The exact expression for the relation between the heat capacities at constant volume and constant pressure is C P - C v = a 2 TV / k , where a is the expansion coefficient, a = ( dV / dT) / V at...

The exact expression for the relation between the heat capacities at constant volume and constant pressure is
C_P

-
C_v

=
a
²
TV/k, where
a
is the expansion coefficient,
a
= (dV/dT)/V
at constant pressure and
k
(kappa) is the isothermal
compressibility,
k
= -(dV/dp)/V.
Confirm that th is general expression
reduces to that for a perfect gas,
C_p,m

-
C_v,m

=
R.