= Ehrenfest entropy-continuity relation
{c}
{title2=$dT/dP=TV\Delta\alpha_T/\Delta C_P$}
= Equal-entropy coexistence slope
{synonym}
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 $dS=(C_P/T)dT-V\alpha_TdP$, 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.
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