are approximately twice as large as in experiment 1, indicating that the reaction is first-order in [H + ]. Working in the same manner, experiments 6 and 7 show that the reaction is also first-order with respect to [C 3 H 6 O], and experiments 6 and 8 show that the rate of the reaction is independent of the [I 2 ]. Thus, the rate law is R = k[C 3 H 6 O][H + ] The value of the rate constant can be determined by substituting the rate, the [C 3 H 6 O], and the [H + ] for an experiment into the rate law and solving for k. Using the data from experiment 1, for example, gives a rate constant of 3.31 · 10 –5 M –1 s –1 . The average rate constant for the eight experiments is 3.49 · 10 –5 M –1 s –1 . 1. Kaufman, D.; Sterner, C.; Masek, B.; et al. J. Chem. Educ. 1982, 59, 885–886. 2. Birk, J. P.; Walters, D. L. J. Chem. Educ. 1992, 69, 585–587. REFERENCES 1400-AP5 9/8/99 4:41 PM Page 754 A solution to the problem of separating solutes with similar distribution ra- tios was introduced by Craig in the 1940s. 1 The technique, which is known as a countercurrent liquid–liquid extraction, is outlined in Figure A6.1 and discussed in detail in this appendix. In contrast to a simple liquid–liquid extraction, in which the phase containing the sample is extracted sequentially, a countercurrent extrac- tion is based on a serial extraction of both the sample and extracting phases. Coun- tercurrent separations are no longer in common use, having been replaced by chromatographic separations. Nevertheless, the theory of countercurrent extrac- tions is instructive because it provides a useful introduction to the theory of chro- matographic separations. To keep track of the progress of a countercurrent liquid–liquid extraction, it is necessary to adopt a labeling convention. Each step in a countercurrent extrac- tion includes the transferring of the upper phase followed by an extraction. Steps are labeled sequentially beginning with zero. Extractions take place in a series of tubes that also are labeled sequentially, starting with zero. The upper and lower phases in each tube are identified by a letter and number. The letters U and L rep- resent, respectively, the upper phase and the lower phase, and the number indi- cates the step in the countercurrent extraction in which the phase was first intro- duced. Thus U 0 is the upper phase introduced at step 0 (the first extraction), and L 2 is the lower phase introduced at step 2 (the third extraction). Finally, partition- ing of the solute in any extraction tube results in a fraction p remaining in the upper phase and a fraction q remaining in the lower phase. Values of q are calcu- lated using equation A6.1 A6.1 and p is equal to 1 – q. Let’s assume that the solute to be separated is present in an aqueous phase of 1 M HCl and that the organic phase is benzene. Because benzene has the smaller density, it is the upper phase, and 1 M HCl is the lower phase. To begin the coun- tercurrent extraction the aqueous sample containing the solute is placed in tube 0 along with a portion of benzene. As shown in Figure A6.1a, initially all the solute is present in phase L 0 . After extracting (Figure A6.1b), a fraction p of the solute is pre- sent in phase U 0 , and a fraction q is in phase L 0 . This completes step 0 of the coun- tercurrent extraction. Thus far there is no difference between a simple liquid–liquid extraction and a countercurrent extraction. At the completion of step 0 the phase U 0 is removed, and a fresh portion of benzene, U 1 , is added to tube 0 (Figure A6.1c). This, too, is just the same as in a ()q V DV V aq 1 0 aq org aq (moles aq) (moles aq) 1 == + 755 AppendixAppendix 6 Countercurrent Separations 1400-AP6 9/8/99 4:42 PM Page 755 756 Modern Analytical Chemistry simple liquid–liquid extraction. The difference between the two extraction techniques, however, occurs at this point when phase U 0 is placed in tube 1 along with a portion of solute-free aqueous 1 M HCl as phase L1 (Figure A6.1c). Tube 0 now con- tains a fraction q of the solute, and tube 1 contains a fraction p of the solute. Carrying out the extraction in tube 0 results in a fraction p of its contents remaining in the upper phase and a fraction q remaining in the lower phase. Thus, phases U 1 and L 0 now contain, respectively, fractions pq and q 2 of the original amount of solute. In the same way it is easy to show that the phases U 0 and L 1 in tube 1 contain, respectively, fractions p 2 and pq of the original amount of solute. This completes step 1 of the extraction (Figure A6.1d). As shown in the remainder of Figure A6.1, the countercurrent extraction continues in this cycle of phase transfers and extractions. In a countercurrent liquid–liquid extraction the lower phase in each tube remains in place, and the upper phase moves from tube 0 to higher numbered tubes. This difference in the movement of the phases is indicated by referring to the lower phase as a stationary phase and the upper phase as a mo- bile phase. With each transfer some of the solute in tube r is moved to tube r + 1, and a portion of the solute in tube r –1 is moved to tube r. As a result, a solute introduced at tube 0 moves with the mobile phase. The solute, however, does not move at the same rate as the mobile phase since, at each step, a portion of the solute is extracted into the stationary phase. A solute that is preferentially extracted into the stationary phase spends proportionally less time in the mobile phase and moves at a slower rate. As the number of steps increases, solutes with different values of q separate into completely different sets of extraction tubes. The effectiveness of a countercurrent extraction can be judged from a histogram indicating the fraction of solute pres- ent in each tube. To determine the total amount of solute in an extraction tube, we add together the fraction of solute present in the tube’s upper and lower phases following each transfer. For example, at the beginning of step 3 (Figure A6.1g) the upper and lower phases of tube 1 contain fractions pq 2 and 2pq 2 of the solute, respectively; thus, the total fraction of solute in the tube is 3pq 2 . Table A6.1 summarizes this for the steps outlined in Figure A6.1. A typical histogram, calculated assum- ing distribution ratios of 5.0 for solute A and 0.5 for solute B, is shown in Figure A6.2. Although four steps is not enough to separate the solutes in this instance, it is easy to see that by ex- tending the countercurrent extraction the solutes will eventu- ally be separated. Figure A6.1 and Table A6.1 show how a solute’s distribution changes during the first four steps of a countercurrent extraction. Now we consider how these re- sults can be generalized to give the distribution of a solute in any tube, at any step during the extraction. You may recognize the pattern of entries in Table A6.1 as fol- lowing the binomial distribution U0 0 L0 1 Extract(a) U0 p L0 q Transfer Step 0 Step 1 Step 2 U1 0 L0 q Extract U1 pq L0 q 2 Transfer U2 0 L0 q 2 Extract U2 pq 2 L0 q 3 Transfer U3 0 L0 q 3 Tube 0 (b) (c) (d) (e) (f) (g) U0 p L1 0 Extract U0 p 2 L1 pq Transfer U1 pq L1 pq Extract U1 2p 2 q L1 2pq 2 Transfer U2 pq 2 L1 2pq 2 Tube 1 U0 p 2 L2 0 Extract U0 p 3 L2 p 2 q Transfer U1 2p 2 q L2 p 2 q Tube 2 U0 p 3 L3 0 Tube 3 Figure A6.1 Scheme for a countercurrent extraction. 1400-AP6 9/8/99 4:42 PM Page 756 Appendix 6 757 Figure A6.2 Progress of a countercurrent extraction after (a) step 0, (b) step 1, (c) step 2, and (d) step 3. The distribution ratio, D, is 5.0 for solute A and 0.5 for solute B. Table A6.1 Fraction of Solute Remaining in Tube r After Extraction Step n for a Countercurrent Extraction n/r 01 2 3 01 1 qp 2 q 2 2pq p 2 3 q 3 3pq 2 3p 2 qp 3 Fraction of solute 012 0 1 Tube number 3 0.2 0.4 0.6 0.8 Solute A Solute B (a) (b) (c) (d) Fraction of solute 012 0 1 Tube number 3 0.2 0.4 0.6 0.8 Fraction of solute 012 0 Tube number 3 0.2 0.4 0.6 0.8 Fraction of solute 012 0 0.6 Tube number 3 0.2 0.1 0.3 0.4 0.5 A6.2 where f(r, n) is the fraction of solute present in tube r at step n of the countercur- rent extraction. After extracting, therefore, the upper phase contains a fraction p · f(r, n), and the lower phase contains a fraction q · f(r, n) of the solute. EXAMPLE A6.1 The countercurrent extraction shown in Figure A6.2 is carried out through step 30. Calculate the fraction of solutes A and B in tubes 5, 10, 15, 20, 25, and 30. SOLUTION The fraction, q, of each solute that remains in the lower phase is calculated using equation A6.1. Since the volumes of the lower and upper phases are equal, we get fr,n n n–r r pq rnr ()= ! ()!! – 1400-AP6 9/8/99 4:42 PM Page 757 758 Modern Analytical Chemistry and Thus, p A is 0.833 and p B is 0.333. For solute A, the fraction present in tubes 5, 10, 15, 20, 25, and 30 after step 30 are The fraction of solute B in tubes 5, 10, 15, 20, 25, and 30 are calculated in the same way, yielding respective values of 0.023, 0.153, 0.025, 0, 0, and 0. A complete histogram of the distribution of solutes A and B is given in Figure A6.3 and shows that the two solutes have been successfully separated. f f f f (5, 30) = (10, 30) = (15, 30) = (20, 30) 30 30 5 5 0 833 0 167 2 1 10 0 30 30 10 10 0 833 0 167 1 4 10 0 30 30 15 15 0 833 0 167 2 2 10 0 5305 15 10 30 10 9 15 30 15 5 ! ()! (. )(. ) . ! ()!! (. ) (. ) . ! ()!! (. ) (. ) . –– –– –– ? =× ≈ ? =× ≈ ? =× ≈ = (25, 30) = (30, 30) = 30 30 20 20 0 833 0 167 0 013 30 30 25 25 0 833 0 167 0 192 30 30 30 30 0 833 0 167 0 004 20 30 20 25 30 25 30 30 30 ! ()!! (. ) (. ) . ! ()!! (. ) (. ) . ! ()!! (. ) (. ) . – – – ? = ? = ? = f f q D B B = + = + = 1 1 1 05 1 0 667 . . q D A A = + = + = 1 1 1 51 0 167. Figure A6.3 Progress of a countercurrent extraction for Example A6.1 after 30 steps. Fraction of solute 0246 1012141618202242628308 0 0.20 Tube number 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 Solute A Solute B 1400-AP6 9/8/99 4:42 PM Page 758 Appendix 6 759 Constructing a histogram using equation A6.2 is tedious, particularly when the number of steps is large. The fraction of solute in most tubes will be approximately zero. Consequently, the histogram’s construction can be simplified by only solving equation A6.2 for those tubes containing an amount of solute exceeding a threshold value. Since the location of the solute obeys a binomial distribution, we can use the distribution’s mean and standard deviation to determine which tubes contain a sig- nificant fraction of solute. The properties of a binomial distribution were covered in Chapter 4, with the mean, m , and standard deviation, s , given as m = np Furthermore, when both np and nq are greater than 5, the binomial distribution is closely approximated by the normal distribution, 2 and the probability tables in Ap- pendix 1A can be used to determine the location of the solute and its recovery. EXAMPLE A6.2 Two solutes, A and B, with distribution ratios of 9 and 4, respectively, are to be separated by a countercurrent extraction in which the volumes of the upper and lower phases are equal. After 100 steps, determine the 99% confidence interval for the location of each solute. SOLUTION The fraction, q, of each solute remaining in the lower phase is calculated using equation A6.1. Since the volumes of the lower and upper phases are equal, we get and, consequently, p A is 0.90 and p B is 0.80. The mean and standard deviation for the distribution of solutes A and B after 100 steps of the countercurrent extraction are The confidence interval for a normally distributed solute is r = m ± zs where r is the number of the tube, and the value of z is determined by the chosen significance level. For a 99% confidence interval, the value of z is 2.58 (Appendix 1B); thus, r A = 90 ± (2.58)(3) = 90 ± 8 r B = 80 ± (2.58)(4) = 80 ± 10 μσ AA A AA BB B B == = = = = == = = = = np np q np np q ( )( . ) ( )( . )( . ) ( )( . ) ( )( . )( . ) 100 0 90 90 100 0 90 0 10 3 100 0 80 80 100 0 80 0 20 4 q D q D A A B B = + = + = = + = + = 1 1 1 91 010 1 1 1 41 020 . . σ= =np p npq(– )1 1400-AP6 9/8/99 4:42 PM Page 759 760 Modern Analytical Chemistry Fraction of solute 60 63 66 69 75 78 81 84 87 90 93 96 9972 0 Tube number 0.14 0.12 0.10 0.08 0.06 0.04 0.02 Solute A Solute B Figure A6.4 Progress of a countercurrent extraction for Example A6.2 after 100 steps. Since the two confidence intervals overlap, a complete separation of the two solutes cannot be achieved in a 100-step countercurrent extraction. The complete distribution of the solutes is shown in Figure A6.4. EXAMPLE A6. 3 For the countercurrent extraction in Example A6.2, calculate the recovery and separation factor for solute A if the contents of tubes 85–99 are pooled together. SOLUTION From Example A6.2 we know that after 100 steps of the countercurrent extraction, solute A is normally distributed about tube 90 with a standard deviation of 3. To determine the fraction of solute in tubes 85–99, we use the single-sided normal distribution in Appendix 1A to determine the fraction of solute in tubes 0–84 and in tube 100. The fraction of solute A in tube 100 is determined by calculating the deviation z (see Chapter 4) and using the table in Appendix 1A to determine the corresponding fraction. For z = 3 this corresponds to 0.135% of solute A. To determine the fraction of solute A in tubes 0–84, we again calculate the deviation z r == = –– –. μ σ 85 90 3 167 z r == = ––μ σ 99 90 3 3 1400-AP6 9/8/99 4:42 PM Page 760 Appendix 6 761 From Appendix 1A we find that 4.75% of solute A is present in tubes 0–84. Solute A’s recovery, therefore, is 100% – 4.75% – 0.135% ? 95% To calculate the separation factor, we must determine the recovery of solute B in tubes 85–99. This is determined by calculating the fraction of solute B in tubes 85–100 and subtracting the fraction of solute B in tube 100. By calculating z and using Appendix 1A, we find that approximately 10.6% of solute B is in tubes 85–100, and that essentially no solute B is in tube 100. The separation factor, S B,A therefore, is S R R BA B A , . .== = 10 6 95 0 112 1. Craig, L. C. J. Biol. Chem. 1944, 155, 519. 2. Mark, H.; Workman, J. Spectroscopy, 1990, 5(3), 55–56. REFERENCES 1400-AP6 9/8/99 4:42 PM Page 761 19. Solution 1: 0.66 M NaCl; Solution 2: 0.1312 M NaCl 20. 0.061 M NO 3 – ; pNO 3 = 1.21 21. 0.069 M Cl – ; pCl = 1.16 22. 8.44 M ethanol Chapter 3 2. (a) 5 ng 3. (a) 1.6 ppm –1 ; (b) 0.55 ppm –1 ; (c) 0.35; (d) analyte; (e) C I /C A < 0.029 4. 2.01 ppm 5. –1.0 6. With ascorbic acid, K A,I = 30; with methionine, K A,I = –2.7 · 10 –3 7. (a) 4.4; (b) ascorbic acid; (c) 4.4 · 10 –3 M 8. (a) 66.5 A/M; (b) –7.9 · 10 –3 ; (c) hypoxanthine; (d) 1.42 · 10 –6 M 9. C C = 0.35 ppm, C D = 0.17 ppm Chapter 4 1. mean = 5.583 g, median = 5.552 g, range = 0.148 g, standard deviation = 0.056 g, variance = 3.1 · 10 –3 2. (a) mean = 243.5 mg, median = 243.4 mg, range = 37.4 mg, standard deviation = 11.9 mg, variance = 141; (b) 29.3% 3. (a) For 100-mg tablets: mean = 95.56 mg, standard deviation = 2.16 g; (b) For 100-mg tablets: 1.97% 4. 98.2% 5. (a) 16.0 g/mol ± 0.4 g/mol; (b) volume in liters and mass in grams 6. (a) 50.0 ppm ± 0.3 ppm; (b) no 7. 0.1175 M ± 0.0001 M 9. 100 mg 10. dilution (b) use 25-mL pipet four times 14. 90 carbons, 36.6% 15. (a) 22.4%; (b) 3.6% Chapter 1 1. (a) quantitative analysis; (b) qualitative or quantitative analysis; (c) qualitative analysis; (d) characterization analysis; (e) fundamental study; (f) quantitative analysis. Chapter 2 1. (a) 3; (b) 3; (c) 5; (d) 3; (e) 4; (f) 3 2. (a) 0.894; (b) 0.893; (c) 0.894; (d) 0.900; (e) 0.0891 3. (a) 12.01 g/mol; (b) 16.0 g/mol; (c) 6.022 · 10 23 mol –1 ; (d) 9.65 · 10 4 C/mol 4. (a) 71.9; (b) 39.8; (c) 6.1 · 10 3 ; (d) 55; (e) 2.57 · 10 –2 ; (f) –4.185; (g) 7.2 · 10 –8 ; (h) 5.30 · 10 –13 5. 1.230% w/w Ni 6. 288.91 g/mol 7. 16.6 mL 8. (a) 0.20 N; (b) 0.10 N; (c) 0.20 N 9. 1.4 · 10 –6 M 10. (a) 12.0 M; (b) 31.1 g, 26.3 mL 11. 2.4 mL 12. 1.80 · 10 –5 % w/v, 0.180 ppm, 1.80 · 10 2 ppb 13. 12 mg NaF/gal 14. 5.160, 1.19 · 10 –9 M 15. (a) moles Mg 2+ =2· moles Mg 2 P 2 O 7 ; (b) moles HCl=2· moles CaCO 3 ; (c) 2 · moles AgCl = moles NH 3 ; (d) moles Fe 2+ =6· moles Cr 2 O 7 2– 16. 0.329 M K 2 Cr 2 O 7 17. Mass (g) Needed to Prepare Compound 0.10 M K + 1.0 · 10 2 ppm K + 1.0% w/v K + KCl 7.5 0.19 19 K 2 SO 4 8.7 0.22 22 K 3 Fe(CN) 6 11 0.28 28 18. Solution A: 4.00 · 10 –3 M; Solution B: 1.00 · 10 –3 M; Solution C: 4.00 · 10 –5 M 762 AppendixAppendix 7 Answers to Selected Problems 1400-AP7 9/8/99 4:43 PM Page 762 16. mean = 16.883, standard deviation = 0.0794, 95% confidence interval = 16.883 ± 0.066% w/w Cr 17. (a) t exp = 2.82 and t(0.05, 8) = 2.31, difference is significant; (b) IUPAC: 2.81 ppTr, LOI: 11.42 ppTr, LOQ: 12.16 ppTr 18. t exp = 3.23 and t(0.01, 6) = 3.71, no evidence that difference is significant 19. t exp = 1.43 and t(0.05, 11) = 2.205, no evidence that difference is significant 20. t exp = 1.07 and t(0.05, 3) = 3.18, no evidence that difference is significant 21. t exp = 4.28 and t(0.05, 9) = 2.26, difference is significant 23. t exp = 0.40 and t(0.05, 12) = 2.18, no evidence that difference is significant 24. t exp = 0.942 and t(0.05, 4) = 2.78, no evidence that difference is significant 25. t exp = 1.91 and t(0.10, 8) = 1.86, difference is significant 26. t exp = 2.17 and t(0.05, 8) = 2.31, no evidence that difference is significant 27. t exp = 1.19 and t(0.05, 11) = 2.205, no evidence that difference is significant 28. t exp = 0.09 and t(0.05, 7) = 2.36, no evidence that difference is significant 29. t exp = 3.69 and t(0.05, 5) = 2.57, difference is significant 30. no evidence for any outliers Chapter 5 1. (a) without correction: 10.011 mL, with correction 10.021 mL; (b) –0.010 mL, –0.10% 2. (a) without correction: 0.2500 g, with correction 0.2501 g; (b) –0.0001 g, –0.040% 3. proportional error 4. 4.94 g/cm 3 5. Serial Dilution Single Dilution Molarity Uncertainty Uncertainty 1.00 · 10 –2 2.94 · 10 –5 2.94 · 10 –5 1.00 · 10 –3 3.64 · 10 –6 6.37 · 10 –6 1.00 · 10 –4 4.23 · 10 –7 6.37 · 10 –7 1.00 · 10 –5 4.75 · 10 –8 1.02 · 10 –7 6. 7.33 ppm 7. 2.20 · 10 –3 % w/w 8. 0.191 ppm 10. 20.8 ppm 11. (a) right; (b) left; (c) left 12. (a) S meas = –2.69 + 0.477[Cd 2+ (nM)], slope = 0.477 ± 0.036, y-intercept = –2.69 ± 2.02; (c) slope = 1.43, approximately three times more sensitive; (d) 49.9 ± 5.9 nM Cd 2+ 13. 12.3 ± 0.2 ppb 14. (a) S A /S IS = 0.3037 ± 0.5576(C A /C IS ), slope = 0.5776 ± 0.0999, y-intercept = 0.3037 ± 0.2484 15. Plotting measured versus accepted gives t exp = 0.679 for the y-intercept and t exp = 0.345 for the slope. Because t(0.05, 3) is 3.18, there is no evidence for a significant difference between the values. 16. m A = 2.61 m A + (14.43 m A ppm –1 ) · (ppm Tl added) Chapter 6 3. 0.626 V 4. (a) 2MnO 4 – (aq)+5H 2 SO 3 (aq)+H 2 O(l) t 2Mn 2+ (aq) + 5SO 4 2– (aq)+4H 3 O + (b) IO 3 – (aq)+ 5I – (aq)+6H 3 O + (aq) t 3I 2 (s)+9H 2 O (c) 3ClO – (aq)+I – (aq) t IO 3 – (aq) + 3Cl – 5. (a) solubility increases; (b) solubility decreases; (c) solubility decreases 6. (a) charge balance: [Na + ]+[H 3 O + ] = [OH – ] + [Cl – ] mass balance: 0.1 M = [Na + ] = [Cl – ] (c) charge balance: [H 3 O + ] = [OH – ]+[F – ] mass balance: 0.1 M = [HF] + [F – ] (e) charge balance: 2[Mg 2+ ]+[H 3 O + ] = [OH – ]+ [HCO 3 – ] + 2[CO 3 2– ] mass balance: [Mg 2+ ]=[H 2 CO 3 ] + [HCO 3 – ] + [CO 3 2– ] (g) charge balance: [Na + ]+[H 3 O + ] = [OH – ]+ [Cl – ] + [NO 2 – ] mass balance: 0.1 M = [Cl – ] mass balance: 0.050 M = [Na + ] = [HNO 2 ] + [NO 2 – ] 7. (b) 6.79; (d) 2.90; (f) 8.61 8. (a) 1.53, 4.15, 9.67 9. (a) 6.7 · 10 –7 M; (b) 3.5 · 10 –9 M; (c) 4.8 · 10 –16 M 10. 2.1 · 10 –4 M, 1.3 · 10 –3 M 11. 710 M (very soluble), 1.2 · 10 –4 M 12. 1.2 · 10 –4 M 13. [Ag + ] = 1.3 · 10 –5 M, [Cl – ] = 1.3 · 10 –5 M, [AgCl(aq)] = 9.0 · 10 –7 M, [AgCl 2 – ] = 9.7 · 10 –10 M 14. (a) 0.050 M; (b) 0.075 M; (c) 0.30 M 15. (a) 6.7 · 10 –7 M; (b) 7.2 · 10 –9 M; (c) 1.7 · 10 –15 M 17. pH levels > 12.35 19. (a) 3.52; (b) 9.10; (c) 10.23 20. (a) 3.27; (b) 8.94; (c) 10.22 21. (a) 3.74; (b) 9.24; (c) 10.24 () [][] [][] . ][ ] ] . () [][ ] [] . – – a NH Cl NH HCl Y e Ba H CO HO K K K ==× = ==× + + + 4 3 9 4 4 2 23 3 8 175 10 28 9 24 10 (c) = [Cd(CN) [CdY ][CN 4 2– 2– – Appendix 7 763 1. 1400-AP7 9/8/99 4:43 PM Page 763 22. pM = 8.18 –log [ML]/[L], 8.18, 8.00 23. 0.761 V, 0.769 V Chapter 7 1. (a) overall variance = 0.9144, method variance = 0.0330, sampling variance = 0.8814; (b) 96.4% 2. 74, 12, 56, 46, 40, 83, 88, 45, 29, 92 3. For a daily cycle, collect samples at least every 12 h, and samples should be collected at least every six months for a yearly cycle; more frequent sampling would be even better. 9. (a) 5.9%; (b) 8 samples 10. 25.7 g, 6.4 g 13. (a) 15 samples; (b) 3.50 g 14. (a) ±0.176; (b) ±0.518; (c) ±0.216 15. sampling plan (a), sampling plan (c) 16. t exp = 1.23 and t(0.05, 7) = 2.36, no evidence for a significant difference 18. (a) 106.3%; (b) 38.6% 19. (a) R Co = 0.991, R Fe = 0.019; (b) 0.0196; (c) 8.72%; (d) –0.73%; (e) 0.57% 20. (a) R Ca = 0.932, R Mg = 0.073; (b) 54.7% 22. Cyanide, CN – , becomes HCN at lower pH levels 24. (a) 88.2% extracted; (b) 95.7% extracted; (c) 98.5% extracted; (d) 99.0% extracted 25. (a) 6600 mL; (b) 408 mL; (c) 123.2 mL; (d) 99.4 mL 26. When using 50.0 mL of the organic solvent, D must be 99.0, but D must be 18.0 when using 25.0 mL of the organic solvent. 27. pH < 4.3 28. three extractions 29. pH > 10.1 30. (a) 4.8% of HA extracted, 99.6% of HB extracted; (b) aqueous; (c) R HA = 95.2%, R HB = 0.4%; (d) 4.2 · 10 –3 ; (e) –2.81% 31. (a) lower concentrations of I – increase extraction efficiency; (b) D = K D /(1 + K f [I – ] aq ) 32. (a) D = K D b 2 [L – ] 2 aq /(1 + b 2 [L – ] 2 aq ); (b) 82.1% extracted 34. 99.7% extracted Chapter 8 2. 4. (a) pH > 4.27; (b) pH > 6.51; (c) pH > 1.99; (d) pH > 10.33; (e) pH > 13.9 5. Occluded MnO 4 – in experiment 1 imparts color to KClO 4 S K KK KK AgCl sp sp sp sp Ag Ag Ag =++ + +++ [] [] [] 1 2 2 3 3 2 23ββ 6. RSS decreases from experiment 1 to experiment 3 7. (b) %w/w Al 2 O 3 reported too high; (d) better sensitivity 8. (b) %w/w Ca reported too low 9. (a) %w/w Fe reported too low; (b) prevent peptization 11. 1.3-g samples 12. 4.17% w/w As 2 O 3 13. 97.65% pure 14. 0.542 g FeSO ? 7H 2 O/tablet 15. 2.44% w/w Fe 2 O 3 , 39.6% w/w CaO, 1.66% w/w MgO 16. three ethoxy groups per molecule 17. 22.22% w/w K 2 SO 4 18. 22.05% w/w Mn, 62.22% w/w Fe 19. 30.72% NaNO 3 20. 54.931 g/mol 21. A is AgNO 3 , B is Pb(NO 3 ) 2 , C is KI, D is Na 2 SO 4 , and E is BaCl 2 23. direct method is more sensitive 24. 0.29-g samples 25. 32 mL 26. (a) 1.560 g PbCrO 4 ; (b) 1.005 mol CrO 4 2– /mol Pb 2+ ; (c) %w/w Pb reported too high 28. 2KO 3 (s) fi 2KO 2 (s)+O 2 (g), 89.3% w/w KO 3 29. 37.65% w/w H 2 O 31. (a) 30.01% w/w Fe; (b) 64.56% w/w C, 5.42% w/w H; (c) FeC 10 H 10 32. (a) Polymer A: mean 30.07%, standard deviation 0.0208, Polymer B: mean 29.79%, standard deviation 0.0200; (b) t exp = 16.81 and t(0.05, 4) = 2.78, difference is significant 33. 20 m mol/m 2 35. 14 mg/m 3 , 1.4 · 10 –2 mg/L 36. (a) 21.28% w/w fat; (b) t exp = 5.85 and t(0.05, 4) = 2.78, difference is significant 37. Average Depth % w/w (cm) Organic Matter 1 14.0 3 16.8 5 34.0 7 26.9 9 40.8 11 54.8 13 65.1 15 9.04 17 not present 38. (a) macro- or mesosample, trace-level analyte; (b) D f = 7.97 + 2.18 · 10 8 [thiourea (M)]; (c) 7.71 · 10 –7 M; (d) ±1.22 · 10 –7 764 Modern Analytical Chemistry 1400-AP7 9/8/99 4:43 PM Page 764 Appendix 7 765 Chapter 9 7. normal, first-derivative and second-derivative titration curves do not show distinct equivalence points, Gran plot shows equivalence point at approximately 5.5 mL; expected equivalence point is 5.07 mL 10. (a) aniline is too weak of a base to titrate in water; (b) report too many moles of aniline; (c) so that no more than one buret-full of titrant is needed 13. titrating three protons, 3 · moles H 4 Y = moles NaOH 15. OH – and HCO 3 – react to produce CO 3 2– and H 2 O 16. (a) only OH – , 1740 ppm OH – ; (b) CO 3 2– and HCO 3 – , 1630 ppm CO 3 2– , 2860 ppm HCO 3 – ; (c) HCO 3 – , 4180 ppm HCO 3 – ; (d) CO 3 2– , 4920 ppm CO 3 2– ; (e) OH – and CO 3 2– , 1240 ppm CO 3 2– , 1390 ppm OH – 17. (a) 0.0585 M HPO 4 2– and 0.0553 M PO 4 3– ; (b) 0.0474 M H 3 PO 4 ; (c) 0.1091 M NaOH; (d) 0.0838 M H 3 PO 4 and 0.0214 M H 2 PO 4 – 18. 15.59% w/w protein 19. 36.7 ppm SO 2 20. 2480 ppm CO 2 21. 97.47% pure 22. 207 23. 138 g/equivalent 24. taurine 25. pK a = 4.83 26. ultramicro sample; major or minor analyte 29. (a) systematic—molarity reported too low; (b) systematic— molarity reported too high; (c) systematic—molarity reported too low; (d) random; (e) systematic—molarity reported too high; (f) no effect; (g) no effect 35. EGTA 36. 12.8 mg Ca 2+ /100 mL 37. 98.08% w/w CaCO 3 38. 54,300 ppm NaCN 39. 53.90% w/w Cd 2+ 40. (b) 0.02483 M Fe 3+ , 0.03536 M Al 3+ 41. (c) 0.05032 M 45. (a) to remove any Sn 2+ which will interfere with the titration (b) no; (c) no 46. (a) “50-mL portion of the resulting solution” and “50 mL of a standard solution of Fe 2+ ”; (b) report too much Cr 3+ ; (c) report too much Cr 3+ 47. (a) report too much H 2 O 2 ; (b) use a reagent blank 48. 98.42% w/w Fe 2 O 3 49. 18.69% w/w Mn 2+ 50. 23.5% w/w U 51. 6.80 · 10 –5 cm 52. 280 ppm CO 53. 6.19 ppm O 2 54. (b) negative determinate error; (c) remove AgCl by filtering; (d) no 58. 26.19% w/w kBr 59. 29.86% BaCl 2 60. 90.9% pure Chapter 10 Wavelength Frequency Wavenumber Energy (m) (s –1 ) (cm –1 ) (J) 4.50 · 10 –9 6.67 · 10 16 2.22 · 10 6 4.42 · 10 –17 2.26 · 10 –7 1.33 · 10 15 4.42 · 10 4 8.81 · 10 –19 3.11 · 10 –6 9.65 · 10 13 3.21 · 10 3 6.39 · 10 –20 2.76 · 10 –7 1.09 · 10 15 3.62 · 10 4 7.20 · 10 –19 [Analyte] (M) Absorbance %T d b 1.40 · 10 –4 0.157 69.7 1120 1.00 7.51 · 10 –4 0.563 27.4 750 1.00 2.56 · 10 –4 0.225 59.6 440 2.00 1.55 · 10 –3 0.167 68.1 21.5 5.00 8.46 · 10 –4 0.478 33.3 565 1.00 4.35 · 10 –3 0.674 21.2 1550 0.100 1.20 · 10 –4 0.0899 81.3 74.9 10.00 3. 59.2% T 4. 19.7% T 5. 3134 cm –1 M –1 Concentration Absorbance Absorbance Absorbance (M) (a) (b) (c) 1.0 · 10 –5 0.020 0.009 0.014 3.0 · 10 –5 0.060 0.035 0.043 5.0 · 10 –5 0.100 0.065 0.071 9.0 · 10 –5 0.180 0.130 0.128 11 · 10 –5 0.220 0.163 0.156 13 · 10 –5 0.260 0.197 0.185 Concentration Absorbance for Absorbance for (M) d￠ =d? = 1000 e￠ = 1900, d? = 100 00 0 2 · 10 –5 0.020 0.020 4 · 10 –5 0.040 0.039 6 · 10 –5 0.060 0.057 8 · 10 –5 0.080 0.074 1 · 10 –4 0.100 0.091 1. 2. 6. 7. (b) 1400-AP7 9/8/99 4:43 PM Page 765 29. 766 Modern Analytical Chemistry [Analyte] (M) Absorbance 0.000 0.00 0.002 0.37 0.004 0.70 0.006 0.97 0.008 1.15 0.010 1.24 9. yes 10. (a) 5 mL thioglycolic acid, 2 mL 20% w/v ammonium citrate and 5 mL 0.22 M NH 3 diluted to 50 mL; (b) no effect if blank is used; (c) to keep absorbance within range of calibration curve; (d) reported results for Fe 3+ are too low 11. dilute to 100 mL 12. (e) 142 ppm P 13. (a) 3.55 · 10 –3 M Cu 2+ ; (b) 6.24 · 10 –3 M Co 2+ , 4.62 · 10 –3 M Cu 2+ ; (c) 3.69 · 10 –3 M Cu 2+ , 1.73 · 10 –2 M Ni 2+ , 9.14 · 10 –3 M Co 2+ 14. 7.44 ppm phenol 15. 0.648 ppm Fe 2+ 16. 0.209 ppm Cl 2 17. 10.6% v/v methanol 18. 0.90 mg barbital/100 mL 19. 220 mg aspirin, 160 mg phenacetin, 32 mg caffeine 20. 0.28 ppm SO 2 21. mean = 18.4% CO; ± 0.83% CO 22. mean = 64.03% w/w polystyrene; t exp = 3.73 and t(0.05, 2) = 4.30, no evidence for a determinate error 23. 613 nm and 658 nm 24. (a) [MnO 4 – ] = 8.2 · 10 –5 M, [Cr 2 O 7 2– ] = 1.8 · 10 –4 M; (b) C V = 58.5 ppm, C Ti = 31.5 ppm; (c) 0.26 ppm Zn 2+ , 0.51 ppm Cu 2+ 25. 1.0 · 10 –5 moles Cu 2+ , 1.30 · 10 –5 moles Ni 2+ 26. ML 2 27. ML 2 28. ML 3 29. 1:1 stoichiometry 30. pK a = 4.31 31. pK a = 4.72 32. (a) 4%; (b) 0.9% 33. 95.5% pure 34. (a) 0.865 ppm Cu; (b) 0.0828 ppm Cr 35. 40.6 ppb Mn 2+ 36. 65.5 ppm Na 37. 0.0110% w/w Pb, 7.61 · 10 –3 % w/w Ni 38. 353 ppm Fe 3+ 39. 3.86 · 10 –5 M 40. 4.82 · 10 –5 M benzo[a]pyrene 41. 64.9% w/w acetylsalicylic acid 42. 2.45 nM Se(IV) 43. 1.38 g fibrinogen/L Chapter 11 1. (a) anode: Pt, Fe 2+ t Fe 3+ + e – , cathode: Ag, Ag + + e – t Ag(s) (b) anode: Ag, Ag(s)+Br – t AgBr(s)+e – , cathode: Cd, Cd 2+ +2e – t Cd(s) (c) anode: Pb, Pb(s)+SO 4 2– t PbSO 4 (s)+2e – cathode: PbO 2 , PbO 2 (s)+SO 4 2– +2e – +4H + t PbSO 4 (s)+2H 2 O 2. (a) –0.059 V; (b) –0.512 V; (c) 2.10 V 3. 0.211 M I – 4. Zn(s)+2H + (aq) t Zn 2+ (aq)+H 2 (g) 5. 7.11 · 10 –3 ; 1.3 · 10 –5 M benzoate 6. the membrane responds only to the protonated form of cocaine 9. (a) 2 · 10 –3 M to 1 · 10 –5 M; (b) E (mV) = 331.4 + 47.76 log [penicillin]; (c) 1.1 · 10 –4 M 10. 4.1 mM K + 11. (a) pH = 3.98; (b) pH = 6.9; (c) pH = 4.67 12. 11.03 ppm NO 3 – 13. (a) 1.20 ppm F – ; (b) 0.0593% w/w F – 15. 98.0% pure 16. 115 ppm H 2 S 18. n =1 19. (b) 0.789 V 20. 82.5 m g S/mL 21. 95.0% pure 22. 30.2 ng Sb 23. 0.863% w/w Tl 24. 29.2 mg caffeine/tablet, 38.8 mg ascorbic acid/tablet 25. 49.8 ppm Sn 4+ 26. 14.0 mg glucose/100 mL 27. 2.0 m g Tl + /mL, 2.0 m g Pb 2+ /mL, 0.89 m g In 3+ /mL Group ppb Cd ppb Pb ppb Cu I 0.03 0.06 0.09 II 0.21 0.33 0.17 III 0 0 0 IV –0.01 0 0.10 V 0 +0.01 –0.015 VI 0.03 –0.02 –0.03 VII 0.02 0.12 0.10 8. 1400-AP7 9/8/99 4:43 PM Page 766 Appendix 7 767 31. n = 2, E 1/2 = –0.390 V, Pb(OH) 3 – , b 3 = 3.71 · 10 12 32. 3.37 ppb Cu 2+ 33. 0.47 m M 34. 3.5 · 10 –8 M Chapter 12 1. (a) N A = 46,000, N B = 48,500, N c = 44,400, average = 46,300; (b) 0.43 mm 2. using equation 12.1: R AB = 1.5, R BC = 1.1; using equation 12.21: R AB = 1.5, R BC = 1.1 3. t r = 350 s, w = 22 s, t r ￠ = 325 s, k￠ = 13, N = 4050, H = 0.49 mm 4. R = 1.0 5. 4.5 m long, 0.167 mm N B ‘ k B ￠ R 100,000 1.05 0.5 1.25 10,000 1.10 1.9 1.50 10,000 1.05 4.0 1.00 38,400 1.05 3.0 1.75 7. (b) B term 5–16 mL/min, A term 16–71 mL/min, C term 71–120 mL/min; (c) 33 mL/min, H = 3.20 mm; (d) 33 mL/min, H = 1.56 mm; (e) 2.1 times as many 8. (a) row 1: v = 3.6, u = 4.1 · 10 –2 cm/s, h = 1.36, H = 7.4 · 10 –6 m; row 2: v = 3.3, u = 3.8 · 10 –2 cm/s, h = 0.97, H = 5.3 · 10 –6 m 9. CHCl 3 , CHCl 2 Br, CHClBr 2 , CHBr 3 11. 1.50 ppb CHCl 3 , 0.754 ppb CHCl 2 Br, 3.49 ppb CHClBr 2 , 1.39 ppb CHBr 3 12. (a) 0.0682% w/w H 2 O; (b) 2.96% w/w H 2 O 13. 1.7 mg H 2 O/g soil 14. 1.9% v/v methyl salicylate 15. 45.8% w/w camphor 16. 7.98 ng heptachlor epoxide/g orange rind 17. I octane = 800, I nonane = 900, I toluene = 842 18. 765 23. 67 mg vitamin C, 19 mg niacinamide, 2.4 mg pyridoxine, 1.4 mg thiamin, 0.46 mg folic acid, 1.5 mg riboflavin, niacin is absent 24. 31.9 mg 25. (a) 228 mg acetylsalicylic acid, 31.5 mg caffeine 26. 0.21 mg vitamin A/100 g 27. (b) 0.83 M HCO 3 – , 0.25 mM Cl – , 0.0030 mM NO 2 – , 0.12 mM NO 3 – , 0.32 mM Ca 2+ , 0.11 mM Mg 2+ , 0.19 mM SO 4 2– ; (c) 0.77 M HCO 3 – ; (d) 1.00 28. 74.5 ppm Cl – , 3.39 ppm NO 3 – , 41.4 ppm SO 4 2– 29. 56,000 30. order is caffeine, aspartame, benzoic acid 32. 436 ppm NO 3 – 34. (a) 3.22 · 10 –4 cm 2 V –1 s –1 ; (b) approximately 290,000 theoretical plates; (c) R = 1.1; (f) m ep =0. Chapter 13 1. 2. 0.74 mM phenylacetate 3. 286 m M H 2 O 2 4. 6.2 · 10 –4 M chromic acid 5. 39.4 ppm glucose, –1.5% error 6. approximately 5 times more dilute 7. k = 0.0486 s –1 , [A] 0 = 1.50 mM 8. 1.2 · 10 –7 M acetylcholine 9. 0.0863 m M fumarase 10. V max = 0.406 M s –1 , K m = 6.49 · 10 –3 M, k 2 = 8.1 · 10 4 s –1 11. competitive inhibition 13. k A /k B = 6900 14. [A] 0 = 0.220 mM, [B] 0 = 0.125 mM, k A = 0.135 min –1 , k B = 0.0332 min –1 Isotope Rate Constant 3 H 5.54 · 10 –2 /year 14 C 1.21 · 10 –4 /year 32 P 4.85 · 10 –2 /day 35 S 7.96 · 10 –3 /day 45 Ca 4.56 · 10 –3 /day 55 Fe 2.38 · 10 –1 /year 60 Co 1.31 · 10 –1 /year 131 I 8.66 · 10 –2 /day 16. 1.7 · 10 –6 M 60 Co 17. 3.42% w/w Ni 18. 5.79 mg vitamin B 12 per tablet 19. 2.7% 20. 7.0 · 10 8 years 21. 11 g C, 1300 min 22. h = 0.762 abs units, t a = 14.1 s, T = 15.8 s, t￠ = 1.7 s, D t = 15.2 s, T￠ = 13.5 s, sensitivity = 7.62 · 10 –3 ppm –1 , about 260–270 samples/h 24. 13,700 ppm Cl – 25. 0.071 M HCl 26. 11.2 mM glucose 27. (a) 2.52%; (b) 68 samples/h; (c) 94.8% w/w cocaine 28. 6.27 · 10 –4 M H 2 SO 4 [] [] –[] – –– A 0 21 12 = ′′ PP tt kt kt ee 6. 15. 1400-AP7 9/8/99 4:43 PM Page 767 768 Modern Analytical Chemistry Chapter 14 3. R = 4.00 + 1.20B* + 0.72A*B*, R = 4.00 – 0.40A + 0.08AB 4. (a) R = 52.610 + 23.755Ca* – 15.875Al* – 6.200Ca*Al*; (b) 16.9 ppm Al 5. (a) R = 38.1 – 3.6X* + 0.1Y* + 8.1Z* + 0.4X*Y*+ 0.9X*Z* – 7.4Y*Z* – 0.6X*Y*Z*; (b) R = 38.1 – 3.6X*+ 8.1Z* – 7.4Y*Z*; (d) catalyst A for high concentrations, catalyst B for low concentrations; (e) 49.2% 6. (a) R = 4.18 + 1.54X* + 0.92Y* – 0.12Z* + 0.29X*Y*– 0.24X*Z* + 0.10Y*Z* + 0.44X*Y*Z*; (b) R = 4.18 + 1.54X* + 0.92Y* 7. (a) R = 8.5 + 2.5X* – 0.5Y* + 3.5Z* – 0.5X*Y* + 0.5X*Z*– 1.5Y*Z* – 0.5X*Y*Z*; (b) R = 8.5 + 2.5X* + 3.5Z*– 1.5Y*Z*; (c) no evidence for curvature 8. (a) R D E = 33.54 – 2.56X* – 1.92Y* + 1.42Z* + 0.62X*Y*+ 0.10X*Z* + 0.54Y*Z* + 0.41X*Y*Z*; R samp = 26.9 + 1.2X* + 4.9Y* – 0.8Z* + 0.5X*Y* – 0.6X*Z*+ Y*Z* + 0.9X*Y*Z*; (b) no 9. (a) R = 0.9 + 0.05* – 0.071B* + 0.039C*+ 0.074D* – 0.15E* – 0.12C*E* + 0.049A*C*E*; (b) yes 11. yes 12. pH and digestion time, 0.72 13. (a) 2; (b) 8; (c) 8; (d) 1 and 10 14. s rand ? 0.055, s syst ? 0.078 15. (a) variability between labs is significant; (b) s rand ? 0.80, s syst ? 1.3 16. (a) 4.9%; (b) yes Chapter 15 3. (d) r = 7.2%, yes 4. s = 0.066; relative standard deviation = 1.3%, precision acceptable 5. average recovery is 99% 6. (a) yes; (b) lab 1400-AP7 9/8/99 4:43 PM Page 768 A absorbance the attenuation of photons as they pass through a sample (A). (p. 373) absorbance spectrum a graph of a sample’s absorbance of electromagnetic radiation versus wavelength (or frequency or wavenumber). (p. 373) accuracy a measure of the agreement between an experimental result and its expected value. (p. 38) acid a proton donor. (p. 140) acid–base titration a titration in which the reaction between the analyte and titrant is an acid–base reaction. (p. 278) acid dissociation constant the equilibrium constant for a reaction in which an acid donates a proton to the solvent (K a ). (p. 140) acidity a measure of a water’s ability to neutralize base. (p. 301) activity true thermodynamic constants use a species activity in place of its molar concentration (a). (p. 172) activity coefficient the number that when multiplied by a species’ concentration gives that species’ activity (g ). (p. 172) adjusted retention time the difference between a solute’s retention time and column’s void time (t r ￠ ). (p. 551) adsorbate a coprecipitated impurity that adsorbs to the surface of a precipitate. (p. 239) aliquot a portion of a solution. (p. 111) alkalinity a measure of a water’s ability to neutralize acid. (p. 300) alpha particle a positively charged subatomic particle equivalent to a helium nucleus (a ). (p. 642) alternative hypothesis a statement that the difference between two values is too great to be explained by indeterminate error; accepted if the significance test shows that null hypothesis should be rejected (H A ). (p. 83) amalgam a metallic solution of mercury with another metal. (p. 509) amperometry a form of voltammetry in which we measure current as a function of time while maintaining a constant potential. (p. 519) amphiprotic a species capable of acting as both an acid and a base. (p. 142) analysis a process that provides chemical or physical information about the constituents in the sample or the sample itself. (p. 36) analysis of variance a statistical method for comparing three or more sets of data. (p. 693) analytes the constituents of interest in a sample. (p. 36) anode the electrode where oxidation occurs. (p. 466) anodic current a faradaic current due to an oxidation reaction. (p. 510) argentometric titration a precipitation titration in which Ag + is the titrant. (p. 355) asymmetry potential the membrane potential when opposite sides of the membrane are in contact with identical solutions yet a nonzero potential is observed. (p. 476) atomization the process of converting an analyte into a free atom. (p. 412) auxiliary complexing agent a second ligand in a complexation titration that initially binds with the analyte but is displaced by the titrant. (p. 316) auxiliary electrode the third electrode in a three-electrode cell that completes the circuit. (p. 463) auxiliary oxidizing agent a reagent used to oxidize the analyte before its analysis by a redox titration. (p. 341) auxiliary reducing agent a reagent used to reduce the analyte before its analysis by a redox titration. (p. 341) B background correction in atomic absorption spectroscopy, the correction of the net absorbance from that due to the sample matrix. (p. 419) back titration a titration in which a reagent is added to a solution containing the analyte, and the excess reagent remaining after its reaction with the analyte is determined by a titration. (p. 275) balance an apparatus used to measure mass. (p. 25) 769 GlossaryGlossary 1400-Glos 9/8/99 4:43 PM Page 769 band broadening the increase in a solute’s baseline width as it moves from the point of injection to the detector. (p. 553) base a proton acceptor. (p. 140) base dissociation constant the equilibrium constant for a reaction in which a base accepts a proton from the solvent (K b ). (p. 141) baseline width the width of a solute’s chromatographic band measured at the baseline (w). (p. 548) Beer’s law the relationship between a sample’s absorbance and the concentration of the absorbing species (A = e bC). (p. 386) beta particle a charged subatomic particle produced when a neutron converts to a proton, or a proton converts to a neutron (b ). (p. 642) binomial distribution probability distribution showing chance of obtaining one of two specific outcomes in a fixed number of trials. (p. 72) bleed the tendency of a stationary phase to elute from the column. (p. 566) blind analysis the analysis of a standard sample whose composition is unknown to the analyst. (p. 683) bonded stationary phase a liquid stationary phase that is chemically bonded to a particulate packing material. (p. 580) breakthrough volume the volume of sample that can be passed through a solid sorbent before the analytes are no longer retained. (p. 196) buffer a solution containing a conjugate weak acid/weak base pair that is resistant to a change in pH when a strong acid or strong base is added. (p. 167) buret volumetric glassware used to deliver variable, but known volumes of solution. (p. 277) C calibration the process of ensuring that the signal measured by a piece of equipment or an instrument is correct. (p. 47) calibration curve the result of a standardization showing graphically how a method’s signal changes with respect to the amount of analyte. (p. 47) capacity factor a measure of how strongly a solute is retained by the stationary phase (k￠ ). (p. 551) capillary column a narrow bored column that usually does not contain a particulate packing material. (p. 562) capillary electrochromatography a form of capillary electrophoresis in which a stationary phase is included within the capillary column. (p. 607) capillary electrophoresis electrophoresis taking place in a capillary tube. (p. 597) capillary gel electrophoresis a form of capillary electrophoresis in which the capillary column contains a gel enabling separations based on size. (p. 606) capillary zone electrophoresis a form of capillary electrophoresis in which separations are based on differences in the solutes’ electrophoretic mobilities. (p. 604) cathode the electrode where reduction occurs. (p. 466) cathodic current a faradaic current due to a reduction reaction. (p. 510) central limit theorem the distribution of measurements subject to indeterminate errors is often a normal distribution. (p. 79) characteristic concentration the concentration of analyte giving an absorbance of 0.00436. (p. 416) characterization analysis an analysis in which we evaluate a sample’s chemical or physical properties. (p. 9) charge balance equation an equation stating that the total concentration of positive charge in a solution must equal the total concentration of negative charge. (p. 159) charging current a current in an electrochemical cell due to the electrical double layer’s formation. (p. 513) chemiluminescence emission induced by a chemical reaction. (p. 374) chromatogram a plot of the detector’s signal as function of elution time or volume. (p. 548) chromatography a separation in which solutes partition between a mobile and stationary phase. (p. 546) chromophore the specific bonds or functional groups in a molecule responsible for the absorption of a particular wavelength of light. (p. 382) coagulation the process of smaller particles of precipitate clumping together to form larger particles. (p. 242) collaborative testing a validation method used to evaluate the sources of random and systematic errors affecting an analytical method. (p. 687) column chromatography a form of chromatography in which the stationary phase is retained in a column. (p. 546) common ion effect the solubility of an insoluble salt decreases when it is placed in a solution already containing one of the salt’s ions. (p. 158) complexation titration a titration in which the reaction between the analyte and titrant is a complexation reaction. (p. 314) composite sample several grab samples combined to form a single sample. (p. 186) concentration an expression stating the relative amount of solute per unit volume or unit mass of solution. (p. 15) concentration techniques a technique in which the signal is proportional to the analyte’s concentration; also called “instrumental” techniques. (p. 38) conditional formation constant the equilibrium formation constant for a metal–ligand complex for a specific set of solution conditions, such as pH. (p. 316) confidence interval range of results around a mean value that could be explained by random error. (p. 75) coning and quartering a process for reducing the size of a gross sample. (p. 199) constant determinate error a determinate error whose value is the same for all samples. (p. 60) continuum source a source that emits radiation over a wide range of wavelengths. (p. 375) 770 Glossary 1400-Glos 9/8/99 4:43 PM Page 770 control chart a graph showing the time-dependent change in the results of an analysis that is used to monitor whether an analysis is in a state of statistical control. (p. 714) convection the movement of material in response to a mechanical force, such as stirring a solution. (p. 512) convenience sampling a sampling plan in which samples are collected because they are easily obtained. (p. 185) coulometric titrations a titration in which the equivalence point is the time required for a constant current to completely oxidize or reduce the analyte. (p. 501) coulometry an electrochemical method in which the current required to exhaustively oxidize or reduce the analyte is measured. (p. 496) countercurrent extraction a liquid–liquid extraction in which solutes are extracted back and forth between fresh portions of two extracting phases. (p. 546) counter electrode the second electrode in a two-electrode cell that completes the circuit. (p. 462) cryogenic focusing the process of concentrating volatile solutes by cooling the column’s inlet below room temperature. (p. 568) cumulative formation constant the formation constant for a metal–ligand complex in which two or more ligands are simultaneously added to a metal ion or to a metal–ligand complex (b i ). (p. 144) current efficiency the percentage of current that actually leads to the analyte’s oxidation or reduction. (p. 496) D dark current the background current present in a photon detector in the absence of radiation from the source. (p. 379) degrees of freedom the number of independent values on which a result is based (n ). (p. 80) desiccant a drying agent. (p. 29) desiccator a closed container containing a desiccant; used to store samples in a moisture-free environment. (p. 29) detection limit a statistical statement about the smallest amount of analyte that can be determined with confidence. (p. 39) determinate error any systematic error that causes a measurement or result to always be too high or too small; can be traced to an identifiable source. (p. 58) determination an analysis of a sample to find the identity, concentration, or properties of the analyte. (p. 36) dialysis a method of separation that uses a semi-permeable membrane. (p. 206) diffusion the movement of material in response to a concentration gradient. (p. 511) diffusion layer the layer of solution adjacent to the electrode in which diffusion is the only means of mass transport. (p. 511) digestion the process by which a precipitate is given time to form larger, purer particles. (p. 239) dilution the process of preparing a less concentrated solution from a more concentrated solution. (p. 31) displacement titration a titration in which the analyte displaces a species, usually from a complex, and the amount of the displaced species is determined by a titration. (p. 275) dissociation constant the equilibrium constant for a reaction in which a metal–ligand complex dissociates to form uncomplexed metal ion and ligand (K d ). (p. 144) distribution ratio a ratio expressing the total concentration of solute in one phase relative to a second phase; all forms of the solute are considered in defining the distribution ratio (D). (p. 216) Dixon’s Q-test statistical test for deciding if an outlier can be removed from a set of data. (p. 93) dropping mercury electrode an electrode in which successive drops of Hg form at the end of a capillary tube as a result of gravity, with each drop providing a fresh electrode surface. (p. 509) duplicate samples two samples taken from a single gross sample and used to evaluate an analytical method’s precision. (p. 708) E effective bandwidth the width of the band of radiation passing through a wavelength selector measured at half the band’s height. (p. 376) electrical double layer the interface between a positively or negatively charged electrode and the negatively or positively charged layer of solution in contact with the electrode. (p. 513) electrode of the first kind a metallic electrode whose potential is a function of the concentration of M n+ in an M n+ /M redox half-reaction. (p. 474) electrode of the second kind a metallic electrode whose potential is a function of the concentration of X in an MX n /M redox half-reaction. (p. 475) electrogravimetry a gravimetric method in which the signal is the mass of an electrodeposit on the cathode or anode in an electrochemical cell. (p. 234) electrokinetic injection an injection technique in capillary electrophoresis in which an electric field is used to inject sample into the capillary column. (p. 603) electromagnetic spectrum the division of electromagnetic radiation on the basis of a photon’s energy. (p. 372) electron capture detector a detector for GC that provides selectivity for solutes with halogen and nitro functional groups. (p. 570) electroosmotic flow the movement of the conductive medium in response to an applied electric field. (p. 598) Glossary 771 1400-Glos 9/8/99 4:43 PM Page 771 electroosmotic flow velocity the velocity with which the solute moves through the capillary due to the electroosmotic flow (n eof ). (p. 599) electropherogram the equivalent of a chromatogram in electrophoresis. (p. 597) electrophoresis a separation technique based on a solute’s ability to move through a conductive medium under the influence of an electric field. (p. 597) electrophoretic mobility a measure of a solute’s ability to move through a conductive medium in response to an applied electric field (m ep ). (p. 598) electrophoretic velocity the velocity with which a solute moves through the conductive medium due to its electrophoretic mobility (n ep ). (p. 598) emission the release of a photon when an analyte returns to a lower-energy state from a higher-energy state. (p. 373) emission spectrum a graph of emission intensity versus wavelength (or frequency or wavenumber). (p. 374) empirical model a model describing a system’s response that is not derived from theoretical principles. (p. 676) end point the point in a titration where we stop adding titrant. (p. 274) enthalpy a change in enthalpy indicates the heat absorbed or released during a chemical reaction at constant pressure. (p. 137) entropy a measure of disorder. (p. 137) enzyme a protein that catalyzes biochemical reactions. (p. 636) enzyme electrodes an electrode that responds to the concentration of a substrate by reacting the substrate with an immobilized enzyme, producing an ion that can be monitored with an ion-selective electrode. (p. 484) equilibrium a system is at equilibrium when the concentrations of reactants and products remain constant. (p. 136) equilibrium constant for a reaction at equilibrium, the equilibrium constant determines the relative concentrations of products and reactants. (p. 138) equivalence point the point in a titration where stoichiometrically equivalent amounts of analyte and titrant react. (p. 274) equivalent the moles of a species that can donate one reaction unit. (p. 17) equivalent weight the mass of a compound containing one equivalent (EW). (p. 17) error a measure of bias in a result or measurement. (p. 64) excitation spectrum a fluorescence or phosphorescence spectrum in which the emission intensity at a fixed wavelength is measured as a function of the wavelength used for excitation. (p. 427) exclusion limit in size-exclusion chromatography, the largest solute that can be separated from other solutes; all larger solutes elute together. (p. 595) external conversion a form of radiationless relaxation in which energy is transferred to the solvent or sample matrix. (p. 425) external standard a standard solution containing a known amount of analyte, prepared separately from samples containing the analyte. (p. 109) extraction the process by which a solute is transferred from one phase to a new phase. (p. 212) F factor a property of a system that is experimentally varied and that may affect the response. (p. 667) factor level a factor’s value. (p. 667) faradaic current any current in an electrochemical cell due to an oxidation or reduction reaction. (p. 510) Faraday’s law the current or charge passed in a redox reaction is proportional to the moles of the reaction’s reactants and products. (p. 496) fiagram a graph showing the detector’s response as a function of time in a flow injection analysis. (p. 650) field blank a blank sample collected in the field. (p. 710) filter a wavelength selector that uses either absorption, or constructive and destructive interference to control the range of selected wavelengths. (p. 376) filter photometer a simple instrument for measuring absorbance that uses absorption or interference filters to select the wavelength. (p. 388) Fisher’s least significant difference a modified form of the t-test for comparing several sets of data. (p. 696) flame ionization detector a nearly universal GC detector in which the solutes are combusted in an H 2 /air flame, producing a measurable current. (p. 570) flow injection analysis an analytical technique in which samples are injected into a carrier stream of reagents, or in which the sample merges with other streams carrying reagents before passing through a detector. (p. 649) fluorescence emission of a photon when the analyte returns to a lower-energy state with the same spin as the higher- energy state. (p. 423) fluorometer an instrument for measuring fluorescence that uses filters to select the excitation and emission wavelengths. (p. 428) formality the number of moles of solute, regardless of chemical form, per liter of solution (F). (p. 15) formal potential the potential of a redox reaction for a specific set of solution conditions, such as pH and ionic composition. (p. 332) formation constant the equilibrium constant for a reaction in which a metal and a ligand bind to form a metal–ligand complex (K f ). (p. 144) formula weight the mass of a compound containing one mole (FW). (p. 17) frequency the number of oscillations of an electromagnetic wave per second (n ). (p. 370) 772 Glossary 1400-Glos 9/8/99 4:43 PM Page 772 fronting a tail at the beginning of a chromatographic peak, usually due to injecting too much sample. (p. 555) F-test statistical test for comparing two variances to see if their difference is too large to be explained by indeterminate error. (p. 87) fundamental analysis an analysis whose purpose is to improve an analytical method’s capabilities. (p. 9) G galvanostat a device used to control the current in an electrochemical cell. (p. 464) gamma ray high-energy radiation emitted by nuclei (g ). (p. 642) gas chromatography a chromatographic technique in which the mobile phase is a gas. (p. 563) gas–liquid chromatography a chromatographic technique in which the mobile phase is a gas and the stationary phase is a liquid coated either on a solid packing material or on the column’s walls. (p. 564) Geiger counter an instrument for counting radioactive particles based on their ability to ionize an inert gas such as Ar. (p. 643) Gibb’s free energy a thermodynamic function for systems at constant temperature and pressure that indicates whether or not a reaction is favorable (?G < 0), unfavorable (?G > 0), or at equilibrium (?G = 0). (p. 137) glass electrode an ion-selective electrode based on a glass membrane in which the potential develops from an ion- exchange reaction on the membrane’s surface. (p. 477) good laboratory practices those general laboratory procedures that, when followed, help ensure the quality of analytical work. (p. 706) good measurement practices those instructions outlining how to properly use equipment and instrumentation to ensure the quality of measurements. (p. 706) grab sample a single sample removed from the target population. (p. 185) gradient elution the process of changing the mobile phase’s solvent strength to enhance the separation of both early and late eluting solutes. (p. 558) Gran plot a linearized form of a titration curve. (p. 293) graphite furnace an electrothermal atomizer that relies on resistive heating to atomize samples. (p. 414) gravimetry any method in which the signal is a mass or change in mass. (p. 233) gross sample the initial sample, collected from the target population without any processing. (p. 193) guard column an inexpensive column used to protect a more expensive analytical column. (p. 579) H half-life the time required for half of the initial number of a radioactive isotope’s atoms to disintegrate (t 1/2 ).(p. 643) hanging mercury drop electrode an electrode in which a drop of Hg is suspended from a capillary tube. (p. 509) headspace sampling the sampling of the vapor phase overlying a liquid phase. (p. 567) Henderson–Hasselbalch equation equation showing the relationship between a buffer’s pH and the relative amounts of the buffer’s conjugate weak acid and weak base. (p. 169) heterogeneous not uniform in composition. (p. 58) high-performance liquid chromatography a chromatographic technique in which the mobile phase is a liquid. (p. 578) histogram a plot showing the number of times an observation occurs as a function of the range of observed values. (p. 77) homogeneous uniform in composition. (p. 72) homogeneous precipitation a precipitation in which the precipitant is generated in situ by a chemical reaction. (p. 241) hydrodynamic injection an injection technique in capillary electrophoresis in which pressure is used to inject sample into the capillary column. (p. 602) hydrodynamic voltammetry a form of voltammetry in which the solution is mechanically stirred. (p. 513) I inclusion a coprecipitated impurity in which the interfering ion occupies a lattice site in the precipitate. (p. 238) inclusion limit in size-exclusion chromatography, the smallest solute that can be separated from other solutes; all smaller solutes elute together. (p. 593) indeterminate error any random error that causes some measurements or results to be too high while others are too low. (p. 62) indicator a colored compound whose change in color signals the end point of a titration. (p. 274) indicator electrode the electrode whose potential is a function of the analyte’s concentration (also known as the working electrode). (p. 462) in situ sampling sampling done within the population without physically removing the sample. (p. 186) intensity the flux of energy per unit time per area (I). (p. 371) interferometer a device that allows all wavelengths of light to be measured simultaneously, eliminating the need for a wavelength selector. (p. 378) internal conversion a form of radiationless relaxation in which the analyte moves from a higher electronic energy level to a lower electronic energy level. (p. 425) internal standard a standard, whose identity is different from the analyte’s, that is added to all samples and standards containing the analyte. (p. 116) intersystem crossing a form of radiationless relaxation in which the analyte moves from a higher electronic energy level to a lower electronic energy level with a different spin state. (p. 425) Glossary 773 1400-Glos 9/8/99 4:43 PM Page 773 ion-exchange chromatography a form of liquid chromatography in which the stationary phase is an ion- exchange resin. (p. 590) ionic strength a quantitative method for reporting the ionic composition of a solution that takes into account the greater effect of more highly charged ions (m ). (p. 172) ionization suppressor a reagent that is more easily ionizable than the analyte. (p. 420) ionophore a neutral ligand whose exterior is hydrophobic and whose interior is hydrophilic. (p. 482) ion-selective electrode an electrode in which the membrane potential is a function of the concentration of a particular ion in solution. (p. 475) ion-suppressor column a column used to minimize the conductivity of the mobile phase in ion-exchange chromatography. (p. 592) isocratic elution the use of a mobile phase whose composition remains constant throughout the separation. (p. 582) isotope dilution a form of internal standardization in which a radioactive form of the analyte serves as the internal standard. (p. 646) isotopes atoms with the same number of protons but different numbers of neutrons are called isotopes. (p. 642) J Jones reductor a reduction column using a Zn amalgam as a reducing agent. (p. 341) Joule heating the heating of a conductive solution due to the passage of an electric current through the solution. (p. 601) judgmental sampling samples collected from the target population using available information about the analyte’s distribution within the population. (p. 184) K Kjeldahl analysis an acid–base titrimetric method for determining the amount of nitrogen in organic compounds. (p. 302) Kovat’s retention index a means for normalizing retention times by comparing a solute’s retention time with those for normal alkanes. (p. 575) L laboratory sample sample taken into the lab for analysis after processing the gross sample. (p. 199) ladder diagram a visual tool for evaluating systems at equilibrium. (p. 150) Le Chatelier’s principle when stressed, a system that was at equilibrium returns to its equilibrium state by reacting in a manner that relieves the stress. (p. 148) leveling acids that are better proton donors than the solvent are leveled to the acid strength of the protonated solvent; bases that are better proton acceptors than the solvent are leveled to the base strength of the deprotonated solvent. (p. 296) lifetime the length of time that an analyte stays in an excited state before returning to a lower-energy state. (p. 423) ligand a Lewis base that binds with a metal ion. (p. 144) limit of identification the smallest concentration or absolute amount of analyte such that the probability of type 1 and type 2 errors are equal (LOI). (p. 95) limit of quantitation the smallest concentration or absolute amount of analyte that can be reliably determined (LOQ). (p. 96) linear regression a mathematical technique for fitting an equation, such as that for a straight line, to experimental data. (p. 118) line source a source that emits radiation at only select wavelengths. (p. 375) Lineweaver–Burk plot a graphical means for evaluating enzyme kinetics. (p. 638) liquid-based ion-selective electrode an ion-selective electrode in which a chelating agent is incorporated into a hydrophobic membrane. (p. 482) liquid junction potential a potential that develops at the interface between two ionic solutions that differ in composition, because of a difference in the mobilities of the ions (E lj ). (p. 470) liquid–solid adsorption chromatography a form of liquid chromatography in which the stationary phase is a solid adsorbent. (p. 590) longitudinal diffusion one contribution to band broadening in which solutes diffuse from areas of high concentration to areas of low concentration. (p. 560) loop injector a means for injecting samples in which the sample is loaded into a short section of tubing and injected onto the column by redirecting the mobile phase through the loop. (p. 584) M manifold the complete system of tubing for merging together samples and reagents in a flow injection analysis. (p. 652) masking a pseudo-separation method in which a species is prevented from participating in a chemical reaction by binding it with a masking agent in an unreactive complex. (p. 207) masking agent the reagent used to bind the species to be masked in an unreactive complex. (p. 208) mass balance equation an equation stating that matter is conserved, and that the total amount of a species added to a solution must equal the sum of the amount of each of its possible forms present in solution. (p. 159) mass spectrum a plot of ion intensity as a function of the ion’s mass-to-charge ratios. (p. 571) 774 Glossary 1400-Glos 9/8/99 4:43 PM Page 774 mass transfer one contribution to band broadening due to the time required for a solute to move from the mobile phase or the stationary phase to the interface between the two phases. (p. 561) mass transport the movement of material toward or away from the electrode surface. (p. 511) matrix all other constituents in a sample except for the analytes. (p. 36) matrix matching adjusting the matrix of an external standard so that it is the same as the matrix of the samples to be analyzed. (p. 110) mean the average value of a set of data ( – X). (p. 54) measurement an experimental determination of an analyte’s chemical or physical properties. (p. 36) measurement error an error due to limitations in the equipment and instruments used to make measurements. (p. 58) median that value for a set of ordered data, for which half of the data is larger in value and half is smaller in value ( – X med ). (p. 55) mediator a species that transfers electrons from the electrode to the analyte. (p. 500) membrane potential a potential developing across a conductive membrane whose opposite sides are in contact with solutions of different composition. (p. 475) meniscus the curved surface of a liquid contained in a tube. (p. 29) metallochromic indicator a visual indicator used to signal the end point in a complexation titration. (p. 323) method a means for analyzing a sample for a specific analyte in a specific matrix. (p. 36) method blank a sample that contains all components of the matrix except the analyte. (p. 45) method error an error due to limitations in the analytical method used to analyze a sample. (p. 58) method of continuous variations a procedure for determining the stoichiometry between two reactants by preparing solutions containing different mole fractions of one reactant; also known as Job’s method. (p. 404) method of standard additions a standardization in which aliquots of a standard solution are added to the sample. (p. 110) micellar electrokinetic capillary chromatography a form of capillary electrophoresis in which neutral solutes are separated based on their ability to partition into a charged micelle. (p. 606) micelle an agglomeration of molecules containing ionic “heads” and hydrophobic “tails,” which form into a structure with a hydrophobic interior and a hydrophilic exterior. (p. 606) Michaelis constant a combination of several rate constants affecting the rate of an enzyme–substrate reaction. (p. 637) migration the movement of a cation or anion in response to an applied potential. (p. 512) mobile phase in chromatography, the extracting phase that moves through the system. (p. 546) molality the number of moles of solute per kilogram of solvent (m). (p. 18) molarity the number of moles of solute per liter of solution (M). (p. 15) mole-ratio method a procedure for determining the stoichiometry between two reactants by preparing solutions containing different mole ratios of two reactants. (p. 406) monochromatic electromagnetic radiation of a single wavelength. (p. 377) monochromator a wavelength selector that uses a diffraction grating or prism, and that allows for a continuous variation of the nominal wavelength. (p. 376) multiple-point standardization any standardization using two or more standards containing known amounts of analyte. (p. 109) N negatron the beta particle formed when a neutron converts to a proton; equivalent to an electron ( 0 –1 b ). (p. 642) nephelometry a method in which the intensity of scattered radiation is measured at an angle of 90° to the source. (p. 442) Nernst equation an equation relating electrochemical potential to the concentrations of products and reactants. (p. 146) neutron activation a means of inducing radioactivity in a nonradioactive sample by bombarding the sample with neutrons. (p. 645) nominal wavelength the wavelength which a wavelength selector is set to pass. (p. 376) nonfaradaic current a current in an electrochemical cell that is not the result of a redox reaction. (p. 512) normal calibration curve a calibration curve prepared using several external standards. (p. 109) normal distribution “bell-shaped” probability distribution curve for measurements and results showing the effect of random error. (p. 73) normality the number of equivalents of solute per liter of solution (N). (p. 16) normal-phase chromatography liquid chromatography using a polar stationary phase and a nonpolar mobile phase. (p. 580) null hypothesis a statement that the difference between two values can be explained by indeterminate error; retained if the significance test does not fail (H 0 ). (p. 83) Nyquist theorem statement that a periodic signal must be sampled at least twice each period to avoid a determinate error in measuring its frequency. (p. 184) Glossary 775 1400-Glos 9/8/99 4:43 PM Page 775 O occlusion a coprecipitated impurity trapped within a precipitate as it forms. (p. 239) Ohm’s law the statement that the current moving through a circuit is proportional to the applied potential and inversely proportional to the circuit’s resistance (E = iR). (p. 463) on-column injection the direct injection of thermally unstable samples onto a capillary column. (p. 568) one-tailed significance test significance test in which the null hypothesis is rejected for values at only one end of the normal distribution. (p. 84) open tubular column a capillary column that does not contain a particulate packing material. (p. 564) outlier data point whose value is much larger or smaller than the remaining data. (p. 93) overpotential the difference between the potential actually required to initiate an oxidation or reduction reaction, and the potential predicted by the Nernst equation. (p. 497) oxidation a loss of electrons. (p. 146) oxidizing agent a species that accepts electrons from another species. (p. 146) P packed column a wide-bore column containing a particulate packing material. (p. 564) paired data two sets of data consisting of results obtained using several samples drawn from different sources. (p. 88) paired t-test statistical test for comparing paired data to determine if their difference is too large to be explained by indeterminate error. (p. 92) particulate gravimetry a gravimetric method in which the mass of a particulate analyte is determined following its separation from its matrix. (p. 234) partition coefficient an equilibrium constant describing the distribution of a solute between two phases; only one form of the solute is used in defining the partition coefficient (K D ). (p. 211) parts per billion nanograms of solute per gram of solution; for aqueous solutions the units are often expressed as micrograms of solute per liter of solution (ppb). (p. 18) parts per million micrograms of solute per gram of solution; for aqueous solutions the units are often expressed as milligrams of solute per liter of solution (ppm). (p. 18) peak capacity the maximum number of solutes that can be resolved on a particular column (n c ). (p. 554) peptization the reverse of coagulation in which a coagulated precipitate reverts to smaller particles. (p. 245) peristaltic pump a device for propelling liquids through flexible tubing. (p. 652) personal error an error due to biases introduced by the analyst. (p. 60) p-function a function of the form pX, where pX = –log(X). (p. 19) pH defined as pH = –log[H 3 O + ]. (p. 142) phosphorescence emission of a photon when the analyte returns to a lower-energy state with the opposite spin as the higher-energy state. (p. 424) photodiode array a linear array of photodiodes providing the ability to detect simultaneously radiation at several wavelengths. (p. 379) photoluminescence emission following absorption of a photon. (p. 374) photon a particle of light carrying an amount of energy equal to hn .(p. 371) pipet glassware designed to deliver a specific volume of solution when filled to its calibration mark. (p. 27) planar chromatography a form of chromatography in which the stationary phase is immobilized on a flat surface. (p. 546) plasma a hot, ionized gas containing an abundance of ions and electrons. (p. 435) polarity index a quantitative measure of a solvent’s polarity. (p. 580) polarography a form of voltammetry using a dropping mercury electrode or a static mercury drop electrode. (p. 515) polychromatic electromagnetic radiation of more than one wavelength. (p. 377) population all members of a system. (p. 71) positron the beta particle formed when a proton converts to a neutron ( 0 –1 b ). (p. 642) potentiometer a device for measuring the potential of an electrochemical cell without drawing a current or altering the cell’s composition. (p. 464) potentiostat a device used to control the potential in an electrochemical cell. (p. 465) power the flux of energy per unit time (P). (p. 371) precipitant a reagent that causes the precipitation of a soluble species. (p. 235) precipitate an insoluble solid that forms when two or more soluble reagents are combined. (p. 139) precipitation gravimetry a gravimetric method in which the signal is the mass of a precipitate. (p. 234) precipitation titration a titration in which the reaction between the analyte and titrant involves a precipitation. (p. 350) precision an indication of the reproducibility of a measurement or result. (p. 39) preconcentration the process of increasing an analyte’s concentration before its analysis. (p. 223) primary reagent a reagent of known purity that can be used to make a solution of known concentration. (p. 106) probability distribution plot showing frequency of occurrence for members of a population. (p. 71) procedure written directions outlining how to analyze a sample. (p. 36) 776 Glossary 1400-Glos 9/8/99 4:43 PM Page 776 proficiency standard a standard sample provided by an external agency as part of certifying the quality of a laboratory’s work. (p. 711) proportional determinate error a determinate error whose value depends on the amount of sample analyzed. (p. 61) protecting agent a reagent that reacts with the analyte, preventing it from transforming into a nonanalyzable form. (p. 420) protocol a set of written guidelines for analyzing a sample specified by an agency. (p. 37) protocol for a specific purpose a precisely written protocol for an analysis that must be followed exactly. (p. 707) purge and trap a technique for separating volatile analytes from liquid samples in which the analytes are subsequently trapped on a solid adsorbent. (p. 214) Q qualitative analysis an analysis in which we determine the identity of the constituent species in a sample. (p. 8) quality assessment the steps taken to evaluate whether an analysis is under statistical control. (p. 708) quality assurance the steps taken during an analysis to ensure that the analysis is under control and that it is properly monitored. (p. 706) quality assurance and quality control those steps taken to ensure that the work conducted in an analytical lab is capable of producing acceptable results; also known as QA/QC. (p. 48) quality control those steps taken to ensure that an analysis is under statistical control. (p. 706) quantitative analysis an analysis in which we determine how much of a constituent species is present in a sample. (p. 9) quantitative transfer the process of moving a sample from one container to another in a manner that ensures all material is transferred. (p. 30) quantum yield the fraction of absorbed photons that produce a desired event, such as fluorescence or phosphorescence (F ). (p. 425) quench to stop a reaction by suddenly changing the reaction conditions. (p. 634) R random sample a sample collected at random from the target population. (p. 183) range the numerical difference between the largest and smallest values in a data set (w). (p. 56) rate the change in a property’s value per unit change in time; the rate of a reaction is a change in concentration per unit change in time. (p. 624) rate constant in a rate law, the proportionality constant between a reaction’s rate and the concentrations of species affecting the rate (k). (p. 624) rate law an equation relating a reaction’s rate at a given time to the concentrations of species affecting the rate. (p. 624) reagent grade reagents conforming to standards set by the American Chemical Society. (p. 107) recovery the fraction of analyte or interferent remaining after a separation (R). (p. 202) redox electrode an inert electrode that serves as a source or sink for electrons for a redox half-reaction. (p. 475) redox indicator a visual indicator used to signal the end point in a redox titration. (p. 339) redox reaction an electron-transfer reaction. (p. 145) redox titration a titration in which the reaction between the analyte and titrant is an oxidation/reduction reaction. (p. 331) reducing agent a species that donates electrons to another species. (p. 146) reduction a gain of electrons. (p. 146) reference electrode an electrode whose potential remains constant and against which other potentials can be measured. (p. 462) relative supersaturation a measure of the extent to which a solution, or a localized region of solution, contains more dissolved solute than that expected at equilibrium (RSS). (p. 241) relaxation any process by which an analyte returns to a lower- energy state from a higher-energy state. (p. 423) releasing agent a reagent whose reaction with an interferant is more favorable than the interferant’s reaction with the analyte. (p. 420) repeatability the precision for an analysis in which the only source of variability is the analysis of replicate samples. (p. 62) reproducibility the precision when comparing results for several samples, for several analysts or several methods. (p. 62) residual current the current in an electrochemical cell that is present even in the absence of the analyte. (p. 513) residual error the difference between an experimental value and the value predicted by a regression equation. (p. 118) resolution in spectroscopy, the separation between two spectral features, such as absorption or emission lines. (p. 376) response the property of a system that is measured (R). (p. 667) response surface a graph showing how a system’s response changes as a function of its factors. (p. 667) retention time the time a solute takes to move from the point of injection to the detector (t r ). (p. 548) retention volume the volume of mobile phase needed to move a solute from its point of injection to the detector (V r ). (p. 548) Glossary 777 1400-Glos 9/8/99 4:43 PM Page 777 reverse-phase chromatography liquid chromatography using a nonpolar stationary phase and a polar mobile phase. (p. 580) robust a method that can be applied to analytes in a wide variety of matrices is considered robust. (p. 42) rugged a method that is insensitive to changes in experimental conditions is considered rugged. (p. 42) ruggedness testing the process of evaluating a method to determine those factors for which a small change in value has a significant effect on the method’s results. (p. 684) S salt bridge a connection between two solutions that allows the movement of current in the form of ionic charge. (p. 466) sample those members of a population that we actually collect and analyze. (p. 71) sampling error an error introduced during the process of collecting a sample for analysis. (p. 58) sampling plan a plan that ensures that a representative sample is collected. (p. 182) saturated calomel electrode reference electrode based on the reduction of Hg 2 Cl 2 to Hg in an aqueous solution saturated with KCl; that is, Hg 2 Cl 2 (s)+2e – t 2Hg(l)+ 2Cl – (aq). (p. 472) scientific notation a shorthand method for expressing very large or very small numbers by indicating powers of ten; for example, 1000 is 1 · 10 3 .(p. 12) scintillation counter an instrument for counting radioactive particles based on their ability to initiate fluorescence in another substance. (p. 643) secondary reagent a reagent whose purity must be established relative to a primary reagent. (p. 107) selectivity a measure of a method’s freedom from interferences as defined by the method’s selectivity coefficient. (p. 40) selectivity coefficient a measure of a method’s sensitivity for an interferent relative to that for the analyte (K A,I ). (p. 40) selectivity factor the ratio of capacity factors for two solutes showing the column’s selectivity for one of the solutes (a ). (p. 552) self-absorption in atomic emission, the decrease in emission intensity when light emitted by excited state atoms in the center of a flame or plasma is absorbed by atoms in the outer portion of the flame. (p. 438) sensitivity a measure of a method’s ability to distinguish between two samples; reported as the change in signal per unit change in the amount of analyte (k). (p. 39) separation factor a measure of the effectiveness of a separation at separating an analyte from an interferent (S I,A ) . (p. 203) signal an experimental measurement that is proportional to the amount of analyte (S). (p. 37) signal averaging the adding together of successive spectra to improve the signal-to-noise ratio. (p. 391) signal processor a device, such as a meter or computer, that displays the signal from the transducer in a form that is easily interpreted by the analyst. (p. 380) signal-to-noise ratio the ratio of the signal’s intensity to the average intensity of the surrounding noise. (p. 379) significance test a statistical test to determine if the difference between two values is significant. (p. 83) significant figures the digits in a measured quantity, including all digits known exactly and one digit (the last) whose quantity is uncertain. (p. 13) silver/silver chloride electrode reference electrode based on the reduction of AgCl to Ag; that is, AgCl(s)+e – t Ag(s)+ Cl – (aq). (p. 473) simplex optimization an efficient optimization method that allows several factors to be optimized at the same time. (p. 672) single-column ion chromatography ion-exchange chromatography in which conditions are adjusted so that an ion-suppressor column is not needed. (p. 593) single-point standardization any standardization using a single standard containing a known amount of analyte. (p. 108) singlet excited state an excited state in which all electron spins are paired. (p. 423) SI units stands for Système International d’Unités. These are the internationally agreed on units for measurements. (p. 12) size-exclusion chromatography a separation method in which a mixture passes through a bed of porous particles, with smaller particles taking longer to pass through the bed due to their ability to move into the porous structure. (p. 206) slope-ratio method a procedure for determining the stoichiometry between two reactants by measuring the relative change in absorbance under conditions when each reactant is the limiting reagent. (p. 407) solid-phase microextraction a solid-phase extraction in which the solid adsorbent is coated on a fused-silica fiber held within a syringe needle. (p. 567) solid-state ion-selective electrode an ion-selective electrode based on a sparingly soluble inorganic crystalline material. (p. 479) solubility product the equilibrium constant for a reaction in which a solid dissociates into its ions (K sp ). (p. 140) spectral searching the matching of a spectrum for an unknown compound to a reference spectrum stored in a computer database. (p. 403) spectrofluorometer an instrument for measuring fluorescence that uses a monochromator to select the excitation and emission wavelengths. (p. 428) spectrophotometer an instrument for measuring absorbance that uses a monochromator to select the wavelength. (p. 389) 778 Glossary 1400-Glos 9/8/99 4:43 PM Page 778 spike recovery an analysis of a sample after spiking with a known amount of analyte. (p. 710) split injection a technique for injecting samples onto a capillary column in which only a small portion of the sample enters the column. (p. 568) splitless injection a technique for injecting a sample onto a capillary column that allows a higher percentage of the sample to enter the column. (p. 568) stacking a means of concentrating solutes in capillary electrophoresis after their injection onto the capillary column. (p. 603) standard deviation a statistical measure of the “average” deviation of data from the data’s mean value (s). (p. 56) standard deviation about the regression the uncertainty in a regression analysis due to indeterminate error (s r ). (p. 121) standard hydrogen electrode reference electrode based on the reduction of H + (aq) to H 2 (g) at a Pt electrode; that is, H + (aq)+e – t 1 ?2H 2 (g). (p. 471) standardization the process of establishing the relationship between the amount of analtye and a method’s signal. (p. 47) standard method a method that has been identified as providing acceptable results. (p. 667) standard operations procedure the procedure followed in collecting and analyzing samples and in interpreting the results of an analysis. (p. 707) standard reference material a material available from the National Institute of Standards and Technology certified to contain known concentrations of analytes. (p. 61) standard state condition in which solids and liquids are in pure form, gases have partial pressures of 1 atm, solutes have concentrations of 1 M, and the temperature is 298 K. (p. 137) static mercury drop electrode an electrode in which successive drops of Hg form at the end of a capillary tube as the result of a mechanical plunger, with each drop providing a fresh electrode surface. (p. 509) stationary phase in chromatography, the extracting phase that remains in a fixed position. (p. 546) steady-state approximation in a kinetic process, the assumption that a compound formed during the reaction reaches a concentration that remains constant until the reaction is nearly complete. (p. 636) stepwise formation constant the formation constant for a metal–ligand complex in which only one ligand is added to the metal ion or to a metal–ligand complex (K i ). (p. 144) stock solution a solution of known concentration from which other solutions are prepared. (p. 30) stopped flow a kinetic method of analysis designed to rapidly mix samples and reagents when using reactions with very fast kinetics. (p. 634) stratified sampling a sampling plan that divides the population into distinct strata from which random samples are collected. (p. 185) stray radiation any radiation reaching the detector that does not follow the optical path from the source to the detector. (p. 387) stripping voltammetry a form of voltammetry in which the analyte is first deposited on the electrode and then removed, or “stripped,” electrochemically while monitoring the current as a function of the applied potential. (p. 516) substrate the specific molecule for which an enzyme serves as a catalyst. (p. 636) supercritical fluid a state of matter where a substance is held at a temperature and pressure that exceeds its critical temperature and pressure. (p. 215) supercritical fluid chromatography a separation technique in which the mobile phase is a supercritical fluid. (p. 596) supernatant the solution that remains after a precipitate forms. (p. 244) support-coated open tubular column an open tubular column in which the stationary phase is coated on a solid support that is attached to the column’s walls. (p. 565) systematic–judgmental sampling a sampling plan that combines judgmental sampling with systematic sampling. (p. 184) systematic sampling samples collected from the target population at regular intervals in time or space. (p. 184) T tailing a tail at the end of a chromatographic peak, usually due to the presence of highly active sites in the stationary phase. (p. 555) technique a chemical or physical principle that can be used to analyze a sample. (p. 36) temperature programming the process of changing the column’s temperature to enhance the separation of both early and late eluting solutes. (p. 558) theoretical model a model describing a system’s response that has a theoretical basis and can be derived from theoretical principles. (p. 675) theoretical plate a quantitative means of evaluating column efficiency that treats the column as though it consists of a series of small zones, or plates, in which partitioning between the mobile and stationary phases occurs. (p. 553) thermal conductivity detector a universal GC detector in which the signal is a change in the thermal conductivity of the mobile phase. (p. 569) thermogram a graph showing change in mass as a function of applied temperature. (p. 256) thermogravimetry a form of volatilization gravimetry in which the change in a sample’s mass is monitored while it is heated. (p. 255) titrant the reagent added to a solution containing the analyte and whose volume is the signal. (p. 274) Glossary 779 1400-Glos 9/8/99 4:43 PM Page 779 titration curve a graph showing the progress of a titration as a function of the volume of titrant added. (p. 276) titration error the determinate error in a titration due to the difference between the end point and the equivalence point. (p. 274) titrimetry any method in which volume is the signal. (p. 274) tolerance the maximum determinate measurement error for equipment or instrument as reported by the manufacturer. (p. 58) total analysis techniques a technique in which the signal is proportional to the absolute amount of analyte; also called “classical” techniques. (p. 38) total ionic strength adjustment buffer a solution containing a relatively high concentration of inert electrolytes such that its composition fixes the ionic concentration of all solutions to which it is added. (p. 487) total Youden blank a blank that corrects the signal for analyte–matrix interactions. (p. 129) tracer a radioactive species used as an internal standard. (p. 646) transducer a device that converts a chemical or physical property, such as pH or photon intensity, to an easily measured electrical signal, such as a voltage or current. (p. 379) transmittance the ratio of the radiant power passing through a sample to that from the radiation’s source (T). (p. 384) trip blank a blank prepared in the laboratory that accompanies a set of sample containers in the field and laboratory. (p. 710) triplet excited state an excited state in which unpaired electron spins occur. (p. 423) t-test statistical test for comparing two mean values to see if their difference is too large to be explained by indeterminate error. (p. 85) turbidimetry a method in which the decrease in transmitted radiation due to scattering is measured. (p. 441) two-tailed significance test significance test in which the null hypothesis is rejected for values at either end of the normal distribution. (p. 84) type 1 error the risk of falsely rejecting the null hypothesis (a ). (p. 84) type 2 error the risk of falsely retaining the null hypothesis (b ). (p. 84) U uncertainty the range of possible values for a measurement. (p. 64) unpaired data two sets of data consisting of results obtained using several samples drawn from a single source. (p. 88) V validation the process of verifying that a procedure yields acceptable results. (p. 47) van Deemter equation an equation showing the effect of the mobile phase’s flow rate on the height of a theoretical plate. (p. 561) variance the square of the standard deviation (s 2 ). (p. 57) vibrational relaxation a form of radiationless relaxation in which an analyte moves from a higher vibrational energy level to a lower vibrational energy level in the same electronic level. (p. 424) void time the time required for unretained solutes to move from the point of injection to the detector (t m ). (p. 549) void volume the volume of mobile phase needed to move an unretained solute from the point of injection to the detector. (p. 549) volatilization gravimetry a gravimetric method in which the loss of a volatile species gives rise to the signal. (p. 234) voltammetry an electrochemical method in which we measure current as a function of the applied potential. (p. 508) voltammogram a plot of current as a function of applied potential. (p. 508) volume percent milliliters of solute per 100 mL of solution (% v/v). (p. 18) volumetric flask glassware designed to contain a specific volume of solution when filled to its calibration mark. (p. 26) W Walden reductor a reduction column using granular Ag as a reducing agent. (p. 341) wall-coated open tubular column an open tubular column in which the stationary phase is coated on the column’s walls. (p. 565) wavelength the distance between any two consecutive maxima or minima of an electromagnetic wave (l ). (p. 370) wavenumber the reciprocal of wavelength ( – n ). (p. 370) weight percent grams of solute per 100 g of solution. (% w/w). (p. 18) weight-to-volume percent grams of solute per 100 mL of solution (% w/v). (p. 18) Z zeta potential the change in potential across a double layer (z ). (p. 599) 780 Glossary 1400-Glos 9/8/99 4:43 PM Page 780 Page numbers followed by italic f and t refer to figures and tables, respectively. A Absorbance, 384–385, 373 concentration and, 385–386, 385f and end point in complexation titration, 324, 324f spectrum, 373, 373f Absorption of electromagnetic radiation, 380–384 filters, 376 of infrared versus UV/visible radiation, 381, 381f lines for copper, 416, 418t for sodium, 384, 384f of photons, energy level diagram for, 372, 372f spectroscopies based on, 373t, 380–388 Accuracy, 38–39 in acid-base titrations, 312 in atomic absorption spectroscopy, 422 in atomic emission spectroscopy, 440 in chemical kinetics methods, 640 in coulometry, 507–508 and experimental errors, 57–62 in gas chromatography, 577 in photoluminescence spectroscopy, 432 in potentiometry, 494–495 in precipitate gravimetry, 254 in ultraviolet/visible spectroscopy, 409 in voltammetry, 531 Acetic acid, acid dissociation constant for, 732t Acid(s), 140 Br?nsted-Lowry definition of, 144 equivalent weights of, 309–310 pH of monoprotic acid, 160–163 polyprotic acid, 163–165 triprotic acids, 165 and sample digestion, 200t strong, 140–141 leveling of, 296 standardizing as titrant, 298–300, 299t titration curves for, 280–281, 281t weak, 140–141 EDTA as, 315 titration curves for, 281–284, 283f, 283t, 288, 288f Acid-base reactions, 140–144 conservation of protons in, 22–23 ladder diagrams for, 150–152, 151f liquid-liquid extractions involving, 219–221, 219f, 221f reaction units in, 22 using coulometric titrations, 504, 504t Acid-base titration(s), 278–314 accuracy in, 312 characterization applications of, 309–311 and equilibrium constants, 310–311, 310f and equivalent weights, 309–310 evaluation of, 311–314 indicators in, pH ranges of, 278, 288–290, 289f–290f, 289t inflection points in, 287–288, 290–291 in nonaqueous solvents, 294f, 295–296, 296f in organic analysis, 302–303, 303t pH and, 279 precision in, 312 qualitative applications involving, 308, 308t quantitative applications involving, 298–308 calculations for, 304–308 representative model of, 296–298 scale of operation in, 311–312, 311f selectivity in, 313, 314f sensitivity in, 312–313 time, cost and equipment in, 313–314 Acid-base titration curves, 279–287 sketching of, 284–287, 285f–286f of strong acids and strong bases, 280–281, 281t of weak acids and weak bases, 281–284, 283f, 283t Acid dissociation constants, 140–141, 732t–739t calculation of, 143 Acidity, 301–302 Activity, 172 versus concentration, in potentiometry, 485–486 Activity coefficient, 172–173 Activity effects, 171–174 Adipic acid, acid dissociation constant for, 732t Adjusted retention time, 551 Adsorbents, for solid-phase extractions, 212–213, 213t Adsorption chromatography, 547, 547f Aerosol, 7 Aerosol pollutants, 7–8 Alanine acid dissociation constant for, 732t ladder diagram for, 163–165, 163f Alcohol, acid-base titrimetric procedures for, 303, 303t Algorithms, searching. See Searching algorithms Aliasing, 184, 185f Aliquot, 111 Alizarin garnet R, 429, 429f Alizarin yellow R, 289t Alkalinity, 300–301, 308 end point volumes and, 301, 302t Alkemade, C. T. J., 412 Alpha particles, 642 Alternative hypothesis, 83 Alumina, 213t Aluminum standard reduction potentials for, 743t in waters and wastewater, 395t Amalgam, 509 Amino acids, aromatic, natural fluorescence and phosphorescence of, 430t Aminobenzene, acid dissociation constant for, 732t Aminocarboxylic acids, 314 Ammonia acid dissociation constant for, 733t as auxiliary complexing agent, 316, 317t Nessler’s method for, 368 in waters and wastewater, 395t Ammonium pyrrolidine dithiocarbamate (APDC), 223–224 Ampere, 12t Amperometric biosensors, 525, 526t Amperometric flow cell, for high-performance liquid chromatography, 585, 585f Amperometry, 519–520, 520f Amphiprotic species, 141–142 IndexIndex 781 1400-Indx 9/27/99 2:17 PM Page 781 Analysis, 36. See also specific types of analysis blind, 683 chemical, versus analytical chemistry, 2, 8 fundamental, 9 inorganic. See Inorganic analysis multielemental, using atomic emission spectroscopy, 436, 436f neutron activation, 645–646 organic. See Organic analysis qualitative, 8 total, 38 Analysis of variance (ANOVA), 693–697, 694t–695t Analyte(s), 36 adjusting concentration of, in gas chromatography, 568 adjustment of oxidation state of, in redox titration, 341–342 inorganic in fluorescence spectroscopy, 429, 429f, 430t in phosphorescence spectroscopy, 429, 429f in photoluminescence spectroscopy, 429, 429f, 430t organic in fluorescence spectroscopy, 429–430 in phosphorescence spectroscopy, 429–430 in photoluminescence spectroscopy, 429–430, 431t radioactive, direct analysis of, 644–645 separation of, from interferents, 201–202 Analytical chemistry analysis, determination and measurement in, 36 basic equipment and instrumentation used in, 25–29 basic tools of, 11–32 versus chemical analysis, 2, 8 common problems in, 8–9 concentration units used in conversion between, 18–19 expression of, 15–20 description of, 2–5 laboratory notebook in, 32 language of, 35–50 numbers in, 12–15, 12t–13t preparation of solutions in, 30–32 stoichiometric calculations in, 20–25 Analytical data. See Data Analytical method(s) accuracy in, 38–39 classification systems for, 622 magnitude of relative error and, 39 equipment, time and cost of, 44 precision in, 39 robustness and ruggedness of, 42 scale of operation in, 42–44, 43f selection of, 38–45 selectivity of, 40–42 sensitivity in, 39–40 Analytical methodology importance of, 48–50 levels of, 36–37 Analytical perspective, 5–8, 6f Analytical separations classifying of, 546–547 overview of, 544–547 problems with, 544 Analyzer(s) centrifuge, 634, 635f Ektachem, 493, 493f stopped flow, 634, 635f Angstrom, 13t Anions, inorganic effective diameters, 173, 173t selected gravimetric method for, 248, 248t Anode, 466 Anodic current, 510 ANOVA. See Analysis of variance (ANOVA) Antabuse, 21, 24 Antinomy, standard reduction potentials for, 743t APDC. See Ammonium pyrrolidine dithiocarbamate (APDC) Apezion L, 566t Applied potential, and faradaic current, 510–511 Argentometric titrations, 355 Arginine, acid dissociation constant for, 733t Aromatic amino acids, natural fluorescence and phosphorescence of, 430t Arsenic standard reduction potentials for, 743t in waters and wastewater, 395t Arsenic acid, acid dissociation constant for, 733t Asparagine, acid dissociation constant for, 733t Aspartic acid, acid dissociation constant for, 733t Asymmetry potential, 476 Atmosphere, 13t Atom(s), UV/visible spectrum for, 383–384 Atomic absorption detection limits, 415, 415t Atomic absorption spectrophotometer, versus molecular absorption, 412 Atomic absorption spectroscopy, 373t, 412–423 cathode lamp used in, 415, 415f chemical interference in, minimization of, 419–420 evaluation in, 422 instrumentation in, 412–415 precision in, 422 preparing sample for, 418 quantitative applications involving, 415–422 representative method in, 421 selecting wavelength and slit width in, 415–418 selectivity in, 422 sensitivity in, 422 spectral interference in, minimization of, 418–419 standardizing method in, 420–421 time, cost and equipment in, 422 Atomic emission detection limits, 437, 437t Atomic emission spectrometer, 436, 436f 782 Index Atomic emission spectroscopy, 373t, 434–441 accuracy in, 440 atomization and excitation in, 435 sources of, 437 chemical interferences in, limiting of, 438 equipment in, 435–436 evaluation in, 440–441 external standards in, 438 flame sources in, 435 multielemental analysis using, 436, 436f plasma sources in, 435–436, 435f precision in, 440 preparing sample in, 437 quantitative applications using, 437–440 representative methods in, 439–440 scale of operation in, 440 selectivity in, 440 sensitivity in, 440 spectra in, 434–435 spectral interferences in, limiting of, 437–438, 438f standardizing method in, 438–440 time, cost and equipment in, 441 wavelength and slit width in, selection of, 437 Atomic fluorescence spectroscopy, 373t Atomization, 412 in atomic emission spectroscopy, 435 sources of, 437 electrothermal versus flame, 416 by miscellaneous methods, 415 Attenuated total reflectance cell (ATR), for infrared spectroscopy, 393–394, 393f Autoprotolysis constant. See Dissociation constant, for solvent Auxiliary complexing agent, 316–317 ammonia as, 316, 317t Auxiliary electrode, 463 Auxiliary oxidizing agent, 341 Auxiliary reducing agent, 341 B Background correction, 419, 437–438, 438f Back titration, 275 Balance calibration of, 105–106 electronic, 25, 25f measurement errors for, 59t single-arm mechanical, 25, 26f Band broadening, 553 longitudinal diffusion and, 560–561, 560f mass transfer and, 560f, 561 multiple paths and, 560, 560f Bandwidth, effective, 376 Barbiturates, serum, UV/visible spectroscopy analysis of, 397t Barium, standard reduction potentials for, 743t Base(s), 140 Br?nsted-Lowry definition of, 144 equivalent weights of, 309–310 pH of polyprotic, 163–165 triprotic, 165 1400-Indx 9/27/99 2:17 PM Page 782 C Cadmium standard reduction potentials for, 743t in waters and wastewater, 395t Calcium, standard reduction potentials for, 743t Calibration, 47 of balance, 105–106 in procedure, 47 of signals, 105–106 Calibration curve, 47 linear regression and, 117–127 for method of standard additions, 114–115, 114f normal, 109, 109f effect of sample’s matrix on, 110, 110f straight-line, linear regression of, 118, 119f Calmagite, 323t Calomel electrodes, saturated, 472, 473f Capacity factor, 550–552 and chromatographic resolution, optimization of, 556–558, 557f–558f Capillary column, 562–563, 564–565, 565f injecting sample in, 568, 569f Capillary electrochromatography (CEC), 607 Capillary electrophoresis, 597–598 applying electric field in, 603–604 detectors in, 604, 604f, 605t efficiency of, 600–601 evaluation in, 609 injecting sample in, 602–603 instrumentation in, 601–604, 601f methods in, 604–607 migration time in, 600 representative methods in, 607–608 resolution in, 601 reversal of electroosmotic flow in, 605, 605f selectivity in, 601 theory of, 598–601 total mobility in, 599–600 types of, 604–607 Capillary gel electrophoresis (CGE), 606–607 Capillary tube, 601, 601f Capillary zone electrophoresis (CZE), 604–606 Carbon, standard reduction potentials for, 743t Carbonate, solubility products for, 731t Carbonic acid, acid dissociation constant for, 733t Catechol, acid dissociation constant for, 733t Catecholamines, natural fluorescence of, 430t Cathode, 466 Cathode lamp, 415, 415f Cathodic current, 510 Cations, inorganic effective diameters, 173, 173t selected gravimetric method for, 248, 248t–249t CEC. See Capillary electrochromatography (CEC) Cell(s) amperometric flow, 585, 585f and sample digestion, 200t strong, 141 standardizing as titrant, 298–300, 299t titration curves of, 280–281, 281t weak, 141 titration curves of, 281–284, 283f, 283t Base dissociation constant, 141 calculation of, 143 Baseline width, 548 Beer’s law, 385–386, 385f limitations to, 386–388, 386f and multicomponent samples, 386 and ultraviolet/visible spectroscopy, 394 Benzoic acid, acid dissociation constant for, 733t Benzylamine, acid dissociation constant for, 733t Berthollet, Claude, 136, 331 Beryllium, standard reduction potentials for, 743t Beta particles, 642 Binomial distribution, 72–73, 73f Biosensor(s) amperometric, 525, 526t potentiometric, 484–485, 485f–486f, 486t Bipyridine, 339t Bismuth, standard reduction potentials for, 743t Blank(s) field, 710 method, 45, 128–129, 128t–129t in quality assessment, 710 total Youden, 129 trip, 710 Bleed, in gas chromatography, 566–567 Blind analysis, of standard samples, 683 Bonded stationary phase, 580 Boric acid, acid dissociation constant for, 733t Boron, standard reduction potentials for, 743t Breakthrough volume, 196 Bromide, solubility products for, 731t Bromine, standard reduction potentials for, 743t Bromocresol green, 289t Bromocresol purple, 289t Bromophenol blue, 289t Bromothymol blue, 289t absorbance spectrum for, 373, 373f Br?nsted-Lowry acids and bases, 144 Buffer solutions, 167–171 ladder diagrams for, 170–171, 171f pH of, 169–170, 492t problems involving, systematic solution to, 168–170 standard buffers, 492t total ionic strength adjustment buffer, 487 Bulk electrochemical methods, 462 Bunsen, Robert, 412 Burets, 277–278, 278f, 278t measurement errors for, 59t Index 783 attenuated total reflectance, 393–394, 393f electrochemical potentiometric. See Potentiometric electrochemical cells voltammetric, 510, 511f in ultraviolet/visible spectroscopy, 391, 392f Cellular components, centrifugation of, 207, 207t Celsius scale, 13t Central composite designs, for response surfaces, 682–683, 683f Central limit theorem, and sample distributions, 77–79, 78t, 79f Central tendency, measures of, 54–55 Centrifugation, 205t, 207, 207t equilibrium-density-gradient, 207, 207f Centrifuge analyzer, 634, 635f Cerium, standard reduction potentials for, 743t CGE. See Capillary gel electrophoresis (CGE) Characteristic concentrations, 416 for copper, 416, 418t Characterization analysis, 9 Characterization applications of acid-base titrations, 309–311 of chemical kinetics methods, 638–639 of coulometry, 506–507 of radiochemical methods, 647–648 of ultraviolet/visible spectroscopy, 403–409 of voltammetry, 527–530 Charge conservation of, 22 SI and non-SI units for, 13t Charge balance equation, 159 Charging current, 513 Chelating agents for fluorometric analysis, 429, 430t liquid-liquid extractions involving, 221–223, 221f–222f Chemical analysis, versus analytical chemistry, 2, 8 Chemical energy, in spectroscopy, 375 Chemical equilibria, 136 Chemical interference in atomic absorption spectroscopy, 419–420 in atomic emission spectroscopy, 438, 440 Chemical kinetics, review of, 750–754 Chemical kinetics methods, 623–642 accuracy in, 640 characterization applications of, 638–639 classification of, 625, 626f curve-fitting methods, 631 direct-computation integral methods, 626–629 direct-computation rate methods, 629–630 evaluation of, 639–642 instrumentation in, 634–636 miscellaneous methods, 632 precision in, 640 quantitative applications using, 636–638 enzyme-catalyzed reactions, 636–637, 637f noncatalytic reactions, 638 nonenzyme-catalyzed reactions, 637–638 representative methods in, 632–633 scale of operation in, 639–640 selectivity in, 640–642 1400-Indx 9/27/99 2:17 PM Page 783 sensitivity in, 640 theory and practice, 623–642 time, cost and equipment in, 642 Chemical limitations, to Beer’s law, 386–387 Chemical oxygen demand (COD), 346, 653 Chemical reactions. See Reaction(s) Chemical state, changes in, separations based on, 210–211, 210t Chemiluminescence, 374 Chemistry analytical. See Analytical chemistry equilibrium, 135–175 Chesapeake Bay monitoring program, 50 Chloride, solubility products for, 731t Chlorine, standard reduction potentials for, 743t Chlorine residual, in waters and wastewater, 395t, 396 Chloroacetic acid, acid dissociation constant for, 734t Chlorophenol red, 289t Chlorpromazine, 526–527 Cholesterol, serum, 397t Chromate, solubility products for, 731t Chromatic columns, in gas chromatography, 564–565 Chromatogram, 548 Chromatographic resolution, 549–550, 549f–550f Chromatographic separations, 215 optimizing of, 556–563 phases in, 546 Chromatography, 546–597 adsorption, 547, 547f capillary electrochromatography, 607 column. See Column chromatography electrophoretic, 547, 547f extraction, 205t gas. See Gas chromatography gas-liquid, 564 high-performance. See High-performance liquid chromatography ion-exchange, 547, 547f, 590–593, 594f micellar electrokinetic capillary, 606 molecular-exclusion. See Size-exclusion chromatography normal-phase, 580 optimizing chromatographic separations in, 556–563 optimizing resolution in capacity factor and, 556–558, 557f–558f column efficiency and, 559–563, 560t column selectivity and, 558, 559f partition, 547, 547f planar, 546 reverse-phase, 580 single-column ion, 593 size-exclusion, 205t, 206, 593–596, 595f–596f superfluid, 596–597, 597t Chromic acid, acid dissociation constant for, 734t Chromium standard reduction potentials for, 744t in waters and wastewater, 395t Chromophore, 382 Citric acid, acid dissociation constant for, 734t Clark, L. C., 519 Classical techniques. See Total analysis techniques Clinical applications, of potentiometry, 492–494 CLP. See Contract Laboratory Program (CLP) Coagulation, 242 Cobalt, standard reduction potentials for, 744t COD. See Chemical oxygen demand (COD) Codeine, 41–42 Coliform bacteria, in drinking water, 36, 37f Collaborative test, 687, 688f analysis of variance and, 693–697, 694t–695t reasonable results of, 698 two-sample, 688–693, 689f, 691f Collection of gas samples, 195–196 of solid samples, 197–198, 197f of solution samples, 193–194 Colorimeter, 388 Colorimetry, 368 Column chromatography, 546–547 capacity factor in, 550–552 column efficiency in, 552–554 general theory of, 547–556, 548f nonideal behavior in, 555, 555f peak capacity in, 554–555 resolution in, 549–550 selectivity factor in, 552 Column efficiency in column chromatography, 552–554 and optimizing resolution in chromatography, 559–563, 560t longitudinal diffusion and, 560–561, 560f mass transfer and, 560f, 561 multiple paths and, 560, 560f Column selectivity, and optimizing resolution in chromatography, 558, 559f Common ion effect, 157–158 Competitive inhibition, of enzymes, 639, 640f Complexation reactions, 144–145 effect on solubility, 165–167 equilibrium constants for, 145 ladder diagrams for, 153–155, 153f–155f reaction units in, 23 separation techniques based on, 207–209 using coulometric titrations, 504, 504t Complexation titrations, 314–331 auxiliary complexing agent in, 316–317 end point in absorbance and, 324, 324f indicators and, 323–324, 323t selection and evaluation of, 322–324 evaluation of, 331, 331f in inorganic analysis, 327–328 quantitative applications involving, 327–331 calculation of, 328–331 representative method of, 324–327 784 Index titrants in, selection and standardization of, 327 titration curves for, 324, 324f, 331, 331f Complexed ions, versus free ions, in potentiometry, 489 Complexing agent, auxiliary, 316–317 Composite sample, 185 Composition, of final precipitate in precipitate gravimetry, 245–246 Concentration. See also Preconcentration, versus separation and absorbance, 385–386, 385f versus activity, in potentiometry, 485–486 adjustment of, in gas chromatography, 568 analysis techniques, 38 determination of by nephelometry, 443 by turbidimetry, 442–443 by voltammetry, 514 in potentiometric electrochemical cells, 468–470 units of, 15–20, 16t Conditional metal-ligand formation constants, for EDTA, 315–316, 316t Confidence intervals for populations, 75–77, 75t for samples, 80–81, 81t Congo red, 289t Coning and quartering, 199, 199f Conservation principles of charge, 22 of electron pairs, 23 of electrons, 23 of mass, 22 and gravimetry, 234–235 of protons, 22–23 use in stoichiometric calculations, 23–25 Constant determinate errors, 60–61, 60t, 61f effect on k in single-point standardization, 118, 118t Continuous extraction, 213–215, 214f of supercritical fluids, 215 of volatile organic compounds, 214 Continuous variations, method of, Color Plate 6, 404–406 Continuum source, 375, 375f and background correction, 419 Contract Laboratory Program (CLP), 48 protocols for trace metals, 48, 49f Control chart(s), 714 interpretation of, 718–721 precision, 717–718, 717t, 718f–719f property, 715–716, 716f, 720f using for quality assurance, 721 Controlled-current coulometry, 499–501. See also Coulometric titrations; Coulometry current-time curve for, 499, 499f–500t end point determination in, 500 instrumentation for, 500–501, 501f maintaining current efficiency in, 499–500, 500f quantitative applications using, 501–503, 502t Controlled-potential coulometry 497–498, 497f. See also Coulometric titrations; Coulometry current-time curve for, 498, 498f 1400-Indx 9/27/99 2:17 PM Page 784 Current efficiency, 496, 499–500, 500f Curve-fitting methods, 631 Curvilinear regression, 127 Cyanide solubility products for, 731t in waters and wastewater, 395t Cyanopropyl, 213t Cyanopropyl-phenylmethyl polysiloxane, 566t Cysteine, acid dissociation constant for, 734t CZE. See Capillary zone electrophoresis (CZE) D Dark current, 379 Data characterizing measurements and results in, 54–57 evaluation of, 53–96 paired, 88, 91–93 statistical analysis of, 82–85 unpaired, 88–91 Debye-Hückel theory, 173–174 Decomposition, of inorganic samples, common fluxes for, 201, 201t Degrees of freedom, 80 Density, separation techniques based on, 206–207 Derivative method, for finding end point, 291–293, 291t, 292f Desiccant, 29 Desiccator, 29, 29f Detection limits, 39, 95–96, 95f–96f atomic absorption, 415, 415t atomic emission, 437, 437t of inductively coupled plasma torch, 437, 437t Detector(s) in capillary electrophoresis, 604, 604f electrochemical, 585, 585f electron capture, 570, 570f flame ionization, 570, 570f in flow injection analysis, 652 in gas chromatography, 569–571 in high-performance liquid chromatography, 584–586 refractor index, 585 spectroscopic, 584–585 in spectroscopy, 379–380 thermal conductivity, 569, 569f Determinate error(s), 58 constant, 60–61, 60t, 61f, 118, 118t identification of, 60–61 proportional, 61, 61t Determination, 36 Dialysis, 205t, 206, 206f Diameters, effective, of selected inorganic cations and anions, 173, 173t Dichloroacetic acid, acid dissociation constant for, 734t Dichromate, 505–506 Diethylamine, acid dissociation constant for, 734t Diffusion, 511 longitudinal, 560–561, 560f instrumentation for, 498 minimizing electrolysis time in, 498 quantitative applications using, 501–503, 502t selecting constant potential in, 497, 497f Convection, 512 Convenience sampling, 185 Conversion external, 425, 425f internal, 425, 425f Copper absorption lines and characteristic concentrations for, Color Plate 1, 416, 418t in aqueous solution, 497, 497f standard reduction potentials for, 744t in waters and wastewater, 395t Coprecipitates, 239, 239f Corer, 197, 197f Coulomb, 13t Coulometric titrations, 501, 501f using acid-base, complexation and precipitation reactions, 504, 504t using redox reactions, 503–504, 503t, 505–506 Coulometry, 235, 496–508 accuracy in, 507–508 characterization applications of, 506–507 controlled-current. See Controlled-current coulometry controlled-potential. See Controlled-potential coulometry evaluation of, 507–508 precision in, 508 representative methods in, 505–506 scale of operation in, 507 selectivity in, 508 sensitivity in, 508 time, cost and equipment in, 508 as total analysis technique, 38 Countercurrent separations, 546, 755–761, 757t, 758t, 760f Counter electrode, 462 Creatinine, in urine, 632–633 Cresol red, 289t Cryogenic focusing, 568 Crystalline solid-state ion-selective electrodes, 479–482, 481t Cumulative formation constant, 144 Cupferron, 249t acid dissociation constant for, 734t Cupron, 249t Current(s) anodic, 510 cathodic, 510 charging, 513 dark, 379 diffusion, 516 in electrochemical methods, 462–465 faradaic, 510–512, 513f limiting, 514 nonfaradaic, 512–513 residual, 513, 521 SI unit for, 12t in voltammetry, 510–513 Index 785 Diffusional microtitration, 312, 312f Diffusion current, 516 Diffusion layer, 511 Digestion and occlusions, 239 of solid samples, 200–201, 200f, 200t Dilution, 31 stock solution preparation by, 31–32 Dimethylamine, acid dissociation constant for, 734t Dimethylgloxime, 4, 249t acid dissociation constant for, 734t Dimethyl yellow, 289t Diol, 213t Diphenylamine, 339t Diphenylamine sulfonic acid, 339t Direct-computation integral methods, 626–629 Direct-computation rate methods, 629–630 Displacement titration, 275 Dissociation, of water, 142 Dissociation constant, 144 for acids, 140–141, 143, 732t–739t for solvent, 295 for water, 142–143 Distance, SI unit for, 12t Distillation, 205t, 209–210, 210f, 210t equipment used in, 209, 210f Distribution(s) binomial, 72–73, 73f Gaussian. See Normal distribution of measurements and results, 70–82 normal, 73–75, 74f–75f, 85–94, 725t–726t probability, 71–75, 77–80 Distribution ratios, 216 Disulfiram. See Antabuse Dithizone, 395 extraction process using, 223, 223f Divinylbenzene, 213t Dixon Q-test, 93–94 critical values for, 728t DME. See Dropping mercury electrode (DME) DMG. See Dimethylgloxime Drinking water Coliform bacteria in, 36, 37f lead in, 36–37, 37f Safe Drinking Water Act, 36 trihalomethanes in, 576–577 Dropping mercury electrode (DME), 509, 509f Drugs, natural fluorescence and phosphorescence of, 430t Dry ashing, 201 Drying equipment, 29, 29f D-tartaric acid, acid dissociation constant for, 738t Duplicate samples, 708 E EDTA, 23, 314 acid dissociation constant for, 734t chemistry and properties of, 315–317, 315f and competition with other ligands, 316–317 conditional metal-ligand formation constants for, 315–316, 316t 1400-Indx 9/27/99 2:17 PM Page 785 ladder diagram for, 315, 315f metal-EDTA formation constants, 315 as titrant, 327 titration curves of, 317–322, 319f calculation of, 317–319, 319f sketching of, 320–322, 320t, 321f–322f as weak acid, 315 Effective bandwidth, 376 Effective diameters, of selected inorganic cations and anions, 173, 173t Ektachem analyzer, 493, 493f Elastic scattering, 441 Electrical double layer, 513 Electric field in capillary electrophoresis, 603–604 of plane-polarized electromagnetic radiation, 369, 369f–370f Electrochemical cell(s) potentiometric. See Potentiometric electrochemical cells voltammetric, 510, 511f Electrochemical detectors, for high- performance liquid chromatography, 585, 585f Electrochemical methods, 461–531 bulk, 462 classification of, 462–465 coulometry, 496–508 current and potential in, control and measurement of, 462–465 interfacial, 462, 463f potentiometry, 465–496 voltammetry, 508–532 Electrochemical reversibility, 527–528 Electrode(s) auxiliary, 463 counter, 462 dropping mercury, 509, 509f enzyme, 484–485, 485f of the first kind, 473–474 gas-sensing, 484, 484f, 485t glass, 477–479, 478f, 479t hanging mercury drop, 509, 509f indicator, 462 metallic, 473–475 ion-selective, 322, 475 crystalline solid-state, 479–482, 481t glass, 477–479, 478f, 479t liquid-based microelectrode, 493, 494f solid-state, 479–482, 481t membrane. See Membrane electrodes redox, 475 reference, 462–463, 471–473, 473f–474f saturated calomel, 472, 473f of the second kind, 474–475 silver/silver chloride, 473, 474f solid, 510, 510f standard hydrogen, 471–472, 472f static mercury drop, 509, 509f Electrodeposition, 205t, 210 Electrogravimetry, 234 Electrokinetic injection, 603 Electrolysis time, in controlled-potential coulometry, 498 Electromagnetic radiation, 369–372. See also specific types of radiation absorption of, 380–384 continuum or line sources of, 375, 375f particle properties of, 371–372 plane-polarized electric field of, 369, 369f–370f magnetic field of, 369, 369f sources of monochromatic versus polychromatic, 377–378 for spectroscopy, 375, 375t spectrum, Color Plate 9, 372, 372f wave properties of, 369–371, 369f Electron, conservation of, 23 Electron capture detector, 570, 570f Electronic transitions, between quantized energy levels, 382, 382t Electron pairs, conservation of, 23 Electron spin resonance spectroscopy, 373t Electron transfer, kinetics of, and faradaic current, 512 Electroosmotic flow, 598, 599f reversal of, in capillary electrophoresis, 605, 605f velocity, 599 Electropherogram, 597–598 Electrophoresis, 597–609. See also Micellar electrokinetic capillary chromatography (MEKC) capillary. See Capillary electrophoresis capillary gel, 606–607 capillary zone, 604–606 Electrophoretic chromatography, 547, 547f Electrophoretic mobility, 598–599 Electrophoretic velocity, 598 Electrothermal atomizers, 36, 37f, 414–415, 415f Contract Laboratory Program protocol for, 48, 49f versus flame atomizer, 416 Elution, isocratic versus gradient, 582 Emission, 373–374. See also Atomic emission entries gamma-ray, 645, 645f of photons, energy level diagram for, 373, 374t spectroscopies involving, 373t, 423 Emission line, intensity of, 434–435 Emission spectrum, 374, 427, 427f versus excitation spectrum, 427 Empirical models, for response surfaces, 675–676 End point, 274 in complexation titration finding by monitoring absorbance, 324, 324f finding with visual indicator, 323–324, 323t selection and evaluation of, 322–324 in controlled-current coulometry, 500 finding of by derivative method 291–293, 291t, 292f 786 Index by monitoring pH, 290–293, 291f–292f by monitoring temperature, 293–295, 294f using indicator, 288–290 in precipitation titrations potentiometrically, 354 using indicator, 354 in redox titrations potentiometrically, 339, 339f with visual indicator, 338–339, 339f, 339t selecting and evaluating of, 287–295 End point volumes for phosphate species, 308t and sources of alkalinity, 301, 302t Energy Gibb’s free, 137–138 SI and non-SI units for, 13t in spectroscopy chemical, 375 thermal, 375 Energy levels diagram of for absorption of photons, 372, 372f for emission of photons, 373, 374t for photon, 372, 372f electronic transitions between, 382, 382f, 382t Enthalpy, 137 Entropy, 137 Environmental pollutants. See Pollutants Environmental Protection Agency, Contract Laboratory Program of. See Contract Laboratory Program (CLP) Enzyme(s), 636 electrode, 484–485, 485f inhibition of, 638–639, 639f–640f turnover number of, 638 Enzyme-catalyzed reactions, 636–637, 637f maximum rate and Michaelis constant for, 638 Equation(s) charge balance, 159 Henderson-Hasselbalch, 169–170 mass balance, 159 Nernst, 146–147, 468–470 regression, 122–124, 124f van Deemter, 561 Equilibrium, 136, 136f chemistry, 135–175 problem-solving in, 156–167 thermodynamics and, 136–138 Equilibrium constant(s), 138 for acid-base titrations, 310–311, 310f and activity effects, 171–174 for chemical reactions, 139–148 for complexation reactions, 145 manipulation of, 138–139 thermodynamic, 172–173 in ultraviolet/visible spectroscopy, 407–409 in voltammetry, 528–530, 528f Equilibrium-density-gradient centrifugation, 207, 207f Equivalence point, 274, 287–288, 287f in redox titrations, 337–338, 338f 1400-Indx 9/27/99 2:17 PM Page 786 F Factor, 667 Factorial designs, for response surfaces, 676–682, 676f, 681f–682f Factor levels, 667 Fahrenheit scale, 13t Fajans’ method, 354, 355t Faradaic current, 510 applied potential and, 510–511 kinetics of electron transfer and, 512 mass transport and, 511–512, 513f Faraday’s law, 496 Fellgett’s advantage, 379 Ferrocyanide, solubility products for, 731t Ferroin, 339t Ferrous ammonium sulfate, 344 FIA. See Flow injection analysis (FIA) Fiagram, 650, 650f–651f Fiber-optic probes, 391, 392f FID. See Flame ionization detector (FID) Field blank, 710 Filter/filtration, 205t absorption, 376 crucible, 244, 244f photometer, 388, 389f in precipitation gravimetry, 243–244, 243f, 263 in spectroscopy, 376 First derivative titration curve, 291, 291t, 292f First-order reactions, 751–752, 753f Fisher’s least significant difference, 696 Fixed-time integral methods, 628 Flame in atomic emission spectroscopy, 435, 435f in spectroscopy, 375 Flame atomizers, 413–414, 414f versus electrothermal atomizers, 416 fuels and oxidants used in, 413, 414t Flame ionization detector (FID), 570, 570f Florence, T. M., 524 Flow injection analysis (FIA) evaluation in, 658 instrumentation in, 651–654 phosphate determination by, 656–657 quantitative analysis using, 655–657 on clinical samples, 656, 656t on environmental samples, 655, 655t representative methods in, 656–657 theory and practice, 649–651, 649f–650f Fluorescence, 423, 425–426 natural, organic compounds with, 429–430, 430t x-ray, 373t Fluorescence spectroscopy atomic, 373t molecular, 373t evaluation in, 432–434 inorganic analytes in, 429, 429f, 430t instrumentation in, 427–428 organic analytes in, 429–430 quantitative applications using, 429–432 Equivalency testing, and standard method, 687 Equivalent, 17 Equivalent weights, 17 acid-base titrations and, 309–310 Eriochrome Black T, 323t Eriochrome Blue Black R, 323t Error(s) in both x and y, weighted linear regression with, 127 determinate. See Determinate error(s) experimental, 57–64 indeterminate. See Indeterminate errors measurement, 58–60, 59t, 495, 495t method, 58 personal, 60 residual, 118, 119f sampling, 58 in significance testing, 84–85 titration, 274 type 1, 84–85 type 2, 84–85 and uncertainty, 64 in y, 119–127, 126f Esters, acid-base titrimetric procedures for, 303, 303t Ethylamine, acid dissociation constant for, 734t Ethylenediamine, acid dissociation constant for, 734t Ethylenediaminetetraacetic acid (EDTA). See EDTA Excitation in atomic emission spectroscopy, 435 sources of, 437 spectrum, versus emission spectrum, 427 thermal, spectroscopies involving, 373t Exclusion limit, 595 Experimental errors and accuracy, 57–62 characterization of, 57–64 Exponential notation, prefixes for, 13t External conversion, 425, 425f External standardization multiple-point, 117, 117t single-point, 117 External standards, 109–110 in atomic emission spectroscopy, 438 in potentiometry, 486–487 Extraction(s), 205t. See also Separation(s) continuous, 213–215, 214f countercurrent, 546 gas-solid, 213 liquid-gas, 214, 214f liquid-liquid. See Liquid-liquid extraction microextractions liquid-liquid, 212, 212f solid-phase, 213, 567 in particulate gravimetry, 263–264, 263f solid-phase, 212–213, 212f, 213t between two phases, 212 using dithizone, 223, 223f of volatile organic compounds, 214 Extraction chromatography, 205t Index 787 radiationless deactivation in, 424–425 spectra in, 424–427 sensitivity in, slit orientation and, 433, 433f standardizing method in, 431–432 Fluoride solubility products for, 731t in toothpaste, 489–490 in waters and wastewater, 395t, 396 Fluorine, standard reduction potentials for, 744t Fluorometer, 433–434 Fluoxetine, in serum, 588–589, 588f Fluxes, for decomposition of inorganic samples, 201, 201t Force, SI and non-SI units for, 13t Formality, 15–16, 16t Formal potential, 332–333 Formation constant(s), 144 cumulative, 144 metal-ligand, 739t–742t conditional, 315–316, 316t stepwise, 144 Formazin, preparation of, 444, 444f Formic acid, acid dissociation constant for, 734t Formula weight, 17 Fourier transform infrared spectrometer (FT-IR), 378, 393 and gas chromatography, 570–571 and polyvinylchloride, 381, 381f Freedom, degrees of, 80 Free ions, versus complexed ions, in potentiometry, 489 Frequency, 370 Fronting, in column chromatography, 555, 555f F-test, 87, 727t–728t comparing two sample variances, 88 FT-IR. See Fourier transform infrared spectrometer (FT-IR) Fumaric acid, acid dissociation constant for, 734t Fundamental analysis, 9 G Gallium, standard reduction potentials for, 744t Galvanostat, 464–465, 465f Gamma ray, 642 emission, 645, 645f Gas(es) properties of, 596t samples collection of, 195–196 preservation and preparation, 196 Gas chromatography, 563–578, 563f accuracy in, 577 and chromatic columns, 564–565 detectors for, 569–571 evaluation in, 577–578 Fourier transform infrared spectrometers and, 570–571 liquid-solid, 590, 590f mass spectrophotometers and, 570–571 mobile phase of, 563 precision in, 577 1400-Indx 9/27/99 2:17 PM Page 787 qualitative applications using, 575 quantitative applications using, 571–574 calculations in, 572–574 clinical analysis, 572 consumer goods, 572 environmental analysis, 571–572, 572f–573f petroleum industry, 572 representative methods in, 576–577 sample introduction in, 567–568 adjusting analyte concentration in, 568 injection of, 568, 569f preparing volatile samples, 567–568 scale of operation in, 577 selectivity in, 578 sensitivity in, 578 stationary phases in, 565–567, 566t, 567f temperature control in, 568–569 time, cost and equipment in, 578 Gas-liquid chromatography (GLC), 564 Gas-sensing electrodes, 484, 484f, 485t Gas-solid extractions, 213 Gaussian distribution. See Normal distribution Gay-Lussac, Joseph Louis, 331 Geiger counter, 643 Gel permeation chromatography. See Size- exclusion chromatography General theory of separation efficiency, 202–205 Gibb’s free energy, 137–138 Glass electrodes, 477–479, 478f, 479t Glass ion-selective electrodes, 477–479, 478f, 479t Glassware, measurement errors for, 59t, 60 GLC. See Gas-liquid chromatography (GLC) GLPs. See Good laboratory practices (GLPs) Glucose, UV/visible spectroscopy analysis of, 397t Glutamic acid, acid dissociation constant for, 734t Glutamine, acid dissociation constant for, 735t Glycine, acid dissociation constant for, 735t Glycolic acid, acid dissociation constant for, 735t GMPs. See Good measurement practices (GMPs) Gold, standard reduction potentials for, 744t Good laboratory practices (GLPs), 706 Good measurement practices (GMPs), 706–707 Grab sample, 185, 197, 197f Gradient elution, 558, 558f versus isocratic elution, 582 Gran plot, 292f, 293, 294t Graphite furnace atomic absorption spectroscopy. See Electrothermal atomizers Gravimetry, 232–265 electrogravimetry, 234 importance of, 235 and nickel, 2–4, 3f–4f overview of, 233–235 particulate. See Particulate gravimetry precipitation. See Precipitation gravimetry thermogravimetry, 255–257, 256f as total analysis technique, 38 types of, 234 volatilization. See Volatilization gravimetry Gross sample, 193 Guard columns, 579 H Half-life, 643 Hanging mercury drop electrode (HMDE), 509, 509f Headspace sampling, 567 Heat, SI and non-SI units for, 13t Henderson-Hasselbalch equation, 169–170 Heterogenous, 58 Heyrovsky, Jaroslav, 508–509 High-performance liquid chromatography (HPLC), 578–589, 579f chromatography columns used in, 578–579 detectors for, 584–586 evaluation in, 589 instrumentation in, Color Plate 8, 583–584 mobile phases in, 580–583, 581t preparing samples for, 586 quantitative applications using, 586–588, 587f calculations in, 586–588 representative method in, 588–589 sample introduction in, 584, 584f stationary phases in, 579–580 Histidine, acid dissociation constant for, 735t Histogram, 77, 79f HMDE. See Hanging mercury drop electrode (HMDE) Homogenous, 72 Homogenous precipitation, 241–242 HPLC. See High-performance liquid chromatography (HPLC) Hydrodynamic injection, 602 Hydrodynamic voltammetry, 513, 516–519 Hydrogen, standard reduction potentials for, 744t Hydrogen cyanide, acid dissociation constant for, 735t Hydrogen fluoride, acid dissociation constant for, 735t Hydrogen peroxide, acid dissociation constant for, 735t Hydrogen sulfide, acid dissociation constant for, 735t Hydrogen thiocyanate, acid dissociation constant for, 735t Hydroxide, solubility products for, 731t–732t Hydroxylamine, acid dissociation constant for, 735t 8-Hydroxyquinoline, acid dissociation constant for, 735t Hypobromous acid, acid dissociation constant for, 735t Hypochlorous acid, acid dissociation constant for, 735t Hypoiodous acid, acid dissociation constant for, 735t 788 Index Hypothesis alternative, 83 null, 83 I ICP torch. See Inductively coupled plasma (ICP) torch Ideal gas law, 149 Identification, limit of, 95–96, 95f IEC. See Ion-exchange chromatography (IEC) Impurities in precipitate gravimetry, 238–240 types of, 238–239 Incident radiation, in spectroscopy based on scattering, 443 Inclusion, 238–239, 239f Inclusion limit, 593, 595 Indeterminate errors, 62 error and uncertainty, 64 evaluation of, 63, 63f, 63t sources of, 62–63, 63f in UV/vis and infrared spectroscopy, 410–411, 410t, 511f Indicator(s), 274 in acid-base titrations, 278 finding end point with, 288–293, 289f–290f, 289t pH ranges of, 288–290, 289f–290f, 289t in complexation titration, 323–324, 323t electrodes, 462 metallic, 473–475 metallochromic, 315, 323, 323t mixed, 289t in precipitation titrations, 354 redox, 338–339, 339f, 339t screened, 289t universal, Color plate 4 Indigo tetrasulfonate, 339t Inductively coupled plasma (ICP) torch, 435, 435f detection limits of, 437, 437t Inflection points, in acid-base titration, 287–288, 290–291 Infrared radiation, versus UV/visible, absorption of, 381, 381f Infrared spectroscopy, 373t, 388–412 attenuated total reflectance cell for, 393–394, 393f instrument designs for, 393–394, 393f and molecules and polyatomic ions, 381, 381f and relative uncertainty in concentration, 410–411, 410t, 411f Inhibition, of enzymes, 638–639, 639f–640f competitive, 639, 640f noncompetitive, 639, 640f uncompetitive, 639, 640f Initial rate method, 753 Injection. See also Flow injection analysis (FIA) electrokinetic, 603 in flow injection analysis, 652 hydrodynamic, 602 loop, 584 on-column, 568 and slow injection analysis, 649–658 1400-Indx 9/27/99 2:17 PM Page 788 Ion-selective electrodes, 322, 475 crystalline solid-state, 479–482, 481t liquid-based, 482–484, 482f, 483t solid-state, 479–482, 481t Ion-suppressor column, 592–593 Iron controlled-current coulometry analysis of, 499, 500f standard reduction potentials for, 744t in waters and wastewater, 395t, 398–399 Isocratic elution, 582 versus gradient elution, 582 Isoleucine, acid dissociation constant for, 735t Isotope, 642 radioactive half-life of, 643 rate of decay for, 643 Isotope dilution, 646–647 Isotope-dilution mass spectrometry, 235 J Jacquinot’s advantage, 378–379 Job’s method. See Method of continuous variations Jones reductor, 341 selected reductions using, 341, 342t Joule, 13t Joule heating, 601–602 Judgmental sampling, 184 K Kelvin, 12t Kilogram, 12t Kinetic methods, 622–658, 751–754 chemical, 623–642 Kirchhoff, Guystav, 412 Kjeldahl analysis, Color Plate 5, 302–303 Kodak Ektachem analyzer, 493, 493f Kovat’s retention index, 575 L Laboratory notebook, 32 Laboratory sample, 199 Ladder diagram(s), 150–155 for acid-base equilibria, 150–152, 151f for alanine, 163, 163f for buffer solutions, 170–171, 171f for complexation equilibria, 153–155, 153f–155f for EDTA, 315, 315f for oxidation-reduction equilibria, 155, 155f–156f Lanthanum, standard reduction potentials for, 744t Large-particle scattering, 441, 441f Law(s) Beer’s, 385–388, 385f–386f Faraday’s, 496 ideal gas, 149 Ohm’s, 463 rate, 624–635, 750–751 split, 568 splitless, 568 Inorganic analysis acid-base titrations in, 300–302, 301f, 302t complexation titrations in, 327–328 precipitate gravimetry and, 247–250, 248t–249t redox titrations in, 344–345 volatilization gravimetry in, 259 In situ sample, 185 Intensity, of emission line, 434–435 Intercellular fluids, analysis of, 493, 494f Interfacial electrochemical methods, 462 family tree for, 462, 463f Interference chemical, 438, 440 compensating for, in development of procedure, 45–46 masking and, 207–208 spectral, 418–419, 437–438, 438f Interferents, separation of analyte from, 201–202 Interferometers, 378–379, 379f Internal conversion, 425, 425f Internal standards, 115–117 in atomic emission spectroscopy, 438 Intersystem crossing, 425, 425f Iodate, solubility products for, 732t Iodic acid, acid dissociation constant for, 735t Iodide, solubility products for, 732t Iodine as oxidizing titrant, 346 protein-bound, 397t standard reduction potentials for, 744t Ion(s) anions effective diameters, 173, 173t selected gravimetric method for, 248, 248t cations effective diameters, 173, 173t selected gravimetric method for, 248, 248t–249t complexed versus free, in potentiometry, 489 inorganic chelating agents for in fluorometric analysis, 429, 430t controlled-potential coulometric analysis of, 501, 502t effective diameters, 173, 173t selected gravimetric method for, 248, 248t–249t metal, chelating agents for in fluorometric analysis, 429, 430t polyatomic, infrared spectrum of, 381 UV/visible spectrum for, 382–383 Ion exchange, 205t, 210 Ion-exchange chromatography (IEC), 547, 547f, 590–593, 594f Ion-exchange resin, 590–593, 591f, 591t Ionic strength, Color Plate 3, 172 Ionization suppressor, 420 Ionophore, 482 Index 789 Lead in drinking water, 36–37, 37f standard reduction potentials for, 744t in waters and wastewater, 395t LeChatelier’s principle, 148–150 Length, SI and non-SI units for, 13t Leucine, acid dissociation constant for, 735t Leveling, 296 Lewis, G. N., 144 Liebig, Justus, 314 Lifetime, 423 Ligand, 144 metal-ligand formation constants, 739t–742t Light. See Electromagnetic radiation Limiting current, 514 Limit of identification, 95–96, 95f Limit of quantification, 96 Linear regression and calibration curves, 117–127 determination of estimated slope and y-intercept by, 119 goal of, 118 residual error in, 118, 119f of straight-line calibration curves, 118, 119f uncertainty in, 120–122, 120f unweighted, with errors in y, 119–124 weighted with errors in both x and y, 127 with errors in y, 124–127, 126f Line sources, 375, 375f Lineweaver-Burk plot, 638, 639f Liquid(s), properties of, 596t Liquid-based ion-selective electrodes, 482–484, 482f, 483t Liquid-based ion-selective microelectrode, 493, 494f Liquid-gas extraction, 214, 214f Liquid junction potential, 470–471, 470f Liquid-liquid extraction, 212, 212f, 215–223, 216f extraction efficiency in, 219, 219f, 221, 221f improved methods for, 544–546, 545f involving acid-base equilibria, 219–221, 219f, 221f involving metal chelators, 221–223, 221f–222f partition coefficients and distribution ratios in, 216 Liquid-liquid microextractions, 212, 212f Liquid-solid adsorption gas chromatography (LSC), 590, 590f Liter, 12t Lithium, standard reduction potentials for, 744t Litmus, 289t Longitudinal diffusion, 560–561, 560f Loop injector, 584 LSC. See Liquid-solid adsorption gas chromatography (LSC) Luminescence, molecular. See Fluorescence spectroscopy, molecular; Phosphorescence spectroscopy, molecular Lysine, acid dissociation constant for, 735t 1400-Indx 9/27/99 2:17 PM Page 789 M Magnesium, standard reduction potentials for, 744t Magnetic field, plane-polarized of electromagnetic radiation, 369, 369f Magnetic resonance spectroscopy. See Nuclear magnetic resonance spectroscopy Maleic acid acid dissociation constant for, 736t titration curve for, 288, 288f Malic acid, acid dissociation constant for, 736t Malonic acid acid dissociation constant for, 736t titration curve for, 288, 288f Manganese standard reduction potentials for, 744t in waters and wastewater, 395t Manifold, in flow injection analysis, 652, 652f–654f Masking, 205t, 207–208 separation techniques based on, 207–209 Masking agents, 208, 208t Mass conservation of, 22 and gravimetry, 234–235 instrumentation for measuring of, 25–26, 25f–26f separation techniques based on, 206–207 SI unit for, 12t using as signal, 233–234 Mass balance equation, 159 Mass spectrometer and gas chromatography, 570–571 for high-performance liquid chromatography, 585–586 Mass spectrum, 571 Mass transfer, and column efficiency, 560f, 561 Mass transport, and faradaic current, 511–512, 513f Matrix, 36 sample, 110, 110f Matrix matching, 110 Maximum holding times, for selected waters and wastewater parameters, 195, 195t Maximum rate, for enzyme-catalyzed reactions, 638 Mean, 54–55. See also t-test of populations, 79 sample, 88–93 Measurement(s), 36 of central tendency, 54–55 characterization of, in analytical data, 54–57 distribution of, 70–82 fundamental units of, 12, 12t–13t of range, 56 of spread, 55–57 of standard deviation, 56–57 uncertainty and in mixed operations, 66–67 in other mathematical functions, 67–68, 68t when adding or subtracting, 65–66 when multiplying or dividing, 66 of variance, 57 Measurement error, 58–60 in potentiometry, 495, 495t for selected equipment, 58, 59t Median, 55 Mediator, 500 MEKC. See Micellar electrokinetic capillary chromatography (MEKC) Membrane potentials, 475–476, 476f selectivity of, 476 Membrane electrodes crystalline solid-state ion-selective electrodes, 479–482, 481t gas-sensing electrodes, 484, 484f, 485t glass ion-selective electrodes, 477–479, 478f, 479t liquid-based ion-selective electrodes, 482–484, 482f, 483t membrane potentials in, 475–476, 476f potentiometric biosensors, 484–485, 485f–486f, 486t in potentiometric electrochemical cells, 475–485 selectivity of membrane in, 476–477, 477f Meniscus, 28f, 29 Mercury standard reduction potentials for, 744t in waters and wastewater, 395t Metal-EDTA formation constants, 315 Metal ions, inorganic, chelating agents for fluorometric analysis, 429, 430t Metallic indicator electrodes, 473–475 Metal-ligand complex, stoichiometry of, 403–407, 406f Metal-ligand formation constants, 739t–742t conditional, for EDTA, 315–316, 316t Metallochromic indicators, 315, 323, 323t Meter, 12t Methionine, acid dissociation constant for, 736t Method, 36 standardization of, 106 Method blank, 45, 128–129, 128t–129t Method error, 58 Method of continuous variations, Color Plate 6, 404–406 Method of standard additions, 110–111, 111f–112f calibration curve for, 114–115, 114f and potentiometry, 488–489 Methylamine, acid dissociation constant for, 736t Methylene blue, 289t, 339t Methyl isobutyl ketone (MIBK), 224 Methyl orange, 278, 289t Methyl-phenyl polysiloxane, 566t Methyl red, 289t MIBK. See Methyl isobutyl ketone (MIBK) 790 Index Micellar electrokinetic capillary chromatography (MEKC), 606 Micelle, 606, 606f Michaelis constant, 637 for enzyme-catalyzed reactions, 638 Microdroplet apparatus, 311, 311f Microextraction(s) liquid-liquid, 212, 212f solid-phase, 213, 567 Microtitration, diffusional, 312, 312f Microwave spectroscopy, 373t Migration, 512 Migration time, in capillary electrophoresis, 600 Mixed indicators, 289t Mixtures, improved separation methods for, 544–546 Mobile phases, 546 in gas chromatography, 563 in high-performance liquid chromatography, 580–583, 581t in superfluid chromatography, 596, 597t Mobility electrophoretic, 598–599 total, 599–600 Mohr method, 354, 355t Molality, 18 Molarity, 15–16, 16t Mole, 12t Molecular absorption spectroscopy versus atomic absorption, 412 instruments used in, 388–394 Molecular-exclusion chromatography. See Size- exclusion chromatography Molecular fluorescence spectroscopy. See Fluorescence spectroscopy, molecular Molecular phosphorescence spectroscopy. See Phosphorescence spectroscopy Molecular photoluminescence spectroscopy. See Photoluminescence spectroscopy Molecules infrared spectrum of, 381, 381f UV/visible spectrum for, 382–383 Mole-ratio method, 406, 406f Molybdenum, standard reduction potentials for, 745t Monensin, 92–93 Monochromatic source, or radiation, 377–378 Monochromator, 376, 378f in infrared spectrometers, 393 wavelength selectors using, 376–378, 378f Monoprotic acid, pH of, 160–163 Mossbauer spectroscopy, 373t Multicomponent samples, Beer’s law and, 386 Multielemental analysis, using atomic emission spectroscopy, 436, 436f Multiple-point standardization, 109 external, 117, 117t versus single-point standardization, 108–109 Multivariate regression, 127 Murexide, 323t 1400-Indx 9/27/99 2:17 PM Page 790 One-tailed significance test, 84, 84f Open tubular columns. See Capillary column Optimization one-factor-at-a-time, 669–671, 669f–671f simplex, 671–674, 671f, 673t, 674f–675f Organic analysis acid-base titrations in, 302–303 precipitate gravimetry and, 250, 250t redox titrations in, 346–347 volatilization gravimetry in, 259–260 Organic compounds, with natural fluorescence or phosphorescence, 429–430, 430t Outlier, 93–94 Overall variance, minimization of, 192–193 Overpotential, 497 Oxalate, solubility products for, 732t Oxalic acid, acid dissociation constant for, 737t Oxidant, used in flame atomizers, 413, 414t Oxidation, 146 Oxidation-reduction reactions. See Redox reactions Oxidation state, of analyte, in redox titration, 341–342 Oxidizing agent, 146 auxiliary, 341 reaction units for, 23 Oxygen, standard reduction potentials for, 745t P Packed columns, 564 Paired data, 88, 91–93 Paired t-test, 92–93 PAN, 323t Particle(s) airborne pollutant, 7–8 alpha, 642 beta, 642 and large-particle scattering, 441, 441f properties of, in electromagnetic radiation, 371–372 radioactive, 642 size in precipitation gravimetry, 240–243, 242f reducing in solid samples, 199–200 separation techniques based on, 205–206 Particulate gravimetry, 234, 262–265 evaluation of, 265 extraction in, 263–264, 263f filtration in, 263 quantitative applications involving, 264–265 calculations in, 264–265 theory and practice, 263–264 Partition chromatography, 547, 547f Partition coefficient, 211–212 in liquid-liquid extractions, 216 Partitioning, between phases, separation techniques based on, 211–215, 212f, 213t Parts per billion, 18 Parts per million, 18 N Negatron, 642 Nephelometry, 441–445 applications of, 444–445 determination of concentration by, 443 precipitates used in, 443–444, 444t versus turbidimetry, 442 Nernst equation, 146–147 and potentiometric electrochemical cells, 468–470 Nessler’s method, for ammonia, 368 Nessler tubes, 368, 388 Neutral red, 289t Neutron activation, 645 analysis, 645–646 Newton, 13t Nickel gravimetric analysis of, 2–4, 3f–4f standard reduction potentials for, 745t Nitrate, in waters and wastewater, 395t Nitrilotriacetic acid, acid dissociation constant for, 736t Nitrogen, standard reduction potentials for, 745t Nitron, 249t Nitrous acid, acid dissociation constant for, 737t Nominal wavelength, 376 Nonaqueous solvents, acid-base titrations in, 294f, 295–296, 296f Noncatalytic reactions, 638 Noncompetitive inhibition, of enzymes, 639, 640f Nonenzyme-catalyzed reactions, 637–638 Nonfaradaic currents, 512–513 Nonideal behavior, in column chromatography, 555, 555f Normal calibration curve, 109, 109f effect of sample’s matrix on, 110, 110f Normal distribution, 73–75, 74f–75f single-sided, 725t–726t statistical methods for, 85–94 Normality, 16–18 Normal-phase chromatography, 580 Nuclear magnetic resonance spectroscopy, 373t Null hypothesis, 83 Numbers, in analytical chemistry, 12–15, 12t–13t Nyquist theorem, 184 O Occlusion, 239, 239f Octadecyl, 213t Octyl, 213t Ohm’s law, 463 On-column injection, 568 One-factor-at-a-time optimization, 669–671, 669f–671f Index 791 Pascal, 13t Peak capacity, 554–555 Peptization, 245 Performance-based approach, to quality assurance, 714–721 Peristaltic pump, 652, 652f Personal error, 60 p-function, 19–20 pH and acid-base titrations, 279 of buffer solutions, 169–170, 492t electrode, 290 of indicators in acid-base titration, 288–290, 289f–290f, 289t and finding end point, 290–293, 291f–292f of monoprotic weak acid, 160–163 potentiometric measurement of, 489–490 and precipitate solubility, 237 scale, 142–143, 143f uncertainty in, 67–68 universal indicator for, Color plate 4 Pharmaceuticals, natural fluorescence and phosphorescence of, 430t Phenol, in waters and wastewater, 395t, 396 Phenolphthalein, 278, 289t Phenol red, 289t Phenylalanine, acid dissociation constant for, 737t Phosphate determination of, by flow injection analysis, 656–657 end point volumes for, 308t solubility products for, 732t in waters and wastewater, 395t Phosphorescence, 424, 426–427 natural, organic compounds with, 429–430, 430t Phosphorescence spectroscopy molecular, 373t evaluation in, 432–434 inorganic analytes in, 429, 429f instrumentation in, 428–429, 428f organic analytes in, 429–430 quantitative applications using, 429–432 radiationless deactivation in, 424–427 spectra in, 424–427 standardizing method in, 431–432 Phosphoric acid, acid dissociation constant for, 737t Phosphorus, standard reduction potentials for, 745t Photodiode array, 379 Photoluminescence, 374 Photoluminescence spectroscopy, 373t, 374, 374f, 423–434 accuracy in, 432 evaluation in, 432–434 excitation versus emission spectra in, 427, 427f 1400-Indx 9/27/99 2:17 PM Page 791 fluorescence and phosphorescence spectra, 424–427 inorganic analytes in, 429, 429f, 430t instrumentation in, 425f–426f, 427–429 organic analytes in, 429–430, 431t precision in, 432 quantitative applications using, 429–432 radiation deactivation in, 424–425, 425f scale of operation in, 432 sensitivity in, 432–433, 433f standardizing methods in, 431–432 time, cost and equipment in, 433–434 Photon(s), 371 absorption of, 372, 372f emission of, 373, 374t as signals, 372–374, 373f, 373t Photon transducers, 379, 380t Phthalic acid, acid dissociation constant for, 737t Physical state, changes in, separations based on, 209–210, 209t Piezoelectric effect, 263 Piperidine, acid dissociation constant for, 737t Pipet measurement errors for, 59t, 60 types of, 27–28, 27f–28f Planar chromatography, 546 Plasma, 435 as energy source, in spectroscopy, 375 sources in atomic emission spectroscopy, 435–436, 435f Plasma torch, inductively coupled, 435, 435f Platinum, standard reduction potentials for, 745t Polarity index, 580, 581t Polarographic half-wave potentials, 747t Polarography, 515–516, 515f–516f Pollutants aerosol, 7–8 natural fluorescence and phosphorescence of, 430t Polyatomic ions, infrared spectrum of, 381 Polychromatic source, of radiation, 377–378 Polydimethyl siloxane, 566t Polyethylene glycol, 566t Polyprotic acid, pH of, 163–165 Polyprotic base, pH of, 163–165 Polyvinylchloride, Fourier transform infrared spectrum of, 381, 381f Population(s), 71 confidence intervals for, 75–77, 75t degrees of freedom for, 80 mean and variance of, 79 probability distributions for, 71–75 Positron, 642 Potassium Kodak Ektachem analyzer for, 493, 493f standard reduction potentials for, 745t Potential(s). See also Controlled-potential coulometry asymmetry, 476 controlling and measuring of, in electrochemical methods, 462–465 formal, 332–333 liquid junction, 470–471, 470f membrane, 475–476, 476f polarographic half-wave, 747t of potentiometric electrochemical cells, 468–470 selection of, in controlled-potential coulometry, 497, 497f SI and non-SI units for, 13t standard reduction, 743t–746t standard-state, determining by voltammetry, 514–515, 515f zeta, 599 Potentiometer, 464, 464f Potentiometric biosensors, 484–485, 485f–486f, 486t Potentiometric electrochemical cells, 466–467, 466f membrane electrodes in, 475–485 metallic indicator electrodes, 473–475 potential and concentration of, 468–470 reference electrodes in, 471–473, 473f–474f shorthand notation for, 467–468 Potentiometric titrations, 494 Potentiometry, 465–496 accuracy in, 494–495 and concentration technique, 38 electrochemical cells in, 466–467, 466f evaluation of, 494–496 measurement error in, 495, 495t and measurement of pH, 489 measurements in, 466–471 precision in, 495, 495t quantitative applications using, 485–494 activity versus concentration, 485–486 clinical applications, 492–494 environmental applications, 494 free ions versus complexed ions, 489 potentiometric titrations in, 494 using method of standard additions, 488–489 representative method for, 489–490 scale of operation in, 494 selectivity in, 496 sensitivity in, 495 time, cost and equipment in, 496 using external standards, 486–487 Potentiostat, 465, 465f Power, SI and non-SI units for, 13t Precipitant, 235 Precipitate, 139 final, composition of, 245–246 in precipitation gravimetry drying of, 245 filtering of, 243–244, 243f rinsing of, 244–245 use in turbidimetry/nephelometry, 443–444, 444t 792 Index Precipitation gravimetry, 234–255 accuracy in, 254 avoiding impurities in, 238–240 controlling particle size in, 240–243, 242f drying precipitate in, 245 evaluation of, 254–255 filtering precipitate in, 243–244, 243f final composition of precipitate in, 245–246 and inorganic analysis, 247–250, 248t–249t and organic analysis, 250, 250t qualitative calculations involving, 254 quantitative applications involving, 247–253 quantitative calculations involving, 250–254 representative method in, 246–247 rinsing precipitate, 244–245 solubility considerations in, 235–238, 236f–237f theory and practice, 235–247 Precipitation reactions, Color Plate 4, 139–140, 205t, 210–211 conservation of charge in, 22 reaction units in, 22 using coulometric titrations, 504, 504t Precipitation titration(s), 350–357 end point in potentiometrically, 354 selection and evaluation of, 354 using indicator, 354 evaluation of, 357 qualitative applications involving, 355–357 quantitative applications involving, 354–357 Precipitation titration curves, 350–353, 357f calculation of, 350–351, 352f, 352t sketching of, 352, 353f Precision control chart construction of, 717–718, 717t, 718f–719f subrange, 718, 719f Preconcentration, versus separation, 223–224 Prescriptive approach, to quality assurance, 712–714, 713f Preservation methods, for selected waters and wastewater parameters, 195, 195t Pressure, SI and non-SI units for, 13t Primary reagents, 106–107 Primary sample. See Gross sample Primary standard buffers, pH of, 492t Primary standards, recommended reagents for, 729–730 Probability distributions for populations, 71–75 for samples, 77–80 Problem-solving, analytical approach to, 5–8, 6f Procedure, 36 calibration and standardization in, 47 compensating for interferences, 45–46 development of, 45–47 protocols in, 48 validation in, 47 Proficiency standards, 711 Proline, acid dissociation constant for, 737t Propagation, of uncertainty, 64–70 1400-Indx 9/27/99 2:17 PM Page 792 isotopes half-life of, 643 rate of decay for, 643 particles, 642 Radiochemical methods, 642–649 characterization applications, 647–648 direct analysis, 644–645 evaluation of, 648–649 instrumentation in, 643–644 isotope dilution, 646–647 neutron activation analysis, 645–646 quantitative analysis using, 644–647 theory and practice in, 643 Raman spectroscopy, 373t Random number table, 728t Random sampling, 183–184 Range, 56 Rate of chemical reactions, 624, 750–751 and direct-computation rate methods, 629–630 and initial rate method, 753 maximum, for enzyme-catalyzed reactions, 638 Rate constant, 624 Rate law, 624–635, 750–751 Rayleigh scattering, 441, 441f Reaction(s) acid-base. See Acid-base reactions complexation. See Complexation reactions enzyme-catalyzed, 636–638, 637f equilibrium constants for, 139–148 first-order, 751–752, 753f ladder diagrams of, 150–155 noncatalytic, 638 nonenzyme-catalyzed, 637–638 oxidation-reduction, 23, 145–148, 748–749 precipitation, 22, 139–140, 504, 504t pseudo-order, 753 rate constant for, 624 rate law for, 624–625 rate of, 624, 750–751 reversible, 136 second-order, 752–753, 753f Reaction unit(s), 21 in acid-base reactions, 22 in complexation reaction, 23 for oxidizing agent, 23 in precipitation reaction, 22 in redox reactions, 23 for reducing agent, 23 Reagent(s) primary, 106–107 recommended for preparing primary standards, 729–730 secondary, 107 used as standards, 106–108 Reagent blank corrections, 128–129, 128t–129t Reagent grade, 107, 107f, 180 Recovery, 202 Propanoic acid, acid dissociation constant for, 738t Propelling unit, in flow injection analysis, 652 Property control chart, construction of, 715–716, 716f, 720f Proportional determinate errors, 61, 61t Propylamine, acid dissociation constant for, 738t Protecting agent, 420 Protein, total serum, UV/visible spectroscopy analysis of, 397t Protein-bound iodine, UV/visible spectroscopy analysis of, 397t Protocol, 37, 48 Protocol for a specific purpose (PSP), 707 Protons, conservation of, 22–23 Pseudo-order reactions, and method of initial rates, 753 PSP. See Protocol for a specific purpose (PSP) Purge-and-trap system, 214, 214f Pyridine, acid dissociation constant for, 738t Q Q-test. See Dixon Q-test Qualitative analysis, 8 Quality assessment, 708–712 external methods of, 711–712 internal methods of, 708–711 analysis of blanks in, 710 analysis of standards in, 710 and spike recoveries, 710–711 Quality assessment limits, for waters and wastewater, 708, 709t Quality assurance, 48, 705–721 components of, 705–708, 706f evaluation of, 712–721 prescriptive approach to, 712–714, 713f performance-based approach to, 714–721 using control chart for, 721 Quality control, 48, 706 Quantification, limit of, 96 Quantitative analysis, 9 Quantitative transfer, 30 Quantum yield, 425 Quartering, coning and, 199, 199f Quench, 634 Quinine, in urine, determination of, 431–432 R Radiation electromagnetic. See Electromagnetic radiation incident, 443 stray, 387–388 Radiationless deactivation, in fluorescence and phosphorescence spectroscopy, 424–427, 425f Radioactive analytes direct analysis of, 644–645 Index 793 Recrystallization, 205t, 210 Redox electrodes, 475 Redox indicators, 338–339, 339f, 339t Redox reactions balancing of, 748–749 ladder diagrams for, 155, 155f–156f reaction units in, 23 Redox titration(s), 331–350 analyte’s oxidation state in, adjustment of, 341–342 end point in finding potentiometrically, 339, 339f selection and evaluation of, 337–339 with visual indicator, 338–339, 339f, 339t equivalence points in, 337–338, 338f evaluation of, 350 in inorganic analysis, 344–345 in organic analysis, 346–347 quantitative applications involving, 341–350 calculation of, 347–350 representative method in, 340 titrants in, selecting and standardizing of, 342–344, 343t Redox titration curves, 332–336, 350f calculation of, 333–334 sketching of, 335, 335f–336f, 335t Reducing agent, 146 auxiliary, 341 reaction units for, 23 Reduction, 146 Reductor column, 341, 341f selected reductions in, 341, 342t Reference electrodes, 462–463, 471–473, 473f–474f Refractor index detector, for high-performance liquid chromatography, 585 Regression curvilinear, 127 equation, 122–124, 124f linear. See Linear regression standard deviation about the, 121 Reilly, C. N., 2 Relative supersaturation (RSS), 241 Relative uncertainty in concentration, in UV/visible and infrared spectroscopy, 410–411, 410t, 411f Relaxation, 423 Releasing agent, 420 Repeatability, 62 Reproducibility, 62 Residual current, 513 correcting in voltammetry, 521 Residual error, 118 in linear regression, 118, 119f Resolution, 376 in capillary electrophoresis, 601 in chromatography, 549–550, 549f–550f optimization of, capacity factor and, 556–558, 557f–558f Resorcinol, acid dissociation constant for, 738t Response, 667 1400-Indx 9/27/99 2:17 PM Page 793 Response surfaces, Color Plate 8, 667–668, 667f–668f central composite designs for, 682–683, 683f empirical models for, 675–676 factorial designs for, 676–682, 676f, 681f–682f mathematical models for, 674–683 searching algorithms for, 668–674, 668f theoretical models for, 675 Results characterization of, in analytical data, 54–57 distribution of, 70–82 Retention time, 548, 548f Retention volume, 548 Reverse-phase chromatography, 580 Reverse-phase separation, 582 solving triangle for, 582, 582f Reversibility, electrochemical, 527–528 Reversible reactions, 136 Riffle, 198, 198f Rinsing, precipitate, in precipitation gravimetry, 244–245 Robust, 42 RSS. See Relative supersaturation (RSS) Rugged, 42 Ruggedness testing, and standard method, 684–687, 684t Ruthenium, standard reduction potentials for, 745t S Safe Drinking Water Act, 36 Salicylic acid, 323t acid dissociation constant for, 738t Salt bridge, 466 Salt substitute, sodium in, 439–440 Sample(s), 71, 71t composite, 185 composition of, and target populations, 182–185 confidence intervals for, 80–81, 81t duplicate, 708 grab, 185, 197, 197f gross, 193 injection of, in capillary electrophoresis, 602 introduction of in gas chromatography, 567–568 in high-performance liquid chromatography, 584, 584f laboratory, 199 matrix, effect on normal calibration curve, 110, 110f means, comparison of, 88–93 multicomponent, 386 Beer’s law and, 386 number of, 191–192 obtaining of, 179–224 preparation of, 179–224 in atomic absorption spectroscopy, 418 in atomic emission spectroscopy, 437 in spectroscopy based on scattering, 443–444 primary. See Gross sample probability distributions for, 77–80 and central limit theorem, 77–79, 78t, 79f in situ, 185 size of, 187–190 solid. See Solid samples solution, 193–195, 195t standard, 683 blind analysis of, 683 types of, 185–186 volatile, 567–568 Sample thief, 198, 198f Sampling, 47 convenience, 185 headspace, 567 importance of, 180–182 judgmental, 184 overall variance attributable to method and, 180, 180f procedure in, 47 random, 183–184 stratified, 185 systematic, 184, 184f systematic-judgmental, 184–185, 186f Sampling error, 58 Sampling plan design of, 182–193 implementation of, 193–201 overall variance in, minimization of, 192–193 Sanders, 465 Saturated calomel electrodes, 472, 473f Scattering elastic, 441 large-particle, 441, 441f origin of, 441 Rayleigh, 441, 441f small-particle. See Rayleigh scattering spectroscopy based on, 441–445 Scientific notation, 12 Scintillation counter, 643–644 SCOT. See Support-coated open tubular columns (SCOT) Screened indicators, 289t Searching algorithms, for response surfaces, 668–674, 668f effectiveness and efficiency, 668–669 Second, 12t Secondary reagents, 107 Second derivative titration curve, 291t, 292, 292f Second-order reactions, 752–753, 753f Sediments, sampling of, 197, 197f Selectivity, of membrane, 476–477, 477f Selectivity coefficient, 40, 477, 477f Selenium, standard reduction potentials for, 745t Self-absorption, 428 Separation(s). See also Extraction(s) analytical, 544–547 chromatographic, 215 optimizing of, 556–563 phases in, 546 794 Index modules in flow injection analysis, 653, 654f versus preconcentration, 223–224 techniques. See Separation techniques, classification of Separation efficiency, general theory of, 202–205 Separation factor, 203 Separation techniques, classification of, 205–215, 205t change of state, 209–211 complexation reactions, 207–209 mass or density, 206–207 partitioning between phases, 211–215, 212f, 213t size, 205–206 Separatory funnel, Color Plate 3, 212, 212f Serine, acid dissociation constant for, 738t Serum, fluoxetine in, 588–589 Serum barbiturates, UV/visible spectroscopy analysis of, 397t Serum cholesterol, UV/visible spectroscopy analysis of, 397t Serum protein, total, 397t SFC. See Superfluid chromatography (SFC) SHE. See Standard hydrogen electrode (SHE) Shorthand notation, for potentiometric electrochemical cells, 467–468 Signal(s), 37 calibration of, 105–106 photons as, 372–374, 373f, 373t in total analysis techniques, 38 using mass as, 233–234 volume as, 274–275 Signal averaging, 391, 391f Signal processors, 380 Signal-to-noise ratio, 379 Sign conventions, in voltammetry, 510–513 Significance testing, 82–83, 82f construction of, 83 errors in, 84–85 one-tailed and two-tailed, 84, 84f Significant figures, 13–15 Silica, 213t Silicon, standard reduction potentials for, 745t Silver and argentometric titrations, 355, 355t standard reduction potentials for, 745t Silver chloride, solubility curve of, 236, 236f Silver/silver chloride electrode, 473, 474f Simplex optimization, 671–674, 671f, 673t, 674f–675f Simulated Rainwater (SRM), 62f Single-arm mechanical balance, 25, 26f Single-column ion chromatography, 593 Single-operator characteristics, 683 Single-point external standardization, 117 Single-point standardization, 108, 108f constant determinate error and, 118, 118t versus multiple-point standardizations, 108–109 Single-sided normal distribution, 725t–726t Singlet excited state, 423, 424f SI units, 12, 12t–13t 1400-Indx 9/27/99 2:17 PM Page 794 Spectrometer(s) atomic emission, 436, 436f Fourier transform infrared, 378, 381, 381f, 393, 570–571 infrared, 393 mass, 570–571, 585–586 Spectrometry, and concentration technique, 38 Spectrophotometers, 389, 389f–390f atomic absorption versus molecular absorption, 412 calibration of, 106 Fourier transform infrared, 570–571 and gas chromatography, 570–571 mass, 570–571 Spectrophotometric titration curves, for complexation titration, 324, 324f, 331, 331f Spectroscopic detectors, for high-performance liquid chromatography, 584–585 Spectroscopy, 368–445 atomic absorption. See Atomic absorption spectroscopy atomic emission. See Atomic emission spectroscopy atomic fluorescence, 373t based on emission, 423 based on scattering, 441–445 electron spin resonance, 373t fluorescence. See Fluorescence spectroscopy graphite furnace atomic absorption, 36, 37f, 48, 49f infrared. See Infrared spectroscopy instrumentation in basic components of, 374–380 detectors, 379–380 signal processors, 380 sources of energy for, 375, 375t transducers, 379–380, 380t wavelength selection, 376–379, 376f microwave, 373t molecular absorption, 388–394 Mossbauer, 373t overview of, 369–374 phosphorescence. See Phosphorescence spectroscopy photoluminescence. See Photoluminescence spectroscopy Raman, 373t signals in, measuring photons as, 372–374 types of, 372–374 involving exchanges of energy, 372, 373t not involving exchanges of energy, 374, 374t ultraviolet/visible. See Ultraviolet/visible spectroscopy x-ray absorption, 373t Spectrum emission, 374, 427, 427f excitation, 427 mass, 571 ultraviolet/visible, Color Plate 6, 382–384, 383f Spike recovery, 710–711 Size-exclusion chromatography, 205t, 206, 593–596, 595f–596f Slit orientation, and sensitivity, in fluorescence spectroscopy, 433, 433f Slit width, selection of in atomic absorption spectroscopy, 415–418 in atomic emission spectroscopy, 437 Slope, estimated, determination by linear regression, 119 Slope-ratio method, 407 Slow injection analysis, 649–658 Small-particle scattering. See Rayleigh scattering SMDE. See Static mercury drop electrode (SMDE) Smith-Hieftje background correction, 419 Sodium absorption spectrum for, 384, 384f in salt substitute, 439–440 standard reduction potentials for, 745t valence energy diagram of, 383, 383f Sodium hydroxide, as indicator for acid-base titrations, 278 Sodium tetraphenylborate, 249t Solid electrodes, in voltammetry, 510, 510f Solid-phase extraction, 212–213, 212f, 213t adsorbents for, 212–213, 213t Solid-phase microextraction (SPME), 213, 567 Solid samples, 196–201 bringing into solution, 200–201, 200f, 200t collection of, 197–198, 197f digestion of, 200–201, 200f, 200t preparation of, 198 preservation of, 198 reducing particle size in, 199–200 Solid-state ion-selective electrodes, 479–482, 481t Solubility, effect of complexation on, 165–167 Solubility product, 140, 731t–732t Solute, total velocity of, in capillary electrophoresis, 599–600 Solution(s) buffer. See Buffer solutions preparation of, 30–32 stock, 30–31 Solution samples, 193–195 collection of, 193–194 preparation of, 195 preservation of, 194–195, 195t Solvent dissociation constant of, 295 nonaqueous, acid-base titrations in, 294f, 295–296, 296f Solvent triangle, 582, 582f SOP. See Standard operations procedure (SOP) Soxhlet extractor, 214, 214f Spectral interference in atomic absorption spectroscopy, 418–419 in atomic emission spectroscopy, 437–438, 438f Spectral searching, 403, 404f Spectrofluorometer, 434 Index 795 Split injection, 568 Splitless injection, 568 SPME. See Solid-phase microextraction (SPME) Spread, measures of, 55–57 Squalane, 566t SRM. See Simulated Rainwater (SRM) Stacking, in capillary electrophoresis, 603, 603f Standard(s) analysis of, in quality assessment, 710 external, 109–110 in atomic emission spectroscopy, 438 in potentiometry, 486–487 internal, 115–117 in atomic emission spectroscopy, 438 primary, recommended reagents for, 729–730 proficiency, 711 reagents used as, 106–108 Standard additions, Color Plate 2, 110–115 Standard buffers, primary, pH of, 492t Standard deviation, 56–57 about the regression, 121 Standard hydrogen electrode (SHE), 471–472, 472f Standardization, 47 external multiple-point, 117, 117t single-point, 117 multiple-point, 109 versus single-point, 108–109 of procedure, 47 single-point, 108–109, 108f, 118, 118t of titrants, 298–300, 299t in complexation titrations, 327 in redox titrations, 342–344, 343t Standard method, 106 in atomic absorption spectroscopy, 420–421 in atomic emission spectroscopy, 438–440 and blind analysis of standard samples, 683 development of, 666–698 and equivalency testing, 687 in fluorescence and phosphorescence spectroscopy, 431–432 optimizing experimental procedure with, 667–683 response surfaces and, 667–674, 667f–668f in phosphorescence spectroscopy, 431–432 in photoluminescence spectroscopy, 431–432 and ruggedness testing, 684–687, 684t single-operator characteristics of, 683 two-sample collaborative test and, 688–693, 689f, 691f validating method as, 687–698 verification of, 683–687 Standard operations procedure (SOP), 707 Standard reduction potentials, 743t–746t Standard reference material, 61 Standard sample, 683 Standard solution, 108 Standard-state conditions, 137 Standard-state potential, determining by voltammetry, 514–515, 515f 1400-Indx 9/27/99 2:17 PM Page 795 Static mercury drop electrode (SMDE), 509, 509f Stationary phase(s), 546 bonded, 580 in gas chromatography, 565–567, 566f, 567f bleed and, 566 in high-performance liquid chromatography, 579–580 Statistical analysis, of data, 82–85 Steady-state approximation, 636 Stepwise formation constant, 144 Stock solution, preparation of, 30–32 Stoichiometry calculations in, 20–25 using conservation principles in, 23–25 of metal-ligand complex, 403–407, 406f Stopped flow analyzer, 634, 635f Straight-line calibration curves, linear regression of, 118, 119f Stratified sampling, 185 Stray radiation, 387–388 Stripping voltammetry, 516–519, 518f, 519t Strontium, standard reduction potentials for, 745t Styrene, 213t Sublimation, 205t, 209 Subrange precision control charts, 718, 719f Substrate, 636 Succinic acid acid dissociation constant for, 738t titration curve for, 288, 288f Sulfate solubility products for, 732t in water, turbidimetric determination of, 445 Sulfide, solubility products for, 732t Sulfur, standard reduction potentials for, 745t Sulfuric acid, acid dissociation constant for, 738t Sulfurous acid, acid dissociation constant for, 738t Supercritical fluids continuous extraction of, 215 critical point properties of, 596, 597t equipment for studying, Color Plate 7 phase diagram of, 596, 596f properties of, 596, 596t Superfluid chromatography (SFC), 596–597 mobile phases for, 596, 597t Supernatant, 244, 244f Support-coated open tubular columns (SCOT), 565 Surface adsorbates, 239, 239f Surfactants, in waters and wastewater, 395t Syringes, types of, 28, 28f Systematic-judgmental sampling, 184–185, 186f Systematic sampling, 184, 184f Système International d’Unités. See SI units T Tailing, in column chromatography, 555, 555f Tare, 26 Target populations, sample composition and, 182–185 TCD. See Thermal conductivity detector (TCD) Technique, 36 classification of, 37–38 Temperature finding end point by, 293–295, 294f programming, 558 scales for, 13t SI and non-SI units for, 13t SI unit for, 12t Tetraethylthiuram disulfide, 21, 24 Thallium, standard reduction potentials for, 745t Theorem central limit, 77–79, 78t, 79f Nyquist, 184 Theoretical models, for response surfaces, 675 Theoretical plate, 553 net height of, 560–563, 562f and resolution in chromatography, 559–560, 560t Thermal conductivity detector (TCD), 569, 569f Thermal energy, sources of, in spectroscopy, 375 Thermal excitation, spectroscopies involving, 373t Thermal ionization mass spectrometry, 7–8 Thermal transducers, 379–380, 380t Thermodynamic equilibrium constant, activity and, 172–173 Thermodynamics, and equilibrium chemistry, 136–138 Thermogram, 256, 256f Thermogravimetry, 255–257, 256f Thermometric titration curve, 293–295, 294f Thiocyanate, solubility products for, 732t Thiosulfate, 344 Thiosulfuric acid, acid dissociation constant for, 738t Threonine, acid dissociation constant for, 738t Thymol blue, 289t Time, SI unit for, 12t TIMS. See Thermal ionization mass spectrometry Tin, standard reduction potentials for, 746t TISAB. See Total ionic strength adjustment buffer (TISAB) Titanium, standard reduction potentials for, 746t Titrant, 274 selection and standardization of, 298–300, 299t in complexation titrations, 327 in redox titrations, 342–344, 343t Titration(s) acid-base. See Acid-base titration(s) argentometric, 355 back, 275 complexation. See Complexation titrations coulometric, 503–504, 503t, 505–506 diffusional microtitration, 312, 312f 796 Index displacement, 275 end-point of, selection and evaluation of, 287–295 instrumentation for, 278, 279f potentiometric, 494 precipitation. See Precipitation titration(s) redox. See Redox titration(s) Titration curve(s), 275–276, 276f–277f acid-base, 279–287, 285f–286f of acids, 280–284, 281t, 283f, 283t, 288, 288f of bases, 280–284, 281t, 283f, 283t for complexation titrations, 324, 324f, 331, 331f of EDTA, 317–322, 319f, 320t, 321f–322f finding end point using, derivative method for, 291t, 292–293, 292f first derivative, 291, 291t, 292f precipitation, 350–353, 352f, 352t, 353f, 357f redox, 332–336, 335f–336f, 335t, 350f second derivative, 291t, 292, 292f thermometric, 293–295, 294f Titration error, 274 Titrimetry, 273–357 overview of, 274–278 as total analysis technique, 38 Tolerance, 58 Toothpaste, fluoride in, 489–490 Total analysis techniques, 38 signals in, 38 Total ionic strength adjustment buffer (TISAB), 487 Total mobility, in capillary electrophoresis, 599–600 Total serum protein, UV/visible spectroscopy analysis of, 397t Total velocity, of solute, in capillary electrophoresis, 599–600 Total Youden blank, 129 Trace metals Contract Laboratory Program protocols for, 48, 49f in environmental samples, voltammetry and, 524–525, 525t Tracer, 646 Transducers photon, 379, 380t thermal, 379–380, 380t Transitions. See Electronic transitions Transmittance, 384 in absorption spectroscopy, 384–385, 384f Transport system, in flow injection analysis, 652 Trichloroacetic acid, acid dissociation constant for, 738t Triethanolamine acid, acid dissociation constant for, 738t Triethylamine, acid dissociation constant for, 738t Trihalomethanes, in drinking water, 576–577 Trip blank, 710 1400-Indx 9/27/99 2:17 PM Page 796 scale of operation in, 409 selectivity in, 412 sensitivity in, 411 time, cost and equipment in, 412 visible spectrum, Color Plate 6 Uncertainty. See also Relative uncertainty in concentration, in UV/visible and infrared spectroscopy in addition or subtraction, 65–66 calculation of, usefulness of, 68–70 error and, 64 in linear regression, 120–122, 120f for mixed operations, 66 in multiplication or division, 66 for other mathematical functions, 67–68, 68t precision versus, 64, 64t propagation of, 64–70 Uncompetitive inhibition, of enzymes, 639, 640f Universal indicator, for pH, Color plate 4 Unpaired data, 88–91 Unweighted linear regression, with errors in y, 119–124 Uranium, standard reduction potentials for, 746t Uric acid, UV/visible spectroscopy analysis of, 397t Urine creatinine in, 632–633 determination of quinine in, 431–432 V Validation, 47 of standard method, 687–698 Vanadium, standard reduction potentials for, 746t van Deemter equation, 561 Variable-time integral methods, 628–629 Variance, 57 analysis of, 693–697 and F-test, 88 measurement of, 57 overall, 192–193 of populations, 79 Velocity electroosmotic flow, 599 electrophoretic, 598 total, 599–600 Verification, of standard method, 683–687 Vibrational relaxation, 424, 425f Visual indicators. See Indicator(s) Vitamin(s), natural fluorescence of, 430t Vitamin B complex, 607–608 VOCs. See Volatile organic compounds (VOCs) Void time, 549, 549f Void volume, 549 Volatile organic compounds (VOCs), extraction of, 214 Volatile sample, preparation of, in gas chromatography, 567–568 Volatilization, 205t Triplet excited state, 423, 424f Triprotic acid, pH of, 165 Triprotic base, pH of, 165 Tswett, Mikhail, 546 t-test, 85–86, 726t paired, 92–93 Tungsten, standard reduction potentials for, 746t Turbidimeter, 442, 442f Turbidimetry, 441–445 applications of, 444–445 determination of concentration by, 442–443 and determination of sulfate in water, 445 versus nephelometry, 442 precipitates used in, 443–444, 444t Turnover number, 638 Two-factor response surface, 669–671, 669f–671f Two-sample collaborative test, 688–693, 689f, 691f Two-tailed significance test, 84, 84f t-table for, 726 Type 1 error, 84–85 Type 2 error, 84–85 U Ultraviolet/visible radiation, versus infrared, absorption of, 381, 381f Ultraviolet/visible spectrophotometry, for high- performance liquid chromatography, 585, 585f Ultraviolet/visible spectroscopy, 373t, 388–412 accuracy in, 409 and analysis of waters and wastewater, 395, 395t, 398–399 for atoms, 383–384 cells used in, 391, 392f characterization applications involving, 403–409 clinical applications involving, 397, 397t determination of equilibrium constants in, 407–409 environmental applications involving, 395–396, 395t evaluation in, 409–412 forensic applications involving, 398 industrial applications involving, 397–398 instrumentation in, 388–394 designs of, 388–393 for molecules and ions, 382–383, 383f precision in, 409–411, 411f qualitative applications involving, 402–403 quantitative applications involving, 391, 392f developing method for single component, 398–399 for mixtures, 400–402 for single analyte, 400 and relative uncertainty in concentration, effect of indeterminate instrument errors on, 410–411, 410t, 411f Index 797 Volatilization gravimetry, 234, 255–262 equipment used in, 257 evaluation of, 262 and inorganic analysis, 259 in organic analysis, 259–260 quantitative applications involving, 259–262 calculations in, 260–262 representative method of, 257–258 theory and practice in, 255–258 Volhard method, 354, 355t Volt, 13t Voltammetric electrochemical cell, 510, 511f Voltammetry, 508–532 accuracy in, 531 characterization applications of, 527–530 and concentration technique, 38 correcting residual current in, 521 current in, 510–513 evaluation of, 531 hydrodynamic, 513, 516–519 precision in, 531 quantitative and qualitative aspects of, 514–515 quantitative applications using, 520–527 for analysis of single components, 521–523 and clinical samples, 525, 526t and environmental samples, 524–525, 525t and miscellaneous samples, 525–527 for multicomponent analysis, 523–524, 523f selecting technique for, 520–521 scale of operation in, 531 selectivity in, 531 sensitivity in, 531 sign conventions in, 510–513 solid electrodes in, 510, 510f stripping, 516–519, 518f, 519t techniques in, 515–520 time, cost and equipment in, 531 voltammetric measurements, 509–510 Voltammogram, 508 effect of complexation on, 528–530, 528f shape of, 513–514, 513f Volume breakthrough, 196 end point for phosphate species, 308t and sources of alkalinity, 301, 302t equipment for measuring of, 26–29, 27f–28f retention, 548 as a signal, 274–275 SI unit for, 12t void, 549 weight-to-volume percent, 18 Volume percent, 18 Volumetric flask, 26, 27f measurement errors for, 59t, 60 1400-Indx 9/27/99 2:17 PM Page 797 W Walden reductor, 341 selected reductions using, 341, 342t Wall-coated open tubular columns (WCOT), 565 Walsh, A., 412 Waters and wastewater. See also Drinking water dissociation of, 142 constant for, 142–143 preservation methods and maximum holding times for, 195, 195t quality assessment limits for, 708, 709t sulfate in, 445 UV/visible spectroscopy of, 395, 395t, 398–399 Watt, 13t Wavelength, 370 nominal, 376 selection of, 376–379, 376f in atomic absorption spectroscopy, 415–418, 418t in atomic emission spectroscopy, 437 in spectroscopy, 376–379, 376f using filters, 376 using monochromators, 376–378, 378f Wavenumber, 370–371 Wave properties, of electromagnetic radiation, 369–371, 369f WCOT. See Wall-coated open tubular columns (WCOT) Weight equivalent, 17, 309–310 formula, 17 Weighted linear regression with errors in both x and y, 127 with errors in y, 124–127, 126f Weight percent, 18 798 Index Weight-to-volume percent, 18 Work, SI and non-SI units for, 13t X X-ray absorption spectroscopy, 373t X-ray fluorescence, 373t Y y-intercept, determination of, by linear regression, 119 Z Zeeman effect background correction, 419 Zeta potential, 599 Zinc standard reduction potentials for, 746t in waters and wastewater, 395t 1400-Indx 9/27/99 2:17 PM Page 798 1400-Indx 9/27/99 2:17 PM Page 799 1400-Indx 9/27/99 2:17 PM Page 800