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It is known that within thermodynamics alone, given the equation of the state of a system, one cannot explicitly determine the heat capacity. What is the mathematical reason for this?

Intuitively, it is clear that the heat capacity contains the information about the degrees of freedom of the system which are not visible to the equation of state and, therefore, we need statistical mechanics to determine them. But how do we see it in the mathematical formalism of thermodynamics?

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The equation of state will provide the property departure from the ideal gas. This means that you will access $Cv-Cv^0$ and $Cp-Cp^0$ with $Cp^0-Cv^0=R$; so you need to know $Cp^0$. – Claude Leibovici Mar 12 at 13:52

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