Helmholtz free energy

Remarks

The notation $A$ is also common. From $dU = T \, dS - p \, dV$ one gets $dF = -S \, dT - p \, dV$, where $p$ is the pressure and $V$ the volume. So the natural variables of $F$ are $T$ and $V$, and $S = -(\partial F / \partial T)_V$, $p = -(\partial F / \partial V)_T$. Example: for an ideal gas of $n$ moles, $F(T, V) = f(T) - nRT \ln V$ for some function $f$, and indeed $-(\partial F / \partial V)_T = nRT / V = p$. Non-example: $U - TV$ is not the Helmholtz free energy; the product must be of the conjugate pair $T$ and $S$, and $U - TV$ has no thermodynamic meaning.