The partition function ZN 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). - (iv) Suggest a simple experimental procedure for the determination of (N). (v) A donor level in a semiconductor can be either empty (ionised) or occupied by an electron with spin-up or spin-down. When the level is occupied, its binding energy is -I. Calculate the grand partition function of the donor atom and the probability that the donor is ionised.