Solubility loop

For the two-component, two-phase liquid system, the question arises as to how much of each of the pure liquid components dissolves in the other at equilibrium. Indeed, some pairs of liquids are so soluble in each other that they become completely miscible with each other when mixed at any proportions. Such pairs, for example, are water and 1-propanol or benzene and carbon tetrachloride. Other pairs of liquids are practically insoluble in each other, as, for example, water and carbon tetrachloride. Finally, there are pairs of liquids that are completely miscible at certain temperatures, but not at others. For example, water and triethylamine are miscible below 18°C, but not above. Such pairs of liquids are said to have a critical solution temperature, For some pairs of liquids, there is a lower (LOST), as in the water-tiiethylamine pair, but the more common behavior is for pairs of liquids to have an upper (UCST), (Fig. 2.2) and some may even have a closed mutual solubility loop [3]. Such instances are rare in solvent extraction practice, but have been exploited in some systems, where separations have been affected by changes in the temperature. 

Thus the enthalpy of mixing is a key quantity for a system to show a UCST. For the other excess functions, e.g. VE and Cp, there are no restrictions, but generally at a UCST, Cf< 0 and VE > 0, while at a LCST, Cp > 0 and VE < 0 (Rowlinson, 1969). If Cf is negative at an LCST and remains so as the temperature increases, then HE and SE may change in such a way that the conditions for a UCST are met. Such systems show a closed solubility loop. The mixture water + nicotine is a classic example of such a system. The behaviour of another example, the mixture water + 2-butoxyethanol, is shown in Fig. 29 (Ellis, 1967). 

The procedures outlined have a practical use. but it should be realized that the subparameter models have some empirical elements. Assumptions such as the geometric mean rule (Eq. 12-6) for estimating interaction energies between unlike molecules may have some validity for dispersion forces but are almost certainly incorrect for dipolar interactions and hydrogen bonds. Experimental uncertainties are also involved since solubility loops only indicate the limits of compatibility and always include doubtful observations. Some of the successes and limitations of various versions of the solubility parameter model are mentioned in passing in the following sections which deal brielly with several important polymer mixtures. 

FIGURE 2-11 Closed solubility loops for some dimethyl pyridines in water. [From Andon and Cox, J. Chem, Soc. 1952, 4601-6.]... 

Di erent polymer-solvent systems may have completely different phase diagrams. For some systems, such as polystyrene-cyclohexanone, UCST < LCST [Fig. 3.13(a)] but for others, e.g., highly polar systems like polyoxyethylene-water, UCST > LCST and closed solubility loop is found [Fig. 3.13(b)]. 

The procedures outlined above have a practical use, but it should be realized that the parametric models are almost entirely empirical. Experimental uncertainties are also involved since solubility measurements are not very accurate. Solubility loops described by the models only indicate the limits of compatibility and always Include doubtful observations. 

The hydrophobic interaction results in the existence of a lower critical solution temperature and in the striking result that raising the temperature reduces the solubility, as can be seen in liquid-liquid phase diagrams (see Figure 5.2a). In general, the solution behaviour of water-soluble polymers represents a balance between the polar and the non-polar components of the molecules, with the result that many water-soluble polymers show closed solubility loops. In such cases, the lower temperature behaviour is due to the hydrophobic effects of the hydrocarbon backbone, while the upper temperature behaviour is due to the swamping effects of the polar (hydrophilic) functional groups. 

Figure 9.14 Examples of binary mixtures that have both a UCST and an LCST. Left Mixtures of nicotine (C10H14N2) and water have a closed solubility loop, with UCST = 233°C and LCST = 61.5°C [13]. Right Mixtures of 1-hexene (C5H12) and methane have a miscibility gap, with UCST = 133.8 K and LCST = 179.6 K [14], Pure hexene solidifies at 133.3 K, so the UCST occurs just above the melting curve of the mixtures.

LCSTs can also exhibit VLLE examples include water -i- 2-butanol and water -t 2-butanone. In such cases, VLLE prevents formation of a closed solubility loop, (ii) Many immiscible liquids form homogeneous azeotropes at high pressures, as in Figure 9.15, but some do not. Those without azeotropes include CO2 + long-chain alkanes, such as n-octane and n-decane. (iii) Often VLLE occurs at heterogeneous azeotropes, as in Figure 9.15, and then the vapor-phase composition lies between the compositions of the two liquid phases. However, VLLE also occurs in some mixtures in which the vapor-phase composition does not lie between the compositions of the liquid phases. Examples of the latter include ammonia + toluene and water + phenol. 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

Figure 10.4 Closed solubility loops in 2,5 lutidine + water and 2,6 lutidine + water.

In some liquid mixtures one may encounter a closed solubility loop between an upper critical solution point with temperature Ju and concentration Xu and a lower critical solution point with temperature Jl and concentration Xl. One can obtain a quantitative representation of such closed solubility loops if the temperature variable lAri is replaced by " At ji = T j-T) T-T IT jTi. This procedure has been applied successfully in the revised-scaling approximation i.e., without a contribution proportional to IATulI ), but with the addition of a correction-to-scaling contribution proportional to 1A7ul as discussed in Section 10.3.5 ... 

From eq 10.59, we see that the relationship of the mole fraction x with the scaling densities tpi and (pi is independent of either or b. Hence, the theoretical expressions for the temperature dependence of the mole fraction along the two phase boundaries, developed in the previous section, remain equally valid for weakly compressible liquid mixtures. This is the physical reason why eq 10.65 yields an excellent representation of the behaviour of the mole fraction x for liquid-liquid equilibria. As an example we show in Figure 10.5 closed solubility loops in 2-butanol + waterAs we explained earlier, closed solubility loops can be represented by the expansion of eq 10.65 provided that Ar is replaced by IATulI in accordance with eq 10.66. The closed solubility loops collapse into a double critical point at P = 85.6 MPa and T = 340 K. The implications of the theory for the behaviour near such a double critical point have been elucidated by Wang et Both near the upper critical solution temperature Ju and near the lower critical solution temperature Tl, Axcxc varies as A7 il in accordance with eq 10.65a. Near the double critical point both 7 j and 7 approach the temperature I d of the double critical point. Hence, near the double critical point... 

Figure 10.5 Closed solubility loops in 2-butanol + water at various pressures.---------,...

Let us now look into the case when there is a solubility loop and focus our attention on the behavior near the lower critical point with temperature c. For the usual Ising model the critical temperature (Tc=l//3c) is dependent on the lattice geometry and the model interactions, i.e., exp(-23cJ)=Kc, with Kc fixed by the lattice geometry, determines I3c. Since in our model, we have from (4) and (5) that JO)=-ln K(, ep)/2, it follows that... 

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