Grand-canonical chemical potential of a classical ideal gas ($\mu =k_{B}T\ln (n{\lambda }^3)$)

For one internal species, the grand canonical partition function of a classical ideal gas satisfies $\ln \Xi =zV/{\lambda }^3$, with quantum gas fugacity $z=e^{\beta \mu }$ and thermal de Broglie wavelength $\lambda =\sqrt{2\pi {\hslash }^2/(m{k}_{B}T)}$. Differentiating the grand potential gives $N=zV/{\lambda }^3$, hence $\mu =k_{B}T\ln (n{\lambda }^3)$. An internal partition factor must be included consistently if internal states are available.