A surface with N0 adsorption centers has N(≤N0) gas molecules adsorbed on it. Show that the chemical potential of the adsorbed molecules is given by
μ=kTln(N0−N)a(T)N,
where a(T) is the partition function of a single adsorbed molecule. Solve the problem by constructing the grand partition function as well as the partition function of the system.
[Neglect the intermolecular interaction among the adsorbed molecules.]