A system has two normal modes of vibration, with frequencies @, and @₂ = 2w₁ . What is the probability that at temperature T, the system has an energy less than 4ħw, ? [In the following x = e-hand Z is the partition function of the system.] 3/2 (a) x³/²(x+2x²)/ Z (b) x³/² (1+x+x²) / Z 3/2 3/2 (c) x ³/² (1+2x²)/Z (d) x³/² (1+x+2x²)/Z

•2.6 (c)-(f) • (c) Show that the grand canonical partition function of an ideal gas can be written as Zg = exp expleßu 23 (d) Use the expression for Zg to calculate the mean value for N, P, S, and show that PV = NkgT and that the Gibbs-Duhem equation gives E = ?NkgT.

Suppose that ak > 0 for all k e N and E ak < 0. For each of the following, prove that the given series converges. ak ( a ) ΣΕΙ 1+ k3ak (b) Lk=1 1+ ak Vak (c) Ek=1 k < 0o.

The partition funetion for the ensemble characterized by constant V, E, and G = µÑ is given to a very good approximation by ø(V, E, µN)=Q(N,V,E)eBHN, where G = µN is the Gibbs energy (µ is the chemical potential and N is the average number of particles). Find an expression for the characteristic thermodynamic function for this ensemble in terms of the partition function ø(V, E, µN).