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,Sivuhin D.V. General physics course. V.2. ''Thermodynamics and molecular physics''. 2005 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 $\left( \right)_P = , \left( \right)_T = - \left( \right)_P$, then the differential of specific entropy also is:

$d = dT - \left( \right)_P dP$,

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: $=$. So,

\left( \right) = \left \rightP

Therefore, the first Ehrenfest equation is:

$$.

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

$$

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

$$.

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.

See also

* Paul Ehrenfest
* Clausius–Clapeyron relation
* Phase transition

References

Thermodynamic equations