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.
Ehrenfest equations are the consequence of continuity of specific entropy
s
v
s
ds=\left({{{\partials}\over{\partialT}}}\right)PdT+\left({{{\partials}\over{\partialP}}}\right)TdP
\left({{{\partials}\over{\partialT}}}\right)P={{cP}\overT},\left({{{\partials}\over{\partialP}}}\right)T=-\left({{{\partialv}\over{\partialT}}}\right)P
d{si}={{ci}\overT}dT-\left({{{\partialvi}\over{\partialT}}}\right)PdP
where
i=1
i=2
{ds1}={ds2}
\left({c2P-c1P}\right){{dT}\overT}=\left[{\left({{{\partialv2}\over{\partialT}}}\right)P-\left({{{\partialv1}\over{\partialT}}}\right)P}\right]dP
Therefore, the first Ehrenfest equation is:
{\DeltacP=T ⋅ \Delta\left({\left({{{\partialv}\over{\partialT}}}\right)P}\right) ⋅ {{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:
{\DeltacV=-T ⋅ \Delta\left({\left({{{\partialP}\over{\partialT}}}\right)v}\right) ⋅ {{dv}\over{dT}}}
The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of
v
P
{\Delta\left({{{\partialv}\over{\partialT}}}\right)P=\Delta\left({\left({{{\partialP}\over{\partialT}}}\right)v}\right) ⋅ {{dv}\over{dP}}}
Continuity of specific volume as a function of
T
P
{\Delta\left({{{\partialv}\over{\partialT}}}\right)P=-\Delta\left({\left({{{\partialv}\over{\partialP}}}\right)T}\right) ⋅ {{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.