Physical Chemistry , 1st ed.

Raising this unit to the power of 3/2 and including the m^3 unit from the vol-

ume term, we have

m

1

 (^2) 
3/2
m^3 
m

1

 3 m

(^3)  1
That is, all of the units cancel, as they should! Combining the numerical part
of the answer with the units part of the answer, we get
q1.90  1029
with no units. This is a very large number!

17.7 State Functions in Terms of Partition Functions

The importance of the partition function in statistical thermodynamics is that

if we know q, we can determine thermodynamic properties. Indeed, almost all

of the thermodynamic state functions can be written in terms of the change of

the partition function as some state variable,Tor V, changes. (Only Aand G

depend directly on q, and on the natural logarithm ofqat that. This fact does

not obviate the discussion to follow.)

Now that we have an expression for q, we can take those derivatives, since

Tand Vare part of the expression for q. We can therefore derive expressions

for various state functions.

We start with E, the total energy of our ensemble. According to equation

17.34,

ENkT^2 





ln

T

q



V

NkT^2 

1

q







T

q



V

The derivative ofqwith respect to T(at constant V) is easy to determine:







T

q







T



2 

h

m

2

kT



3/2

V



2 

h

m

2

k



3/2

V





T

T3/2



2 

h

m

2

k



3/2

V

3

2

T1/2

If we take this expression and divide by qitself, almost all of the terms will

cancel:

The only thing remaining is 3/2 1/T. Therefore, for the energy of the parti-

cles in the system:

ENkT^2 

3

2



T

1



E

3

2

NkT (17.56)



2 

h

m

2

k



3/2

V

3

2

T1/2





2 

h

m

2

kT



3/2

V

608 CHAPTER 17 Statistical Thermodynamics: Introduction