Chapter 3 Statistical thermodynamics

Content 3.1 IntroductionIntroduction 3.2 Boltzmann statisticsBoltzmann statistics 3.3 Partition functionPartition function 3.4 Calculation of partition functionCalculation of partition function 3.5 Contribution of Q to thermo_functionContribution of Q to thermo_function 3.6 Calculation of ideal gas functionCalculation of ideal gas function

3.1 Introduction Method and target According to the Stat. Unit mechanic properties (such as rate, momentum, vibration) are related with the system microcosmic and macrocosmic properties, work out the thermo- dynamics properties through the Stat. average. According to some basic suppositions of the substance structure,

and the spectrum data which get from the experiments, we can get some basic constant of the substance structure, such as the space between the nucleus, bond angle, vibration frequency and so on to work out the molecule partition function. And then according to the partition function we can work out the substance’s thermo- dynamics properties.

3.1.2 Advantage Related with the system microcosmic and macrocosmic properties, it is satisfied for some results we get from the simple molecule. No need to carry out the complicated low temperature measured heat experiment, then we can get the quite exact entropy.

The structure model must be supposed when calculating, certain approximate properties exist; for large complicated molecules and the agglomerated system, it still has some difficulties in calculating Disadvantage

3.1.4 Localized system Particles can be distinguished from each other. For example, in the crystal, particles vibrate in the local crystal position, every position can be imagined to have different numbers to be distinguished, so the micro-cosmic state number of localized system is very large.

3.1.5 Non-localized system Basic particles can not be distinguished from each other. Such as, the gas molecule can not be distinguished from each other. When the particles are the same,its micro- cosmic state number is less than the localized system.

3.1.6 Assembly of independent particles The reciprocity of the particles is very faint, therefore it can be ignored, the total energy of the system is equal to the summation of every particles energy, that is:

3.1.7 Kinds of statistical system MaxwellMaxwell-Boltzmann statistics usually called Boltzmann statisticsBoltzmann Bose-Einstein statisticsEinstein FermiFermi-Dirac statisticsDirac

3.2 Boltzmann Statistics Microcosmic state number of localized system Most probable distribution of localized system Degeneration Degeneration and Microcosmic state number Most probable distribution of non-localized system The other form of Boltzmann formula Entropy in Helmholz free energy expression

3.2.1 Microcosmic state number of localized system One macrocosmic system which is consisted by N independent particles which can be extinguished, it has many different partition forms in the quantitative level. Suppose one of the partition forms is: energy level ε 1 ， ε 2 ， · · ·, ε i one distributed form N 1 ， N 2 ， · · · ， N i

The microcosmic state number of this partition is:

There are many forms of partition, the total microcosmic state number is: No matter what partition, it has to satisfy the following two conditions:

3.2.2 Most probable distribution of localized system The Ω i of every distribution is different but there is a maximal value Ω max among them, in the macrocosmic system which has enough particles, the whole microcosmic number can approximately be replaced by Ω max, this is the most probable distribution.

The problem is how to find out a appropriate distribution N i under two limit conditions to make Ω the max one, in mathematics, this is the question how to work it out under the conditional limit of formula (1). That is: ( work out the extreme, make

Firstly, outspread the factorial by the String formula, then use the methods of Lagrange multiply gene, the most probable distribution we get : The α and β in the formula is the non-fixed gene which are brought in by the methods of Lagrange multiply gene.

Most probable distribution of localized system Work it out by the mathematics methods: So the most probable distribution formula is: or

3.2.3 Degeneration Energy is quantitative, but probably several different quanta state exist in every energy level, the reflection on spectrum is that the spectrum line of certain energy level usually consisted by several very contiguous exact spectrum line. In the quanta mechanics,the probable micro- cosmic state number of energy level is called the degeneration of that energy level, we use g i to stand for it.

For example, the translation energy formula of the gas molecule is: The n x, n y and n z are the translation quantum numbers which separately in

so n x =1, n y =1 and n z =1, it only has one probable state, so g i =1, it is non- degeneration. the x, y and z axis,

At this moment, under the situation ε i are the same, it has three different microcosmic states, so g i =0. When

3.2.4 Degeneration and Microcosmic state number Suppose one distribution of certain localized system which has N particles: Energy level ε 1 ， ε 2 ， · · · ， ε i Every energy level degeneration g 1 ， g 2 ， · · · ， g i One distributed form N 1 ， N 2 ， · · · ， N i

Choose N 1 particles from N particles and then put them in the energy level ε 1, there are C N N1 selective methods; But there are g 1 different state in the ε 1 energy level, every particle in energy level ε 1 has g 1 methods, so it has g N1 1 methods; Therefore, put N 1 particles in energy level ε 1, it has g N1 1 C N N1 microcosmic number. Analogy in turns, the microcosmic number of this distribution methods is:

Because there are many distribution forms, under the situation which U, V and N are definite, the total microcosmic state numbers are: The limit condition of sum still is:

Use the most probable distribution principle, ΣΩ1≈Ω max, use the Stiring formula and Lagrange multiply gene method to work out condition limit, when the microcosmic state number is the maximal one, the distribution form N* i is:

Compare with the most probable distribution formula when we do not consider degeneration, it has an excessive item g i.

3.2.5 Most probable distribution of non-localized system Because particles can not be distinguished in the non-localized system, the microcosmic number which distribute in the energy level is less than the localized system, so amend the equal particle of the localized system microcosmic state number formula, that is the calculation formula divides N!

3.2.5 Most probable distribution of non-localized system Therefore, under the condition that U, V and N are the same, the total microcosmic state number of the non- localized system is:

Most probable distribution of non-localized system Use the most probable distribution principle, use the Stiring formula and Lagrang multiply gene method to work out the condition limit,when the micro- cosmic state number is the maximal one, the distribution form N* i (non- localized) is:

Most probable distribution of non-localized system It can be seen that the most probable distribution formula of the localized and non-localized system.

3.2.6 The other form of Boltzmann formula (1) Compare the particles of energy level i to j, use the most probable distribution formula to compare, expurgated the same items, then we can get:

The other form of Boltzmann formula (2) Degeneration is not considered in the classical mechanics, so the formula above is Suppose the lowest energy level is ε 0, ε i - ε 0 =Δε i, the particles in ε 0 energy level is N 0, omit “*”, so the formula above can be written as:

This formula can be used conveniently, such as when we discuss the distribution of pressure in the gravity field, suppose the temperature is the same though altitude changes in the range from 0 to h, then we can get it The other form of Boltzmann formula

3.2.7 Entropy in Helmholz free energy expression According to the Boltzmann formula which expose the essence of entropy (1) for localized system, non-degeneration

Entropy in Helmholz free energy expression Outspread of Stiring formula:

Entropy in Helmholz free energy expression

Entropy in Helmholz free energy expression (2) for localized system, degeneration is g i The deduce methods is similar with the previous one,among the results we get, the only excessive item than the result of (1) is item g i.

Entropy in Helmholz free energy expression (3) for the non-localized system Because the particles can not be distinguished, it need to equally amended, divide N! in the corresponding localized system formula, so:

3.3 Partition function definition According to Boltzmann the most probable distribution formula (omit mark “.”) Cause the sum item of the denominator is:

3.3 Partition function q is called molecule partition function, or partition function, its unit is 1. The e -ε i /kT in the sum item is called Boltzmann gene.The partition function q is the sum of every probable state Bolzmann gene of one particle in the system, so q is also called state summation.

Definition The comparison of any item in q; The comparison of any two items in q.

3.3.2 Separation of partition function The energy of one molecule is considered as the summation of the Translation energy of whole particles motion and the inner motion energy of the molecule. The inner energy concludes the Translation energy (ε r ), Vibration energy (ε v ), electron energy (ε e ) and atom nucleus energy (ε n ), all of the energy can be considered to be independent.

Separation of partition function The total energy of molecule is equal to the summation of every energy Every different energy has corresponding degeneration, when the total energy is ε i, the total degeneration is equal to the product of every energy degeneration, that is:

Separation of partition function According to the definition of partition function, put the expressions of ε i and g i into it, then we can get:

Separation of partition function It can be proved in the mathematics, the product summation of several independent variables is equal to the separate product summation, so the formula above can be written as :

Separation of partition function q t, q r, q v, q e and q n are separately called Translation, Turn, Vibration, Electron, and atomic nucleus partition functions.

Separation of partition function Suppose the total particles is N (1) Helmholz free energy F

3.3.3 Relation between Q and thermodynamics function (2) entropy S Or we can get the following formula directly according to the entropy expression which was get before.

Relation between Q and thermodynamics function (3) thermodynamic energy U Or the formula can be get from the comparison of two expressions of S (non-localized)

Relation between Q and thermodynamics function (4) Gibbs free energy G according to definition, G=F+pV, therefore:

Relation between Q and thermodynamics function (5) enthalpy H (6) heat capacity under constant volume

Relation between Q and thermodynamics function According to the expressions above, only if the partition function is known, the value of the thermo- dynamics function can be worked out.

3.3.4 Relation between Q and thermodynamics function According to the method which the relationship of non-localized system and thermodynamics function is the same, we can get:

Relation between Q and thermodynamics function

Relation between Q and thermodynamics function It can be seen from the formulas above U, H and the expression of Cv are the same in the localized and non-localized system; However, in the expressions of F, S and G, compared with the localized system, it lacks the relational 1/N! constant, but it can be expurgated each other when we calculate the change of the functions. This chapter mainly discusses non-localized.

3.4 Calculation of partition function Atomic nucleus partition function Electron partition function Translation partition function Turn partition function Vibration partition function

3.4.1 Partition function of atomic nucleus The ε n,0 ε n,1 in the formula separately stand for the atom nucleus energy which is in the ground and the first excited state, g n,0 g n,1 separately stand for the degeneration of the corresponding level.

Partition function of electrons Because in the chemical reaction, nucleus is always in the ground state, otherwise the energy level interval between the ground and the first excited state is very large,so commonly all the items after the second one in the bracket are ignored, so:

Partition function of electrons If the energy of the nucleus ground state energy level is chose as zero: That is the atom nucleus partition function is equal to the ground state degeneration, it comes from the nucleus spin effect. S n in the formula is the nucleus spin quantum number.

3.4.2 Partition function of electrons The electron energy interval is also very large, ( ε e,1 -ε e,0 )=400 kJ.mol -1, except for F, Cl minority elements, the second item in the bracket is also be ignored.

3.4.2 Partition function of electrons Though the temperature is very high, the electron is also probably be excited, but usually the electron is not excited, the molecule has been decomposed.Therefore, usually the electron is always in the ground state, so:

Partition function of electrons If ε e,0 is considered as zero, therefore q e =g e,0 =2j+1, j in the formula is electron total momentum quantum number.Electron total momentum distance which moves around nucleus is also quantitative,

Partition function of electrons the heft along certain chosen axis probably has 2j+1 tropism. Some freeness atom and steady ionic j=0, g e,0 =1, are non-degeneration. If there is a non-match electron, it probably has two different spin, such as Na, its j=1/2, g e,0 =2.

Suppose the particle which quality is m moves in the cubic system which volume is a. b. c, according to the Translation energy expression which is get from the fluctuation equation: Translation partition function

h in the formula is plank constant, n x, n y, n z is the Translation quantum number which are in the x, y, z axis, its value is positive integer 1, 2, …, ∞.

Translation partition function Put ε i,t into it:

Translation partition function Because for all quantum number work out the summation from 0 ~ ∞, it concludes all of the states, the item g i,t will not appear in the formula. The Translation partition function in the three axes is analogous, here we just explain one q t,x of them, others can be analogy.

Translation partition function Because α 2 is a very little value, the mark of sum can be replaced by the mark of integral, so:

Translation partition function Cite the integral formula: q t,y and q t,z have the same expressions, just a is turned to b or c, so: Then the formula turns to:

3.4.4 Turn partition function The Turn partition function of single atom molecule is zero, q r of different nucleus double atoms molecules, the same nucleus double atoms molecules and linearity multi-atom molecules have analogous forms, but the q r expression of non-linearity multi-atom molecules is more complicated.

3.4.4 Turn partition function (1) The q r of different nucleus double atoms molecule, suppose it is a rigid rotor and turns around the centroid, its energy level formula is:

3.4.4 Turn partition function J in the formula is the Turn energy level quantum number, I is the Turn inertia,suppose the double atoms quantity are m 1, m 2, r is nucleus interval.

Turn partition function The tropism of Turn angel momentum is also quantitative, so the energy level degeneration is

Turn partition function Q r is called Turn character temperature, because the right side of the formula has the dimension of temperature. Put Q r into q r expression, then we can get: Make

Turn partition function Work out the Q r from the Turn inertia I. Except H 2, the Q r of most molecules is very small, Q r T<<1, therefore we use the mark of integral instead of the mark of summation, and make x=J(J+1), dx=(2J+1)dJ, put them into it, then we can get:

Turn partition function

Turn partition function (2) The q r of some nucleus double atoms and linearity multi-atom molecules ( σ is symmetry number, the microcosmic state repeated time when it spins 360°)

Turn partition function (3) The q r of non-linearity multi-atom molecules I x, I y and I z separately are Turn inertia in the three axes.

3.4.5 Vibration partition function (1) The q v of double atoms molecule suppose the molecule only does one kind of simple Vibration which rate is V, the Vibration is non-degeneration, g i,v =1,its vibration energy is υ in the formula is Vibration quantum number, when υ =0, ε v,0 is called zero Vibration energy.

Vibration partition function Cause Q v =hv/k, Q v is called the Vibration character temperature,it also has temperature dimension, so:

Vibration partition function Vibration character temperature is one of the important properties, the higher Q v is, the smaller percentage of the excited state is, the second item and the items after it in the q v expression can be ignored. The Q v of some molecule are lower, such as iodine Q v =310K,

Vibration partition function therefore the item υ =1 can not be ignored. Under the condition of low temperature, Cite the mathematic similar formula:

Vibration partition function So the expression of q v is: We regard the zero Vibration energy as zero, that is ε v,0 =1/2hv=0, so:

Vibration partition function (2) q v of the multi-atom molecule The Vibration liberty degree f v of multi- atom molecule is: f t is Translation liberty degree, f r is Turn liberty degree, n is total atom. Therefore, the q v of the linearity multi- atom molecule is:

Vibration partition function The q v of non-linearity multi-atom molecule only need change (3n-5) to (3n-6).

3.5 Contribution of Q to thermodynamic function Contribution of atomic nucleus partition function Contribution of atomic nucleus partition function Contribution of electron partition function Contribution of Turn partition function

3.5.1 atomic nucleus’ Usually in the chemical reaction, nucleus is always in ground state, S n is the nucleus spin quantum number, it has nothing to do with the system temperature and volume. If the ground state energy is chose as zero, so:

atomic nucleus’ q n has no contributions to thermo- dynamic energy, enthalpy and molar heat capacity under constant volume, that is:

atomic nucleus’ q n has little contributions to F n, S n and G n, that is: F n =-NkTInq n S n =NkInq n G n =-NkTInq n

atomic nucleus’ When we calculate the changing value, this item will be expurgated, so we will ignore the contribution of q n. Only when we calculate the prescribed entropy, the contribution of q n has to be considered.

3.5.2 Electrons’ Usually electron is in ground state, and we choose the ground energy as zero, so: Because the total angle momentum quantum number j of electron has nothing to do with temperature and volume, q e has no contribution to thermodynamics, enthalpy and isometric heat capacity, that is:

Election’ q e has little contributions to F e, S e, G e, that is: F e (non-localized)=-NkTInq e S e (non-localized)=NkInq e G e (non-localized)=-NkTInq e

Election’ Except for S e, when we calculate the changing value of F e and G e, this item also can be expurgated commonly if the first excited state of election can not be ignored and the ground state is not equal to zero,so the whole expression of q e must be put it into to calculate.

3.5.3 Turn’ Because the interval of Turn energy level is very little, Turn partition function has great contributions to thermodynamics function, such as entropy and so on. As it is known For the non-localized system which has N particles,calculate the contribution which is done to thermodynamics function by q t.

Turn’ (1) Turn Helmholtz free energy

Turn’ (2) Turn entropy Because This is called Sackur-Tetrode formula.

Turn’ (3) Turn thermodynamic energy (4) Turn isometric heat capacity

Turn’ (5) Turn enthalpy and turn Gibbs free energy Put in the corresponding expressions U t, F t then we can get turn enthalpy and turn Gibbs free energy.

3.6 Calculation of thermodynamic function for Single atom ideal gas Because the inter motion of single atom molecule has no Translation and Vibration, only the atom nucleus, electron and outer Turn have contributions to thermodynamics. Ideal gas is localized system, so a series of its thermodynamics are showed by the partition function calculation formulas as following:

3.6.1 Helmholtz free energy

Helmholtz free energy Both of the 1,2 items can be expurgated, when ∆F is being calculation.

3.6.2 S This formula is also called Sachur-Tetrode formula

3.6.3 U Because q n, q e are no useful for thermo- dynamics, only Turn energy has contribution to it, so:

3.6.4 C v The conclusion is the same with the result of the classical energy share theory, single atom molecule only has three translation liberty degree, every liberty degree contribute 1/2k, then N particles total have 3/2Nk.

3.6.5 State equation of ideal gases Put the expression of F into it, because other items have nothing to do with volume only one item has relationship with V in translation item, put it in, and then we can get the state equation of ideal gas.

3.6.5 State equation of ideal gases The equation of ideal gas can be educed by the stat. Thermodynamics methods, this is the classical thermo- dynamics which it can not do. ending