# Ehrenfest equations

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[according to whom?] 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 ${\displaystyle v}$ и ${\displaystyle P}$:

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

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

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

## Application

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