[vdemas]

Q : Use [dU/dV]_T = T [dp/dT]_V - p to derive the equation mu = [T(dV/dT)_p - V] / C_p.


All I have thus far is :

T = [dU/dS]_V and  -p = [dU/dV]_S

[Hunt]

What does "mu" represent ?

[Hunt]

I suppose "mu" is the Joule-Thomson coefficient.

The key is to express the differential of H as a function of "mu" .

dH = (dH / dP) dP + (dH / dT) dT

Euler's chain rule : (dH / dP)(dP / dT )( dT / dH ) = -1

mu = - (dH / dT)(dP / dH)


dH = - u Cp dP + Cp dT

This is a standard result.

H= U + PV

Then , dH = dU + PdV + VdP

- u Cp dP + Cp dT = dU + PdV + VdP

Divide by dV keeping T constant :

- u Cp (dP/dV)_T = (dU/dV)_T + P + V(dP/dV)_T

Substitute from the given the value of (dU/dV)_T to get :

- mu Cp (dP/dV)_T = T (dP/dT)_V + V (dP/dV)_T


mu = - 1 / Cp [ T (dP/dT)_V (dV/dP)_T + V ]

Using euler's chain rule for P,V, and T :

(dP/dT)_V  (dV/dP)_T = - (dV / dT)_P


mu = 1 / Cp [ T (dV/dT)_P - V ]