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Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: vdemas on June 24, 2008, 06:05:05 AM
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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
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What does "mu" represent ?
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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 ]