Physical Chemistry Third Edition

2.1 Work and the State of a System 43

(T 2 ,V 2 )

V 1 (T 1 ,V 1 )

V 2

T 1 T 2

T

V

Figure 2.3 A Curve in State Space.

Work Done on an Ideal Gas in a Reversible Process

The ideal gas is a convenient system for example calculations, since it has simple

properties but illustrates the principles involved. However, you should be careful not

to apply ideal gas equations to other systems. For a finite reversible change in volume

ofnmoles of an ideal gas,

wrev

∫

c

dwrev−

∫

c

PdV−nR

∫

c

T

V

dV (ideal gas) (2.1-12)

Because the integrand in this integral depends onTandVwe must specify howT

depends onVin order to carry out the integration. This dependence corresponds to a

curve in state space, and the lettercunder the integral stands for this curve. This kind

of integral is called aline integral.

If the temperature is constant the process is calledisothermal. For an isothermal

process,

wrev

∫

c

dwrev−

∫V 2

V 1

nRT

V

dV−nRT

∫V 2

V 1

1

V

dV

wrevnRTln

(

V 2

V 1

)

(closed ideal gas,

isothermal process)

(2.1-13)

whereV 1 is the initial value of the volume andV 2 is the final value.