Physical Chemistry , 1st ed.

as long as the other relevant conditions apply to the system. Statistical ther-

modynamics’ definition of entropy is thus consistent with the third law of ther-

modynamics as developed by phenomenological thermodynamics.

Knowing the relationship between qand S, it is simple mathematics to de-

termine what the Helmholtz energy,A, and the Gibbs energy,G, are in terms

of the partition function. They are

ANkTln 

N

q

 (17.44)

GNkT ln 

N

q 1

 (17.45)

Notice that both Aand Gare directlyrelated to q, rather than related to a de-

rivative ofq. Although this is an artifact of the mathematics and the definition

of the ensemble, there should be some wonder that the important state func-

tions (Gand A) are so intimately related to the partition function q, which be-

comes the central focus in statistical thermodynamics.

Finally, since the chemical potential ifor the ith chemical species is the

basic focus in chemical equilibrium, we can easily define iin terms ofq:

i





N

G

i



Therefore, from equation 17.45:

ikTln 

N

q

i

 (17.46)

Chemical potential is also directly related to q.

17.6 The Partition Function: Monatomic Gases

The previous section made it clear that all thermodynamic state functions are

in some way related to the partition function q. This means that in order to

know these state functions, we need to know what qis. How?

First, recall that qis simply a sum of negative exponentials of the discrete

energy levels:

q

i

gie   i/kT

Technically this is an infinite sum, because there are an infinite number of pos-

sible energy levels for any particle (which is a general conclusion of both clas-

sical and quantum mechanics). However, because qis defined in terms ofneg-

ativeexponentials, each successive term gets smaller, so the potentially infinite

number of terms in the summation does not automatically imply that q.

Second, if the energy levels are close enough together, then each term in the

summation is infinitesimally close to the previous term, and also infinitesi-

mally close to the next term. It can be well approximated that rather than a

sum of discrete terms,qcan be written as an integral of a continuous function:

q

i

gie i/kT→



i 0

gie   i/kTdi (17.47)

In order to determine a theoretical value of a partition function, we need an

expression for the energy levels (^) i.
604 CHAPTER 17 Statistical Thermodynamics: Introduction