Chapter 9: Other Topics in Phase Equilibria This chapter deals with relations that derive in cases of equilibrium between combinations of two co-existing phases other than vapour and liquid, i.e., liquid-liquid, solid-liquid, and solid- vapour. Each of these phase equilibria may be employed to overcome difficulties encountered in purification processes that exploit the difference in the volatilities of the components of a mixture, i.e., by vapour-liquid equilibria. As with the case of vapour-liquid equilibria, the objective is to derive relations that connect the compositions of the two co-existing phases as functions of temperature and pressure. 9.1 Liquid-liquid Equilibria (LLE) 9.1.1 LLE Phase Diagrams Unlike gases which are miscible in all proportions at low pressures, liquid solutions (binary or higher order) often display partial immiscibility at least over certain range of temperature, and composition. If one attempts to form a solution within that certain composition range the system splits spontaneously into two liquid phases each comprising a solution of different composition. Thus, in such situations the equilibrium state of the system is two phases of a fixed composition corresponding to a temperature. The compositions of two such phases, however, change with temperature. This typical phase behavior of such binary liquid-liquid systems is depicted in fig. 9.1a. The closed curve represents the region where the system exists Fig. 9.1 Phase diagrams for a binary liquid system showing partial immiscibility in two phases, while outside it the state is a homogenous single liquid phase. Take for example, the point P (or Q). This point defines a state where a homogeneous liquid solution exists at a temperature TTPQ(or ) with a composition xx11PQ(or ). However, if one tries to form a solution of composition x1Y (at a temperatureTY ) the system automatically splits into I II two liquid phases (I and II) with compositions given by xx11and .The straight line MYN represents a tie-line, one that connects the compositions of the two liquid phases that co-exist in equilibrium. Any attempt to form a solution at TY with a composition corresponding to any I II point within MYN always results in two phases of fixed compositions given by xx11and . The total mass (or moles) of the original solution is distributed between the two phases accordingly. Note, however, that the compositions at the end tie lines (horizontal and parallel to MYN) change as the temperature changes. Indeed as one approaches either point U or L, the tie line reduces to a point, and beyond either point the system exists as a single homogeneous solution. The temperatures corresponding to points U and L, i.e., TTUCSTand LCST are called the Upper Consolute Solution Temperature (UCST) and Lower Consolute Solution Temperature (LCST) respectively. They define the limit of miscibility of the components of the binary solution. Not all liquid-liquid systems, however, depict the behaviour described by fig. 9.1a. Other variants of phase behaviour are shown in figs. 9.1b – 9.1c. In each case there is at least one consolute temperature. 9.1.2 Phase Stability Criteria The passage of a liquid solution from a state of a single, homogeneous phase to a biphasic LLE state occurs when a single phase is no longer thermodynamically stable. It is instructive to derive the conditions under which this may take place. Let us consider a binary liquid mixture for example for which the molar Gibbs free energy of mixing ()∆Gmix is plotted as function of mole fraction ()x1 of component 1 for two temperaturesTT12and (fig. 9.2). At temperatureT1 the value of ∆Gmix is always less than zero and also goes through a minimum. This signifies that the mixture forms a single liquid phase at all compositions at the given temperature. The mathematical description of this condition is given by the two following equations: ∂G mix = 0 ..(9.1) ∂ x1 TP, And: ∂2G mix > 0 ..(9.2) ∂ 2 x1 TP, Alternately: ∂∆G mix = 0 ..(9.3) ∂ x1 TP, And: ∂∆2 G mix > 0 ..(9.4) ∂ 2 x1 TP, However, at certain other temperature T2 the value of ∆Gmix , while being always less than zero, passes through a region where it is concave to the composition axis. Consider a solution with a composition given by the point ‘A’. At this composition the Gibbs free energy of the mixture is given by: AA A A Gmix = xG11 + xG 22 +∆ Gmix ..(9.5) Fig. 9.2 Molar Gibbs free energy mixing vs. mole fraction x1 for a binary solution Let the points ‘B’ and ‘C’ denote the points of at which a tangent BC is drawn on the ∆Gmix curve for the isotherm T2 , and ‘D’ the point vertically below ‘A’ on the straight line BC (which is a tangent to the ∆Gmix curve at points B and C). If the solution at ‘A’ splits into two BC phases with compositions characterized by xx11and then its molar Gibbs free energy is given by: BA A D Gmix = xG11 + xG 22 +∆ Gmix ..(9.6) D A DA Clearly, since ∆Gmix <∆ G mix,( or G mix< G mix )a phase splitting is thermodynamically favoured A over a single phase solution of composition x1 . Point ‘D’ represents the lowest Gibbs free A energy that the mixture with an overall composition x1 can have at the temperature T2. In A other words, at the temperatureT2 a solution with an overall composition x1 exists in two BC phases characterized by compositions xx11and . If nnBCand represent the quantities of A BC solution in each phase then the following equation connects x1 to xx11and : A BC x1=++( nxB 11 nx C) () n BC n ..(9.7) The above considerations allow one to formulate a phase separation criteria for liquid solutions. Essentially, for phase splitting to occur, the ∆Gmix curve must in part be concave to the composition axis. Thus in general mathematical terms a homogeneous liquid solution becomes unstable if: ∂∂<22G x 0; (where,xx ≡ ) ..(9.8) ()mix TP, i It follows that the alternate criterion for instability of a single phase liquid mixture is: ∂∆22Gx ∂ <0 ..(9.9) ( mix )TP, It may be evident that at T2 the above conditions hold over the composition range BC x1<< xx 11. Conversely, a homogeneous liquid phase obtains for compositions over the B C ranges 0 <<xx11and xx11<<1. In these ranges, therefore, the following mathematical condition applies: ∂∂>22Gx 0 ..(9.10) ()mix TP, Or: ∂∆22Gx ∂ >0 ..(9.11) ( mix )TP, Referring again to fig. 9.2 it may be evident that at points ‘X’ and ‘Y’ the following mathematical condition holds: ∂∆22Gx ∂ =0 ..(9.12) ( mix )TP, A series of ∆Gmix curves at other temperatures (say between TT12and ) may be drawn, each showing different ranges of unstable compositions. All such curves may be more concisely expressed by the Tx− plot in fig. 9.2. There is an absolute temperature T C above which the mixture is stable at all compositions since the condition described by eqn. 9.11 applies at all compositions. The binodal curve in fig. 9.2 represents the boundary between the single phase region and the two phase regions. Within the two phase region the spinodal curve represents the locus of points at which ∂∆22Gx ∂ =0. As may be evident from the Tx− plot in ( mix )TP, 22 fig. 9.2, it is the boundary between the unstable (∂∆Gxmix ∂) <0 and metastable TP, 22 (∂∆Gxmix ∂) >0 regions. TP, The above mathematical conditions may be used to explain the phase behaviour depicted in fig. 9.1. Since the binary mixture displays both UCST and LCST, it follows that at TT> UCST and TT< LCST the mixture forms single phase at all compositions. Thus for such temperatures the ∆G curve is described by eqn. 9.11, i.e., ∂∆22Gx ∂ >0 for all mix ( mix )TP, compositions. Conversely for TLCST << TTUCST the ∆Gmix curve displays a form corresponding to that for temperature T2 in fig. 9.2. That is, for TLCST << TTUCST there is partial immiscibility of the mixture. The mathematical conditions for the existence of the consolute temperatures may then be surmised by the following relations. For the existence of UCST: ∂2G = = 2 0 for some value of x1 at TTUCST ∂x1 and ∂2G >> 2 0 for all values of x1 at TTUCST ∂x1 For the existence of UCST: ∂2G = = 2 0 for some value of x1 at TTLCST ∂x1 and ∂2G >< 2 0 for all values of x1 at TTLCST ∂x1 The foregoing mathematical relations are more conveniently expressed in terms of the excess molar Gibbs free energy function as follows: E id GG=mix − G mix ..(9.13) Or: E G=−++ Gmix xGxGRTx1122( 1ln xx 12 + ln x 2) ..(9.14) Thus: E Gmix =+++ G xGxGRTx1122( 1ln xx 12 + ln x 2) ..(9.15) On applying the criterion for instability given by eqn. 9.8, an equivalent criterion obtains as follows: ∂2G E 11 +RT +<0 ∂ 2 xTP, xx12 ∂2GE RT +<0 ..(9.16) ∂ 2 xTP, xx12 For an ideal solution G E = 0; hence the condition in eqn. 9.16 can never hold. Thus, and ideal solution can never display phase-splitting behaviour. Consider now the case of a binary mixture described by the excess Gibbs free energy E function: G= α xx12. Thus: ∂2G E = −2α ..(9.17) ∂ 2 x TP, On applying the condition given by eqn. 9.16 one obtains the following relation: RT −+20α < ..(9.18) xx12 Or: RT 2α > ..(9.19) xx12 The maximum value of the term xx12= 0.25; hence the values of α that satisfies the inequality 9.19 is given by: α ≥ 2RT ..(9.20) An equivalent criterion of instability may be derived using activity coefficient as a parameter in place of the excess Gibbs free energy function.