Page 86 - Modern physical chemistry

4.11 Relating Differentials and Partial Derivatives 75

TABLE 4.1 Constant-Volume Heat Capacities
Cv , J Kl rrwt 1 at
Gas 150K 300K 500K 700K 900K
O 2 20.84 21.05 22.76 24.69 26.02
H2 18.24 20.54 20.96 21.13 21.55
H 2 O - 25.27 26.90 29.16 31.67
N2 20.79 20.79 21.25 22.43 23.77
NO 22.84 21.55 22.18 23.72 25.10
CO 20.79 20.84 21.46 22.84 24.27
CO2 - 28.91 36.32 41.25 44.73

Example 4.6
Estimate the translational, rotational, and vibrational contributions to heat capacity
Cv of H 20 at 500 K.
For the three translational degrees of freedom of the H 20 molecule, we have

Cv tr = ~R =~(8.3145 J K· 1 mOI- 1 )= 12.47 J K- 1 mol- 1 .
, 2 2
Since H20 is nonlinear, there are three rotational degrees of freedom. We expect these
to be excited classically at 500 K; so

Cv rot = ~R = 12.47 J K- 1 mol- 1 .
, 2
Subtracting the sum of these contributions from the value in table 4.1 yields

CV,vib = 26.90 - 24.94 = 1.96 J K"l mOrl.
Consistent with the data in example 3.7, the three vibrational degrees of freedom are
only weakly excited.

4. 11 Relating Differentials and Partial Derivatives

Because of its importance in thermodynamics, let us here establish the fundamental
relationship linking differentials with the pertinent partial derivatives.
Consider a functionjin which x and y vary independently:
j = j{X,y). [4.71]

For a change tlx in x and y in !J.y, we find injthe change

~f = f(x+ ~,y + ~y)- f(x,y) = f(X +~,y + ~y)- f(X,y+ ~y)+ f(X,Y+ ~y)- f(X,y)
f(X+~,y+~y)- f(X,y+~y) f(X,y+~y)- f{X,y) [4.72]
= ~+ ~y.
~ ~y
By definition, the limit of each ratio of changes is a derivative. Here we have