Asymptotic homogenization

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, 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: $ds={\left(\frac{\partial s}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$. As ${\left(\frac{\partial s}{\partial T}\right)}_P=\frac{c_P}{T},{\left(\frac{\partial s}{\partial P}\right)}_T=-{\left(\frac{\partial v}{\partial T}\right)}_P$, then differential of specific entropy also is:

$d{s}_i=\frac{c_{iP}}{T}dT-{\left(\frac{\partial v_i}{\partial T}\right)}_{P}dP$

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: $d{s}_1=d{s}_2$. So,

$\left(c_{2P}-c_{1P}\right)\frac{dT}{T}=\left[{\left(\frac{\partial v_2}{\partial T}\right)}_P-{\left(\frac{\partial v_1}{\partial T}\right)}_P\right]dP$

Therefore, the first Ehrenfest equation:

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

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

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

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

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

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

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

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

• Paul Ehrenfest
• Clausius–Clapeyron relation
• phase transition