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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?

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For a single phase system of constant composition at thermodynamic equilibrium, the thermodynamic state of the system is completely determined by specifying any two intensive properties. This follows from the phase rule, which is an empirical relationship developed by Gibbs. In this case, the two intensive properties are T and P.

This is not how I would have derived the desired relationship. I would have started with $$dH=C_pdT+\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]dP$$

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  • $\begingroup$ 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 $T,P$? $\endgroup$
    – JonTrav1
    Commented Nov 12, 2016 at 14:13
  • $\begingroup$ I'm not sure about case of a magnetic field. In this problem, the choice of T,P is not arbitrary. It is the form that is needed to derive the desired relationship. $\endgroup$
    – Chet Miller
    Commented Nov 12, 2016 at 14:19

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