La diferencia entre $C_p$ y $C_v$ viene dada por: \begin{equation} C_p-C_v=\left[\left(\frac{\partial U}{\partial V}\right)_T+P\right]\left(\frac{\partial V}{\partial T}\right)_P \end{equation} Sustituyendo en la ecuación anterior, $\left(\frac{\partial U}{\partial V}\right)_T=\frac{\alpha}{\kappa}T-P$, nos da: \begin{equation} C_p-C_v=\left[\frac{\alpha}{\kappa}T-P+P\right]V\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P \end{equation} Simplificando: \begin{equation} C_p-C_v=\frac{TV\alpha^2}{\kappa} \end{equation}