Virial osmotic

Deszczynski, M., Harding, S.E., Winzor, D.J. (2006). Negative second virial coefficients as predictors of protein crystal growth evidence from sedimentation equilibrium studies that refutes the designation of those light scattering parameters as osmotic virial coefficients. Biophysical Chemistry, 120, 106-113. 

The major theories developed to predict phase separation and biomolecule partitioning in aqueous two-phase systems are mostly extensions of the widely known polymer solution theories of Flory and Huggins and the osmotic virial expansion. 

Where R is the gas constant, T the absolute temperature, and M the molecular weight of the polymer. This series is usually called the osmotic virial expansion, with A (i = 2, 3,...) being referred to as the i-th virial coefficient of the... 

First, it is necessary to establish the phase diagram of the aqueous two-phase system formed by water and two water-soluble polymers. Second, a method must be established for calculating the distribution coefficient of a biomolecule that partitions between the two aqueous phases. A simple molecular-thermodynamic description is provided by the osmotic virial... 

Ternary diagram calculated using osmotic virial equation Figure 3. Two-phase aqueous extraction system for separating a mixture of biomolecules. 

Reminiscent of Eqs. (3.11) and (3.12), tliis series is called an osmotic virial expansion. Show tlrat the second osmotic virial coefficieirt B is ... 

Section 2 brings the cluster development for the osmotic pressure. Section 3 generalizes the approach of Section 2 to distribution functions, including a new and simple derivation of the cluster expansion of the pair distribution function. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 contains an application of our general solution theory to compact macromolecular molecules. Section 6 contains the second osmotic virial coefficient of flexible macromokcules, followed in Section 7 by concluding remarks. 

Table I. Composition, droplets radius (R) and osmotic virial coefficients for several microemulsions series.

Our purpose in this section is to derive a set of useful expressions for the chemical potentials starting with the principles of statistical mechanics. The expressions we shall obtain take the form of virial expansions similar to those of the Edmond and Ogston (6) but having a very different theoretical basis. Our model parameters are isobaric-isothermal virial coefficients which are about an order of magnitude smaller than the osmotic virial coefficients in the Edmond and Ogston model. We shall develop the theory neglecting the effect of polydispersity because we empirically did not find this to be very important at the level of accuracy commonly attainable in experimental phase diagrams for these systems. 

The inverse of Equations 5 give activity as a power series in molality. Taking this inverse and collecting terms in like powers of the molalities up to first order we obtain Equations 6 which give the solute chemical potential as a power series in solute molality with osmotic virial coefficients that are functions of temperature and pressure. 

Equations 6 and 7 are the fundamental expressions giving the chemical potentials as functions of solute molalities and Hill osmotic virial coefficients. 

In this section we "semi-empirically" adapt some scaling ideas from the Group Renormalization theory (12, 15) of polymer solutions to obtain expressions for the osmotic virial coefficients of Equations 6 and 7 in terms of the degree of polymerization. In the following discussion we will occasionally omit the indices on the osmotic virial coefficients for the sake of simplicity. 

At this point we note that Equation 13 is the McMillan-Mayer (16) expansion for the osmotic compressibility factor which is fundamentally different from the analogous expansion that was obtained from the formalism of Hill (Equation 10). We also identify B as a McMillan-Mayer osmotic virial coefficient. 

We have shown that there is a scaling relation for B of the form given in Equation 13. However, we have not shown that an analogous relation exists for the Hill osmotic virial coefficients (C). We start the proof with the exact relation between B and C shown in Equation 14. 

Second, if McMillan-Mayer (16) osmotic virial coefficients (Bip are available from light scattering or some other experiment for several molecular weights of the same polymer, one could use Equation 20,... 

McMillan-Mayer osmotic virial coefficient for components i and j, L/Mole. 

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... 

Interpretation of the Second Osmotic Virial Coefhcient Depolarization of Scattered Light Polydisperse Samples Turbidimetry... 

Generally speaking, positive values of A2 mean net repulsion between the particles, while negative values of A2 correspond to attraction. Eor more detailed analysis of the values of the second osmotic virial coefficient, the use of other definitions of the particle concentration is more convenient. The common virial expansion"... 

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