Question

3. In a second-order phase transition, there is no difference in entropy or volume between the two phases and therefore the Clausius-Clapeyron equation cannot be applied. Two Ehrenfest equations can, however, be found. They follow from the the equality in entropy or volume across the phase transition, and they relate dp/dT to differences in α and C, or K between the two phases (a) By setting dSdS2, find dp/dT in terms of the differences in a and Cp between the two phases. This is the first Ehrenfest equation. (b) By setting dVi-dV2, find dp/dT in terms of the differences in α and K between the two phases. This is the second Ehrenfest equation. Hint: Treat S as a function of p, T, write out dSi dS2 and crush the relevant derivatives (and do the same for V).]

Answer 1

Solution:

Ehrenfest equations are equations which describe changes in specific heat capacity and derivatives of specific volume in second order phase transitions.

I am doing this derivation with basic considerations and assumptions. You can manipulate it according to your given parameters.

If one considers specific entropy s as a function of temperature and pressure, then its differential is:

$ds=\left ( \partial s/\partial T \right )_{P}dT+\left ( \partial s/\partial P \right )_{T}dP$

As $\left ( \partial s/\partial T \right )_{P}=c_{P}/T,\left ( \partial s/\partial P \right )_{T}=-\left ( \partial v/\partial T \right )_{P}$ , Then the differential of specific entropy will also be:

$ds_{i}= c_{iP}/TdT-\left ( \partial v_{i} /\partial T\right )_{P}dP$ 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:

$ds_{1}=ds_{2}$ . So,

$\left ( c_{2P}-c_{1P} \right )dT/T=\left [ \left ( \partial v_{2}/\partial T \right ) _{P}-\left ( \partial v_{1}/\partial T \right )_{P}\right ]dP$

Therefore, the first Ehrenfest equation is:

$\Delta c_{P}=T.\Delta \left ( \left ( \partial v/\partial T \right )_{P} \right ).dP/dT$

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

$\Delta c_{V}=-T.\Delta \left ( \left ( \partial P/\partial T \right )_{v} \right ).dv/dT$

I hope this concept would help you.

Thank you.