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,[1] as both specific entropy and specific volume do not change in second-order phase transitions.

## Contents

* 1 Quantitative consideration
* 2 Limitations
* 3 See also
* 4 References

## 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=(∂s∂T)PdT+(∂s∂P)TdP. As (∂s∂T)P=cPT, (∂s∂P)T=−(∂v∂T)P, then the differential of specific entropy also is:

dsi=ciPTdT−(∂vi∂T)PdP,

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: ds1=ds2. So,

(c2P−c1P)dTT=[(∂v2∂T)P−(∂v1∂T)P]dP

Therefore, the first Ehrenfest equation is:

ΔcP=T⋅Δ((∂v∂T)P)⋅dPdT.

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

ΔcV=−T⋅Δ((∂P∂T)v)⋅dvdT

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

Δ(∂v∂T)P=Δ((∂P∂T)v)⋅dvdP.

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

Δ(∂v∂T)P=−Δ((∂v∂P)T)⋅dPdT.

## 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

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

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