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

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

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,

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

Therefore, the first Ehrenfest equation is:

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

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

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

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

$${\Delta \left( \right)_P  = \Delta \left( {\left(  \right)_v } \right) \cdot {{dv} \over {dP}}}$$.

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

$${\Delta \left( \right)_P  =  - \Delta \left( {\left(  \right)_T } \right) \cdot {{dP} \over {dT}}}$$.

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