Discussion
Which of the following relationship between the internal energy \(U\) and the Helmholtz’s free energy \(F\) is true?
Answer: A
The correct relation is \(U = -T^{2}\left[\frac{\partial (F/T)}{\partial T}\right]_{V}\), the Gibbs-Helmholtz equation.
Since \(F = U - TS\) and \(S = -(\partial F/\partial T)_{V}\), we have \(U = F - T(\partial F/\partial T)_{V}\). Now differentiate \(F/T\):
\[ \begin{aligned} \frac{\partial (F/T)}{\partial T} &= \frac{1}{T}\frac{\partial F}{\partial T} - \frac{F}{T^{2}} \\ -T^{2}\frac{\partial (F/T)}{\partial T} &= F - T\frac{\partial F}{\partial T} = U \end{aligned} \]
The '+' version has the wrong sign. \(\pm T(\partial F/\partial T)_{V}\) equals \(\mp TS\), which is only the entropy term, not the internal energy.
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