Dependence of U with respect to V

Let's calculate $\left(\frac{\partial U}{\partial V}\right)_T$ using the Gibbs equation $dU=TdS-PdV$.

We divide the Gibbs equation by dV keeping the temperature constant :

\begin{equation} \left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial S}{\partial V}\right)_T-P \end{equation}

Using Maxwell's relation, $\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial P}{\partial T}\right)_V$ , we get:

\begin{equation} \left(\frac{\partial U}{\partial V}\right)_T=T\left(\frac{\partial P}{\partial T}\right)_V- P=\frac{\alpha T}{\kappa}-P \end{equation}

In the above equation we used the relation: $\left(\frac{\partial P}{\partial T}\right)_V=\frac{\alpha}{\kappa}$

Dependence of U with respect to T

Keeping the volume constant, the heat exchanged coincides with the change in internal energy and therefore:

\begin{equation} \left(\frac{\partial U}{\partial T}\right)_V =C_V \end{equation}

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