Ehrenfest equations

Ehrenfest equations (named after Paul Ehrenfest) are equations which describe changes in specific heat capacity and derivatives of specific volume in second-order phase transitions. The Clausius–Clapeyron relation does not make sense for second-order phase transitions,^[1] as both specific entropy and specific volume do not change in second-order phase transitions.

Ehrenfest equations are the consequence of continuity of specific entropy ${\displaystyle s}$ and specific volume ${\displaystyle v}$, which are first derivatives of specific Gibbs free energy – in second-order phase transitions. If one considers specific entropy ${\displaystyle s}$ as a function of temperature and pressure, then its differential is: $${\displaystyle ds={\left(\frac{\partial s}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP.}$$ As ${\displaystyle {\left(\frac{\partial s}{\partial T}\right)}_P=\frac{c_P}{T}}$, ${\displaystyle {\left(\frac{\partial s}{\partial P}\right)}_T=-{\left(\frac{\partial v}{\partial T}\right)}_P}$, then the differential of specific entropy also is:

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

where ${\displaystyle i=1}$ and ${\displaystyle i=2}$ are the two phases which transit one into other. Due to continuity of specific entropy, the following holds in second-order phase transitions: ${\displaystyle d{s}_1}=d{s}_2$. So,

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

Therefore, the first Ehrenfest equation is:

$${\displaystyle \mathrm{\Delta }{c}_P=T\cdot \mathrm{\Delta }\left({\left(\frac{\partial v}{\partial T}\right)}_P\right)\cdot \frac{dP}{dT}}.$$

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

$${\displaystyle \mathrm{\Delta }{c}_V=-T\cdot \mathrm{\Delta }\left({\left(\frac{\partial P}{\partial T}\right)}_v\right)\cdot \frac{dv}{dT}}$$

The third Ehrenfest equation is got in a like manner, but specific entropy is considered as a function of ${\displaystyle v}$ and ${\displaystyle P}$:

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

Continuity of specific volume as a function of ${\displaystyle T}$ and ${\displaystyle P}$ gives the fourth Ehrenfest equation:

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

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

• Paul Ehrenfest
• Clausius–Clapeyron relation
• Phase transition

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