Physical Chemistry , 1st ed.

where the second equation is simply an algebraic rearrangement of the first

one. Rearranging by applying the properties of logarithms (see the footnotes

earlier in this chapter):

Sk ln N! 

j

ln gjNj

j

ln Nj!

We can now apply Stirling’s approximation to the ln N! and ln Ni! terms:

Sk Nln NN

j

ln gjNj

j

(Njln NjNj)

Distributing the last summation through both terms:

Sk Nln NN

j

ln gjNj

j

Njln Nj

j

Nj

We recognize that Njis equal to N, the total number of particles. Therefore,

this summation cancels with the Nterm earlier in the brackets. So

Sk Nln N

j

ln gjNj

j

Njln Nj (17.40)

We can further simplify the above equation by combining the two remaining

summations algebraically, once again taking advantage of the properties of log-

arithms. We get

Sk Nln N

j

Njln 

N

gj

j

 (17.41)

The term ln(gj/Nj) can be expressed in terms of the Boltzmann distribution,

equation 17.20, if we take the logarithm of that equation. This introduces a

term in the energies (^) i.Using the fact that (^) i (^) iE, the total energy of the sys-
tem, and recognizing that we have an expression for Ein terms ofq, we can
show that equation 17.41 is equivalent to
SNk T





ln

T

q



V

ln q (17.42)

However, in the case of entropy the identity of the particles is a factor. In

section 17.2 we assumed that we could tell the difference between individual

particles; that is, we assumed they were distinguishable. In fact, at the atomic

level we cannot distinguish between individual, identical particles; atoms and

molecules are macroscopically indistinguishable.This means that we are over-

counting the total number of possible distributions for. The factor that fixes

this overcounting is a factor ofN! in the denominator of. (That is, there are

1/N! times fewer distributions for indistinguishable particles than for distin-

guishable particles.) When this factor is considered, the equations become

(^) indist
N

1

!



j

gjNj

and the final expression for entropy becomes

SNk T





ln

T

q



V

ln 

N

q

 1     (17.43)

This is the more accurate expression for the entropy,S.

It is the statistical thermodynamical approach to entropy that relates this

state function to the well-known and classic relationship with disorder.Disorder

N!





j

Nj!

602 CHAPTER 17 Statistical Thermodynamics: Introduction