5 7.10 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 thecanonical probability distribution:exp(-BE,)P, =where B = 1/(k T), andZ =Eexp(-B E,)is the partition function.(a) Demonstrate that the entropy can be writtenS=ΣΡ In P,(b) Demonstrate that the mean Helmholtz free energy is related to the partition functionaccording toZ = exp(-8F).

It is given that, Pr=exp-βErZ...........(1)Z=∑rexp-βEr...........(2) Where, β=1kT.

7.10 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: P, exp(-BE,) where B = 1/(k T), and Z = exp(-B E,) is the partition function. (a) Demonstrate that the entropy can be written S = -k P, In P,. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF). %3D

7.10 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: P, exp(-BE,) where B = 1/(k T), and Z = exp(-B E,) is the partition function. (a) Demonstrate that the entropy can be written S = -k P, In P,. (b) Demonstrate that the mean Helmholtz free energy is related to the partition function according to Z = exp(-BF). %3D