Consider the mixing of two gases. We have a box of total volume 2V divided by a non-insulating wall evenly into two chambers each with volume V. The chambers are in thermal contact with each other and the whole box is in thermal contact with a reservoir at temperature T. The left chamber contains NA particles of an ideal gas of species A and the right chamber contains an equal number NB particles of an ideal gas of a different species B. At some point we remove the wall dividing the two chambers. (a) Find the partition function for this system both before and after the wall is removed. Then determine the change in entropy between the two configurations (having the wall vs. not having the wall). [Note: You will need to use Stirling's approximation. You do not need to keep track of the factors of V2TN for this part but you will need it for the next part.] Hint (highlight to reveal): [Don't overthink the first part! You will use Eq. 1 and the result from Problem 7.2(a) .] Hint (highlight to reveal): [The most important feature here is that Z1 x V.] Hint (highlight to reveal): [To get the entropy first find the Helmholtz free energy.] (b) Repeat the previous part but now assume that both sides of the box contain the same species of particle. Hint (highlight to reveal): [You will need to use the factor of V27N in Stirling's approximation to get a non-zero result.] Hint (highlight to reveal): [After removal, all 2/ particles are now indistinguishable.] [Supplementary Part (Not for Credit): Try to explain the physical origin and implications of this (tiny) increase in entropy.] Commentary: The additional entropy generated in the above two parts is called the entropy of mixing