Physical Chemistry , 1st ed.

This is similar to the expression we get using the kinetic theory of gases (which
we will consider in a later chapter). Statistical thermodynamics therefore yields
the same answer as other physical chemical theories.
We can also determine an expression involving pressure, since we know that

pNkT 





ln

V

q



T

NkT 

1

q







V

q



T

The derivative ofqwith respect to volume,q/V, is simply (2mkT/h^2 )3/2.
Substituting:

pNkT

pN

V

kT (17.57)

This rearranges to

pVNkT (17.58)

Compare this to the ideal gas law: it is identical provided that

R Nk (17.59)

where Rrepresents the ideal gas law constant. Boltzmann’s constant therefore
provides a statistical thermodynamic foundation for the ideal gas law constant.
In fact, ifNNA(Avogadro’s number),

RNAk (17.60)

and we have the molar ideal gas law pV RT. Of course, for nmoles of gas,
this becomes the most general form of the ideal gas law,pVnRT. The rela-
tionship between Rand kis also illustrated by the units used to describe the
two constants.

Example 17.5

Verify equation 17.60, using SI units of energy for Rand k, and then deter-

mine the value ofkin units of Latm/K.

Solution

We can use the values of any two of the variables in equation 17.60 and cal-

culate the third and compare our result to a tabulated value. For this exam-

ple, let us pick values for kand NA:k1.381  10 ^23 J/K and NA

6.022  1023 /mol. We find that

R(6.022  1023 /mol) 1.381  10 ^23 

K

J



R8.316 

mo

J

lK



which is off by only 0.02% from the accepted value.

In order to determine kin units of Latm/K, we will need to use the value

for Rthat has Latm units. Using R0.08205 Latm/(molK), we substitute

for different constants this time:



2 

h

m

2

kT



3/2





2 

h

m

2

kT



3/2

V

17.7 State Functions in Terms of Partition Functions 609