Conjugate coexistence curve

The values of (f> and 4>" for a given mother solution vary with T. The curves obtained when T is plotted against 4> and (j>" are called conjugate coexistence curves. They chringe as the eomposition of the mother solution is varied. 

Given f P), these q I equations can, in principle, be solved for Tp, ", and f Py as functions of . The resulting relation between Tp and gives the cloud-point curve, while that between Tp and " gives a line called the shadow curve. The latter cannot be determined experimentally, since the second phase is too small in volume to be analyzed for the composition. It can be shown that the cloud-point and shadow curves coincide with the conjugate coexistence curves when and only when the solution is strictly binary. This fact is important, because some authors make no distinction between cloud-point curve and coexistence curve in describing phase equilibria of polydisperse solutions. 

Fig. 9-22. Calculated and observed phase diagrams for PS f4 + PS fl28 (the weight fraction of the latter is 0.05) + CH on the T — ( > plane. Thick solid line, calculated cloud-point curve. Dot-dashed line, calculated shadow curve. Dashed lines, calculated two-phase conjugate coexistence curves for the indicated polymer volume fractions. Thin solid line, three-phase coexistence curve. Unfilled circle, calculated critical point. Filled circles, measured cloud points. Filled triangles, measured polymer volume fractions in three separated phases. 

At the critical pohit (and anywhere in the two-phase region because of the horizontal tie-line) the compressibility is infinite. However the compressibility of each conjugate phase can be obtained as a series expansion by evaluating the derivative (as a fiuictioii of p. ) for a particular value of T, and then substituting the values of p. for the ends of the coexistence curve. The final result is... 

T, X curve entirely analogous to the T, Vm, (or T, p) curve for a pure fluid. A constant-temperature cross-section yields a similar p, x coexistence curve. One can take these constant-pressure or -temperature sections including two-phase regions without complication because these variables are fields , variables that have the same values in the conjugate phases on opposite sides of the coexistence surface. In contrast a section at constant x would have no simple interpretation because the curve representing the intersection of the constant-jc plane with the coexistence surface represents points on tie-lines that do not lie in the plane and that terminate at a second phase with a different value of x. 

This method introduces some difficulties into the analysis (1) one does not have experimental points x and x" at the same temperature for both branches of the coexistence curve and (2) one has to infer from the set of experimental points the best co-ordinates of the critical point (T , x ). The second difficulty is probably inherent in any method, but the first difficulty could be avoided by sampling isothermal conjugate phases and analysing them for x and x" by any appropriate method (e.g. density or refractive index). In fact this has rarely been done. Thompson and Rice and Wims, McIntyre, and Hynne measured... 

An issue relevant for potential antifoam behavior for compositions within the coexistence curve concerns the physical state of the two conjugate solutions. Essentially they form an emulsion, the continuous phase of which is in part determined by the relative amounts of each phase. Consider then the tie line shown in Figure 4.33 joining compositions Ti and it2. The relative amounts of each phase L( iti) and at the overall composition xj/j on the tie line is then given by the lever rule so that... 

Formation of foam usually involves intense agitation, arguably so intense that metastable supersaturated states are not likely to be present and any phenomena associated with phase separation across the equilibrium coexistence curve will be realized. However, direct independent measurement of points on that curve, such as the so-called cloud point, sometimes appear to involve minimal agitation. In which case it is possible that antifoam effects due to the relevant conjugate phase can be interpreted as evidence for antifoam effects in homogeneous systems. We will keep this possibility in mind when reviewing the antifoam effects associated with partial miscibility. 

The two liquid phases can be regarded, the one as a solution of the component I. in component II., the other as a solution of component II. in component I. If the vapour phase be removed and the pressure on the two liquid phases be maintained constant (say, at atmospheric pressure), the system will still be univariant, and to each temperature there will correspond a definite concentration of the components in the two liquid phases and addition of excess of one will merely alter the relative amounts of the two solutions. As the temperature changes, the composition of the two solutions will change, and there will therefore be obtained two solubility curves, one showing the solubility of component I. in component IL, the other showing the solubility of component II. in component I. Since heat may be either evolved or absorbed when one liquid dissolves in another, the solubility may diminish or increase with rise of temperature (theorem of Lc Chatelier, p. i8). The two solutions which at a given temperature coexist in equilibrium are known as conjugate solutions. 

---