How do I solve this?

Before the invention of Quantum Mechanics, thermal physicists had derived the total partition function for an ideal gas of N particles of mass m at temperature T in terms of the single particle partition function as ZT : 21'" where Z; is the total partition function of the system. They also had gured out the correct formula for Z1 : VnQ with HQ : (mar/27mg)? While they had no clue what it: was: the formula for 21 was correct. But (1) they didn't initially include the N! in the denominator of ZT because they didn't appreciate the identical particle issue. W'e'll investige the repercussions of this omission below. (a) (3 points) Using the above formula for ZT (without the NE). calculate an expression for the entropy of an ideal gas. We will use this expression for part (b) (b) (3 points) Now consider the following simple experiment. We start with an ideal gas of N particles in volume V at fundamental temperature T. The entropy for this is obviously just what you derived in part (a). Now install a barrier right down the middle of the box so that we now have two identical boxes with N1 : N2 : N/2 and V] : V2 : V/2 with both at temperature 7'. Find the total entropy of this system by adding the entropies of the two boxes and compare to the initial entropy before installing the barrier, using your result from part (a). Show that this change in entropy violates the 2'\" law of thermodynamics (Yikesl). (c) (3 points) Now include the N l and you will get the formula that we derived in class (and text). Use this correct formula to again calculate the total change in entropy after adding the barrier and show that the 2\"'1 law is not violated. (W'hewl). (d) (1 point) What does the result that you got for part (c) tell you about the process of installing the barrier