(e) Using the characteristic function, show thata(N)(N²). = k„T(f) Show that fluctuations in the number of adsorbed particles satisfy(N*),(N)?1-fNf 9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst.(a) Show that the chemical potential of the gas particles is related to their temperatureand pressure via µ=k,T[In (P/T*2) + A], where A, is a constant.(b) If there are N distinct adsorption sites on the surface, and each adsorbed particlegains an energy e upon adsorption, calculate the grand partition function for thetwo-dimensional gas with a chemical potential u.(c) In equilibrium, the gas and surface particles are at the same temperature and chemicalpotential. Show that the fraction of occupied surface sites is then given by f(T, P) =P/(P+Po(T)). Find Po(T).(d) In the grand canonical ensemble, the particle number N is a random variable. Cal-culate its characteristic function (exp(-ikN)) in terms of Q(Bu), and hence showthat(N"), =-(kgT)"-1where G is the grand potential.

Since we answer up to three subparts we will answer the first three. Please resubmit the question…

9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst (a) Show that the chemical potential of the gas particles is related to their temperature and pressure via μ = k&T [În (P/T³/²) + A₁], where A, is a constant. (b) If there are N distinct adsorption sites on the surface, and each adsorbed particle gains an energy & upon adsorption, calculate the grand partition function for the two-dimensional gas with a chemical potential μ. (c) In equilibrium, the gas and surface particles are at the same temperature and chemica potential. Show that the fraction of occupied surface sites is then given by f(T, P) = P/(P+Po(T)). Find Po(T).

9. Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst (a) Show that the chemical potential of the gas particles is related to their temperature and pressure via μ = k&T [În (P/T³/²) + A₁], where A, is a constant. (b) If there are N distinct adsorption sites on the surface, and each adsorbed particle gains an energy & upon adsorption, calculate the grand partition function for the two-dimensional gas with a chemical potential μ. (c) In equilibrium, the gas and surface particles are at the same temperature and chemica potential. Show that the fraction of occupied surface sites is then given by f(T, P) = P/(P+Po(T)). Find Po(T).