
Now, from the two-property rule, T ¼ T (p,v) and hence
vp
vv
T
¼
vp
vT
v
vT
vv
p
(7.73)
Thus
c
p
c
v
¼
vp
vv
s
vp
vv
T
¼ v
vp
vv
s
v
vp
vv
T
(7.74)
The denominator of Eqn (7.74), v (vp/vv)
T
, is the reciprocal of the isothermal compressibility, k.By
analogy, the numerator can be written as 1/k
s
, where k
s
¼ adiabatic, or isentropic, compressibility.
Thus
c
p
c
v
¼
k
k
s
¼ k: (7.75)
If the slopes of isentropes in the p–v plane, are compared with the slopes of isotherms, see Fig. 7.3,
it can be seen that c
p
/c
v
> 1 for a gas in the superheat region.
7.5 THE CLAUSIUS–CLAPEYRON EQUATION
From the Maxwell relationships
vs
vv
T
¼
vp
vT
v
: (7.19c)
Pressure, p
Specific volume, v
Isentrope
Isotherm ...