Consider a system of n distinguishable particles, each of which has two energy levels, e and e (a) (2 points) evaluate the partition function z(β,n) and the free energy f(t n). (b) (2 points) evaluate the entropy s(t, n) and its limit for e = 0, so(t, n). (c) (6 points) parameterize the states by n and n+ (the number of particles with e- and evaluate the grand partition function 2(b, using binomial expansion and geometric progression. don't forget to account for the degeneracy of states with given n and n (d) (1+2+2 points) using the result of (c), evaluate the grand potential φ(t, μ) and s(t, μ), n(t, μ). write the expressions for so(t, p) and no(t, μ) corresponding to (-0. ) (6 points) to determine the chemical potential μ = μ(t, n) for ε = 0, one should set no(t, μ) to be equal to n and solve for μ. the resulting expression is very complicated, but in thermodinamic lit (n1) one should only keep the first two terms in the expansion: μ(n, t)-co(t) + c (22 + 0(n-2) start with expression no(t,μ) found in part (d), set no(t, a) resulting equation to determine co(t) and ci(t) n, and solve the (f) (4 points) substitute the result for μ(n, t) found in part (e) into (t, μ) found in part (d), keeping only the terms which survive in the large nliit i. e. dropping the terms which scale as 1/n, but keeping terms which are linear in n as we as all constants). make sure that your result, so(n, t), agrees with part (b).