1 Chapter 14 Kinetic theory The kinetic theory is a special aspect of the statistical mechanics of large number of particles. Suitable averages of the physical characteristics and motions of individual particles provide information about the macroscopic behavior of the system as a whole. The dynamics of many-particle systems is called statistical mechanics. 2 § 14.1 The ideal gas 1. Newton’s model of gas—static model The corpuscular particles of gas occupy fixed positions and filled the entire space between them. Repel force ↓∝↑?∝ VPVrF 1,1 2 2. Kinetic model (Daniel Bernoulli, James Clerk Maxwell, Ludwig Boltzmann and others) 1gas is composed of many tiny particles, freely moving at high speed. 2the pressure arises from the innumerable collisions of particles with each other and with the walls of the container. The pressure of a gas in thermal equilibrium inside a container never runs down or decreases with time, the collisions must be modeled as completely elastic, the motion is perpetual. 3 The forces involved in the collisions of the particles of the gas with each other and with walls of the container must conservative, briefly acting only during the intervals of the collisions and essentially zero otherwise. § 14.1 The ideal gas 3 § 14.1 The ideal gas 4gases can be compressed easily. The space of the particles in gas is the order of 10 times larger than those of liquids and solids. 3. Our purpose Find: 1the connection between temperature T and pressure P and their microscopic essentials; 2the average velocity of the particles; 3specific heat—the microscopic essentials. § 14.1 The ideal gas 4 4. The properties of ideal gas (model) The ideal gas equation of state: nRTPV = 2NV particle <<V, the volume occupied by the particles themselves is a negligibly small fraction of the volume containing the gas. 1the number of particles N in the gas is very large, for instance 112312 106.02mol/1002.6mol10 ×=×× ? 3all the particles are in random motion and obey Newton’s law of motion. § 14.1 The ideal gas 4the particles are equally likely to be moving in any direction--symmetry. 5the gas particles interact with each other and with the walls of the container only via elastic collisions. No force act on a particle except during a collision. All collisions elastic and of negligible duration. 6the gas is in thermal equilibrium with its surroundings. 7the particles of the gas are identical and indistinguishable. § 14.1 The ideal gas 5 § 14.2 the pressure and temperature of an ideal gas 1. The pressure of an ideal gas 1the probability that particles in element volume ?V will collide with the wall L tv L L AL LA p x ? = ? = ? = 2 1 2 1 2 1 2the change of the momentum of one particle i ? j ? m m kmvjmvimvv kmvjmvimvv iziyix iziyix ??? ??? afti bfri ++?= ++= r r L? A imvppp ixi ? 2 bftiafti ?=?=? rrr 3the impulse given to the wall by the particles in the element volume ?V it L mv imv L tv imvptF ix ix ix ixii ?? 2 2 ? 2 2 ?= ? =??=? r r 4the total impulse delivered to the wall during the interval ?t by all the particles that collide with the wall itvN L m itv L m tF xix i ?? )( 22 ave ???=?=? ∑ r Total number of particles § 14.2 the pressure and temperature of an ideal gas 6 5the symmetry implies that ??= ??=?? ??=?++?=?? ++==? ??=??=?? 2 ave 22 22222 2222 222 3 1 3 1 3 vN L m F vv vvvvv vvvvvv vvv x xzyx zyx zyx rr 6pressure P exerted by the gas on the wall ??=??=??== 222 ave 333A v V nM vN V m vN AL mF P § 14.2 the pressure and temperature of an ideal gas 2. The absolute temperature an ideal gas 1the microscopic interpretation of the absolute temperature ??=??= 22 2 1 3 2 3 vm V N v V Nm PQ NkTv Nm PV =??=∴ 2 3 ??= 2 3 v k m T kTKEKE k T vmKE 2 3 or 3 2 2 1 aveave 2 ave ==∴ ??=Q § 14.2 the pressure and temperature of an ideal gas 7 kTKE 2 3 ave = Conclusions: a. The temperature of all the gas is a manifestation of the average translational kinetic energy of each particle. b. A measurement of temperature is a measurement of the average translational kinetic energy of any particle of the gas. c. Two different gases at same temperature have the same average translational kinetic energy per particle. § 14.2 the pressure and temperature of an ideal gas 2the rms speed m kT vv k m T 3 3 22 =??∴??=Q 212121 rms ) 3 () 3 () 3 ( M RT mN RT m kT v A === The average speed rms 21 2 2 rms )( vv N v vv N v v i i i i <??∴ =??==?? ∑∑ Q § 14.2 the pressure and temperature of an ideal gas 8 补充内容：统计方法的一般概念 一、统计规律——大量偶然事件整体所遵从的规律 不能预测 多次重复 掷骰子 抛硬币 伽尔顿板实验例： 伽尔顿板实验 每个小球落入哪个槽是偶然的 少量小球按狭槽分布有明显偶然性 大量小球按狭槽分布呈现规律性 掷骰子 每掷一次出现点数是偶然的 掷少数次，点数分布有明显偶然性 掷大量次数，每点出现次数约 1/6，呈现规律 抛硬币 每抛一次出现正反面是偶然的 抛少数次，正反数分布有明显偶然性 抛大量次数，正反数约各 1/2，呈现规律性 补充内容：统计方法的一般概念 9 共同特点： 1、群体规律： 只能通过大量偶然事件总体显示出 来， 对少数事件不适用。 近似规律统计规律 ≠ 个体规律简单叠加统计规律 ≠ 2、量变 —质变： 整体特征占主导地位 例： 理想气体实验定律， 传真照片 …... 3、与宏观条件相关 如：伽尔顿板中钉的分布 4、伴有涨落 补充内容：统计方法的一般概念 二、统计规律的数学形式——概率理论 1、定义： 总观测次数 N 出现结果 A次数 A N A出现的概率 N N W A A lin= ∞→N 2、意义： 描述事物出现可能性的大小 两类物理定律 第一类： 约束不可能事件 第二类： 约束可能性小事件 违反能量守恒定律的事件不可能发生 不违反能量守恒定律的事件并不都能发生 例： 一壶水在火上 会沸 腾？ 会结冰？ 补充内容：统计方法的一般概念 10 3、性质 1)叠加定理 不可能同时出现的事件 ——互斥事件 出现几个互斥事件的总概率等于每个事件单独出 现的概率之和： BABA WWW += + 出现所有可能的互斥事件的总概率为1 归一化条件： 1= ∫ +∞ ∞? Wd 出现例：掷骰子 6 1 3 6 1 2 3 2 = = W W : : 3 1 32 = + W 出现 1—6： w =1 补充内容：统计方法的一般概念 2) 乘法定理 同时发生两个相容独立事件的概率是两个事件单 独发生时的概率之积 BABA WWW ×= + 相容统计独立事件： 彼此独立，可以同时发生的事件 例： 同时掷两枚骰子 其一出现 2： 6 1 2 =W 另一出现 3： 6 1 3 =W 同时发生 36 1 6 1 6 1 32 =×= + W 补充内容：统计方法的一般概念 11 三、几个基本概念 1、分布函数 例： 伽尔顿板实验 槽: 1, 2, 3, …... 粒子数: N 1 , N 2 , N 3 …... ∑ = i i NN 1,2,3,4,... 粒子出现在第 i 槽内的概率为 ： N N W i i = 该槽内小球数 小球总数 成正比变化，与槽宽随槽的位置 xx N N w i i ?= 小球在 x 附近，单位宽度区间出现的概率 xN N i ? = 概率 密度 补充内容：统计方法的一般概念 概率密度 xN N i ? 是 x 的函数 ——分布函数 曲线下窄条面积 W N N xxfS d d d ==?=? )( 曲线下总面积 ∫∫∫ ==?= L LL Wx x W xxf 0 00 1dd d d d)( xN N x W xf d d d d ==)( N N W d d =一般情 况： 分布曲线 L f(x) ox x xx ?+ x xx d+ 补充内容：统计方法的一般概念 12 类 比： 人口数量按年龄分布 考试成绩按分数分布 大气中尘埃按直径分布 星系中恒星按大小分布 树上苹果按大小分布 河床中卵石按尺度分布 雹粒按尺度分布 麦克斯韦分子运动速率分布（将 学） …... 颗粒度 问题 L f(x) ox x xx ?+ x xx d+ 补充内容：统计方法的一般概念 2、统计平均值 平均分 ∑∑ == g g g g g N N gN N g 1 平方平均分 22 g N N g g g ∑ = 总人数 ∑ = g g NN 人数按分数的分布 N g 得分数 g 的概率 N N g 图示 100人参加测试的成绩分布（满分 50）例： 补充内容：统计方法的一般概念 13 一般情况 测量值： LL i MMM ,, 21 出现次数： LL i NNN ,, 21 总次数： LL ++++= i NNNN 21 出现概 率： LL ,,, N N W N N W N N W i i === 2 2 1 1 统计平均 值： LL LL ++++ ++++ = i ii NNN NMNMNM M 21 2211 LL ++++= ii WMWMWM 2211 ii i WMM ∑= ii i WMM 22 ∑=同 理 ： 补充内容：统计方法的一般概念 ( ) 变量间隔分布函数物理量 ××=== ∫ ∫∫ xxMfWMM dd 3、涨落 实际出现的情况与统计平均值的偏差 例 伽尔顿板：某槽中小球数各次不完全相同，在平均 值附近起伏。 掷骰子：出现 4，概率 1/6，每掷 600次，统计平均 实 际： LL,,,, 次次次次 9810210099 4 =N 次100 4 =N ;, 很大时，涨落可忽略，涨落 NN ↓↑ 意义。太小时，统计规律失去，涨落 NN ↑↓, 补充内容：统计方法的一般概念 14 1. Maxwell’s speed distribution law kTmv RTMv ev kT m ev RT M vN N vP 2223 2223 2 2 ) 2 (4 ) 2 (4 d d )( ? ? = == π π π π It is a probability distribution function. N N vvP d d)( = § 14.3 the distribution of molecular speed and mean free path --the fraction of molecules whose speeds lie in the interval of with dv centered on v . vvP d)( --the fraction of molecules with speeds in an interval of v 1 to v 2 . ∫ 2 1 d)( v v vvP --all molecules fall into this category, so the value of this total area is unity. 1d)( 0 = ∫ ∞ vvP § 14.3 the distribution of molecular speed and mean free path 15 2. Average, RMS, and most probable speeds 1 Average speed We weight each value of v in the distribution, then we add up all these values of vP(v)dv ∫ ∞ =>=< 0 8 d)( M RT vvvPv π 2the root-mean-square speed--RMS ∫ ∞ =>=< 0 22 3 d)( M RT vvPvv M RT vv averms 3 )( 2 == § 14.3 the distribution of molecular speed and mean free path 3the most probable speed The most probable speed is the speed at which P(v)is maximum. M RT v v vP P 2 0 d )(d =?= It is obvious that rmsaveP vvv << § 14.3 the distribution of molecular speed and mean free path 16 M RT v P 2 = The area of the distribution curves always have a value of unity. Rain? Small numbers of very fast molecules with speeds far out in the tail of the curve evaporate, making clouds and rain a possibility. § 14.3 the distribution of molecular speed and mean free path § 14.3 the distribution of molecular speed and mean free path 3. Mean free path and the mean frequency of collision 17 The cross-section area of cylinder sweeps by the molecule between successive collisions 2 dπ The volume of the zigzags cylinder in the time interval ?t tvd ?π ><)( 2 The number of molecules in the cylinder tvdVN ?π ><))(/( 2 § 14.3 the distribution of molecular speed and mean free path The mean frequency of collision： The number of collisions in unit time >< vdVN ))(/( 2 π § 14.3 the distribution of molecular speed and mean free path The mean free path： VNdVtNvd tv t t rev / 1 / incollisionsofnumber duringpaththeoflength 22 π?π ? ? ? λ = >< ≈ = 18 § 14.4 the internal energy and the specific heat of a monatomic ideal gas 1. The internal energy of a monatomic ideal gas There is no internal motion for a monatomic ideal gas nRTkTnNkTNU A 2 3 ) 2 3 () 2 3 ( === one mole of monatomic ideal gas RTU 2 3 = 2. The molar specific heat of a monatomic ideal gas 1molar specific heat at constant volume for a monatomic ideal gas RnRT TnT U n c UVPUWUQ T Q n c V VV 2 3 ) 2 3 ( d d1 ) d d ( 1 dddddd ) d d ( 1 === =+=+= = TncU V ?=? Conclusion: The internal energy of a gas depends on the temperature of the gas but not on its pressure or density. § 14.4 the internal energy and the specific heat of a monatomic ideal gas 19 2 molar specific heat at constant pressure for a monatomic ideal gas ]) d d () d d [( 1 ddddd ) d d ( 1 PPP PP T V P T U n c VPUWUQ T Q n c += +=+= = PV PPP T V P n c T V P T U n c ) d d ( 1 ]) d d () d d [( 1 += += § 14.4 the internal energy and the specific heat of a monatomic ideal gas RRcc P nR T V VPP 2 5 ) d d ( =+=∴=Q c P is greater than c V , because of energy must now be supplied not only to raise temperature of the gas but also for the gas to do work. Consider three paths between the two isotherms in P-V diagram, no matter what path is actually taken between T and T+?T, we can always use path 1 and ? U=nc V ?T to compute the change of internal energy. Note: § 14.4 the internal energy and the specific heat of a monatomic ideal gas 20 § 14.4 the internal energy and the specific heat of a monatomic ideal gas The complications indicate that the internal energy of diatomic and polyatomic gases also is a function of absolute temperature, but not same as the function of monatomic gases. 3. Complications arise for diatomic and polyatomic gases Problem: The diatomic and polyatomic gases have molar specific heats whose values are quite different from those of monatomic gases. Why? § 14.4 the internal energy and the specific heat of a monatomic ideal gas 21 Heat transfer Kinetic energy of center of mass Kinetic energy of rotations and vibrations For diatomic gases and polyatomic gases, both c V and c P is greater than monatomic gases. It indicated that: § 14.4 the internal energy and the specific heat of a monatomic ideal gas § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 1. Degrees of freedom Degrees of freedom is a pure number that quantifies the number of distinct quadratic generalized coordinates needed to specify the microscopic total mechanical energy of one particle of the system. The sum of the microscopic total mechanical energies of each of the particles is the total internal energy U of the many-particle system. 22 2. Quadratic generalized coordinates 1A particle in one-dimension 2 2 1 x mvKE = iv x ? 2A rigid body in one-dimension iv x ? ω r 22 2 1 2 1 ω CMx ImvKE += 3A particle in two-dimension jvivv yx ?? += r 222 2 1 2 1 2 1 yx mvmvmvKE +== § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 4A rigid body spins about its center of mass in two-dimension jvivv yx ?? += r ω r 222 2 1 2 1 2 1 ωImvmvKE yx ++= 5rigid diatomic molecules y x z 22 222 2 1 2 1 2 1 2 1 2 1 yyxxzyx IImvmvmvKE ωω ++++= § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 23 6rigid body in three dimensions x z o y 222 222 2 1 2 1 2 1 2 1 2 1 2 1 zzyyxx zyx III mvmvmvKE ωωω +++ ++= 7polyatomic molecules The kinetic energy translation, rotation vibration § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 3. The equipartition of energy theorem kTKE 2 3 ave = The kinetic energy of one monatomic gas particle For a system of particles in thermal equilibrium, the average kinetic energy associated with each active degree of freedom of a particle is the same. kT 2 1 § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 24 4. Some failures of classical kinetic theory 1the size of the particles of monatomic gas 3 translational degrees of freedom 3 rotational degrees of freedom 6=f 2what about common diatomic gases? R R f T U n c nRT fkTfNU V 3 2d d1 2 )] 2 1 ([ === == Theory: Experiment: Rc V 2 3 = § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid Rotational degrees of freedom: f=2 Vibrational degree of freedom: f=2 Translational degrees of freedom: f=3 Experiment: Rc V 2 5 = K)J/(mol1.29 2 7 d d1 2 7 ) 2 1 (7 ?=== == R T U n c nRTkTNU V Theory: Model: dumbbells § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 25 The molar specific heat at constant volume of a diatomic gas as a function of temperature Only quantum mechanics can explain the results. § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid Translational Energy level Rotational Energy level vibrational Energy level E E E § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 26 4. Specific heat of a solid 1model of an ideal solid 2the microscopic mechanical energy 22 2 1 2 1 xkmvPEKE xx +=+ § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid The number of degrees of freedom: 3×2=6 The total internal energy of a solid system of atoms nRTNkTkTNU 33)] 2 1 (6[ === 3specific heat of solid R T nRT nT U nT Q n c UVPUQ 3 d )3(d1 d d1 d d1 dddd ==≈= ≈+= ----law of Dulong and Petit § 14.5 degrees of freedom , the equipartition of energy theorem and specific heat of a solid 27 § 14.6 apply first law of thermodynamics to some special processes for an ideal gas 1. at constant volume(isochoric process) i V P V TncUWUQ V ?=?=+?= According to the first law of thermodynamics Monatomic molecules: Rc V 2 3 = Rigid diatomic molecules: Rc V 2 5 = 0dd === ∫∫ f i V V VPWW TncQ V ?= f i f i T T P P = 2. at constant pressure (isobaric process) i V P V f V TncTnRTnc VVPTncWUQ PV ifV ?=?+?= ?+?=+?= )( According to the first law of thermodynamics VPVP ccRcc =+= γ )(dd if V V VVPVPWW f i ?=== ∫∫ TncQ P ?= f i f i T T V V = § 14.6 apply first law of thermodynamics to some special processes for an ideal gas 28 3. at constant temperature (isothermal process) i V P V f V According to the first law of thermodynamics i f V V nRTWWUQ ln==+?= i f V V V V V V nRTV V nRT VPWW f i f i lnd dd == == ∫ ∫∫ 0=?=? TncU V iiff VPVP = § 14.6 apply first law of thermodynamics to some special processes for an ideal gas ∞= T c 4. in thermal isolation (adiabatic process) i V P V f V TncVPU VPUQ V ddd 0ddd =?= =+= V c V RPTnRPVVP nRTPV d ddd ?==+ = VPVP c cc PVVP V VP d)1(ddd ??= ? ?=+ γ γγ γ γγ iiff P P V V VPVP V V P P V V P P VPPV f i f i =??= ?=??= ∫∫ dd dd dd § 14.6 apply first law of thermodynamics to some special processes for an ideal gas 29 γ γ γ γ V VP PPVVP ii ii == )( 1 dd 11 γγ γ γ γ γ ?? ? ? === ∫∫ fi ii V V ii V V VV VP V V VP VPW f i f i )( 1 1 ]1)[( 1 1 ffii f iii VPVP V VVP W ? ? =? ? = ? γγ γ 0=Q WU ?=? According to the first law of thermodynamics 0) d d ( 1 == T Q n c Q § 14.6 apply first law of thermodynamics to some special processes for an ideal gas 5. free expansion process 0 0 0 =? = = U Q W 6. cyclical processes 0 total =? = U QW total WUQ +?= § 14.6 apply first law of thermodynamics to some special processes for an ideal gas