Physical Chemistry Third Edition

25.2 The Probability Distribution for a Dilute Gas 1051

Exercise 25.8

Find the percent error in approximating Eq. (25.2-10) by Eq. (25.2-11) ifgj1000000000 and

Nj3.

For a system of bosons, we imagineNjobjects ingjcompartments in a long box,

withgj−1 movable partitions between the compartments. Since there is no limit

on the number of objects in a compartment, all possible orders (permutations) of the

objects and partitions are allowed. Then realize that the partitions are indistinguishable

from each other, and the objects are indistinguishable from each other. The version

of Eq. (25.2-10) for bosons is therefore

tj

(gj+Nj−1)!

(gj−1)!Nj!

(bosons) (25.2-12)

There areNjfactors in the quotient (gj+Nj−1)!/(gj−1)! that do not cancel. In

the dilute occupation caseNj gj, so that Eq. (25.2-11) also applies to a system of

boson molecules.

Exercise 25.9

Show that in the dilute occupation case, Eq. (25.2-11) also applies to boson molecules.

We will now apply Eq. (25.2-11) to our dilute gas, whether our molecules are bosons

or fermions. For two independent energy levels the total number of ways to choose

states is the product of the number of ways of choosing states for each level. For the

entire distribution the number of system microstates is given by a product containing

one factor as in Eq. (25.2-11) for each molecular energy level:

W({N})

∏∞

j 1

tj≈

∏∞

j 1

g

Nj

j

Nj!

(25.2-13)

There are infinitely many molecule energy levels, as shown in the product limits. From

now on we will omit the upper limits on sums or products over the energy levels, but

will understand the product to include all possible molecular energy levels.

Only those distributions can be considered that conform to the specified values of

N,E, andV. These constraints correspond to

∑

j

NjN (25.2-14a)

∑

j

NjεjE (25.2-14b)

εjmust correspond to the correct volume for everyj (25.2-14c)

The problem of finding the most probable distribution is now the problem of finding the

distribution that corresponds to the largest value ofW({N}) subject to the constraints

given by Eqs. (25.2-14a) and (25.2-14b).

In order to use the methods of calculus, we assume that theNj’s are able to take on

any real values instead of taking on only integral values. A value ofNjcorresponding to