Can The Joule Coefficient be Derived Using the First Law and Maxwell Relation?

[lee403]

The Joule coefficient, μ_J, is defined as μ = (∂T/∂V)_U. Show that μJ CV = p – αT/κ

Relevant equations
C_v=(∂U/∂T)_v
κ_T=-1/V*(∂V/∂p)_T
α=1/V*(∂V/∂T)_p
dV(T,p)=(∂V/∂T)_pdt+(∂V/∂p)_TdpWhat I have attempted was to make μ = (∂T/∂V)_U=(∂T/∂U)_V(∂U/∂V)_T=(∂U/∂V)_T*C_v^-1.

Then α/κ_T=(∂V/∂T)_p/(∂V/∂p)_T because the 1/V cancels.

Since (∂V/∂T)_p=(∂V/∂p)_T *(∂p/∂T)_V
p – αT/κ becomes p-T(∂p/∂T)_V

so μ_J*C_v = (∂U/∂V)_T (because C_v and C_v^-1 cancel)
(∂U/∂V)_T= p-T(∂p/∂T)_V

I try to make the assumption that it is an ideal gas so (∂p/∂T)_V becomes nR/V
so then p-T(nR/V)= p-p= 0. But that would mean (∂U/∂V)_T = 0.

I don't know that this statement is true or if it is safe to assume an ideal gas. I am afraid that maybe I missed a negative or that I used properties of partial differentials incorrectly. Can anyone provide assistance?

 

[TSny]

Hello. Welcome to PF.

You don't want to assume an ideal gas.

You've gotten to μ = (∂U/∂V)_T*Cv^-1. So, you need to deal with the factor (∂U/∂V)_T. The first law will be helpful. Then, maybe use a Maxwell relation.