Physical Chemistry Third Edition

1126 27 Equilibrium Statistical Mechanics. III. Ensembles

In order to investigate the parameterβ, we assert thatβhas the same properties for

any system and consider a dilute gas withNmolecules. In a dilute gas the molecules

are independent of each other and the system energy is a sum of the energies of the

molecules:

Eiε 1 i+ε 2 i+ε 3 i+ε 4 i+ ··· +εNi (27.1-23)

where the subscript 1istands for the quantum numbers of molecule number 1 that

correspond to system state numberi, and so on. The canonical partition function of a

dilute gas is given by a sum over all of these quantum numbers:

Z

∑

1 i

∑

2 i

∑

3 i

···

∑

Ni

exp[−β(ε 1 i+ε 2 i+ε 3 i+ ··· +εNi)] (27.1-24)

If each of the sums were independent of the others every molecule state would occur

in every sum, and the multiple sum could be factored:

Z

∑

1 i

e−βε^1 i

∑

2 i

e−βε^2 i

∑

3 i

e−βε^3 i···

∑

Ni

e−βεNi

(

independent

sums

)

(27.1-25)

Identical molecules are indistinguishable from each other and the sums are not

independent of each other, so that Eq. (27.1-25) is not usable. There are two reasons for

this. First, if the molecules are fermions, any term in which two or more molecules are

in the same state must be deleted from the sums. If the volume of the gas is very large

and if the gas is dilute, the total number of molecular states available to be occupied

can be much larger than the number of molecules in the system. We call this the case of

dilute occupationof molecule states, and have assumed this case to apply in Chapters

25 and 26. In this case the forbidden sets of quantum numbers for a system of fermions

occur in only a small fraction of the terms of the sum, and these terms can be left in the

sum without serious error. This approximation also removes the difference between

fermion and boson molecules.

The second reason that we cannot use Eq. (27.1-25) is that we must take account

of the indistinguishability of the particles, whether they are fermions or bosons. If the

molecules were distinguishable from each other we could assign a set ofNmolecules to

Nstates inN! ways, whereN! representsNfactorial, equal to (N)(N−1)(N−2)···

(3)(2)(1), the product of all integers starting withNand ranging down to 1. Summing

independently over all values of the molecule quantum numbers denoted by 1i,2i, etc.,

includesN! terms that represent the same system state. We can make correction for the

overcounting of states by dividing byN!:

Z

1

N!

∑

1 i

e−βε^1 i

∑

2 i

e−βε^2 i

∑

3 i

e−βε^3 i···

∑

Ni

e−βεNi (27.1-26)

Since every sum is over the same set of molecule states, these sums over molecule

states are identical:

Z

zN

N!

(gas of noninteracting molecules) (27.1-27)

wherezis a sum over molecule states, themolecular partition function:

z

∑

k

e−βεk

(

definition ofz,

the molecular partition function

)

(27.1-28)