How Do You Calculate Phase Transition Thermodynamics?

[Daria]

Homework Statement

The free energy per unit volume of a liquid is measured to be
FL/V=a(T)ρ2/2
Where ρ is a number density, a(T)=α/T with α a constant.
The free energy of the solid phase is
Fs/V=b(T)ρ3/3 with b(T)= β/T with β a constant.

At a given temperature, the pressure of the liquid can be increased to reach a value Ps at which the liquid solidifies. The density of the liquid is ρL at the transition, the density of the coexisting solid is ρS

1) Determine ρL, ρS and Ps as a function of temperature.
2) What is the change in the molar entropy across the transition?
3) Use Clapeyron’s relation to obtain the slope of the solidification curve. Is this consistent with the result obtained in 1) for P?

Relevant Equations

dP/dT = ΔS/ ΔV = ΔH/(TΔV)
F=U-TS
dF=-PdV-SdT

Hello! I'm struggling with this exercise for three days already. Would appreciate any help. Thanks in advance. This is my thoughts...

So a number density ρ=N/V. As I understand N doesn’t change through transition, but V changes

The free energy F=U-TS or dF=-PdV-SdT

As the liquid coexists with the solid, the liquid and solid free energy functions have the common tangent. I found derivatives for FL and Fs (I thought it is connected with a common tangent, but it didn’t help me with anything)

Clapeyron’s relation: dP/dT = ΔS/ ΔV = ΔH/(TΔV)

This is everything I came up with during the brainstorm and I don’t know if any of this is right, I don’t understand what to do next and what direction to go :’(

 

[jbsteele]

The first step is to define your system and the parameters of interest. What are you trying to solve for? Are you looking for a numerical value or an equation? Are you looking to understand the thermodynamics of a phase transition, such as melting or boiling? Once you have clearly defined your problem, you can then begin to think about the equations and tools you will need to solve it. You have identified some of the key concepts and equations that might be useful in understanding the thermodynamics of a phase transition. The number density ρ=N/V is related to the number of particles N and the volume V, and the free energy F=U-TS contains the internal energy U, temperature T, and entropy S. Clapeyron's relation connects pressure, temperature, and volume through the enthalpy ΔH. To use these equations, you need to determine the values for each of the parameters. For example, you need to know the initial temperature and pressure in order to calculate the enthalpy difference across the phase transition. You also need to know the change in volume and entropy across the transition in order to calculate the free energy. Once you have determined the values of these parameters, you can then use the equations to calculate the free energy of the system. This will allow you to understand the thermodynamics of the phase transition and the driving forces behind it.