Cross virial coefficients

Tsonopoulous, C. "Second Virial Cross Coefficients Correlation and Prediction of k..," Adv. in Chem., in press(1979). 

Second Virial Cross-Coefficients Correlation and Prediction of kij... 

Second virial cross efficients between the polar chemical and the light gases appear in the literature less often than second virial coefficients for pure polar substances. However, these coefficients (B ) can be used to determine AP(TC) of the polar solute. By substituting Equations 5, 12, and 13 in Equation 16 and by using the definition of the second virial cross coefficient (3) ... 

Equations 17 and 14 can be fit to the experimental second virial cross coefficients to find AP(TC). 

Water Solute in Hydrocarbon-Rich Vapor and Liquid. The pure component parameters of water solute AP(TC) and a were determined by using Equations 5 and 16 to fit the gas-phase volumetric properties of steam (5) and the second virial cross coefficients of steam and light gases such as methane, ethane, and nitrogen (6). The least-squares minimization technique was used to find the parameters that gave the minimum deviations between calculated and experimental pressures and second virial cross coefficients. (Table I lists the parameters for pure steam and of other compounds used in this study.)... 

Methanol in Hydrocarbon-Rich Vapor and Liquid. The volumetric properties of methanol gas (12) and the second virial cross coefficients of methanol and light gases (13) were used to determine the pure-component parameters AP(TC) and a for methanol. Table II shows the enthalpy departure of gaseous methanol from ideal gas at three temperatures and several pressures. For comparison, the experimental values (14) and the values calculated by the Soave equation (1) are also shown. Table II indicates that the Won modified equation of state predicts the enthalpy departure of methanol very well at low temperatures and fairly well at high temperatures, but that the original Soave equation considerably underestimates the enthalpy departure at all temperatures and pressures. Since the original Soave equation was meant to be applied only to hydrocarbons, we are not surprised at this result. Comparison of calculated and experimental second virial cross coefficients between methanol and methane (and also C02) is presented elsewhere (15).)... 

Tessier, P.M., Sandler, S.I., Lenhoff, A.M. Direct measurement of protein osmotic second virial cross coefficients by cross-interaction chromatography. Protein Sci. 13, 1379-1390 (2004)... 

The coefficient Bij characterizes a bimolecular interaction between molecules i and J, and therefore Bij = Bji. Two lands of second virial coefficient arise Bn and By, wherein the subscripts are the same (i =j) and Bij, wherein they are different (i j). The first is a virial coefficient for a pure species the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients Cm, Cjjj, and are for the pure species and Qy = Cyi = Cjn, and so on, are cross coefficients. 

Values of the pure-species virial coefficients Bj, B, etc., can be determin from the generalized correlation represented by Eqs, (3.47) through (3.49). cross coefficients Bik, By, etc., are found from an extension of the same correlatic For this purpose, Prausnitzf has rewritten Eq. (3.47) in the more general for... 

Here CiU and C222 are the third virial coefficients for pure species 1 and 2 whereas C,12 and C122 are cross-coefficients. Published generalized correlation for third virial coefficients ) are based on a very limited supply of experimen data. Consistent with the mixing rules of Eq. (11.44) and (14.1), the temperatu ( derivatives of B and C are given exactly by... 

Two types of virial coefficients have appeared Bh and B22, for which successive subscripts are the same, and B12, for which the two subscripts different. The first type represents the virial coeffident of a pure species second is a mixture property, known as a cross coefficient. Both are functions temperature only. 

Although developed for pure materials, these correlations can be extended to gas or vapor mixtures. Basic to this extension are the mixing rules for the second virial coefficient and its temperature derivative as given by Eqs. (4-60) and (4-62). Values for the cross coefficients Bp with i and their derivatives are provided by Eq. (4-72) written in extended form ... 

The second virial pure component and cross coefficients are next calculated ... 

For gases at low and moderate pressures, it is often preferable to use the virial expansion that provides successive corrections to the ideal-gas law (see Chapter 4, Thermodynamics ). The first correction (called the second virial coefficient, B) has been derived from volumetric data for many pure fluids. Higher virial coefficients are much less well known, as are the cross-coefficients for interactions between unlike molecules. Estimation techniques for second (and to a lesser extent third) virial coefficients exist [15] and work reasonably well for many fluids, especially organic compounds of low polarity. Dymond et al. have compiled extensive experimental data for virial coefficients [32]. 

We use mixing rules to extend equations of state to mixtures. The mixing rules allow us to extrapolate these equations to mixtures, from mosdy pure component data. Mixing rules for van der Waals-type parameters a and b were developed based on a two-body attractive interaction and a hard sphere repulsion, respectively. The binary interaction parameter allows us to better describe the cross coefficient, ai however, data from the mixture are needed. Mixing rules for the viral coefficients arise from a theoretical basis. Mixing rules for the second virial coefficient, B, are based on two-body interactions for the third virial coefficient, C, on three-body interactions and so on. Finally, Kay s rules were presented, from which we can find the psuedocritical properties of the mixture from the pure component properties. These values allow us to apply the generalized compressibility charts to mixtures. 

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... 

---