Problem-TH-04001 Fiirst Law Relation between \(C_p, C_v\).

 Starting from the first law of thermodynamics show that
 \[C_p - C_v = \Big[P +\Big(\frac{\partial U}{\partial V}\Big)_T\Big]
\Big(\frac{\partial V}{\partial T}\Big)_P\]
In the above \(C_p\) : heat capacity at constant pressure; \(C_v\) :
heat capacity at constant volume; \(U\) : internal energy: \(V\) is
the volume. For an ideal gas show that the above reduces
to \(C_p -C_v = Nk_B\) , where \(N\) is the number of molecules and
\(k_B\) is the Boltzmann constant.