# Phase transition between two phases with different Cv

• Baibhab Bose
In summary, the conversation discusses the concept of phase transitions and how to determine the order of a transition based on the presence or absence of latent heat. It also delves into the thermodynamic functions and entropy involved in second order phase transitions, and how to mathematically approach a problem involving these conditions. The conversation ends with a clarification on the use of 0K as a reference point for calculating absolute entropy.
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..!

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

Lord Jestocost said:
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!

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?

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?

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:
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!

Lord Jestocost said:
##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?

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'##

## 1. What is a phase transition?

A phase transition is a physical or chemical process in which matter changes from one state to another, such as from solid to liquid or liquid to gas.

## 2. What is Cv in relation to phase transition?

Cv, or specific heat capacity, is a measure of the amount of heat needed to raise the temperature of a substance by 1 degree. In the context of phase transition, it represents the amount of energy needed to change the phase of a substance without changing its temperature.

## 3. How does Cv affect the phase transition between two phases?

The phase transition between two phases with different Cv occurs when there is a change in the amount of energy required to maintain the substance in its current phase. This change in energy leads to a change in the physical properties of the substance, such as density and thermal conductivity.

## 4. What is the relationship between Cv and the critical point of a phase transition?

The critical point of a phase transition is the temperature and pressure at which the two phases become indistinguishable. This point is often characterized by a sharp increase or decrease in Cv, indicating a dramatic change in the behavior of the substance.

## 5. How is the phase transition between two phases with different Cv studied?

The phase transition between two phases with different Cv can be studied through various techniques, such as differential scanning calorimetry, which measures the heat flow during a phase transition, or by observing changes in the physical properties of the substance, such as its density or refractive index.

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