Mixed Second Virial Coefficients

Little mention has been made of the mixed second virial coefficient which is often determined simultaneously with the activity coefficients. Although the medium-high-pressure g.l.c. technique used for determining is very restrictive in regard to the systems that can be studied, the method is important as it is independent of Bn or Bg2. Apart from the Bristol group, Dantzler et and Vigdergauz and Semkin have used the method for B i determinations. 

Prigogine, The Molecular Theory of Solutions , North Holland, Amsterdam, 1957. 

It is only within the past 10 years that the main limitations and problems associated with this technique have been appreciated or solved. Already new applications and uses of the values of yf, determined by the g.I.c. method are being found, and as more reliable measurements are produced, these applications will increase. 

Provided that these points are observed, the g.I.c. method for determining yi8 is capable of 0.2 per cent precision and the overall accuracy, when compared with static measurement results, is better than 0.5 per cent. The results can therefore be used with confidence when testing solution theories. This rapid and easy-to-operate method could have important bearings on theoretical studies of mixtures, and as more and more data are produced, a rapid growth in our knowledge of intermolecular forces can be expected. 

I wish to acknowledge the helpful discussions I have had personally with Dr J. D. Bradley and Dr F. Marsicano, and in correspondence with Mr A. J. B. Cruickshank and Dr C. P. Hicks. 

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The term involving the mixed second virial coefficients was not used because of the uncertainties in the values of the second virial coefficients of the pure components. 

Further limitations are that the choice of carrier gas is restricted to those which are insoluble or nearly insoluble in the stationary phase and the range of mixtures to those in which adsorption effects of any kind are negligible. For systems which fall within these limitations accurate activity coefficients at infinite dilution can be obtained, which would be very difficult to determine in other ways. Furthermore, if the retention volumes are measured as a function of pressure, the mixed second virial coefficients, J i2, of carrier gas (referred to in this work as component 2) and solute (component 1) can be obtained simultaneously with the activity coefficient, yfj (where 3 refers to the solvent). 

First-order perturbation calculations of the mixed second virial coefficients A have been carried out for chains differing in molecular weight but having the same architecture and chemical structure (so that the segment-segment excluded volume parameter is the same for all polymer species), and for some combinations of topologically different chains, e,g. linear chains with rings, stars or combs, stars with stars of different functionality, stars with combs, The simplest result is that for... 

Mixing mles for the parameters in an empirical equation of state, eg, a cubic equation, are necessarily empirical. With cubic equations, linear or quadratic expressions are normally used, and in equations 34—36, parameters b and 9 for mixtures are usually given by the following, where, as for the second virial coefficient, = 0-. 

Although developed for pure materials, this correlation can be extended to gas or vapor mixtures. Basic to this extension is the mixing rule for second virial coefficients and its temperature derivative ... 

Special care has to be taken if the polymer is only soluble in a solvent mixture or if a certain property, e.g., a definite value of the second virial coefficient, needs to be adjusted by adding another solvent. In this case the analysis is complicated due to the different refractive indices of the solvent components [32]. In case of a binary solvent mixture we find, that formally Equation (42) is still valid. The refractive index increment needs to be replaced by an increment accounting for a complex formation of the polymer and the solvent mixture, when one of the solvents adsorbs preferentially on the polymer. Instead of measuring the true molar mass Mw the apparent molar mass Mapp is measured. How large the difference is depends on the difference between the refractive index increments ([dn/dc) — (dn/dc)A>0. (dn/dc)fl is the increment determined in the mixed solvents in osmotic equilibrium, while (dn/dc)A0 is determined for infinite dilution of the polymer in solvent A. For clarity we omitted the fixed parameters such as temperature, T, and pressure, p. 

For accuracy in light-scattering measurement the proper choice of solvent is necessary. The difference in refractive index between polymer and solvent should be as large as possible. Moreover, the solvent should itself have relatively low scattering and the polymer-solvent system must not have too high a second virial coefficient as the extrapolation to zero polymer concentration becomes less certain for high A2. Mixed solvent should be avoided unless both components have the same refractive index. 

The mixed electrolyte terms in 0 and account for differences among interactions between ions of like sign. The defining equations for the second virial coefficients, 0,.. > are given by Equations (13),... 

The intrinsic viscosity of PVB is shown as a function of solvent composition for various MIBK/MeOH mixtures in Figure 6. Since [ij] increases with a (see Equation 8), the higher [ly] the better the solvent. Apparently, most mixtures of MIBK and MeOH are better solvents for PVB than either pure solvent. Based on Figure 6, PVB should have a weak selective adsorption of MIBK in a 1 1 solvent mixture and weak adsorption of MeOH in a 3 1 MIBK/MeOH solvent mix. These predictions are in accord with light scattering data discussed previously. The intrinsic viscosity data is also consistent with the second virial coefficient data in Table II in indicating that the 1 1 and 3 1 MIBK/MeOH mixtures are nearly equally good solvents for PVB, the 9 1 mix is a worse solvent, but still better than pure MeOH. 

The unconventional applications of SEC usually produce estimated values of various characteristics, which are valuable for further analyses. These embrace assessment of theta conditions for given polymer (mixed solvent-eluent composition and temperature Section 16.2.2), second virial coefficients A2 [109], coefficients of preferential solvation of macromolecules in mixed solvents (eluents) [40], as well as estimation of pore size distribution within porous bodies (inverse SEC) [136-140] and rates of diffusion of macromolecules within porous bodies. Some semiquantitative information on polymer samples can be obtained from the SEC results indirectly, for example, the assessment of the polymer stereoregularity from the stability of macromolecular aggregates (PVC [140]), of the segment lengths in polymer crystallites after their controlled partial degradation [141], and of the enthalpic interactions between unlike polymers in solution (in eluent) [142], as well as between polymer and column packing [123,143]. 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

Let us now consider some actual numerical data for specific mixed biopolymer systems. Table 5.1 shows a set of examples comparing the values of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated theoretically on the basis of various simple excluded volume models using equations (5.32) to (5.35). For the purposes of this comparison, the experimental data were obtained under conditions of relatively high ionic strength (/ > 0.1 mol dm- ), i.e., under conditions where the contribution of the electrostatic term (A if1) is expected to be relatively insignificant. 

However, these new mixing rules (based both to infinite- or zero pressure limit) give, for the composition dependence of the second virial coefficient, results that are inconsistent with those obtained from statistical mechanics. 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The parameter is best obtained by fitting the equation for to the experimental heats of mixing of analogous materials as reported elsewhere It can also be obtained from any other binary quantity such as the second virial coefficient, the thermal expansion coefficient of mixture, or the volume change on mixing. is assumed to be independent of temperature but as we described in the previous section this may not be valid. At present there is no way of predicting the temperature variation and one can only use empirical expressions or assume a constant value most appropriate for the temperature range of interest. 

If the excess volume of mixing is zero, then all the volume fraction multipliers cancel out in these forms, because then (f)i = (1 — 2)- Though the present derivation should provide a basis for doing better, this assumption of zero volume of mixing will be our second assumption here. Then Eqs. (4.44) describe a scaling of second virial coefficients that is natural even though special ... 

Eisenberg, H. Multicomponent polyelectrolyte solutions IV Second virial coefficient in mixed polyvinylsulfonate KCl systems near the 0 temperature. J. Chem. Phys. 44, 137 (1966). 

Keywords Protein solubility Osmotic second virial coefficient Preferential binding parameter Mixed solvent Salting-in and salting-out Contents... 

The osmotic second virial coefficient was used to examine the crystallization of proteins and their solubility in water and in aqueous mixed solvents. 

Using classical thermodynamics, another relation between the osmotic second virial coefficient and the aqueous solubility was established [35]. In that paper, the aqueous mixed solvent is treated as a single component and the obtained relation contains two adjustable parameters. This equation was used to correlate the osmotic second virial coefficient and the aqueous protein solubility in the systems water-lysozyme-salt (NaCl) and water-ovalbumin-salt ((NH4)2S04). 

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