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For either phase, the entropy differential at fixed composition isThe pressure derivative uses the Maxwell relation from $dG=-SdT+VdP$. Differentiating the equality of entropies along coexistence gives the general relationFor a differentiable coexistence curve with pressure varying, equality of Gibbs free energies also implies $-\Delta SdT+\Delta VdP=0$. Since $\Delta S=0$, the two phases then share $V$. Thus, with $\Delta C_P\ne 0$,This is the Ehrenfest entropy-continuity relation. Entropy equality alone would give $T\Delta (V{\alpha }_T)/\Delta C_P$ without the shared-volume condition; a constant-pressure boundary or vanishing discontinuities needs separate limiting analysis.