I am wondering how one would construct the grand partition function of a composite system of solid and gas with the same chemical potential energy. I would think to begin with the partition function for a single particle and sum over it's energy states (available in the solid and the gas). Then to generalize to N particles by raising it to the Nth power. However, I run into trouble when considering whether the particles are distinguishable or not. In the gas phase they are not, where as in the solid phase they are. Thus, how would one account for the "over-counting". You certainly could not divide by simply N Factorial. Perhaps my approach is wrong altogether?