In this problem, we show that for an isolated system, equilibrium is attained by maximizing the entropy (this is the second law of thermodynamics and is equivalent to stating that equilibrium is attained by minimizing the free energy, assuming constant energy). Imagine a box that is isolated from the external environment, meaning that the total energy of the contents within the box is constant. The box contains a partition that divides it into two subspaces. The partition is constructed such that it can slide freely along the length of the box, similar to a plunger in a syringe; however, it does not allow energy to transfer between the two subspaces. We now fill both sides of the partition with some amount of gas. The partition slides around until it comes to rest at its equilibrium position. How would you expect the pressures to be related at equilibrium? Now formalize this intuition, by showing that entropy is maximized when the pressures are equal. Hint: we know that the total entropy of the system is Stot=S1(N1,V1,E1)+S2(N2,V2,E2) You will need the thermodynamic definition of pressure to complete the derivation, Tp=(VS)E,N