I'm supposed to derive the following thermodynamic identity:
$$
– \left( \frac{\partial U}{\partial V} \right)_{T,N} + T \left( \frac{\partial P}{\partial T} \right)_{V,N} = P
$$
…by starting with the Helmholtz free energy $F = U – TS$.
$\ $
How do I begin this problem? To me it seems like the only way I would be able to relate $U, T, S$ to the variables $P,V$ would be through the use of some kind of equation of state, but I'm not explicitly given one.
Is it enough to use the first law of thermodynamics? I'm also confused as to why I would need the Helmholtz free energy in the first place? A prod in the right direction would be much appreciated.
Best Answer
Start from:
$$dU = TdS - PdV$$
Now "divide by dV"
$$\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial S}{\partial V}\right)_T - P$$
Now you use one of the Maxwell relations:
$$\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$$
and you have the result:
$$\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial P}{\partial T}\right)_V - P$$