Physical Chemistry , 1st ed.

q 4  1  4 2.80  10 ^11  4 7.84  10 ^22  4 2.19  10 ^32

q 4

This second example shows how sensitive qis to the values of the energy levels.

For the second set of energy levels in the above example, the fact that qis

approximately equal to the number of levels suggests a molecular interpreta-

tion of the partition function:qis a measure of the number of energy states

that are available to a particle at any particular temperature. Thus, for low-

energy states at a given temperature, many of those states can be populated by

thermal energy. In the above example, approximately 12 of the lower-energy

states (recall that each level is fourfold degenerate) could be populated at

298 K. But if the higher-energy states are considered, only the ground state

(degeneracy 4) is generally populated, so a partition function value of 4 is

consistent with this interpretation ofq.

17.5 Thermodynamic Properties from Statistical Thermodynamics

Now that we have established the complete form of our partition function,

how can we determine thermodynamic properties from it? We will start with

energy. The total energy of the ensemble is given by equation 17.10:

E

i

Ni i

Substituting for Nifrom equations 17.20 and 17.31:

E

i



N

q

gie  i/kT   i

EN^1

q



i

gie    i/kT   i (17.33)

Consider briefly the derivative of equation 17.32 with respect to temperature:







T

q







T



i

gie    i/kT



i

gi





T

e i/kT



i

gie   i/kT

k

T

i

 2







T

q



kT

1

 2 

i

gie    i/kT   i

If we divide both sides by q,we get



1

q







T

q



1

q



kT

1

 2 

i

gie    i/kT   i

According to the rules of calculus, the left side of the above equation is

(ln q/T). Moving the kT^2 term to the left side, we have

kT^2 





ln

T

q



1

q



i

gie    i/kT   i

600 CHAPTER 17 Statistical Thermodynamics: Introduction