Page 161 - Advanced thermodynamics for engineers

7.2 USES OF THE THERMODYNAMIC RELATIONSHIPS 147

7.2.2 SPECIFIC HEAT AT CONSTANT PRESSURE, C P , AS A FUNCTION OF
PRESSURE
Similarly, the variation of the specific heat capacity at constant pressure with pressure can be inves-
tigated by differentiating the specific heat capacity with respect to pressure, giving

!
2
vc p v vs v vs v v
¼ T ¼ T ¼ T (7.30)
vp vp vT vT vp vT 2
T p T p
T
This equation can be used to see if the specific heat capacity at constant pressure of gases obeying
the ideal gas law and those obeying van der Waals equation are functions of pressure. This is
done below.
Ideal gas
pv ¼ RT (7.26)

Hence

2
vv R v v
¼ ; and ¼ 0: (7.31)
vT p vT 2
p p
This means that the specific heat capacity at constant pressure for a gas obeying the ideal gas law is
not a function of pressure, i.e. c p sf(p) for an ideal gas. This conclusion is in agreement with the
Joule–Thomson experiment for superheated gases.
van der Waals gas
RT a
p ¼ 2 : (7.28)
v b v
Equation (7.28) can be rewritten as
a

p þ ðv bÞ¼ RT;
v 2
which expands to
a ab
pv þ pb ¼ RT:
v v 2
Differentiating implicitly gives

vv a vv ð 2Þab vv
p þð Þ ¼ R;
vT p v 2 vT p v 3 vT p
which can be rearranged to give

vv a 2ab
p þ ¼ R;
vT v 2 v 3
p