# What is the sign of chemical potential of a noninteracting classical ideal gas obeying MB distribution?

The chemical potential of a noninteracting Bose gas can never be negative while that of a noninteracting Fermi gas can be both positive or negative. What can be said about the chemical potential of noninteracting classical ideal gas obeying MB distribution?

The simplest way to compute this is through the grand canonical ensemble. The partition function for a single gas molecule is $$Z_1 = V/\lambda^3$$, where $$\lambda$$ is the thermal de Broglie wavelength. Then the grand partition function is $$\mathcal{Z} = \sum_N e^{\beta \mu N} \frac{Z_1^N}{N!} = \exp\left( \frac{e^{\beta \mu} V}{\lambda^3}\right).$$ The particle number is found by differentiating, $$N = \frac{1}{\beta} \frac{\partial \log \mathcal{Z}}{\partial \mu} =\frac{e^{\beta \mu} V}{\lambda^3}.$$ Therefore, solving for $$\mu$$, we have $$\mu = k_B T \log \frac{\lambda^3 N}{V}.$$ The ideal gas only behaves classically if the occupancy of each state is small, so we must have $$\lambda^3 \ll V/N$$. Then the logarithm is negative, so for a classical ideal gas $$\mu < 0$$.