Question

Obtain ehrenfests theorem describing second order phase transition

Answer 1

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. 

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. 

The first Ehrenfest equation is:

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

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

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

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

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