# Ehrenfest equations

Ehrenfest equations — equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions. Clausius–Clapeyron relation does not make sense for second-order phase transitions,[1] as both specific heat capacity and specific volume do not change in second-order phase transitions.

## Quantitative consideration

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

$d {s_i} = {{c_{i P} } \over T}dT - \left( {{{\partial v_i } \over {\partial T}}} \right)_P dP$

Where $i=1$ and $i=2$ are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions: ${ds_1} = {ds_2}$. So,

$\left( {c_{2P} - c_{1P} } \right){{dT} \over T} = \left[ {\left( {{{\partial v_2 } \over {\partial T}}} \right)_P - \left( {{{\partial v_1 } \over {\partial T}}} \right)_P } \right]dP$

Therefore, the first Ehrenfest equation:

${\Delta c_P = T \cdot \Delta \left( {\left( {{{\partial v} \over {\partial T}}} \right)_P } \right) \cdot {{dP} \over {dT}}}$

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

${\Delta c_V = - T \cdot \Delta \left( {\left( {{{\partial P} \over {\partial T}}} \right)_v } \right) \cdot {{dv} \over {dT}}}$

The third Ehrenfest equation is got in a like manner, but specific entropy is considered as function of $v$ и $P$.

${\Delta \left( {{{\partial v} \over {\partial T}}} \right)_P = \Delta \left( {\left( {{{\partial P} \over {\partial T}}} \right)_v } \right) \cdot {{dv} \over {dP}}}$

Continuity of specific volume as of function of $T$ and $P$ gives the fourth Ehrenfest equation:

${\Delta \left( {{{\partial v} \over {\partial T}}} \right)_P = - \Delta \left( {\left( {{{\partial v} \over {\partial P}}} \right)_T } \right) \cdot {{dP} \over {dT}}}$

## Application

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