Consider the system described in Exercise 4.15. The canonical partition function is                                                             Where N is the total number of molecules, and q is...

Consider the system described in Exercise 4.15. The canonical partition function is

Where N is the total number of molecules, and
q
is the Boltzmann weighted sum over all single molecule states, both those associated with isomers of type
A
and those associated with isomers of type
B.

(a)   Show that one may partition the sum and write

With

Where
is over all the partitions of N molecules into N_A
 molecules of type A and N_B
molecules of type B, q_A
is the Boltzmann weighted sum over states of isomer A, and q_B
is similarly defined.

(b)   Show that the condition of chemical equilibria is identical to finding the portioning that minimizes the Helmholtz free energy