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Inspection of these results verifies that xi = z, = 1 for T = Tm, Moreover, analysis shows that both xi and zi vary monotonically with T Hence systems described by Eqs (1485) exhibit lens-shaped SLE diagrams, as shown on Fig 1421(a), where the upper line is the freezing curve and the lower line is the melting curve The liquid-solution region lies above the freezing curve, and the solid-solution region lies below the melting curve Examples of systems exhibiting diagrams of this type range from nitrogenlcarbon monoxide at low temperature to copperlnickel at high temperature Comparison of this figure with Fig (1012)
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CHAPTER 14 Touics in Phase Eauilibria
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Figure 1421 T x z diagrams(a) Case I, ideal liquid and solid solutions; (b) Case II, ideal liquid solution; immiscible solids
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suggests that Case I-SLE behavior is analogous to Raoult's-law behavior for VLE Comparison of the assumptions leading to Eqs (1485) and (101) confirms the analogy As with Raoult's law, Eq (1485) rarely describes the behavior of actual systems However, it is an important limiting case, and serves as a standard against which observed SLE can be compared
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The two equilibrium equations resulting from Eq (1477) are here:
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where Q1 and $2 are given as functions solely of temperature by Eqs (1488) Thus xl and x2 are also solely functions of temperature, and Eqs (1489) and (1490) can apply simultaneously only for the particulartemperaturewhere $2 = 1 and hence xl +x2 = 1 This is the eutectic temperature T, Thus, three distinct equilibrium situations exist: one where Eq (1489) alone applies, one where Eq (1490) alone applies, and the special case where they apply together at T,
Equation (1489) alone applies By this equation and Eq (1488a),
This equation is valid only from T = Tm, where xl = 1, to T = T,, where xl = xl,, , the eutectic composition (Note that xl = 0 only for T = 0) Equation (1491) therefore applies where a liquid solution is in equilibrium with pure species 1 as a solid phase This
147 Solid/Vapor Equilibrium (SVE)
is represented by region I on Fig 1421(b), where liquid solutions with compositions xl given by line BE are in equilibrium with pure solid 1 Equation (1490) alone applies By this equation and Eq (1488b), with x2 = 1 - X I :
This equation is valid only from T = Tm,,where xl = 0, to T = Te, where xl = xl,, the eutectic composition Equation (1492) therefore applies where a liquid solution is in equilibrium with pure species 2 as a solid phase This is represented by region I1 on Fig 1421(b), where liquid solutions with compositions xl given by line AE are in equilibrium with pure solid 2 Equations (1489) and (1490) apply simultaneously, and are set equal since they must both give the eutectic composition xl, The resulting expression,
is satisfied for the single temperature T = T, Substitution of T, into either Eq (1491) or (1492) yields the eutectic composition Coordinates Te and xle define a eutectic state, a special state of three-phase equilibrium, lying along line CED on Fig 1421(b),for which liquid of composition xle coexists with pure solid 1 and pure solid 2 This is a state of solidsolid/liquid equilibrium At temperatures below Te the two pure immiscible solids coexist Figure 1421(b), the phase diagram for Case 11, is an exact analog of Fig 1420(a) for immiscible liquids, because the assumptions upon which its generating equations are based are analogs of the corresponding VLLE assumptions
147 SOLID/VAPOR EQUILIBRIUM (SVE)
At temperatures below its triple point, a pure solid can vaporize Solidvapor equilibrium for a pure species is represented on a P T diagram by the sublimation curve (see Fig 31); here, as for VLE, the equilibrium pressure for a particular temperature is called the (solidlvapor) saturation pressure P We consider in this section the equilibrium of a pure solid (species 1) with a binary vapor mixture containing species 1 and a second species (species 2), assumed insoluble in the solid phase Since it is usually the major constituent of the vapor phase, species 2 is conventionally called the solvent species Hence species 1 is the solute species, and its mole fraction yl in the vapor phase is its solubility in the solvent The goal is to develop a procedure for computing yl as a function of T and P for vapor solvents Only one phase-equilibrium equation can be written for this system, because species 2, by assumption, does not distribute between the two phases The solid is pure species 1 Thus,
Equation (1141) for a pure liquid is, with minor change of notation, appropriate here:
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