Nonideal mixed associations

The analysis of mixed associations by light scattering and sedimentation equilibrium experiments has been restricted so far to ideal, dilute solutions. Also it has been necessary to assume that the refractive index increments as well as the partial specific volumes of the associating species are equal. These two restrictions are removed in this study. Using some simple assumptions, methods are reported for the analysis of ideal or nonideal mixed associations by either experimental technique. The advantages and disadvantages of these two techniques for studying mixed associations are discussed. The application of these methods to various types of mixed associations is presented. 

Figure 1. Curve 1 could represent an ideal polymer solution containing A and B but undergoing no association the nonideal counterpart of this is shown in curve 2. An ideal mixed association between A and B, such as described by Equation 1 might be described by curve 3, whereas, curve 4 could represent a nonideal, mixed association.

The analysis of nonideal, mixed associations from conventional sedimentation equilibrium experiments is a very difficult matter, and at present seems to be an impasse. These difficulties arise because the redistribution of the reactants is combined with nonideal behavior. We can illustrate the difficulties with the following example. The equation for reactant A can be written as... 

Equations 6-8 have been used to obtain Equation 9 Equation 9 is a general equation which can be applied to all mixed associations. The final form of Equation 9 indicates that there are only two solute components (independent variables) involved in the mixed association. When no association occurs, then (dc/dCi)T>c — 1. The superscript eq is used to indicate a mixed association is present. Now that the quantity cMweq has been defined, how do we obtain Mweq, and how do we use it (or its analogs) to obtain the equilibrium constant or constants and the nonideal terms if they are present ... 

The sedimentation equilibrium experiment requires much smaller volumes of solution, about 0.15 ml. With six-hole rotors and multichannel centerpieces (41) it is potentially possible to do fifteen experiments at the same time. For situations where the photoelectric scanner can be used one might (depending on the extinct coefficients) be able to go to much lower concentrations. Dust is no problem since the centrifugal field causes it to go to the cell bottom. For conventional sedimentation equilibrium experiments, the analysis of mixed associations under nonideal conditions may be virtually impossible. Also, sedimentation equilibrium experiments take time, although methods are available to reduce this somewhat (42, 43). For certain situations the combination of optical systems available to the ultracentrifuge may allow for the most precise analysis of a mixed association. The Archibald experiment may suffer some loss in precision since one must extrapolate the data to the cell extremes (rm and r6) to obtain MW(M, which must then be extrapolated to zero time. Nevertheless, all three methods indicate that it is quite possible to study mixed associations. We have indicated some approaches that could be used to overcome problems of nonideality, unequal refractive index increments, and unequal partial specific volumes. 

Van Laar [2] developed an activity-coefficient equation for liquid solutions based on the vdW eos. This equation has found wide use. Based on the observation that strongly nonideal solutions were associated with large heats of mixing, it was conjectured that nonideality was due to heat of mixing only. Neglecting... 

The residence time distribution measures features of ideal or nonideal flows associated with the bulk flow patterns or macromixing in a reactor or other process vessel. The term micromixing, as used in this chapter, applies to spatial mixing at the molecular scale that is bounded but not determined uniquely by the residence time distribution. The bounds are extreme conditions known as complete segregation and maximum mixedness. They represent, respectively, the least and most molecular-level mixing that is possible for a given residence time distribution. 

The conversion of A in the nonideal reactor is lower than in the ideal PFR, as expected. However, the difference is not large. The mixing associated with the broadened residence time distribution shown in Figure 10-6 is not sufficient to cause a significant conversion difference between the nonideal reactor and the ideal PFR. 

SOLUTION In this example, we assume that all the nonideality is associated with a difference in energetics between the species in the mixture and the pure species that is, the excess entropy is zero. This assumption is valid for species of roughly the same size. Let s consider the difference in energetics between a andfc in a mixture vs. a andb as pure species. Figure E7.9A illustrates the possible interactions in the mixture and of pure a and pure b. Pure a and b exhibit only a-a interactions and h-h interactions, respectively. The mixture contains not only these like a-a and h-h interactions but also unlike a-h interactions. In fact, when species a and h are mixed, we can view the process in terms of intermolecular interactions. Some a-a interactions of pure a are replaced by a-h interactions, while some h-h interactions of pure h are also replaced by a-h interactions. To quantify the difference in energy of the mixture relative to the pure species and, therefore, the nonideality of the mixture, we compare the magnitude of the interactions in the mixture with those that existed as pure species. 

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