Asymptotic homogenization: Difference between revisions

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'''[[Paul Ehrenfest|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'',<ref>Sivuhin D.V. General physics course. V.2. Thermodynamics and molecular physics. 2005</ref> as both specific [[heat capacity]] and  [[specific volume]] do not change in second-order phase transitions.


==Quantitative consideration==
Ehrenfest equations are the consequence of continuity of specific entropy <math>s</math> and specific volume <math>v</math>, which are first derivatives of specific [[Gibbs free energy]] - in second-order phase transitions. If we consider specific entropy <math>s</math> as a function of [[temperature]] and [[pressure]], then its [[differential of a function|differential]] is:
<math>ds = \left( {{{\partial s} \over {\partial T}}} \right)_P dT + \left( {{{\partial s} \over {\partial P}}} \right)_T dP</math>.
As <math>\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 differential of specific entropy also is:


<math>d {s_i}  = {{c_{i P} } \over T}dT - \left( {{{\partial v_i } \over {\partial T}}} \right)_P dP</math>
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Where <math>i=1</math> and <math>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>{ds_1} = {ds_2}</math>. So,
 
<math>\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:
 
<math>{\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 function of temperature and specific volume:
 
<math>{\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 function of <math>v</math> и <math>P</math>.
 
<math>{\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 of function of <math>T</math> and <math>P</math> gives the fourth Ehrenfest equation:
 
<math>{\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>
 
==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==
* [[Paul Ehrenfest]]
* [[Clausius–Clapeyron relation]]
* [[phase transition]]
 
==References==
 
<references />
 
[[Category:Thermodynamic equations]]

Revision as of 13:43, 7 February 2014


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