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 [math]\displaystyle{ s }[/math] and specific volume [math]\displaystyle{ v }[/math], which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy [math]\displaystyle{ s }[/math] as a function of temperature and pressure, then its differential is: [math]\displaystyle{ ds = \left( {{{\partial s} \over {\partial T}}} \right)_P dT + \left( {{{\partial s} \over {\partial P}}} \right)_T dP }[/math]. As [math]\displaystyle{ \left( {{{\partial s} \over {\partial T}}} \right)_P = {{c_P } \over T} , \left( {{{\partial s} \over {\partial P}}} \right)_T = - \left( {{{\partial v} \over {\partial T}}} \right)_P }[/math], then the differential of specific entropy also is:

[math]\displaystyle{ d {s_i} = {{c_{i P} } \over T}dT - \left( {{{\partial v_i } \over {\partial T}}} \right)_P dP }[/math],

where [math]\displaystyle{ i=1 }[/math] and [math]\displaystyle{ i=2 }[/math] are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions: [math]\displaystyle{ {ds_1} = {ds_2} }[/math]. So,

[math]\displaystyle{ \left( {c_{2P} - c_{1P} } \right){{dT} \over T} = \left[ {\left( {{{\partial v_2 } \over {\partial T}}} \right)_P - \left( {{{\partial v_1 } \over {\partial T}}} \right)_P } \right]dP }[/math]

Therefore, the first Ehrenfest equation is:

[math]\displaystyle{ {\Delta c_P = T \cdot \Delta \left( {\left( {{{\partial v} \over {\partial T}}} \right)_P } \right) \cdot {{dP} \over {dT}}} }[/math].

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

[math]\displaystyle{ {\Delta c_V = - T \cdot \Delta \left( {\left( {{{\partial P} \over {\partial T}}} \right)_v } \right) \cdot {{dv} \over {dT}}} }[/math]

The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of [math]\displaystyle{ v }[/math] and [math]\displaystyle{ P }[/math]:

[math]\displaystyle{ {\Delta \left( {{{\partial v} \over {\partial T}}} \right)_P = \Delta \left( {\left( {{{\partial P} \over {\partial T}}} \right)_v } \right) \cdot {{dv} \over {dP}}} }[/math].

Continuity of specific volume as a function of [math]\displaystyle{ T }[/math] and [math]\displaystyle{ P }[/math] gives the fourth Ehrenfest equation:

[math]\displaystyle{ {\Delta \left( {{{\partial v} \over {\partial T}}} \right)_P = - \Delta \left( {\left( {{{\partial v} \over {\partial P}}} \right)_T } \right) \cdot {{dP} \over {dT}}} }[/math].

## 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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