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.
Quantitative consideration
[edit]
Ehrenfest equations are the consequence of continuity of specific entropy
and specific volume
, which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy
as a function of temperature and pressure, then its differential is:
.
As
, then the differential of specific entropy also is:
,
where
and
are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions:
. So,
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
and
:
.
Continuity of specific volume as a function of
and
gives the fourth Ehrenfest equation:
.
Derivatives of Gibbs free energy are not always finite. Transitions between different magnetic states of metals can't be described by Ehrenfest equations.
- ^ Sivuhin D.V. General physics course. V.2. Thermodynamics and molecular physics. 2005