1. (15 pts.) One mole of argon (an ideal gas) is compressed isothermally from 1.0 x 10-2m3 and 2 atm to 16 atm. Calculate the amount of heat flowing out of the gas during this process. Cy = 3R/2. 2. (30 pts.) Consider the following fundamental relation: S M2 U(S, Mx, N) = NRTo exp NR + N2 Here, To, R, and Co are constants and M, is the magnetic moment in the x-direction. Remembering that the expression for magnetic work is oWmag = HxdMy, write the expressions for the following (we are looking for more than the definition!): a) (10 pts.) Any two equations of state b) (10 pts.) The Helmholtz Free Energy, F(T, Mx, N) c) (10 pts.) The isothermal magnetic susceptibility, XT = (aHx/ T OMx 3. (35 pts.) Consider a surface with area A and surface tension o, onto which a number of particles N can adsorb, in equilibrium with a particle reservoir at a chemical potential u. The expression for area-surface tension work is SWs = odA. a) (5 pts.) If you are in equilibrium with a particle reservoir, and are at constant area and temperature, write an expression for the potential I that is extremized in equilibrium and its differential form dr. b) (20 pts.) You are interested the surface tension changes upon undergoing an isothermal process where you rapidly increase the area of the surface, such that exchange of the adsorbed species with the reservoir is slow and thus negligible. You have access to the following material properties: and = a (partial molar area) (@o) = XA (from experiment) A aN = BA Ce-Bu A do L(1+Ce-Bu) (from a stat mech adsorption model) Here, do, C, a, XA and S are constants. Determine an expression for the change in surface tension (from on to of) upon changing the surface area from An to A1. c) (5 pts.) Write a Clapeyron equation that describes the phase boundary in the u-o plane. d) (5 pts.) Sketch this phase boundary if the characteristics of the two phases are:a: Large area, not many particles adsorbed ,8: Small area, many particles adsorbed 4. [20 pts.] a] [5 pts.] The first postulate is: "There exist particular states [called equilibrium states] of simple systems that, macroscopically, are characterized completely by the internal energy U, the volume V, and the mole numbers N1, N2, ofthe chemical components." Describe briey how this postulate leads to the first law of thermodynamics. b] [6 pts.] Consider a system with two subsystems, separated by a semipermeable membrane as shown below: There are three different types of particles (A, B, and C] in the two compartments [a and ,8] with the membrane being permeable to A and B but not C. Derive the equilibrium condition for this system. c] [3 pts. each] A system containing one mol of a non-ideal gas can be expanded through two processes, from an initial T1 and V1 to a nal T2 and V2 one reversible [A] and one irreversible [B]. i] Is the entropy change of the system in process A greater than, less than, or equal to the entropy change of the system in process E. ii] ls the entropy change of the surroundings in process A greater than, less than, or equal to the entropy change of the system in process B. iii] Can you say if the entropy in the nal state [2] is greater than the initial state [1] for process A