Mayer's relation

Remarks

For an ideal gas the internal energy $U$ depends only on the temperature $T$, and $pV = nRT$. So the enthalpy $H = U + pV = U(T) + nRT$ also depends only on $T$. Differentiating gives $C_p = dH/dT = dU/dT + nR = C_V + nR$. For a monatomic ideal gas, $C_V = \tfrac{3}{2} nR$ and $C_p = \tfrac{5}{2} nR$, and their difference is $nR$. For an arbitrary simple system the general identity is $C_p - C_V = T V \alpha^2 / \kappa_T$. Here $\alpha$ is the thermal expansion coefficient and $\kappa_T$ the isothermal compressibility. For an ideal gas $\alpha = 1/T$ and $\kappa_T = 1/p$, which gives back $nR$. The relation fails for real gases, where $U$ depends also on the volume.

Going deeper

For a student

The relation is the first place where the two heat capacities are compared, and it uses both the equation of state and the fact that $U$ depends only on $T$. Together with the heat capacity ratio $\gamma = C_p / C_V$ it gives $C_V = nR/(\gamma - 1)$ and $C_p = \gamma nR/(\gamma - 1)$. Historically it allowed Mayer to estimate the mechanical equivalent of heat from gas data. The general form $C_p - C_V = TV\alpha^2/\kappa_T$ is derived from the Maxwell relations and shows $C_p \ge C_V$ for every stable system.