Physical Chemistry Third Edition

26.1 The Statistical Thermodynamics of a Dilute Gas 1083

In a dilute gas the individual molecules independently occupy molecule energy

levels. Each system microstate corresponds to a distribution, which is a list of the

numbers of molecules occupying each molecular energy level. However, a given dis-

tribution can correspond to many different system microstates. The total number of

possible system microstates is equal to the sum of the numbers of microstates corre-

sponding to each distribution:

Ω

∑

{N}

W({N}) (26.1-2)

whereW({N}) is the number of system microstates corresponding to the distribution

denoted by {N}. There is one term in the sum for each possible distribution. We now

make what seems like a drastic approximation. We replace the entire sum shown in

Eq. (26.1-2) by its largest term,Wmp, the term corresponding to themost probable

distribution. In our model system of four oscillators, this replacement would amount

to replacingΩ35 byΩ12. The replacement ofΩbyWmpis a much worse

approximation for a dilute gas of many molecules, but is a very good approximation

for the logarithm of the sum.

This strange situation can be illustrated as follows: Assume that we want to have

12 significant digits when we approximate ln(Ω)byln(Wmp). Assume also thatΩhas

roughly the value as in Example 26.1, so that ln(Ω)≈ 1025. We let

ΩxWmp (26.1-3a)

or

ln(Ω)ln(Wmp)+ln(x) (26.1-3b)

Since ln(Ω) is roughly equal to 10^25 , to have ln(Ω) and ln(Wmp) agree to 12 significant

digits, ln(x) must obey

ln(x)≤ 1013 ,orx≤e^10

13

An error in approximatingΩis permissible that makes it incorrect by a factor ofe^10

13

.

The exponent in this expression is roughly equal to the federal debt of the United

States in U.S. dollars, anderaised to that power is beyond imagination. However, this

tremendous error inΩstill produces a value of ln(Ω) that is correct to 12 significant

digits. Such is the strange behavior of very large numbers, due to the fact that the

logarithm is a very slowly varying function for large values of its argument. Although

we will not prove this fact, the largest term in the sum shown in Eq. (26.1-2) is smaller

than the entire sum by roughly the same factor as that shown in Eq. (26.1-3), giving 12

significant digits in approximating ln(Ω)byln(Wmp).^1

Exercise 26.2

Find the value ofdln(x)/dx

a.forx 1

b.forx 1. 00 × 105

c.forx 1. 00 × 1010

5

(^1) One year, the author’s physical chemistry class showed up for their final examination wearing identical
T-shirts. On the T-shirts were written “The top 10 things I learned in p-chem.” One of the statements was
“12 is equal to 35, but only to 12 significant digits.”