Virial-type expansion

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 theory leads to an osmotic virial type expansion and gives a fundamental interpretation of the coefficients appeEiring in this expansion in terms of forces between the species. The expansion reduces to the Edmunds-Ogston expression only when certain assumptions are made -namely that the fluids are incompressible and that the solvent is... 

As T — 1, the fraction on the right-hand side of Equation 6 becomes equal to unity, and a(T) RT. Thus, a sT-> 1, the differential Equation of State 1 evolves continuously to a virial-type expansion... 

Numerous models of ionic solutions have been put forward in the existing body of literature. The most important of these models, whieh is actually found to be included in all the others, is Debye and Htiekel s, which attributes the imperfection solely to the eleetrostatie forces between the ions but, in spite of this, is acceptable only for fiilly-dissoeiated strong electrolytes and very-dilute solutions. Then, we shall eite Pitzer s model (1973), which combines Debye and Hiickel s model with a virial-type expansion, and is therefore able to extend the range of concentrations examined. Beuner and Renon s model (1978) builds on Pitzer s, extending it to solutions containing neutral molecules such as SO2, NH3, CO2 or H2S. The most recent models take account of the concept of local composition (see section 3.2). In this category, we can cite the NRTL electrolyte model introduced by Chen (1979). Finally, other models have expressed the intermolecular interactions over short distances by a... 

Attempts were made, unsuccessfully, to apply virial type gas phase solubility equationsto the data reported here. It may be that this approach will be useful with equations extended by additional virial expansion terms. 

We shall not dwell any further on the applications of Eq. (148) and its experimental verification (see Refs. 11 and 8, and references quoted in the latter). We just wish to end this section with a remark which will be relevant later the result (148) could have been obtained formally if we had taken a virial expansion of the type (115) limited to the second order but calculated with an effective potential ... 

The LCM is a semi-theoretical model with a minimum number of adjustable parameters and is based on the Non-Random Two Liquid (NRTL) model for nonelectrolytes (20). The LCM does not have the inherent drawbacks of virial-expansion type equations as the modified Pitzer, and it proved to be more accurate than the Bromley method. Some advantages of the LCM are that the binary parameters are well defined, have weak temperature dependence, and can be regressed from various thermodynamic data sources. Additionally, the LCM does not require ion-pair equilibria to correct for activity coefficient prediction at higher ionic strengths. Thus, the LCM avoids defining, and ultimately solving, ion-pair activity coefficients and equilibrium expressions necessary in the Davies technique. Overall, the LCM appears to be the most suitable activity coefficient technique for aqueous solutions used in FGD hence, a data base and methods to use the LCM were developed. 

To seek a reasonable accurate analytical approximation for the available area, as a function of 6S = Nsnr /A and 6y = Nynr /A one should have accurate values for a reasonable number of coefficients in the low-density expansion of the binaiy RSA model, which is not a trivial task. Even for binaiy mixtures of disks at equilibrium, a problem that received much more attention than RSA, analytical expressions are known only for the first three terms of the virial expansion [21], The values of the fourth and fifth terms, obtained using laborious numerical calculations, were reported only for a few values of y and molar fractions of the two types of disks [22], In the non-equilibrium RSA of binaiy particles, one should take into account, when calculating the higher terms of the series, not only various y and molar fractions, but also the order of deposition of particles. Furthermore, as already noted, it is not clear whether the involved calculations needed to obtain the next unknown terms of the low-density expansion would improve much the accuracy of estimating the jamming coverage. 

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 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-... 

It was shown by Koschine and Lehrmann [92-kos/leh] that a better procedure was to use a system of two cells of matching volumes, each containing a capacitor - they had a parallel-plate type. Gas pressure and capacity were measured in one cell. The gas was then expanded into the other previously-evacuated cell, and a second set of measurements made. Finally, the first cell was evacuated and the measurements repeated after expansion into this volume. Their scheme for data analysis gives values for both the density and dielectric virial coefficients. 

To proceed further, we shall use the Lie equations, and hence need an initial approximation for the excess viscosity (valid for small volume fractions) from which estimates of the infinitesimal generators can be constructed. At the present time, very little information of this type exists. Indeed, reliable values are available only for the second- and third-order coefficients V2 and 3 appearing in the viscosity virial expansion... 

The third type of experimental evidence relevant to the problem of HI comes from the analysis of the second virial coefficient in the density expansion of the osmotic pressure, carried out by Kozak et al. 

The ASF function can be evaluated analytically in terms of power series of , analogous to virial expansions [127-130] for some well-defined particle configurations. The expansions are rather cumbersome especially for higher particle coverage [127-130]. Therefore, the ASF is usually determined by numerical simulations of the Monte Carlo type [15,127,128,131], which... 

Here a z), < s z) and A(z) = zh(z) are the conventional expansion and second virial factors of the continuum Edwards model, and K and K2 are nonuniversal scale factors. Thus, the continuum Edwards model is a correct theory for a certain limiting regime in the molecular-weight/tempera-ture plane—but this regime is not the one previously thought. The explanation of eq. (2.104) relies on a Wilson-deGennes-type renormalization group see Ref. 222 for details, and Ref. 223 for further discussion. 

The cs are often referred to as virial coefficients, and this is the type of predictive relationship one seeks. By specifying n, P, and T, Equation 2.29 yields Vfor the real gas being studied. The accuracy of the value predicted for V depends on the accuracy of the laboratory data used in the fitting (finding the virial coefficients) and on the degree to which the truncated expansion can represent the data. Usually, this means that the function is reliable in the temperature, volume, and pressure regions for which there were data. 

Upon decrease of temperature, and it reaches jc = 1 at the critical temperature of the solvent. The behavior in Figs. 31 and 32 is characteristic of a type I phase diagram, and this behavior agrees with the predictions of the virial expansion of the SCF calculations (Sect. 3) and the more accurate TPTl (Sect. 4) for = 1, but it disagrees with the phase behavior of hexadecane and carbon dioxide. 

Equation (8.4.2) is valid only at infinite dilution. For finite concentrations, the use of a virial expansion of the type introduced in Eq. (8.3.22) leads to... 

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