Asymptotic homogenization: Difference between revisions

Revision as of 08:27, 28 January 2014

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

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: $d s = {(\frac{\partial s}{\partial T})}_{P} d T + {(\frac{\partial s}{\partial P})}_{T} d P$ . As ${(\frac{\partial s}{\partial T})}_{P} = \frac{c_{P}}{T}, {(\frac{\partial s}{\partial P})}_{T} = - {(\frac{\partial v}{\partial T})}_{P}$ , then differential of specific entropy also is:

$d s_{i} = \frac{c_{i P}}{T} d T - {(\frac{\partial v_{i}}{\partial T})}_{P} d P$

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: $d s_{1} = d s_{2}$ . So,

$(c_{2 P} - c_{1 P}) \frac{d T}{T} = [{(\frac{\partial v_{2}}{\partial T})}_{P} - {(\frac{\partial v_{1}}{\partial T})}_{P}] d P$

Therefore, the first Ehrenfest equation:

$Δ c_{P} = T \cdot Δ ({(\frac{\partial v}{\partial T})}_{P}) \cdot \frac{d P}{d T}$

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

$Δ c_{V} = - T \cdot Δ ({(\frac{\partial P}{\partial T})}_{v}) \cdot \frac{d v}{d T}$

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

$Δ {(\frac{\partial v}{\partial T})}_{P} = Δ ({(\frac{\partial P}{\partial T})}_{v}) \cdot \frac{d v}{d P}$

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

$Δ {(\frac{\partial v}{\partial T})}_{P} = - Δ ({(\frac{\partial v}{\partial P})}_{T}) \cdot \frac{d P}{d T}$

Application

Derivatives of Gibbs free energy are not always finite. Transitions between different magnetic states of metals can't be described by Ehrenfest equations.

References

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

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

[1]

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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>
+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]]

Asymptotic homogenization: Difference between revisions

Revision as of 08:27, 28 January 2014

Contents

Quantitative consideration

Application

See also

References

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Asymptotic homogenization: Difference between revisions

Revision as of 08:27, 28 January 2014

Quantitative consideration

Application

See also

References

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