Theoretical treatment using virial

Theoretical Treatment of Aqueous Two-Phase Extraction by Using Virial Expansions... 

A theoretical treatment of aqueous two-phase extraction at the isoelectric point is presented. We extend the constant pressure solution theory of Hill to the prediction of the chemical potential of a species in a system containing soivent, two polymers and protein. The theory leads to an osmotic virial-type expansion and gives a fundamentai interpretation of the osmotic viriai coefficients in terms of forces between species. The expansion is identical to the Edmunds-Ogston-type expression oniy when certain assumptions are made — one of which is that the solvent is non-interacting. The coefficients are calculated using simple excluded volume models for polymer-protein interactions and are then inserted into the expansion to predict isoelectric partition coefficients. The results are compared with trends observed experimentally for protein partition coefficients as functions of protein and polymer molecular weights. 

The discussion of the Joule-Thonison effect in the previous section clearly showed that it is advantageous in theoretical treatments of confined fluids to tackle a given physical problem by a combination of different methods. This was illustrated in Section 5.7 whore wo employed a virial expansion of the equation of state, a van der Waals type of equation of state, and MC simulations in the specialized mixed isostress isostrain ensemble to investigate various aspects of the impact of confinement on the Joule-Thomson effect. The mean-field approach was particularly useful because it could predict certain trends on the basis of analytic equations. However, the mean-field treatment developed in Sections 4.2.2 and 5.7.5 is hampered by the assump-... 

The theoretical treatments of Ajg discussed here were motivated by the question of the effects of molecular weight dispersion on measured second virial coefficients. Once Ajf Mjy Mj ) is available it is obvious in principle how to obtain A 2 or. 4 2 for any desired form of distribution. Detailed calculations using the hard-spherelike interaction model with the familiar Schulz-Zimm distribution indicate that the ratio of virial coefficients increase wi out limit with the... 

An important series of papers by Professor Pitzer and colleagues (26, 27, 28, 29), beginning in 1912, has laid the ground work for what appears to be the "most comprehensive and theoretically founded treatment to date. This treatment is based on the ion interaction model using the Debye-Huckel ion distribution and establishes the concept that the effect of short range forces, that is the second virial coefficient, should also depend on the ionic strength. Interaction parameters for a large number of electrolytes have been determined. 

It is helpful to contrast the view we adopt in this book with the perspective of Hill (1986). In that case, the normative example is some separable system such as the polyatomic ideal gas. Evaluation of a partition function for a small system is then the essential task of application of the model theory. Series expansions, such as a virial expansion, are exploited to evaluate corrections when necessary. Examples of that type fill out the concepts. In the present book, we establish and then exploit the potential distribution theorem. Evaluation of the same partition functions will still be required. But we won t stop with an assumption of separability. On the basis of the potential distribution theorem, we then formulate additional simplified low-dimensional partition function models to describe many-body effects. Quasi-chemical treatments are prototypes for those subsequent approximate models. Though the design of the subsequent calculation is often heuristic, the more basic development here focuses on theories for discovery of those model partition functions. These deeper theoretical tools are known in more esoteric settings, but haven t been used to fill out the picture we present here. 

The virial equation is useful for many cases but, when there is strong association in the vapor phase, the theoretical basis of the virial equation is not valid and we must resort to what is commonly called a "chemical treatment", utilizing a chemical equilibrium constant for dimerization. Dimerization in the vapor phase is especially important for organic acids and even at low pressures, the vapor-phase fugacity coefficients of mixtures containing one (or more) organic acid are significantly removed from unity. 

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