Physical Chemistry , 1st ed.

The derivative with respect to xof the derivative ofFwith respect to yis equal

to the derivative with respect to yof the derivative ofFwith respect to x.If

this is the case, then the original differential dFin equation 4.28 satisfies one

requirement of an exact differential: the value of the multiple differential does

not depend on the order of differentiation.* Equation 4.29 is known as the

cross-derivative equality requirementof exact differentials. In the application

of the double derivatives in equation 4.29 to real thermodynamic equations,

the partial derivatives may have some other expression, as the following ex-

ample shows.

Example 4.6

Is the following expression considered an exact differential?

dT

R

p

dV 

V

R

dp

Solution

Using equation 4.28 as a template, we can figure by analogy that



V

T

pR

p

and that



T

p



V



V

R



Taking the derivative of the first partial with respect to p,we get



p



V

T

pR

(^1) 
and taking the derivative of the second partial with respect to Vwe get

V


T

p



V



R

1



By definition, the original differential is an exact differential. Therefore, it

doesn’t matter in which order we differentiate T(p,V), since the double de-

rivative gives us the same value either way.

In the evaluation of exact differentials, the order of differentiation does not

matter. For state functions, the path of change does not matter. All that mat-

ters is the difference between the initial and final conditions. We submit that

the conditions are parallel and that the conclusions are transferable: the dif-

ferential forms of the natural variable equations for the thermodynamic ener-

gies are exact differentials. Therefore, the two ways of taking the mixed second

derivatives ofU,H,G, and Amust be equal. That is,



p



H

S



pS



S



H

p



Sp

(4.30)

100 CHAPTER 4 Free Energy and Chemical Potential

*This is equivalent to saying that the value of an integral of a state function is path-

independent, an idea used in Chapter 2.