Ehrenfest equations are the consequence of continuity of specific entropy ${\displaystyle s}$  and specific volume ${\displaystyle v}$ , which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy ${\displaystyle s}$  as a function of temperature and pressure, then its differential is: ${\displaystyle ds=\left({{\partial s} \over {\partial T}}\right)_{P}dT+\left({{\partial s} \over {\partial P}}\right)_{T}dP}$ . As ${\displaystyle \left({{\partial s} \over {\partial T}}\right)_{P}={{c_{P}} \over T},\left({{\partial s} \over {\partial P}}\right)_{T}=-\left({{\partial v} \over {\partial T}}\right)_{P}}$ , then the differential of specific entropy also is:

${\displaystyle d{s_{i}}={{c_{iP}} \over T}dT-\left({{\partial v_{i}} \over {\partial T}}\right)_{P}dP}$ ,

where ${\displaystyle i=1}$  and ${\displaystyle i=2}$  are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions: ${\displaystyle {ds_{1}}={ds_{2}}}$ . So,

${\displaystyle \left({c_{2P}-c_{1P}}\right){{dT} \over T}=\left[{\left({{\partial v_{2}} \over {\partial T}}\right)_{P}-\left({{\partial v_{1}} \over {\partial T}}\right)_{P}}\right]dP}$ 

Therefore, the first Ehrenfest equation is:

${\displaystyle {\Delta c_{P}=T\cdot \Delta \left({\left({{\partial v} \over {\partial T}}\right)_{P}}\right)\cdot {{dP} \over {dT}}}}$ .

The second Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of temperature and specific volume:

${\displaystyle {\Delta c_{V}=-T\cdot \Delta \left({\left({{\partial P} \over {\partial T}}\right)_{v}}\right)\cdot {{dv} \over {dT}}}}$ 

The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of ${\displaystyle v}$  and ${\displaystyle P}$ :

${\displaystyle {\Delta \left({{\partial v} \over {\partial T}}\right)_{P}=\Delta \left({\left({{\partial P} \over {\partial T}}\right)_{v}}\right)\cdot {{dv} \over {dP}}}}$ .

Continuity of specific volume as a function of ${\displaystyle T}$  and ${\displaystyle P}$  gives the fourth Ehrenfest equation:

${\displaystyle {\Delta \left({{\partial v} \over {\partial T}}\right)_{P}=-\Delta \left({\left({{\partial v} \over {\partial P}}\right)_{T}}\right)\cdot {{dP} \over {dT}}}}$ .

Derivatives of Gibbs free energy are not always finite. Transitions between different magnetic states of metals can't be described by Ehrenfest equations.

• Paul Ehrenfest
• Clausius–Clapeyron relation
• Phase transition