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,^[1] 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 ${\displaystyle s}$ 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:

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

Limitations

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