'Second' Ehrenfest equation for second order phase transition under hydrostatic pressure

It is shown that the fundamental conditions for the second-order phase transitions Delta V = 0 and Delta S = 0, from which the two Ehrenfest equations follow (the 'usual' and the 'second' ones), are realised only at zero hydrostatic pressure (P = 0). At P > 0 the volume jump Delta V at the transition is proportional to the pressure and to the jump of the compressibility Delta zeta(V), whereas the entropy jump Delta S is proportional to the pressure and to the jump of the thermal expansion coefficient Delta alpha(V). This means that at non-zero hydrostatic pressure the phase transition is of the first order and is described by the Clausius-Clapeyron equation. At small pressure this equation coincides with the 'second' Ehrenfest equation dT(c)/dP = Delta zeta(V)/Delta alpha(V). At high P, the Clausius-Clapeyron equation describes qualitatively the caused by the crystal compression positive curvature of the T-C(P) dependence.