[物理化学英语课件]统计热力学Elementary_statistical_thermodynamics.ppt

1、Chapter 8 Elementary statistical thermodynamics8.1 IntroductionStatistical thermodynamics,or statistical mechanics,is the study of the microscopic behaviors of thermodynamic systems using statistical methods and probability theory.The essential problem in statistical thermodynamics is to determine t

2、he distribution of a given amount of energy E over N particles in a system.The macroscopic properties,such as thermodynamic energy,heat capacity,etc.,can be calculated in terms of partition functions.Statistical thermodynamics is a bridge of connecting between macroscopic and microscopic properties

3、of a system.Definition of statistical thermodynamicsThere are two kinds of systemsInteracting system (相倚子系统)Non-interacting system(独立子系统，for instance,ideal gas)Only the latter will be introduced in this chapter.Two kinds of particlesIdentical particles,or indistinguishable particles (such as gaseous

4、 molecules),is also called non-localized particles.Distinguishable particles (Such as the atoms in crystal),is also called localized particles.8.2 Energy level and its degeneracy0 1 2 3 4 5 Energy levels are said to be degenerate,if the same energy level is obtained by more than one quantum mechanic

5、al state.They are then called degenerate energy levels.The number of quantum states at the same energy level is called the degree of degeneracy.A molecular energy state is the sum of an electronic(e),nuclear(n),vibrational(v),rotational(r)and translational(t)component,such that:The degree of freedom

6、 of movementTranslation:x,y,z F=3RotationFor linear molecules,F=2For non-linear molecules,F=3VibrationA polyatomic molecule containing n atoms has 3n degrees of freedom totally.Three of these degrees of freedom can be assigned to translational motion of the center of mass,two or three to rotational

7、motion.3n-5 for a linear molecule;3n-6 for a nonlinear molecule CO2 has 33-5=4 degrees of freedom of vibration;nonlinear molecule of H2O has 33-6=3 degrees of freedom of vibration.8.2.1 Translational particleThe expression for the allowed translational energy levels of a particle of mass m confined

8、within a 3-dimensional box with sides of length a,b,c isWhere h6.62610-34Js，nx,ny,nz are integrals called quantum numbers.The number of them is 1,2,.If a=b=c,equation becomes all energy levels except ground energy level are degenerate.Example At 300K,101.325 kPa,1 mol of H2 was added into a cubic bo

9、x.Calculate the energy level t,0 at ground state,and the energy difference between the first excited state and ground state.Solution Take the H2 at the condition as an ideal gas,then the volume of it isThe mass of hydrogen molecule isthe energy difference is so small that the translational particles

10、 are excited easily to populate on different excited states,and that the energy changes of different energy levels can be think of as a continuous change approximately.8.2.2 Rigid rotator(diatomic)The equation for rotational energy level of diatomic molecules is：where J is rotational quantum number,

11、I is the moment of inertia(转动惯量)is the reduced mass(折合质量),The degree of degeneracy is 8.2.3 One-dimensional harmonic oscillatorWhere v quantum number，when v=0，the energy is called zero point energy.One dimensional harmonic vibration is non-degenerate.8.2.4 Electron and atomic nucleus The differences

12、 between energy levels of electron motion and nucleus motion are big enough to keep the electrons and nuclei stay at their ground states.Both degree of degeneracy,ge,0,for electron motion at ground state and degree of degeneracy,gn,0,for nucleus motion at ground state are different for different sub

13、stances,but they are constant for a given substance.8.3 Distribution and microstate8.3.1 Distribution of energy levelsWe call the occupation number ni the number of distribution in energy level i.For example,a distribution of 6 identical particles among 9 units of energy must satisfy with the condit

14、ions The total number of ways of distribution is 26.8.3.2 Distribution of states The number of particles occupied in a microscopic quantum state is called the number of distribution of states.One distribution D of energy levels has a certain number of microstates WD,the sum of all WD is the total nu

15、mber of microstates of a system.That is.There are 16 ways of distribution of four distinguishable particles in two identical boxes.8.3.3 Distinguishable particles Consider N distinguishable or localized particles distribute into N nondegenerate energy levels.Now consider another kind of distribution

16、 that the numbers of particles occupied in different energy levels are denoted as n1,n2,ni.All the energy levels are still nondegenerate.three different distributions of six particles.The exchanges of particles in the same energy level do not create new microstate because every energy level has only

17、 one quantum state.The numbers of microstates for three distributions areWe now consider that the degree of degeneracy of energy levels is g1,g2,gi.Suppose number of quantum states is unconstrained.Consider ni particles occupy energy level i,every particle can chose one from all quantum states in th

18、e energy level.Hence the ways of selection for ni particles are For all energy levels,the number of microstates caused by the degeneracy of levels is the number of microstates for a certain distribution D can be written as 8.3.4 Identical particlesassume that there is no restriction on the number of

19、 particles which can occupy a given energy level and that energy levels is nondegenerate.there is only one way for ni particles to occupy the energy level i.Therefore,the number of microstates of a distribution D for a system is WD=1.If energy level is degenerate,It is easy to see that there is(2+1)

20、ways of distributing 2 particles in two quantum states which can be written as Suppose 8 identical particles populate in 4 quantum states in an energy level.This is equivalent to the permutation of the sum of 8 persons and(4-1)dividing walls,both persons and dividing walls are indistinguishable.the

21、number of microstates for ni particles distributing in gi quantum states in an energy level is one kind of distribution is the products of the number of microstates for every level multiplied by one another.That is If nigi,this equation can be simplified into Compare this equation with equation,we c

22、an see that under the same conditions of N,ni,and gi,the number of microstates of a distinguishable-particle system is N!times that of an identical-particle system.8.4 The most probable distribution,equilibrium 8.4 The most probable distribution,equilibrium distribution,and Boltzmann distributiondis

23、tribution,and Boltzmann distribution8.4.1 The principle of equal a priori probabilities Statistical thermodynamics is based on the fundamental assumption that all possible configurations of a given system,which satisfy the given boundary conditions such as temperature,volume and number of particles,

24、are equally likely to occur.Example Consider the orientations of three unconstrained and distinguishable spin-1/2 particles.What is the probability that two are spin up and one spin down at any instant?Solution Of the eight possible spin configurations for the system,The second,third,and fourth comp

25、rise the subset two up and one down.Therefore,the probability for this particular configuration isP=3/88.4.2 The most probable distributionThe probability for distribution D is the microstates of three harmonic oscillators which are distinguishable particles with total vibrational energy of=WI+WII+W

26、III=3+6+1=10 Which distribution is the most probable distribution?WD is called thermodynamic probability of distribution D 8.4.3 Equilibrium distributionIn a system with large number of N,the most probable distribution may represent all distributions.Stirlings approximation a more accurate formConsi

27、der a system consisting of N localized particles which distribute over two degenerate quantum states,A and B.M denotes for the number of particles in state A and(N-M)in state B.the number of microstates for this distribution can be expressed asWhen M=N/2,WD has a maximum value.()()!2/!2/!NNNWB=Every

28、 particle has two possibilities to populate on the quantum states,state A or state B.The total number of microstates for the system would be 2N.The probability for the most probable distribution is When the number of particles in a system is about 1024,the probability is thenWe consider another dist

29、ribution that has a distribution number deviating m from N/2,its probability would beWhen mN,in terms of Stirlings approximation this equation can be converted intoThe probability for all distributions ranging from is the summation of their probabilities.By using error function we obtain the distrib

30、ution at equilibrium is most certainly going to be the most probable distribution,or at the very least,with these kind of numbers,something very close to it.8.4.4 Boltzmann distributionFor a large number of noninteracting particles =1.3810-23 J K-1,Boltzmann constant.is proportional coefficient.The

31、population can also be expressed in the form of energy level distribution The total number is thenDefine the particle partition functionthenThe distribution that obeys these equations is called the Boltzmann distribution.The equations are also known as Boltzmann distribution law.for any two levels：T

32、he ratio to total number：Boltzmann distribution is the most probable distribution.The maximum value can be derived by using Lagranges method of undetermined multipliers 8.5 Computations of the partition function8.5.1 Some features of partition functions(1)at T=0,the partition function is equal to th

33、e degeneracy of the ground state.(2)When T is so high that for each term i/kT=0,(3)factorization property If the energy is a sum of those from independent modes of motion,thenThe partition functions for 5 mode motions are expressed as8.5.2 Zero-point energyzero-point energy is the energy at ground state or the energy as the temperature is lowered to absolute zero.Suppose some energy level of groun