Physical Chemistry Third Edition

25.2 The Probability Distribution for a Dilute Gas 1053

For ln(N!)

ln(N!)≈

1

2

ln(2πN)+Nln(N)−N (25.2-20)

Stirling’s approximation is more nearly correct for larger values ofN. Because we

assume thatNjis a large number, the first term is much smaller than the other terms.

We omit it and write

ln(N!)≈Nln(N)−N (25.2-21)

You can see why we assumed that our system is very large. Not only did we require that

gjNjfor a typical energy level, we also need to assume thatNjis a large number

in order to use this approximation.

Exercise 25.10

Using a table of factorials or a calculator that computes factorials, find the percent error in the

results of the expression of Eq. (25.2-20) and that of Eq. (25.2-21) forN10,N60, and

N200. Find the difference between the results of Eq. (25.2-20) and of Eq. (25.2-21) for

N 1 × 109.

Substitution of Eqs. (25.2-18) and (25.2-21) into Eq. (25.2-17) gives a set of simul-

taneous equations (one equation for each value ofi):

∂

∂Ni

(∑

j

[Njln(gj)−Njln(Nj)+Nj+αNj−βNjεj]−αN+βE

)

 0

The terms−αNandβEare independent of theN’s and have zero derivatives. Because

N 1 ,N 2 ,N 3 , and so on, are all independent variables and since all of theNvariables

except forNiare held fixed in the differentiation, only the term in the sum withji

has a nonzero derivative, and we have

ln(gi)−ln(Ni)− 1 + 1 +α−βεi0(i1, 2, 3,...) (25.2-22)

Fortunately, each equation contains only one variable. We solve Eq. (25.2-22) forNi:

Nigieαe−βεi (25.2-23)

The probability that a randomly selected molecule is found in energy leveljis equal

to the fraction of molecules in that energy level:

pj

Nj

N



1

N

gjeαe−βεj (probability of levelj) (25.2-24)

where we return to the use of the subscriptjto designate a molecule energy level. All

of the states of a level are equally probable since they have the same energy, so the

probability of a molecular state is

pi

Ni

N



1

N

eαe−βεi (probability of statei) (25.2-25)

where we now use the subscriptito designate a molecule state.