Page 159 - Advanced Thermodynamics for Engineers, Second Edition

146 CHAPTER 7 GENERAL THERMODYNAMIC RELATIONSHIPS

7.2.1 SPECIFIC HEAT AT CONSTANT VOLUME, C V , AS A FUNCTION OF VOLUME
Hence, if data are available for a substance in the form of a p–v–T surface (e.g. Fig. 2.5), or a
2
2
mathematical relationship, it is possible to evaluate (v p/vT ) v and then (v c v /v v) T . By integration, it is
then possible to obtain the values of c v at different volumes. For example, consider whether the specific
heat capacity at constant volume is a function of the volume of both an ideal gas and a van der
Waals gas.
Ideal gas
The gas relationship for an ideal gas is
pv ¼ RT (7.26)
and

vp R v p
2
¼ ; and ¼ 0: (7.27)
vT v vT 2
v v
Hence
c v sfðvÞ
for an ideal gas.
This is in agreement with Joule’s experiment for assessing the change of internal energy, u, with
volume.
van der Waals gas
The van der Waals gas is discussed in much greater depth in Section 8.2; the equation of state will
just be introduced here and some of the concepts developed above will be investigated.
The equation of state of a van der Waals gas is

RT a
p ¼ 2 ; (7.28)
v b v
where a and b are constants (described in Section 8.2).
Hence

2
vp R v p
¼ ; and ¼ 0: (7.29)
vT v v b vT 2 v
Again,

c v sfðvÞ
for a van der Waals gas.
So, for ‘gases’ obeying these state equations c v s f(v), but in certain cases and under certain
conditions c v could be a function of the volume of the system and it would be calculated in this way. Of
course, if these equations are to be integrated to evaluate the internal energy it is necessary to know the
value of c v at a datum volume and temperature.