Ehrenfest equations

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Ehrenfest equations — equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions. Clausius–Clapeyron relation does not make sense for second-order phase transitions,[1] as both specific heat capacity and specific volume do not change in second-order phase transitions.

Quantitative consideration[edit]

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 we consider specific entropy s as a function of temperature and pressure, then its differential is: ds = \left( {{{\partial s} \over {\partial T}}} \right)_P dT + \left( {{{\partial s} \over {\partial P}}} \right)_T dP. As \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 , then differential of specific entropy also is:

d {s_i}  = {{c_{i P} } \over T}dT - \left( {{{\partial v_i } \over {\partial T}}} \right)_P dP

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: {ds_1} = {ds_2}. So,

\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

Therefore, the first Ehrenfest equation:

{\Delta c_P  = T \cdot \Delta \left( {\left( {{{\partial v} \over {\partial T}}} \right)_P } \right) \cdot {{dP} \over {dT}}}

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

{\Delta c_V  =  - T \cdot \Delta \left( {\left( {{{\partial P} \over {\partial T}}} \right)_v } \right) \cdot {{dv} \over {dT}}}

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

{\Delta \left( {{{\partial v} \over {\partial T}}} \right)_P  = \Delta \left( {\left( {{{\partial P} \over {\partial T}}} \right)_v } \right) \cdot {{dv} \over {dP}}}

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

{\Delta \left( {{{\partial v} \over {\partial T}}} \right)_P  =  - \Delta \left( {\left( {{{\partial v} \over {\partial P}}} \right)_T } \right) \cdot {{dP} \over {dT}}}

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