Physical Chemistry , 1st ed.

The right side of this equation is most of the right side of equation 17.33.

Substituting:

ENkT^2 



ln

T

q

V (17.34)

where we are indicating the constant-volume condition explicitly. Equation

17.34 is an amazing result: if we know how the logarithm ofqvaries with tem-

perature, we can calculate the energy of our system. This expression demon-

strates the central role that the partition function plays in statistical thermo-

dynamics.

We have already introduced the pressure as a thermodynamic variable. In a

fashion similar to how we got equation 17.34, it can be shown that

pNkT





ln

V

q



T

(17.35)

From the first-law relationship between the internal energy and H, the enthalpy:

HEpV

HENkT

HNkT^2 





ln

T

q

VNkT

from which we can get

HNkTT





ln

T

q

 (^1)  (17.36)
Statistical thermodynamics therefore gives expressions for all of the basic ther-
modynamic state functions, and they all depend on the partition function q.
In order to get expressions for Gibbs free energy Gand the Helmholtz en-
ergy A, we will need an expression for the entropy,S. The statistical thermo-
dynamic approach for Sis somewhat different. Rather than derive a statistical
thermodynamic expression for S(which can be done but will not be given
here*), we present Ludwig Boltzmann’s 1877 seminal contribution relating en-
tropy Sand the distribution of particles in an ensemble :
S ln (17.37)
The proportionality constant is also Boltzmann’s constant,k(the same kused
to define ). This definition of entropy becomes
Skln (17.38)
This postulate is so important in the development of statistical thermody-
namics that it is carved on Boltzmann’s tombstone (Figure 17.10).
Using equation 17.38 as a starting point, we can substitute for from equa-
tion 17.9 and get
Skln


j
gjNj

kln N! 
j

N
gjN
j!
j
 (17.39)

N!





j

Nj!

17.5 Thermodynamic Properties from Statistical Thermodynamics 601

*Interested readers can find details in D. McQuarrie,Statistical Thermodynamics,

University Science Books, Mill Valley, Calif., 1973.

Figure 17.10 Boltzmann’s assumption that

entropy is proportional to the number of possi-

ble arrangements is so important to statistical

thermodynamics that it is engraved on Boltz-

mann’s tombstone in Vienna.

Courtesy of Frantisek Zboray, Vienna.