Ehrenfest equations

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.

Contents

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 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 %2B \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_P = - 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

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