Question

7. (Denbigh, ch. 9) Consider a two-component "regular solution:" RT log 71 = b(p)x2, RT log 72 = b(p)x2, (a) Verify that the activity coefficients satisfy the Gibbs-Duhem equation. (b) Show that Asmix is the same as for an ideal solution, and show that Ahmix = b(p)x1x2, unlike an ideal solution which has Ahmix = 0. b/RT=4 ₡1 Apmix 02 0.4 0.6 0.8 O RT b/RT=2 0.5 b/RT=0 -1.0- b/RT =- 2 02 0.4 0.6 0.8 -0.05- locally unstable b/RT=2.5 Aplmix RT globally unstable -0.15 -0.20 b/RT=2 -0.25 -0.30- b/RT=1.5 Figure 2: Molar Gibb's free energy of mixing versus composition for different values of B = b/RT. Wide range of B = b/RT (top graph); narrower range of B = b/RT closer to stability transition (bottom graph). -0.10 T2 500 480 460 440 T 420 400 Tı 380 360 0.0 0.2 0.4 0.6 0.8 1.0 21 Figure 3: Binary phase diagram showing miscible liquid phase and immiscible solid phase. (c) Calculate the molar Gibbs free energy of mixing Aftmix. A graph Aftmix is provided in figure 2. Show that the boundary of the two-phase region for T, P fixed is 2Bx1(1-21)=1, where B = b/RT. Note that a single phase exists for 2Bx1(1 - 21) < 1; two phases coexist for 2Bx1(1 -21) > 1. 8. (Denbigh ch. 9) Consider a binary mixture that has an azeotrope. Show that T is an extremum (minimum or maximum) under constant pressure conditions and P is an extremum under constant temperature conditions at the azeotrope 21 = y1 (equal vapor and liquid phase compositions). 9. (Denbigh ch. 9) Consider two metals that form an ideal solution (i.e. are completely miscible) in the liquid phase but are completely immiscible in the solid phase, as shown in figure 3. The melting points of the pure phases are T1 and T2; the heats of fusion, Ah1 and Ah2, are constant. (a) Prove that this system forms a minimum melting temperature eutectic point as seen in Figure 2. To do this, show that the slope dT/dx of the solid-liquid phase boundary is negative for small x and and the slope of the phase boundary is positive for large x. (b) For the special case Ah1 = Ah2 = Ah and T1 = T2 =To, show that the eutectic composition and temperature are given by x = 1 2' T = Ah + RTo log 2 AhTo

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