Physics: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 for second-order phase transitions,[1] 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 [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

References

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