Define Entropy of Ideal Gas?

A vessel of volume V₁ contains N molecules of an ideal gas held at temperature T and pressure P_1. The energy of a molecule may be written in the form         

Where ?_k denotes the energy levels corresponding to the internal states of the molecules of the gas.

a) Calculate the free energy F. Explicitly display the dependence on the volume V_1.

At this time consider another vessel, as well at temperature ?? containing the same number of molecules of the identical gas held at pressure P_2.

b) Give an phrase for the total entropy of the two gases in terms of P_1, P_2, ??, N.

c) The vessels are then connected to allow the gases to mix without doing work. Calculate explicitly the change in entropy of the system. Ensure whether your answer makes sense by considering the special case V₁₌V₂ (P₁ =P₂).