1. The partition function Zy for a system of N independent particles is given as ZN = (n₁V)N/N!. (i) Find the Gibbs sum Z and mean occupancy (N) for an ideal gas of identi- cal atoms. (ii) Show that the probability there are N atoms in the gas in volume V in dif- fusive contact with a reservoir is P(N): = (N) exp(-(N)) N! (iii) Calculate ((N — (N))²).