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I want to know how to derive pressure $p$ from a partial differential of Helmholtz free energy $F$. I want to start from

$F(T, V, N) = U(S(T,V,N),V, N) - TS(T, V, N)$,

where $U$ and $V$ are internal energy and volume, respectively. Then partial differentiate it and get,

$\left(\frac{\partial F}{\partial V}\right)_{T,N} = \left(\frac{\partial U}{\partial V}\right)_{T,N} - T\left(\frac{\partial S}{\partial V}\right)_{T,N}$.

The LHS should be the $p(T,V,N)$. But I stuck here. Because $\left(\frac{\partial U}{\partial V}\right)_{T,N}$ and $\left(\frac{\partial S}{\partial V}\right)_{T,N}$ is not allowed to express any thermodynamic quantities. (Of course if we fixed $S$ instead of $T$, $\left(\frac{\partial U}{\partial V}\right)_{S,N} = -p(S, V, N)$.) How to go forward?

Thanks in advance!

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  • $\begingroup$ are you looking for something like $dF = dU- d(TS) = dU -TdS -SdT \implies dF=TdS-PdV-TdS-SdT $ (As $dU = TdS - PdV$) $\implies dF=-PdV-SdT \implies \frac{\mathrm{dF} }{\mathrm{d} V} = -P $ $\endgroup$
    – Gowtham
    Feb 2, 2015 at 7:49
  • $\begingroup$ Thanks Gowtham. That is obviously correct. But I rather want to know how does $(\partial U/\partial V)_{T,N}$ and $(\partial S/\partial V)_{T,N}$ transform. $\endgroup$
    – rhtica
    Feb 2, 2015 at 8:51
  • $\begingroup$ $\left(\frac{\partial F}{\partial V}\right)_{T, N} = \left(\frac{\partial U}{\partial S}\right)_{V, N} \left(\frac{\partial S}{\partial V}\right)_{T, N} + \left(\frac{\partial U}{\partial V}\right)_{S, N} - T\left(\frac{\partial S}{\partial V}\right)_{T, N}$, right? See also physics.stackexchange.com/questions/148724/…, "Question 4" $\endgroup$
    – alarge
    Feb 2, 2015 at 9:28

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