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Bill_K
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#2
Apr29-11, 04:32 PM
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TobyC, you're absolutely correct! And the derivation in Wikipedia is not. Try this:

dU = T dS - P dV
(∂U/∂V)T = T (∂S/∂V)T - P

We need a Maxwell relation:

A = U - TS
dA = - S dT - P dV
⇒ (∂S/∂V)T = (∂P/∂T)V

Putting that in:
(∂U/∂V)T = T (∂P/∂T)V - P

Let u = U/V be the energy density. Then P = u/3 and ∂U/∂V = u gives you a differential equation

u = T/3 du/dT - u/3

whose solution is u = C T4