Page 159 - Advanced thermodynamics for engineers

7.2 USES OF THE THERMODYNAMIC RELATIONSHIPS 145

Thus, equating the two areas gives

vT vp
dv ds ¼ ds dv: (7.21)
vv vs
s v
and hence
vp vT

¼ : (7.19a)
vs v vv s

7.2 USES OF THE THERMODYNAMIC RELATIONSHIPS
When performing certain types of calculation it is useful to have data on the values of the specific heat
capacities of the substance under consideration and also the variation of these specific heat capacities
with the other properties. The specific heat capacities themselves are first derivatives of the internal
energy (u) and the enthalpy (h), and hence the variations of specific heat capacities are second de-
rivatives of the basic properties.
By definition, the specific heat capacity at constant volume is

vu vu vs
c v ¼ ¼ : (7.22)
vT vs vT
v v v
But

vu vs
¼ T; and thus c v ¼ T : (7.23)
vs vT
v v
Similarly

vh vh vs vs
c p ¼ ¼ ¼ T : (7.24)
vT p vs p vT p vT p
Consider the variation of the specific heat capacity at constant volume, c v , with specific volume, v,
if the temperature is maintained constant. This can be derived in the following way.

2
vc v v vs v s v vs
¼ T ¼ T ¼ T :
vv vv vT vvvT vT vv
T v T
Now, from Eqn (7.19c),
vs vp

¼ ;
vv T vT v
giving

2 2
vc v v s v p
¼ T ¼ T (7.25)
vv T vTvv vT 2 v