= Heat-capacity difference and mechanical stability
{title2=$C_p-C_V=TV\alpha_V^2/\kappa_T\geq0$}
For a fixed-composition <simple compressible system>, a <Maxwell relation> gives $C_p-C_V=T(\partial p/\partial T)_V(\partial V/\partial T)_p$. At fixed <volume>, the differential <equation of state> gives $(\partial p/\partial T)_V=\alpha_V/\kappa_T$, proving the displayed result. Positive absolute <temperature> and positive <isothermal compressibility> imply a nonnegative difference. It vanishes when <thermal expansion> does, so strict inequality requires a nonzero expansion coefficient. This stability argument includes nonideal internal-energy changes, unlike a purely ideal-gas expansion-work explanation.
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