Consider a cylinder containing an internal movable piston. Let there be N_1 molecules (not moles) to the left and N_2 molecules (not moles) to the right of the piston. Initially in the state a the piston is held fixed by an internal constraint which is then removed and the piston begins to move. We wish to find the position of the piston in the new equilibrium state b such that the entropy is maximized in this stale subject to the restrictions placed by b. Consider the interior of the cylinder to be subdivided into a very large number, M, of boxes. The microstate of the system is specified by the boxes (cells) occupied by molecules at a given time. We are not interested in spelling out the details of each microstate but simply in their number. Let the total volume of the cylinder be V_0 = V_1 + V_2. Once the internal restraint, present in the state a, has been removed, then the volumes V_1 and V_2 are no longer fixed to their initial values but can change to new values subject to the condition V_0 = V_1 + V_2 (a) For a given value of V_1 and V_2, write an expression for the number of cells to the left and right of the piston in terms of V_1, V_2, and M (total number of cells). (b) Assuming that there are no restrictions on the number of molecules that can be placed in a cell and each molecule is numbered, write an expression for the number of ways W_1 and W_2 that N_1 and N_2 molecules can be distributed in to the left and right of the piston. (c) Write an expression for the total number of ways W_T in which the molecules are distributed through the whole system. (d) Use W_T to calculate the entropy from the Boltzmann equation. (e) Maximize this entropy and show that: V_1/V_0 = N_1/N_1 + N_2 (f) Do you think that this result agrees with what would be predicted from common sense? What does it say about the validity of the Boltzmann equation?