Physical Chemistry Third Edition

1048 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

The molecules in our system are indistinguishable from each other in the same way

that electrons are indistinguishable from each other. If the total number of electrons,

protons, and neutrons in a molecule is an odd number, the molecule is a fermion, because

exchanging two molecules changes the sign of the wave function an odd number of

times. The system wave function must be antisymmetrized and the Pauli exclusion

principle applies. No two fermion molecules in the system can occupy exactly the same

state. If the molecule contains an even number of electrons, protons, and neutrons it

is a boson, and the wave function must be symmetrized. The Pauli exclusion principle

does not apply.

Exercise 25.7

Identify each of the following as a fermion or boson:

a.A^13 C atom

b.F 2 if both atoms are^19 F

c.NO if both atoms are the common isotopes

d.CH 4 if all atoms are the common isotopes

e.CH 4 if the C atom is^13 C

The Average Distribution

We want to find the probability that a randomly chosen molecule is in a given molecule

state. We first consider the average distribution. If the system is in microstatek,

let the number of molecules in leveljbe denoted byNj(k). The set of numbers

N 1 (k),N 2 (k),N 3 (k), and so forth, is a molecular distribution that specifies the occupa-

tion of all of the molecular energy levels if the system is in system microstatek.We

denote this distribution by the single symbol{N(k)}. This distribution is analogous to

the distribution on a single line of Table 25.1 for our model system of four oscillators.

We must average over all system microstates that haveEequal toUandNequal

toNAvn, and correspond to the correct value ofV. We denote the number of such

microstates byW. This degeneracy of a system energy level is analogous to the degen-

eracy of 35 for the energy levelE 4 hνin the model system of four harmonic oscil-

lators. For a many-particle system,Wwill be a very large number. The number of

molecules in molecule energy leveljaveraged over all of these system microstates is:

N ̄j^1

Ω

∑Ω

k 1

Nj(k) (25.2-4)

We call the distribution{N ̄}theaverage distribution. Unfortunately, the average dis-

tribution in Eq. (25.2-4) cannot be computed becauseΩis a very large number.

The Most Probable Distribution

Since we cannot find the average distribution, we will seek the most probable distri-

bution.^2 In our model system of four oscillators, we saw that the average distribution

(^2) Our discussion follows that of N. Davidson,Statistical Mechanics, McGraw-Hill, New York, 1962,
Chapter 6.