Physical Chemistry Third Edition

48 2 Work, Heat, and Energy: The First Law of Thermodynamics

Exact and Inexact Differentials

The differential of a function of two or more independent variables is called anexact

differential.Iffis a function ofxandy, the differential

df

(

∂f

∂x

)

y

dx+

(

∂f

∂y

)

x

dx (exact differential) (2.1-19)

is an exact differential. There is an important theorem of mathematics concerning

the line integral of an exact differential:A line integral of an exact differential is

equal to the function evaluated at the final point of the integration curve minus the

function evaluated at the initial point of the curve. The line integral of an exact

differential therefore depends only on the end points, and not on the curve connect-

ing them.The line integral of an exact differential is said to bepath-independent.

The converse of this theorem is also true.If the integral of a differential is path-

independent for all paths between the same end points, the differential must be an exact

differential.

Since the pressurePis a state function,dPis an exact differential. The line integral

ofdPin part b of Example 2.2 is equal to the value ofPat the end of the process minus

the value ofPat the beginning of the process:

∫

c

dP∆PP(final)−P(initial)

Exercise 2.3

Verify that∆Pin Example 2.2 is equal to the final pressure minus the initial pressure.

Work Is an Inexact Differential

A differential that is not exact is called aninexact differential. The differential

duM(x,y)dx+N(x,y)dy (2.1-20)

is an inexact differential ifMandNare not the appropriate partial derivatives of

the same function. The line integral of an inexact differential depends on the path of

integration as well as on the initial point and the final point. We will show thatdwis an

inexact differential by showing that two processes with the same initial and final states

can correspond to different amounts of work done on the system.

EXAMPLE 2.7

Consider a reversible process with the same initial and final states as the process of

Example 2.2, but with a different path. Calculate the work done on the ideal gas system

of Example 2.2 if it is reversibly cooled at constant volume of 5.000 L from 298.15 K