The Joule coefficient, μJ, is defined as μ = (∂T/∂V)U. Show that μJ CV = p – αT/κ Relevant equations Cv=(∂U/∂T)v κT=-1/V*(∂V/∂p)T α=1/V*(∂V/∂T)p dV(T,p)=(∂V/∂T)pdt+(∂V/∂p)Tdp What I have attempted was to make μ = (∂T/∂V)U=(∂T/∂U)V(∂U/∂V)T=(∂U/∂V)T*Cv-1. Then α/κT=(∂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 μJ*Cv = (∂U/∂V)T (because Cv and Cv-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?