# Determine Joule-Kelvin coefficient for gas given equations of state

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Homework Statement
The equations of state of a gas are ##P=\frac{U}{V}## and ##T=3B(U²/NV)^{1/3}##. Determine ##\alpha## and ##\mu##.
Relevant Equations
##U##: internal energy; ##T##: temperature; ##\mu##: Joule-Kelvin coefficient; ##B##: positive constant; ##V##: volume; ##N##: number of moles; ##\alpha##: coefficient of thermal expansion; ##P##: pressure; ##c_P##: heat capacity at constant pressure.
Hi

##\mu=\frac{\alpha TV–V}{N c_P}##. So, firstly, I have to calculate ##\alpha## and ##c_P##. So ##\alpha=\frac{1}{V} \frac{\partial V}{\partial T}## at constant ##P##. I can write ##U=PV##, then I replace it in the equation of ##T##, solve for ##V## and then I differentiate with respect to ##T##.

Then, ##c_P=\frac{T}{N} \frac{\partial S}{\partial T}## at constant ##P##. Do I have to find the fundamental equation for ##S## using Euler and Gibbs-Duhem relations, or is there an easier way?

Then, ##c_P=\frac{T}{N} \frac{\partial S}{\partial T}## at constant ##P##. Do I have to find the fundamental equation for ##S## using Euler and Gibbs-Duhem relations, or is there an easier way?
Try using ##c_P = \frac 1 N \left( \frac{dQ}{dT} \right)_P## along with the first law.

Try using ##c_P = \frac 1 N \left( \frac{dQ}{dT} \right)_P## along with the first law.
Thanks! I have arrived to ##c_P=\frac{2T^2}{9B^3P}## and ##\alpha=\frac{NT^2}{9B^3P^2V}##. But when I replace this identities in the expression for ##\mu## I get ##\mu=0##

Thanks! I have arrived to ##c_P=\frac{2T^2}{9B^3P}## and ##\alpha=\frac{NT^2}{9B^3P^2V}##.
I believe these are correct. They will simplify nicely if you use ##P=\frac{U}{V}## and ##T=3B(U^2/NV)^{1/3}## to express ##B^3## in terms of ##P##, ##V##, ##T##, and ##N##.

But when I replace this identities in the expression for ##\mu## I get ##\mu=0##
I don't get ##\mu = 0##.