Final temperature of a gas passed through a porous plug

[Msilva]

Homework Statement

A gas has the following equations of state: $$P=\frac{U}{V} \,\,\,and \,\,\,T=3B\frac{U^{\frac{2}{3}}}{N^\frac{1}{3}V^\frac{1}{3}}$$
where B is a positive constant. The system obeys the Nernst Postulate (S tends to zero as T tends to zero). The gas, at a initial temperature $T_i$ and initial pressure $P_i$, is passed through a porous plug in a Joule-Thomson Process. The final pressure is $P_f$. Calculate the final temperature.

This question is from Callen, Thermodynamics (1985). Question 6.3-2

Homework Equations

The fundamental equation of Joule-Thomson effect is $dT=\frac{v}{c_p}(T\alpha-1)dP$ (*)
Alpha is the coefficient of thermal expansion and Cp is the heat capacity at constant pressure.
I think that may be useful know the differential of enthalpy $dH=TdS+VdP$ (assuming N constant)

The Attempt at a Solution

I tried to use the equations of state and write $T=3B(\frac{P^2V}{N})^\frac{1}{3}$
Then, isolating V in this expression, I tried to find alpha: $\alpha=\frac{1}{v} \frac{\partial v}{\partial T}$ and replacing theese two values in the expression (*). My problem is that I didn't find the Cp.
Is my logic right? I can conclude by this way or am I completely wrong?

 

[TSny]

You can work it this way, but as you noticed, it can take some work to get a useful expression for C_p.

Instead, you can try using what you know about enthalpy for the porous plug experiment. Can you use your equations of state to express enthalpy as a function of P and T?