Multicomponent systems, solid solubilities

As will be evident from the examples cited below, the S-S theory for multicomponent systems has been successful in describing thermodynamics (e.g., phase equilibria, CED, solubility) and PVT behavior of polymer mixtures with gases, liquids, or solids. For binary systems, Eqs. (6.49) and (6.51) might be written as... 

It is often possible to remove the impurities from a crystalline mass by dissolving the crystals in a small amount of fresh hot solvent and cooling the solution to produce a fresh crop of purer crystals, provided that the impurities are more soluble in the solvent than is the main product. This step may have to be repeated many times before a yield of crystals of the desired purity is obtained, depending on the nature of the phase equilibria exhibited by the particular multicomponent system (Chapter 4). A eutectic system, for example, can yield a near-pure crystal product in one step (see Figure 4.4) whereas a solid solution system would need many (see Figure 4.7). 

As could be expected, challenges facing the pharmaceutical industry contribute to the advancement of the computational soUd-state chemistry. For example, some of the virtual screening and other CPSSC methods were developed specifically to help address issues of the pharmaceutical industry. Significant progress has been made recently in many traditional applications (e.g., solubility prediction [55], CSP [56], and morphology prediction [25, 57, 58]) in order to accommodate predictions for complex pharmaceutical systems (solid and liquid multicomponent phases of relatively large and flexible molecules). 

It is more complicated to calculate LLE in multicomponent systems accurately than to describe vapor-liquid or solid-hquid equilibria. The reason is that in the case of LLE the activity coefficients have to describe not only the concentration dependence but also the temperature dependence correctly, whereas in the case of the other phase equilibria (VLE, SLE) the activity coefficients primarily have to describe the deviation from ideal behavior (Raoult s law resp. ideal solid solubility), and the temperature dependence is mainly described by the standard fugacities (vapor pressure resp. melting temperature and heat of fusion). 

The section on solubility will be completed with a discussion of a slightly more general case where the solid phase (3 is a binary substitutional solution. Jesser et al. [9] have shown that the equivalent of Equation (6.33) for a multicomponent system is... 

Hardly any phase diagram studies were found to concentrate on the boron-rich corner of multicomponent systems and many confusing results exist about boron-rich transition metal borides, which more recently turned out to merely represent a solid solution of metal atoms in ) -rhombohedral boron. Only few reliable data exist on the solubility limits of metals in the different boron modifications as a function of temperature as well as on the (stabilizing) influence of T metals on the polymorphic transition itself, Thus metal solubilities in B indicated throughout this publication are the ss of M in f -rh. B . 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

These three approaches have found widespread application to a large variety of systems and equilibria types ranging from vapor-liquid equilibria for binary and multicomponent polymer solutions, blends, and copolymers, liquid-liquid equilibria for polymer solutions and blends, solid-liquid-liquid equilibria, and solubility of gases in polymers, to mention only a few. In some cases, the results are purely predictive in others interaction parameters are required and the models are capable of correlating (describing) the experimental information. In Section 16.7, we attempt to summarize and comparatively discuss the performance of these three approaches. We attempt there, for reasons of completion, to discuss the performance of a few other (mostly) predictive models such as the group-contribution lattice fluid and the group-contribution Flory equations of state, which are not extensively discussed separately. 

In the last one and a half to two decades, the KB theory of solutions was successfully used to predict the solubility in numerous systems. Indeed, it is quite impressive that one method was successful for such different systems as those examined in this chapter the solubility of gases and drugs and hydrophobic organic pollutants in binary and multicomponent mixed solvents and the solubility of protein. In addition, this method was successfully applied to the solubility of solids in supercritical fluids, to the solubility of mixed gases in individual solvents, to the explanation of salting-out phenomena (Ruckenstein and Shulgin 2009). In some cases (multicomponent mixtures) the KB theory of solutions constitutes the most powerful theoretical method for investigating the thermodynamic properties. 

Chemical Description. The constituents of the system include atoms, ions, molecules as well as molecular and nonmolecular solids. The homogeneous catalytic reaction of interest occurs in the multicomponent liquid phase Pl- By definition, all the molecular and ionic species S involved directly in the catalysis are soluble in this liquid phase. In particular, one or more organic reagents must be present in Pl- The liquid phase must contain numerous soluble transition-metal... 

Another potential problem in water-soluble binder systems is the achievement of easily processable viscosities. Celluloses are used as thickeners in some industries (food, pharmaceutical), and they behave similarly in tape casting. PVA slips also tend toward higher viscosities, but not to the extent of celluloses. Reported slip recipes have addressed this thickening effect by using additional water to lower slip viscosities. This results in castable slips, but it also results in wet dry ratios of 6.5 1 or 8 1, or z-dimension shrinkage of 77%P The additional water also results in comparatively low solids to solids plus solvent ratios. Some reported formulations display viscosities of 37,500 mPa-s (cP) at 38.6 wt% solids,>17,800 mPa-s (cP) at 43 wt%, or solids loading as low as 17.3 wt%. The role of pH on the viscosity of the multicomponent slip is not widely reported. 

Strictly speaking, a phase is defined as a region in a material in which the structure and composition are homogeneous. This assumes that the system is in chemical equilibrium, a situation which would eventually occur because of diffusion. However, for reasons that will become clear shortly, such a situation rarely occurs in a multicomponent solid, even if the system is isomorphous (aU components completely soluble) because of the slow inter-diffusion of components within a solid. Therefore, practically speaking, we generally define a phase as a region in which the crystal structure is the same and the components are the same but allow for the fact that chemical equilibrium may not yet have been reached. 

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