Ehrenfest equations
Introduction '''Ehrenfest equations''' (named after Paul Ehrenfest) are equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions. The Clausius–Clapeyron relation does not make sense for second-order phase transitions, as both specific entropy and specific volume do not change in second-order phase transitions.
Quantitative consideration Ehrenfest equations are the consequence of continuity of specific entropy  and specific volume , which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy  as a function of temperature and pressure, then its differential is:.As , then the differential of specific entropy also is:,where  and  are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions: . So,Therefore, the first Ehrenfest equation is:.The second Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of temperature and specific volume:The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of  and :.Continuity of specific volume as a function of  and  gives the fourth Ehrenfest equation:.
Limitations Derivatives of Gibbs free energy are not always finite. Transitions between different magnetic states of metals can't be described by Ehrenfest equations.
See also * Paul Ehrenfest* Clausius–Clapeyron relation* Phase transition
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