Electroanalytical
Chemistry
Lecture #4
Why Electrons Transfer?
The Metal Electrode
EF
E
?Ef = Fermi level;
highest occupied
electronic energy
level in a metal
Why Electrons Transfer
EF
Eredox E
F
Eredox
?Net flow of electrons from M
to solute
?Ef more negative than Eredox
?more cathodic
?more reducing
Reduction Oxidation
?Net flow of electrons from
solute to M
?Ef more positive than Eredox
?more anodic
?more oxidizing
E E
The Kinetics of Electron
Transfer
?Consider:
O + ne- = R
?Assume:
?O and R are stable,soluble
?Electrode of 3rd kind (i.e.,inert)
?no competing chemical reactions occur
kR
ko
Equilibrium for this Reaction
is Characterised by...
?The Nernst equation:
Ecell = E0 - (RT/nF) ln (cR*/co*)
?where:
cR* = [R] in bulk solution
co* = [O] in bulk solution
?So,Ecell is related directly to [O] and [R]
Equilibrium (cont’d)
?At equilibrium,
no net current flows,i.e.,
?E = 0 ? i = 0
?However,there will be a dynamic
equilibrium at electrode surface:
O + ne- = R
R - ne- = O
both processes will occur at equal rates
so no net change in solution composition
Current Density,I
?Since i is dependent on area of electrode,
we,normalize currents and examine
I = i/A
we call this current density
?So at equilibrium,I = 0 = iA + iC
? ia/A = -ic/A = IA = -Ic = Io
which we call the exchange current
density
?Note,by convention iA produces positive
current
Exchange Current Density
?Significance?
?Quantitative measure of amount of
electron transfer activity at equilibrium
?Io large ? much simultaneous ox/red
electron transfer (ET)
? inherently fast ET (kinetics)
?Io small ? little simultaneous ox/red
electron transfer (ET)
? sluggish ET reaction (kinetics)
Summary,Equilibrium
?Position of equilibrium characterized
electrochemically by 2 parameters:
?Eeqbm - equilibrium potential,Eo
?Io - exchange current density
How Does I vary with E?
?Let’s consider:
?case 1,at equilibrium
?case 2,at E more negative than Eeqbm
?case 3,at E more positive than Eeqbm
Case 1,At Equilibrium
E = Eo - (RT/nF)ln(CR*/CO*)
E - E0 = - (RT/nF)ln(CR*/CO*)
E = Eo so,CR* = Co*
I = IA + IC = 0 no net current flows
IA
IC
O R
Reaction Coordinate
?G
Case 2,At E < Eeqbm
?E - Eeqbm = negative number
= - (RT/nF)ln(CR*/CO*)
? ln(CR*/CO*) is positive
? CR* > CO* ? some O converted to R
? net reduction
? passage of net reduction current
IA
IC
O
R
Reaction Coordinate
?G
I = IA + IC < 0
Case 2,At E > Eeqbm
?E - Eeqbm = positive number
= - (RT/nF)ln(CR*/CO*)
? ln(CR*/CO*) is negative
? CR* < CO* ? some R converted to O
? net oxidation
? passage of net oxidation current
IA
IC O
R
Reaction Coordinate
?G
I = IA + IC > 0
Overpotential,?
?Fast ET = current rises almost vertically
?Slow ET = need to go to very positive/negative
potentials to produce significant current
?Cost is measured in overpotential,? = E - Eeqbm
?
fast slow
Cathodic Potential,VEeqbm
Ca
thodi
c
Cu
rre
nt,
A
Edecomp
Can We Eliminate ??
What are the Sources of ?
?? = ?A + ?R + ?C
??A,activation
an inherently slow ET = rate determining step
??R,resistance
due to finite conductivity in electrolyte
solution or formation of insulating layer on
electrode surface; use Luggin capillary
??C,concentration
polarization of electrode (short times,stirring)
Luggin Capillary
?Reference electrode
placed in glass
capillary containing
test solution
?Narrow end placed
close to working
electrode
?Exact position
determined
experimentally
Reference
Luggin
Capillary
Working
Electrode
The Kinetics of ET
?Let’s make 2 assumptions:
?both ox/red reactions are first order
?well-stirred solution (mass transport plays no
role)
?Then rate of reduction of O is:
- kR co*
where kR is electron transfer rate constant
The Kinetics of ET (cont’d)
?Then the cathodic current density is:
?IC = -nF (kRCO*)
?Experimentally,kR is found to have an
exponential (Arrhenius) potential
dependence:
kR = kOC exp (- ?CnF E/RT)
?where ?C = cathodic transfer coefficient
(symmetry)
?kOC = rate constant for ET at E=0 (eqbm)
?,Transfer Coefficient
O
R
Reaction Coordinate
?G
? - measure of symmetry of
activation energy barrier
? = 0.5 ? activated complex
halfway between reagents/
products on reaction coordinate;
typical case for ET at type III
M electrode
The Kinetics of ET (cont/d)
Substituting:
?IC = - nF (kR co*) =
= - nF c0* kOC exp(- ?CnF E/RT)
Since oxidation also occurring
simultaneously,
?rate of oxidation = kA cR*
?IA = (nF)kACR*
?kA = kOA exp(+ ?AnF E/RT)
?So,substituting
IA= nF CR* kOA exp(+?AnF E/RT)
?And,since I = IC + IA then:
?I = -nF cO*kOC exp(- ?CnF E/RT) +
nF cR*kOA exp(+ ?AnF E/RT)
I = nF (-cO*kOC exp(- ?CnF E/RT) +
cR*kOA exp(+ ?AnF E/RT))
The Kinetics of ET (cont’d)
The Kinetics of ET (cont’d)
?At equilibrium (E=Eeqbm),recall
Io = IA = - IC
?So,the exchange current density is given
by:
nF cO*kOC exp(- ?CnF Eeqbm/RT) =
nF cR*kOA exp(+ ?AnF Eeqbm/RT) = I0
The Kinetics of ET (cont’d)
?We can further simplify this expression by
introducing ? (= E + Eeqbm):
?I = nF [-cO*kOC exp(- ?CnF (? +
Eeqbm)/RT) + cR*kOA exp(+ ?AnF (? +
Eeqbm)/ RT)]
?Recall that ea+b = eaeb
?So,
I = nF [-cO*kOC exp(- ?CnF ?/RT) exp(-
?CnF Eeqbm/RT) + cR*kOA exp(+ ?AnF ?/
RT) exp(+ ?AnF Eeqbm/ RT)]
?So,
I = nF [-cO*kOC exp(- ?CnF ?/RT) exp(-
?CnF Eeqbm/RT) + cR*kOA exp(+ ?AnF ?/
RT) exp(+ ?AnF Eeqbm/ RT)]
?And recall that IA = -IC = I0
So,
I = Io [-exp(- ?CnF ?/RT) +
exp(+ ?AnF ?/ RT)]
This is the Butler-Volmer equation
The Kinetics of ET (cont’d)
The Butler-Volmer
Equation
?I = Io [- exp(- ?CnF ?/RT) +
exp(+ ?AnF ?/ RT)]
?This equation says that I is a function of:
??
?I0
??C and ?A
The Butler-Volmer
Equation (cont’d)
?For simple ET,
?C + ?A = 1 ie.,?C =1 - ?A
?Substituting:
I = Io [exp((?A - 1)nF ?/RT) + exp(?AnF
?/ RT)]
Let’s Consider 2 Limiting
Cases of B-V Equation
?1,low overpotentials,?< 10 mV
?2,high overpotentials,? > 52 mV
Case 1,Low Overpotential
?Here we can use a Taylor expansion to
represent ex:
ex = 1 + x +,..
?Ignoring higher order terms:
I = Io [1+ (?A nF ?/RT) - 1 - (?A- 1)nF ?/
RT)] = Io nF?/RT
?I = Io nF?/RT
so total current density varies linearly with
? near Eeqbm
Case 1,Low Overpotential
(cont’d)
?I = (Io nF/RT) ?
intercept = 0
slope = Io nF/RT
?Note,F/RT = 38.92 V-1 at 25oC
Case 2,High
Overpotential
?Let’s look at what happens as ? becomes
more negative then if IC >> IA
?We can neglect IA term as rate of
oxidation becomes negligible then
I = -IC = Io exp (-?CnF ?/RT)
?So,current density varies exponentially
with ?
Case 2,High
Overpotential (cont’d)
?I = Io exp (-?CnF ?/RT)
?Taking ln of both sides:
ln I = ln (-IC) = lnIo + (-?CnF/RT) ?
which has the form of equation of a line
?We call this the cathodic Tafel equation
?Note,same if ? more positive then
ln I = ln Io + ?A nF/RT ?
we call this the anodic Tafel equation
Tafel Equations
?Taken together the equations form the
basis for experimental determination of
?Io
??c
??A
?We call plots of ln i vs,? are called Tafel
plots
?can calculate ? from slope and Io from y-
intercept
Tafel Equations (cont’d)
?Cathodic,ln I = lnIo + (-?CnF/RT) ?
y = b + m x
?If ?C = ?A = 0.5 (normal),
for n= 1 at RT
slope = (120 mV)-1
Tafel Plots
In real systems often see large negative deviations
from linearity at high ? due to mass transfer
limitations
?,VEeqbm
ln |i|
_ +
Cathodic Anodic
ln Io
High overpotential:
ln I = lnIo + (?AnF/RT) ?
Mass transport
limited current
Low overpotential:
I = (Io nF/RT) ?
EXAMPLE:
?Can distinguish simultaneous vs,
sequential ET using Tafel Plots
?EX,Cu(II)/Cu in Na2SO4
?If Cu2+ + 2e- = Cu0 then slope = 1/60 mV
?If Cu2+ + e- = Cu+ slow?
Cu+ + e- = Cu0 then slope = 1/120 mV
?Reality,slope = 1/40 mV
viewed as ?n = 1 + 0.5 = 1.5
?Interpreted as pre-equilibrium for 1st ET
followed by 2nd ET
Effect of ? on Current
Density
??A = 0.75 oxidation is favored
??C = 0.75 reduction is favored
Homework:
?Consider what how a Tafel plot changes
as the value of the transfer coefficient
changes.
Chemistry
Lecture #4
Why Electrons Transfer?
The Metal Electrode
EF
E
?Ef = Fermi level;
highest occupied
electronic energy
level in a metal
Why Electrons Transfer
EF
Eredox E
F
Eredox
?Net flow of electrons from M
to solute
?Ef more negative than Eredox
?more cathodic
?more reducing
Reduction Oxidation
?Net flow of electrons from
solute to M
?Ef more positive than Eredox
?more anodic
?more oxidizing
E E
The Kinetics of Electron
Transfer
?Consider:
O + ne- = R
?Assume:
?O and R are stable,soluble
?Electrode of 3rd kind (i.e.,inert)
?no competing chemical reactions occur
kR
ko
Equilibrium for this Reaction
is Characterised by...
?The Nernst equation:
Ecell = E0 - (RT/nF) ln (cR*/co*)
?where:
cR* = [R] in bulk solution
co* = [O] in bulk solution
?So,Ecell is related directly to [O] and [R]
Equilibrium (cont’d)
?At equilibrium,
no net current flows,i.e.,
?E = 0 ? i = 0
?However,there will be a dynamic
equilibrium at electrode surface:
O + ne- = R
R - ne- = O
both processes will occur at equal rates
so no net change in solution composition
Current Density,I
?Since i is dependent on area of electrode,
we,normalize currents and examine
I = i/A
we call this current density
?So at equilibrium,I = 0 = iA + iC
? ia/A = -ic/A = IA = -Ic = Io
which we call the exchange current
density
?Note,by convention iA produces positive
current
Exchange Current Density
?Significance?
?Quantitative measure of amount of
electron transfer activity at equilibrium
?Io large ? much simultaneous ox/red
electron transfer (ET)
? inherently fast ET (kinetics)
?Io small ? little simultaneous ox/red
electron transfer (ET)
? sluggish ET reaction (kinetics)
Summary,Equilibrium
?Position of equilibrium characterized
electrochemically by 2 parameters:
?Eeqbm - equilibrium potential,Eo
?Io - exchange current density
How Does I vary with E?
?Let’s consider:
?case 1,at equilibrium
?case 2,at E more negative than Eeqbm
?case 3,at E more positive than Eeqbm
Case 1,At Equilibrium
E = Eo - (RT/nF)ln(CR*/CO*)
E - E0 = - (RT/nF)ln(CR*/CO*)
E = Eo so,CR* = Co*
I = IA + IC = 0 no net current flows
IA
IC
O R
Reaction Coordinate
?G
Case 2,At E < Eeqbm
?E - Eeqbm = negative number
= - (RT/nF)ln(CR*/CO*)
? ln(CR*/CO*) is positive
? CR* > CO* ? some O converted to R
? net reduction
? passage of net reduction current
IA
IC
O
R
Reaction Coordinate
?G
I = IA + IC < 0
Case 2,At E > Eeqbm
?E - Eeqbm = positive number
= - (RT/nF)ln(CR*/CO*)
? ln(CR*/CO*) is negative
? CR* < CO* ? some R converted to O
? net oxidation
? passage of net oxidation current
IA
IC O
R
Reaction Coordinate
?G
I = IA + IC > 0
Overpotential,?
?Fast ET = current rises almost vertically
?Slow ET = need to go to very positive/negative
potentials to produce significant current
?Cost is measured in overpotential,? = E - Eeqbm
?
fast slow
Cathodic Potential,VEeqbm
Ca
thodi
c
Cu
rre
nt,
A
Edecomp
Can We Eliminate ??
What are the Sources of ?
?? = ?A + ?R + ?C
??A,activation
an inherently slow ET = rate determining step
??R,resistance
due to finite conductivity in electrolyte
solution or formation of insulating layer on
electrode surface; use Luggin capillary
??C,concentration
polarization of electrode (short times,stirring)
Luggin Capillary
?Reference electrode
placed in glass
capillary containing
test solution
?Narrow end placed
close to working
electrode
?Exact position
determined
experimentally
Reference
Luggin
Capillary
Working
Electrode
The Kinetics of ET
?Let’s make 2 assumptions:
?both ox/red reactions are first order
?well-stirred solution (mass transport plays no
role)
?Then rate of reduction of O is:
- kR co*
where kR is electron transfer rate constant
The Kinetics of ET (cont’d)
?Then the cathodic current density is:
?IC = -nF (kRCO*)
?Experimentally,kR is found to have an
exponential (Arrhenius) potential
dependence:
kR = kOC exp (- ?CnF E/RT)
?where ?C = cathodic transfer coefficient
(symmetry)
?kOC = rate constant for ET at E=0 (eqbm)
?,Transfer Coefficient
O
R
Reaction Coordinate
?G
? - measure of symmetry of
activation energy barrier
? = 0.5 ? activated complex
halfway between reagents/
products on reaction coordinate;
typical case for ET at type III
M electrode
The Kinetics of ET (cont/d)
Substituting:
?IC = - nF (kR co*) =
= - nF c0* kOC exp(- ?CnF E/RT)
Since oxidation also occurring
simultaneously,
?rate of oxidation = kA cR*
?IA = (nF)kACR*
?kA = kOA exp(+ ?AnF E/RT)
?So,substituting
IA= nF CR* kOA exp(+?AnF E/RT)
?And,since I = IC + IA then:
?I = -nF cO*kOC exp(- ?CnF E/RT) +
nF cR*kOA exp(+ ?AnF E/RT)
I = nF (-cO*kOC exp(- ?CnF E/RT) +
cR*kOA exp(+ ?AnF E/RT))
The Kinetics of ET (cont’d)
The Kinetics of ET (cont’d)
?At equilibrium (E=Eeqbm),recall
Io = IA = - IC
?So,the exchange current density is given
by:
nF cO*kOC exp(- ?CnF Eeqbm/RT) =
nF cR*kOA exp(+ ?AnF Eeqbm/RT) = I0
The Kinetics of ET (cont’d)
?We can further simplify this expression by
introducing ? (= E + Eeqbm):
?I = nF [-cO*kOC exp(- ?CnF (? +
Eeqbm)/RT) + cR*kOA exp(+ ?AnF (? +
Eeqbm)/ RT)]
?Recall that ea+b = eaeb
?So,
I = nF [-cO*kOC exp(- ?CnF ?/RT) exp(-
?CnF Eeqbm/RT) + cR*kOA exp(+ ?AnF ?/
RT) exp(+ ?AnF Eeqbm/ RT)]
?So,
I = nF [-cO*kOC exp(- ?CnF ?/RT) exp(-
?CnF Eeqbm/RT) + cR*kOA exp(+ ?AnF ?/
RT) exp(+ ?AnF Eeqbm/ RT)]
?And recall that IA = -IC = I0
So,
I = Io [-exp(- ?CnF ?/RT) +
exp(+ ?AnF ?/ RT)]
This is the Butler-Volmer equation
The Kinetics of ET (cont’d)
The Butler-Volmer
Equation
?I = Io [- exp(- ?CnF ?/RT) +
exp(+ ?AnF ?/ RT)]
?This equation says that I is a function of:
??
?I0
??C and ?A
The Butler-Volmer
Equation (cont’d)
?For simple ET,
?C + ?A = 1 ie.,?C =1 - ?A
?Substituting:
I = Io [exp((?A - 1)nF ?/RT) + exp(?AnF
?/ RT)]
Let’s Consider 2 Limiting
Cases of B-V Equation
?1,low overpotentials,?< 10 mV
?2,high overpotentials,? > 52 mV
Case 1,Low Overpotential
?Here we can use a Taylor expansion to
represent ex:
ex = 1 + x +,..
?Ignoring higher order terms:
I = Io [1+ (?A nF ?/RT) - 1 - (?A- 1)nF ?/
RT)] = Io nF?/RT
?I = Io nF?/RT
so total current density varies linearly with
? near Eeqbm
Case 1,Low Overpotential
(cont’d)
?I = (Io nF/RT) ?
intercept = 0
slope = Io nF/RT
?Note,F/RT = 38.92 V-1 at 25oC
Case 2,High
Overpotential
?Let’s look at what happens as ? becomes
more negative then if IC >> IA
?We can neglect IA term as rate of
oxidation becomes negligible then
I = -IC = Io exp (-?CnF ?/RT)
?So,current density varies exponentially
with ?
Case 2,High
Overpotential (cont’d)
?I = Io exp (-?CnF ?/RT)
?Taking ln of both sides:
ln I = ln (-IC) = lnIo + (-?CnF/RT) ?
which has the form of equation of a line
?We call this the cathodic Tafel equation
?Note,same if ? more positive then
ln I = ln Io + ?A nF/RT ?
we call this the anodic Tafel equation
Tafel Equations
?Taken together the equations form the
basis for experimental determination of
?Io
??c
??A
?We call plots of ln i vs,? are called Tafel
plots
?can calculate ? from slope and Io from y-
intercept
Tafel Equations (cont’d)
?Cathodic,ln I = lnIo + (-?CnF/RT) ?
y = b + m x
?If ?C = ?A = 0.5 (normal),
for n= 1 at RT
slope = (120 mV)-1
Tafel Plots
In real systems often see large negative deviations
from linearity at high ? due to mass transfer
limitations
?,VEeqbm
ln |i|
_ +
Cathodic Anodic
ln Io
High overpotential:
ln I = lnIo + (?AnF/RT) ?
Mass transport
limited current
Low overpotential:
I = (Io nF/RT) ?
EXAMPLE:
?Can distinguish simultaneous vs,
sequential ET using Tafel Plots
?EX,Cu(II)/Cu in Na2SO4
?If Cu2+ + 2e- = Cu0 then slope = 1/60 mV
?If Cu2+ + e- = Cu+ slow?
Cu+ + e- = Cu0 then slope = 1/120 mV
?Reality,slope = 1/40 mV
viewed as ?n = 1 + 0.5 = 1.5
?Interpreted as pre-equilibrium for 1st ET
followed by 2nd ET
Effect of ? on Current
Density
??A = 0.75 oxidation is favored
??C = 0.75 reduction is favored
Homework:
?Consider what how a Tafel plot changes
as the value of the transfer coefficient
changes.