Concise Physical Chemistry

c03 JWBS043-Rogers September 13, 2010 11:24 Printer Name: Yet to Come

THERMODYNAMIC CYCLES 37

so the condition

∂

∂x

N(x,y)=

∂

∂y

M(x,y)

is a test for exactness.

3.2 THERMODYNAMIC CYCLES

The problem of inexact and exact differentials can be expressed in several ways.

Taking work as an example, workwisnot a thermodynamic function. The differential

of the workdwisnot exact. The work done in a thermodynamic process depends

upon the path. The integral

w=

∫V 2

V 1

f(T)pdv

is aline integral. The line integral cannot be evaluated until we know f(T), an

arbitrary function of the temperature. There is an infinite number of paths (ways of

getting fromV 1 toV 2 ), hence an infinite number of solutions to the work integral.

Consider two ways of taking a gas from thermodynamic state A with molar volume

V 1 to state B with molar volumeV 2. Let the substance be in the gaseous state. Let the

first path be an isothermal compression from 1.00 bar to 10.0 bars at 290 K followed

by anisobaric(constant pressure) temperature rise from 290 K to 310 K. The second

path will start with the isobaric temperature rise from 290 K to 310 K, followed by

an isothermal compression from 1.00 bar to 10.0 bars at 310 K. The beginning and

end points of the process are the same for both paths. For simplicity, assume that the

gas is nitrogen, which is nearly ideal over this temperature and pressure range. The

two paths are shown in Fig. 3.1.

The volumeV=RT/pcan be calculated after each step. The results are shown

at each corner of the rectangular diagram below. The volumes at each corner of the

diagram are larger on the left (before compression) than on the right and larger at the

top (after heating) than on the bottom.

2441 244.1

2660 266.0

A

B