Question

Consider a gas of N molecules of mass m, occupying a volume V at temperature T and characterized by a fugacity f(T, V, N)--יג, where-2TmRT bound to a surface that has a total of No. The partition function of the bound gas molecules is Ž(T) At equilibrium, some of these molecules are a and only depends on T since gas molecules are bound to the surface. Using the grand canonical ensemble, find the ave to zero and the temperature is high.

Answer 1

Single particle grand canonical partition function Z = Sigma_r, s re^-beta (E_s - mu N_r) = Sigma f^Nr Z_1 (beta) where z_1 (beta) is the single particle canonical partition function, f is fugacity As particles are indistinguishable Z = Sigma_r f^Nr (Z_1)^Nr/Nr! = Sigma (fZ)^Nr/Nr! = e^fZ_1 As f = N lambda^3/V, lambda = z_n/squareroot 2 pi m k_B T Z = e^fz_1 Z_1 = v/h^3 integral_0^infinity e^-p^2/2mk_B T. 4 pi p^2 dp = v (2 pi k_B T/h^2)^3/2 [For ideal gas] Now, langle N rangle = f partial differential/partial differential f ln z [Because ln z = fz_1 and partial differential/partial differential d lnz = z_1 and f partial differential/partial differential f ln z = fz_1 = N lambda^3/v.v/lambda^3 = v] langle N rangle = e^my beta V/lambda^3 Therefore partial differential langle N rangle/partial differential mu = PV/(k_B T)^2 Therefore (Delta N)^2 = langle N rangle = > Delta N^2 = (squareroot N)^2 Let P_L = probability first i-th can is filled by 3 molecules and