could please solve these problems including explanation so I can understand the process?

(2) Starting with the expression m (S) = -KB M pi In pi i=1 show that F(T) = -kBTInZ and e -BF(T) = M e-BE i=1 Also, establish the relation e-BF(T ) = e-BF(E). ALL E Note that F(E) is the generalized Helmholtz free energy while F(T) is the Helmholtz free energy.(4) A polymer is a molecule consisting of chemical units called monomers covalently linked together to form a chain. Consider such a polymer in a solvent at temperature T. The angles between successive monomers are free to fluctuate so the chain takes a contorted shape that keeps fluctuating due to the random buffeting of the chain by packets of thermal energy of size kBT, which are always present in the solution in thermal equilibrium. The exchange of a k T of energy between the chain and the solvent is sufficient to randomly reorient the bond connecting two neighboring monomers. Many such random events will deform the chain producing a continually changing zig-zag type of chain conformation. The overall size of the chain / will fluctuate and we want to determine its most likely value R in equilibrium. The Flory free energy gives the free energy of a polymer chain of size r and is given by F(T,r) 372 WN2 KBT 2 Nb2 + 7-3 (a) What are the units of F? Here, w and b are constants. What are the units of w and b? (b) Show that 1, 2 R = W565NS. (5) Find the distribution over {0,1,2, ..., /} with maximum Shannon entropy. Go through each step of the calculation. Solve explicitly for the Lagrange multiplier in terms of N