Consider a system in thermal equilibrium with a heat bath held at absolute temperature T. The probability of observing the system in some state r of energy E, is is given by the canonical probability distribution: where = 1/(k T), and Pr = exp(- Er) Z z = exp(-E,) r is the partition function. (a) Demonstrate that the entropy can be written S = -kP, In P. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF).