Physical Chemistry Third Edition

1050 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

for the first pick, there areg−1 choices for the second pick, until there areg−(N−1)

choices for theNth pick.

Number of ways of choosingP(g,N)g(g−1)(g−2)···(g−(N−1))



g!

(g−N)!

(25.2-9)

The quantityP(g,N) is called the number of permutations ofNobjects chosen from

gobjects. However, in our case the order in which theNoccupied states are chosen

is immaterial, so we must divide byN!, the number of ways of permutingNobjects.

The result is called the number ofcombinationsofgobjects takenNat a time.

tC(g,N)

g!

(g−N)!N!

(25.2-10)

The expression in Eq. (25.2-10) is also the expression for abinomial coefficient, which

is the coefficient of theaNbg−Nterm in the expression for (a+b)g.

We now apply Eq. (25.2-10) to energy levelj. There areNjfactors in the quotient

gj!/(gj−Nj)! that do not cancel. Since in our case of dilute occupationgjNj, all

of the factors in the product are nearly equal togj, and the product of these factors is

approximately equal tog

Nj

j. To a good approximation for fermion molecules

ti≈

g

Nj

j

Nj!

(25.2-11)

This expression givestj1 forNj0 (an unoccupied level) since 0! is defined to

equal unity and since any number raised to the zero power equals unity.

EXAMPLE25.2

Find the percent error in approximating Eq. (25.2-6) by Eq. (25.2-11):

a.Forgj100 andNj 3

b.Forgj1000000 andNj 3

Solution

a.Forgj100 andNj 3

tj

(100)(99)(98)

100!



970200

100!

tj(approx)

(100)^3

100!



1000000

100!

3 .07% error

We do not evaluate 100! because it is a large number.

b.Forgj1000000 andNj 3

tj

(1000000)(999999)(999998)

1000000!



9. 9999700 × 1017

1000000!

tj(approx)

(1000000)^3

1000000!



1. 00000 × 1018

1000000!

0 .00030% error