Physical Chemistry , 1st ed.

Note the presence of the liter-to-m^3 conversion and the joule-to-Latm con-

version in the appropriate places. Evaluating this lengthy but straightforward

expression, we get

S126.07 

mo

J

lK



This calculated value ofSis virtually the same as the experimental value. (The

variance is 0.02%.)

The experimental value ofSand its calculated value using statistical ther-

modynamics in the above example are virtually identical! At this point in our

development of statistical thermodynamics, absolute entropy is our best evi-

dence that the ideas behind statistical thermodynamics are valid and useful in

understanding the thermodynamic behavior of systems (at least systems of

gases). Table 17.1 compares experimental values with calculated values ofSfor

several monatomic gases. You can see that the agreement is very, very good.

Now that an expression for Shas been determined (and verified), we can

derive expressions for Gand Ain terms of the partition function q. Without

going through the derivations, we have

ANkTln 

2 

h

m

2

kT



3/2



N

V

 (17.62)

GNkT ln 

2 

h

m

2

kT



3/2



N

V

  (^1)  (17.63)
Notice that of the two, the expression for Ais simpler. This is a consequence
of the fact that we have defined our system in terms of microstates that have
the same volume,V, and temperature,T. These two variables are the natural
variables for A, the Helmholtz energy. It is not surprising, then, that the ex-
pression for Ain statistical thermodynamics is relatively simple. If we instead
defined our ensemble in terms of microstates that have the same pressure and
temperature, we would find that the expression for G, the Gibbs free energy, is
relatively straightforward because pand Tare the natural variables for G.
Finally, we define a new parameter. It is not a state function, but a parameter
that has parallels in quantum mechanics. Notice that the part ofq in the term


2 

h

m

2

kT



3/2

has SI units of 1/m^3. Therefore, the term



2 

h

m

2

kT



1/2

(which is the cube root of the previous expression) has units of 1/m. The re-

ciprocalof this expression therefore has units of length, meters. We define the

reciprocal of this expression as (the Greek capital letter lambda), the ther-

mal de Broglie wavelength:



2 

h

m

2

kT



1/2

(17.64)

The original de Broglie wavelength was defined in terms of the momentum,

p, of a particle:



h

p



m

h

v

 (17.65)

17.7 State Functions in Terms of Partition Functions 611

Table 17.1 Comparison of calculated and
experimental entropy,S,for
monatomic gasesa
Gas Scalc Sexpt
He 126.07 126.04
Ne 146.22 146.22
Ar 154.74 154.73
Kr 163.98 163.97
Xe 169.58 169.57
aAll values are in units of J/(molK) at conditions of
298 K, 1.00 atm.