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Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: vdemas on June 24, 2008, 06:05:05 AM

Title: Equation of state
Post by: vdemas on June 24, 2008, 06:05:05 AM
Q : Use [dU/dV]T = T [dp/dT]V - p to derive the equation mu = [T(dV/dT)p - V] / Cp.


All I have thus far is :

T = [dU/dS]V and  -p = [dU/dV]S
Title: Re: Equation of state
Post by: Hunt on July 17, 2008, 12:47:20 PM
What does "mu" represent ?
Title: Re: Equation of state
Post by: Hunt on July 18, 2008, 01:27:22 PM
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 ]