Second virial cross-coefficient

In the generalized method of Hayden and O Connell (1975), the pure-component and cross second virial coefficients are given by the sum of two contributions... 

Second Virial Coefficient. A group contribution method including polar and nonpolar contributions has been proposed for second virial coefficients (241). This method has been appHed to both pure components and mixtures, the latter through prediction of cross-second virial coefficients. 

Here, as previously, the solvent is taken as component 1, one of the biopolymers is the /-component, while another is the /-component m, mr and m,j are the concentrations (moles per kg of water) of the components An and Ajj are the second virial coefficients (m3/mol) characterizing the like pair interactions of the types biopolymer -biopolymer and biopolymer,-biopolymer, respectively and Ay is the cross second virial coefficient corresponding to the biopolymer -biopolymer pair interaction. 

A2 from equation (5.16) or the cross second virial coefficient from equation (5.17). In turn, this knowledge of the second virial coefficients and their temperature dependence allows calculation of the values of the chemical potentials of all components of the biopolymer solution or colloidal system, as well as enthalpic and entropic contributions to those chemical potentials. On the basis of this information, a full description and prediction of the thermodynamic behaviour can be realised (see chapter 3 and the first paragraph of this chapter for the details). 

The quantity A i represents the cross second virial coefficient characterizing the interaction between the fraction k of the /-component and the fraction l of the /-component. 

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. 

Table 5.1 Comparison of the cross second virial coefficients obtained experimentally by static laser light scattering with those calculated from theory on the basis of the excluded volume contribution only.

We can compare experimentally measured values of the cross second virial coefficient (Ay ) with theoretical ones (Ay exc) based on the contri-... 

In the above equations denotes the vapor mole fraction of component i, Pf is the vapor pressure of the pure component i, Bu is the second virial coefficient of component i, dn = 2Bn — Bn — B22 and B 2 is the crossed second virial coefficient of the binary mixture. The vapor pressures, the virial coefficients of the pure components and the crossed second virial coefficients of the binary mixtures were taken from [32], The Wilson [38], NRTL [39] and the Van Ness-Abbott [40] equations were used for the activity coefficients in Eq. (17). The expressions for the activity coefficients provided by these three methods were differentiated analytically and the obtained derivatives were used to calculate D = 1 -I- Xj(9 In Yil 2Ci)pj. There is good agreement between the values of D obtained with the three expressions for the systems V,V-dimethylformamide-methanol and methanol-water. For the system V,V-dimethylformamide-water, the D values calculated with the Van Ness-Abbott equation [40] were found in good agreement with those obtained with the NRTL equation, but the agreement with the Wilson expression was less satisfactory. 

This investigation has enhanced our understanding of the factors which contribute to the molecular weight dependence of protein partitioning. The molecular weight dependence of the protein partition coefficient results from a competition between two terms in the partition coefficient expansion, namely the crossed second virial coefficient and the differences between the polymer concentrations in the top and bottom phases. While the trend in binodal concentrations tends (in part) to favor the trends observed experimentally, the trend in the second virial coefficient tends to oppose the experimental trends. 

Substituting the values of R2, R3, R4, L and into the expressions for the cross second virial coefficient allows us to determine how the... 

Combining rules are needed to estimate the cross second virial coefficients B j that are required in applying the virial equation to mixtures. Equation (4.220) is used for this purpose by treating the cross-interaction quantities as though they are properties of a real substance that follow the principle of corresponding states ... 

Here 5n and B22 are the second virial coefficients for pure species 1 and pure species 2, respectively, and 5)2 is the cross second virial coefficient. For a binary mixture... 

The virial coefficients of the pure substances as well as the cross virial coefficients depend only on temperature, not on concentration. The cross second virial coefficients Bij and Bji are identical, as they represent the same interactions between molecule i and molecule j. Thus, the second virial coefficient of a binary mixture is given by... 

Cross second virial coefficients B j can often be found in the literature [6, 7], whereas hardly any data are available for third cross virial coefficients. 

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