Concise Physical Chemistry

c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

114 A STATISTICAL APPROACH TO THERMODYNAMICS

TABLE 8.1 Thermodynamic Functions

(Irikura, 1998).

S=NkB

[

∂

∂T

(TlnQ)−lnN+ 1

]

CV=NkBT

∂^2

∂T^2

(TlnQ)

Cp=CV+R

H(t)−H(0)=

∫T

0

CpdT=

RT^2

Q

∂Q

∂T

+RT

∂

∂T

(TlnQ)=lnQ+

T

Q

∂Q

∂T

∂^2

∂T^2

(TlnQ)=

2

Q

∂Q

∂T

+

T

Q

∂^2 Q

∂T^2

−

T

Q^2

(

∂Q

∂T

) 2

∂Q

∂T

=

1

kBT^2

∑

i

εie−εi/kBT

∂^2 Q

∂T^2

=

− 2

T

∂Q

∂T

+

1

k^2 BT^4

∑

i

ε^2 ie−εi/kBT

The essential thermodynamic functions are expressed above the line in Table 8.1,

and the derivatives are given below the line for somewhat simpler and more practical

evaluation. Along with the equations in Table 8.1, “excess” free energy and enthalpy

functions are defined as (G◦−H 0 )/Tand (H◦−H 0 )/T. The excess thermodynamic

property is the amount of that quantity above the value per degree K the system would

have if all its parts were in the lowest possible energy state. Excess free energy and

enthalpy functions vary in a gradual way over a wide range of temperatureT.This

enables one to make accurate interpolations and to extrapolate from data one has to

get information one does not have.

Now we need to find out how to calculate the partition function.

8.6 THE PARTITION FUNCTION OF A SIMPLE SYSTEM

Spectroscopy gives information as to the intervals between energy levels in molecules,

hence it is a gateway to molecular partition functions. As a simple example, consider

a noninteracting^2 system of chemical bonds treated as harmonic oscillators (Chap-

ter 18). A harmonic oscillator has the peculiar property that its energy levels are

(^2) Strictly, the system isweakly interactingbecause particle transfer from one level to another is a necessary
condition of the statistical treatment.