Ehrenfest entropy-continuity relation ($dT/dP=TV\Delta {\alpha }_T/\Delta C_P$)

If two equilibrium phases have continuous entropy and volume but jumps in second derivatives of Gibbs free energy, differentiate their entropy equality along the transition line. Since $dS=(C_P/T)dT-V{\alpha }_{T}dP$, one obtains the title slope, for nonzero $\Delta C_P$. Before imposing equal volume, the numerator is $T\Delta (V{\alpha }_T)$. Equality of Gibbs free energies with equal entropy implies equal volume along a differentiable coexistence curve on which pressure varies. A constant-pressure curve or zero numerator and denominator needs separate analysis. This entropy-continuity condition is one Ehrenfest relation; volume continuity gives another independent relation.