(4) See figure below. Two systems are brought into contact and allowed to exchange energy. We want to find the distribution of energy between the systems in equilibrium, E, and E2 . By the second law, this will be the distribution that maximizes the entropy of the combined system after contact. The maximization has to be done subject to the constraint that the total energy is fixed: E1 + E2 = E , where E is a constant. Thus, we want to maximize S1 (E1, V1, N1) + 52(E2, V2, N2) subject to the constraint E1 + E2 = E. This is can be done using Lagrange multipliers. Form the auxiliary function S with multiplier B: S(E1, E2) = S1 + 52 - B(E1 + E2 - E) and maximize $ by settingand maximize S by setting as_ as_ (We have suppressed the dependence in E- on V and N as these do not fluctuate here.) (a) Show that this leads to the equilibrium condition 631 _ 6'32 351 _ 652' This equation tells us that equilibrium will be reached when the payoff in terms of increased entropy from moving a small parcel ofenergy into 1 from 2 equals the payoff from moving a bit of energy into 2 from 1. In equilibrium, there is no net payoff in entropy terms from moving energy in one direction (towards system 1) or the other (towards system 2.} Note that if we had specified explicit functional forms for 51 and 5'2, we could compute the derivatives and find the energies E; and E5: the equilibrium condition gives us one equation and the energy constraint equation gives us the other. m m (1': Jr: in m Eir V1\"): Hill (b) Now, assume that the partition between 1 and 2 is movable and permeable, thereby allowing exchanges of volumes and particles (and energy, as before.) Show that the equilibrium conditions the conditions that maximize the entropy under appropriate constraints are 651 _ 632 & 631 _ 652 & 351 _ 632 6E1 ' 652 6V1 ' 6V2 6N1 ' 6N2' Empirically, we find that as_1 & as_P & as_ p 613'? av'r 6N__T' The equilibrium conditions (fora simple fluid) can then be expressed as T1:T2 3' P1:P2 31 #1:#2