Meyer's Relation

Meyer's relation connects $C_p$ and $C_v$ for an ideal gas. Heating $\mu$ moles at constant volume by $dT$ does no work, so all the heat raises internal energy: $dU=\mu C_v\,dT$. Heating the same gas at constant pressure by the same $dT$ instead requires heat $Q=\mu C_p\,dT$, part of which does expansion work $dW=P\,dV$; by the first law, $\mu C_p\,dT=\mu C_v\,dT+P\,dV$. Differentiating the equation of state $PV=\mu RT$ at constant pressure gives $P\,dV=\mu R\,dT$, so substituting and cancelling $\mu\,dT$ throughout gives $C_p-C_v=R,$ Meyer's relation: the molar specific heat capacity at constant pressure always exceeds that at constant volume by exactly the universal gas constant, $R=8.314\ \text{J mol}^{-1}\text{K …