2.1 Work and the State of a System
43
(T 2 ,V 2 )
(T 1 ,V 1 )
V 1
V 2
T 2
T 1
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
of n moles of an ideal gas,
w rev
c
dw rev −
c
PdV −nR
c
T
V
dV (ideal gas)
(2.1-12)
Because the integrand in this integral depends on T and V we must specify how T
depends on V in order to carry out the integration. This dependence corresponds to a
curve in state space, and the letter c under the integral stands for this curve. This kind
of integral is called a line integral.
If the temperature is constant the process is called isothermal. For an isothermal
process,
w rev
c
dw rev −
V 2
V 1
nRT
V
dV −nRT
V 2
V 1
1
V
dV
w rev nRT ln
V 2
V 1
(closed ideal gas,
isothermal process)
(2.1-13)
where V 1 is the initial value of the volume and V 2 is the final value.