Grand potential

Remarks

From $dU = T \, dS - p \, dV + \mu \, dN$ one gets $d\Omega = -S \, dT - p \, dV - N \, d\mu$, where $p$ is the pressure and $V$ the volume. So the natural variables of $\Omega$ are $T$, $V$ and $\mu$, and $N = -(\partial \Omega / \partial \mu)_{T, V}$. For a homogeneous system, the Euler relation $U = TS - pV + \mu N$ gives $\Omega = -pV$. Example: for a classical ideal gas, $\Omega = -pV = -NkT$, where $k$ is the Boltzmann constant. Non-example: $F + \mu N$ is not the grand potential, since the sign of $\mu N$ is wrong and its natural variables would not be $T, V, \mu$.