Physical Chemistry , 1st ed.

SkT





ln

T

Q



V

ln Q (^1)  (18.48)
AkT(ln Q 1) (18.49)
GkTln Q (18.50)
Notice that each expression involves the natural logarithm ofQ(or a deriva-
tive of the natural logarithm ofQ). Recall also that Qfor a molecule is defined
as the product of five independent q’s. Logarithms of products can be rewrit-
ten as the sum of logarithms of the individual terms in the product; that is,
ln Qln (qtransqelectqvibqrotqnuc)
ln Qln qtrans ln qelect ln qvib ln qrot ln qnuc (18.51)
Equation 18.51 implies that if the partition function Qcan be separated into
the sum of individual logarithm terms, then the thermodynamic functions in
equations 18.45 through 18.50 can also be separated into sums of individual
energy, enthalpy, entropy, and so on. For example, we can rewrite the total (that
is, internal) energy Eas
EtransNkT^2 
ln

q
T
trans
V
EelectNkT^2 ln
T
qelect
V
EvibNkT^2 
ln
T
qvib

V

(18.52)

ErotNkT^2 

ln

T

qrot



V

EnucNkT^2 

ln

T

qnuc



V

so that

EEtrans Eelect Evib Erot Enuc (18.53)

and similarly for each of the other thermodynamic functions.

Since we have derived expressions for each of the partition functions of a

molecule, we can evaluate the expressions in equations 18.52, and similarly for

each of the other thermodynamic functions, for each part of the overall mo-

lecular partition function. These expressions are given in Table 18.5. You should

be able to derive most of the expressions in Table 18.5 by simply performing the

appropriate derivation given in equations 18.52 and the equivalent for the other

thermodynamic properties. (Remember that [(ln q)/T] (1/q)(q/T).)

Heat capacities are defined in terms of the change in Eor Hwith respect to

temperature at constant volume or pressure:

CV





T

E



V

Cp





H

T



p

(Again, remember that we are using the variable Eto stand for the internal en-

ergy in this chapter.) From Table 18.5, we can take the derivative of each term

638 CHAPTER 18 More Statistical Thermodynamics