Physical Chemistry , 1st ed.

1.7 More on Derivatives

The above examples of taking partial derivatives of equations of state are rel-

atively straightforward. Thermodynamics, however, is well known for using

such techniques extensively. We therefore devote this section to a discussion of

partial derivative techniques that we will use in the future. The expressions that

we derive in thermodynamics using partial derivation can be extremely useful:

the behavior of a system that cannot be measured directly can instead be cal-

culated through some of the expressions we derive.

Various rules about partial derivatives are expressed using the general vari-

ables A,B,C,D,...instead of variables we know. It will be our job to apply

these expressions to the state variables of interest. The two rules of particular

interest are the chain rule for partial derivatives and the cyclic rule for partial

derivatives.

First, you should recognize that a partial derivative obeys some of the same

algebraic rules as fractions. For example, since we have determined that



T

p



V, n



n

V

R



we can take the reciprocal of both sides to find that



T

p



V, n



n

V

R



Note that the variables that remain constant in the partial derivative stay the

same in the conversion. Partial derivatives also multiply through algebraically

just like fractions, as the following example demonstrates.

IfAis a function of two variables Band C, written as A(B,C), and both

variables Band Care functions of the variables Dand E , written respectively

as B(D,E) and C(D,E), then the chain rule for partial derivatives* is



A

B



C



D

A

E

D

B



C

^

A

E



D



B

E

C (1.24)

This makes intuitive sense in that you can cancel Din the first term and E

in the second term, if the variable held constant is the same for both partials

in each term. This chain rule is reminiscent of the definition of the total de-

rivative for a function of many variables.

In the cases ofp,V, and T, we can use equation 1.24 to develop the cyclic

rule.For a given amount of gas, pressure depends on Vand T, volume depends

on pand T, and temperature depends on pand V. For any general state vari-

able of a gas F, its total derivative (which is ultimately based on equation 1.12)

with respect to temperature at constant pwould be



T

F



p



T

F



V



T

T



p



V

F



T



V

T



p

The term ( T/ T)pis simply 1, since the derivative of a variable with respect

to itself is always 1. IfFis the pressure p, then ( F/ T)p( p/ T)p0, since

pis held constant. The above expression becomes

0 

T

p

V

V

p

T

V

T



p

18 CHAPTER 1 Gases and the Zeroth Law of Thermodynamics

*We present the chain rule here, but do not derive it. Derivations can be found in most

calculus books.