Ehrenfest equations

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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[according to whom?] 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 и :

.

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

.

Application[edit]

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[edit]

References[edit]

  1. ^ Sivuhin D.V. General physics course. V.2. Thermodynamics and molecular physics. 2005