= Solution
For either <phase>, the <entropy> differential at fixed composition is
$$
dS=\frac{C_P}{T}\,dT-\left(\frac{\partial V}{\partial T}\right)_P dP
=\frac{C_P}{T}\,dT-V\alpha_T\,dP.
$$
The pressure derivative uses the <Maxwell relation> from $dG=-S\,dT+V\,dP$. Differentiating the equality of <entropies> along coexistence gives the general relation
$$
\frac{\Delta C_P}{T}\,dT=\Delta(V\alpha_T)\,dP.
$$
For a differentiable coexistence curve with pressure varying, equality of <Gibbs free energies> also implies $-\Delta S\,dT+\Delta V\,dP=0$. Since $\Delta S=0$, the two <phases> then share $V$. Thus, with $\Delta C_P\ne0$,
$$
\boxed{\frac{dT}{dP}=\frac{TV\Delta\alpha_T}{\Delta C_P}.}
$$
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.
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