Physical Chemistry , 1st ed.

Example 18.9

Calculate the constant-volume heat capacity of 1 mole of gaseous water at

727°C and 1 bar pressure. Compare this to a value of 33.0 J/(molK) at 727°C

and about 1 atm pressure (where 1 atm 1.01325 bar) and comment on the

difference.

Solution

The three vibrational temperatures for H 2 O are 2287, 5163, and 5350 K. (See

Example 18.5.) Our expression for CVbecomes

640 CHAPTER 18 More Statistical Thermodynamics

CV (6.02 

1023 mol^1 )1.381 

10 ^23 

K

J







3

2

 

3

2

 (^) 

2

1

2

0

8

0

7

0

K

K



2



exp

2

1

2

0

8

0

7

0

K

K





^1 exp

2

1

2

0

8

0

7

0

K

K



2

We g e t

CV(8.314

m

J

ol

)(3 0.658 0.154 0.137)

CV32.8

mo

J

lK



This is a very good agreement with experiment! The variance comes from the

fact that H 2 O is certainly not acting as an ideal gas, even at these tempera-

tures and pressures. At higher temperatures and lower pressures, predicted

values are much closer to experiment. Finally, notice how much the vibra-

tions contribute to the heat capacity. The lower-energy vibration, which has

the lower value ofv, contributes much more than the two higher vibrations

combined.

The point of this section is that statistical thermodynamics can derive ex-

pressions for thermodynamic properties of molecules. Many computer pro-

grams are available that use the expressions in Table 18.5 to calculate thermo-

dynamic properties of molecules, given their energy levels (which can be

determined spectroscopically or theoretically). Application of these equations

gives the physical chemist a powerful tool for understanding the thermody-

namic properties of molecules.

18.9 Equilibria

Classical thermodynamics is very useful when applied to chemical or physical

processes that are in a state of equilibrium. How well does statistical thermo-

dynamics apply to equilibrium?

Let us assume a balanced, gas-phase equilibrium:

(^) AA (^) BB (^) CC (^) DD (18.55)
where A and B represent reactants, C and D are the products, and (^) A, (^) B,
(^) C,and (^) Dare the molar coefficients of the balanced chemical reaction.

JQPJ

(^) 

5

1

1

0

6

0

3

0

K

K



2

   

5

1

3

0

5

0

0

0

K

K



2





exp^5

1

3

0

5

0

0

0

K

K





^1 exp

5

1

3

0

5

0

0

0

K

K



2

exp^5

1

1

0

6

0

3

0

K

K





^1 exp

5

1

1

0

6

0

3

0

K

K



2