# Difference between heat capacities $C_p - C_v$

I am trying to prove the following relation between heat capacities: $$C_p - C_v = T\left(\frac{\partial P}{\partial T}\right)_{V}\left(\frac{\partial V}{\partial T}\right)_{P}$$

In Wikipedia, they use the fact that $$dV = \left(\frac{\partial V}{\partial T}\right)_{P}dT + \left(\frac{\partial V}{\partial P}\right)_{T}dP$$

But do not explain how to derive this fact. How can we know that $V$ is not also a function of entropy? Maybe a function of energy?

This is not how I would have derived the desired relationship. I would have started with $$dH=C_pdT+\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]dP$$
• So the choice of $T,P$ is arbitrary? If magnetic field was around, magnetization and for example pressure would have determined the same system as $T,P$? – JonTrav1 Nov 12 '16 at 14:13