# Phase transition between two phases with different Cv

Baibhab Bose
Homework Statement:
A many-body system undergoes a phase transition between two phases A and B at a
temperature ܶTc . The temperature-dependent specific heat at constant volume Cv of the two
phases are given by Cv(A)=aT^3+bT and Cv(B)=cT^3.Assuming negligible volume change of
the system, and no latent heat generated in the phase transition, ܶTc is...?
Relevant Equations:
Clausius Clapeyron equation maybe.
I actually can't figure out what kind of phase transition it is and how to proceed through..!

Gold Member
What does it mean for the order of a phase transition if there is no latent heat generated at the transition temperature Tc?

Baibhab Bose
What does it mean for the order of a phase transition if there is no latent heat generated at the transition temperature Tc?
Since 2nd order phase transition is accompanied by no heat change and 1st order transition does, that indicates this is a 2nd order transition!

Gold Member
The next question is: Is there a thermodynamic function which shows a characteristic behavior at the transition temperature ##T_c## and which can be related to the heat capacities in both phases?

Baibhab Bose
In second order phase transition, Gibbs free energy remains constant.
dG=dU-Tds-SdT+PdV+VdP=0
TdS=0 (since no heat change)
SdT=0 (process at same temp)
PdV=0('negligible volume change)
so that leaves us
dG=dU+VdP
but if we write dU=CvdT then again dT=0.
So what to do?

Gold Member
When dealing with a second order phase transition, the entropy is continuous at ##T_c## (it has a kink as the heat capacities of both phases differ from each other). See the figures on page 20 in
[PDF]
Phase Transitions and Phase Diagrams - Virginia

The temperature dependence of the entropy in both phases can be calculated as the temperature dependence of the heat capacities (##C_{V,A}(T)=aT^3+bT## and ##C_{V,B}(T)=cT^3##) in both phases are given:

##S_A (T) = \int_0^T \frac {C_{V,A}(T')} {T'} \, dT'##
##S_B (T) = \int_0^T \frac {C_{V,B}(T')} {T'} \, dT'##

(assuming ##S_A(0)=0## and ##S_B(0)=0##).

As the entropy is continuous at ##T_c##, one has ##S_A(T_c) = S_B(T_c)## . This equation can now be used to determine ##T_c##.

Last edited:
Baibhab Bose
Okay, now I have learned how to use these conditions on phase transitions mathematically to approach a problem. And I have got the answer. thank you so much!

Baibhab Bose
##S_A (T) = \int_0^T \frac {C_{V,A}(T')} {T'} \, dT'##
##S_B (T) = \int_0^T \frac {C_{V,B}(T')} {T'} \, dT'##

I have one confusion here, you have defined the entropy at Tc by integrating Cv(T)/T from 0 to Tc. I don't understand why should T=0 be taken here as the lower limit. The transitions are not taking place around T=0, and T=0 is not of any special relevance here, then why?

Gold Member
When dealing with a second order phase transition, the absolute values of the entropy of both phases are equal to each other at the phase transition point ##T_c##. To calculate the absolute entropy of a material on base of experimental heat capacity data, one needs a reference point which is taken as the entropy ##S (T)## at ##T=0K##. Under constant volume conditions, one thus gets for the absolute entropy at a certain temperature ##T##:

##S(T)-S(0) = \int_0^T \frac {C_{V}(T')} {T'} \,dT'##