Ehrenfest's equations
(i) Show that, for a second-order transition in a $p V$ system, not only are the chemical potentials equal, but so are the volume and entropy per particle.
(ii) Let $v$ be the volume per particle. Using $d v_1=d v_2$ for a small change along the phase boundary, obtain
$$
\frac{\mathrm{d} p}{\mathrm{~d} T}=\frac{\beta_2-\beta_1}{\kappa_2-\kappa_1}=\frac{\Delta \beta}{\Delta \kappa}
$$
(iii) Treating the entropy per particle, obtain
$$
\frac{\mathrm{d} p}{\mathrm{~d} T}=-\frac{1}{v T} \frac{c_{p 2}-c_{p 1}}{\beta_2-\beta_1}=\frac{\Delta C_p}{V T \Delta \beta} .
$$
These are called Ehrenfest's equations. They apply to second-order phase transitions as long as the quantities involved are all well behaved. (Since for many continuous phase transitions some or all of the response functions diverge, these equations are less useful in practice than the Clausius-Clapeyron equation. However the corresponding results for magnetic properties are useful in the study of superconductivity.)