Ehrenfest equations

Remarks

In each phase $ds = \frac{c_p}{T}\, dT - v \alpha_p\, dp$, using the Maxwell relation $(\partial s/\partial p)_T = -(\partial v/\partial T)_p$. Since $s_\alpha = s_\beta$ all along the curve, the differentials agree along it, and subtracting gives $\frac{\Delta c_p}{T}\, dT = v\, \Delta\alpha_p\, dp$, the first equation. In the same way $dv = v\alpha_p\, dT - v\kappa_T\, dp$ and $v_\alpha = v_\beta$ give $\Delta\alpha_p\, dT = \Delta\kappa_T\, dp$, the second equation. They play the role of the Clausius–Clapeyron relation when the latent heat and the volume jump vanish. Many real continuous transitions, such as the lambda transition of liquid helium, have a heat capacity that diverges instead of jumping. There the hypotheses fail and the equations do not apply.

Going deeper

For a student

The derivation copies the proof of the Clapeyron equation one derivative higher: instead of equal chemical potentials, one uses equal entropies and equal volumes along the curve. Combining the two equations gives the consistency relation $\Delta c_p = T v \frac{(\Delta\alpha_p)^2}{\Delta\kappa_T}$, which can be tested on experimental data. Landau theory predicts finite jumps of this kind, but for many real systems fluctuations change the behaviour near the transition.