Virial coefficients for mixture

Knobler and his co-workers have measured the interaction virial coefficient, for mixtures of n-alkanes and for n-alkane + perfluoro-n-alkane... 

In order to obtain interaction second virial coefficients for mixtures, some method is required for determining the acentric factor a>y and the pseudo-critical constants T ij and pertaining to the unlike interactions. In the present case, extended van der Waals one-fluid mixing rules are applied in terms of which... 

Lets now consider application of mixing rules to the virial equation. Since there is a sound theoretical basis for the virial coefficients in terms of intermolecular interactions, we can relate the virial coefficients for mixtures in terms of intermolecular potentials via Equation (4.29) with no arbitrary assumptions that is, these mixing rules are rigorous results from statistical mechanics. 

Subroutine BIJS2. This subroutine calculates the pure-component and cross second virial coefficients for binary mixtures according to the method of Hayden and O Connell (1975). 

CALCULATE EFF SECOND VIRIAL COEFFICIENT FOR COMP I IN MIXTURE, SS(I)... 

BUS calculated second virial coefficients for pure compoments and all binary pairs in a mixture of N components (N 20) at specified temperature. These coefficients are placed in common storage /VIRIAL/. 

Gas mixtures are subject to the same degree of non-ideality as the one-component ( pure ) gases that were discussed in the previous section. In particular, the second virial coefficient for a gas mixture can be written as a quadratic average... 

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. 

The theoretical foundations of these rules are, however, rather weak the first one is supposed to result from a formula derived by London for dispersion forces between unlike molecules, the validity of which is actually restricted to distances much larger than r the second one would only be true for molecules acting as rigid spheres. Many authors tried to check the validity of the combination rules by measuring the second virial coefficients of mixtures. It seems that within the experimental accuracy (unfortunately not very high) both rules are roughly verified.24... 

The importance of the virial-coefficient equations is especially great for mixed electrolytes. Of the needed virial coefficients for a complex mixture such as sea water, most are determined by the pure electrolyte measurements and all the others of any significance are determined from data on simple mixtures such as NaCl-KCl, NaCl-MgC, NaCl-Na.SO, etc., which have been measured. The effect of the terms obtained from mixtures is very small in any case and these terms can be ignored for all but the most abundant species. 

For a multicomponent mixture, a virial coefficient is needed to account for each possible interaction. The second virial coefficients for a two component mixture are Bpi, Bp2> and B22 where B- represents the interaction between two molecules of component 1, Bp2 represents the interaction between a molecule of 1 and a molecule of 2, and B22 represents interaction between two molecules of 2. A tabulation of some compounds whose virial coefficients have been measured by GC is given in Table 11.6. 

Interaction parameter in Flory-Huggins treatment of polymer mixtures after normalization on a per monomer basis this becomes Xay also susceptibilities in discussion of density functional theories. Applied external field acting on species a in conformation X". Applied external field as a function of position acting on species a. Contribution of attractive interactions to the second virial coefficient for species pair ay also van der Waals coefficient. 

Equation (5) is an equation-of-state for the adsorption of a pure gas as a function of temperature and pressure. The constants of this equation are the Henry constant, the saturation capacity, and the virial coefficients at a reference temperature. The temperature variable is incorporated in Equation (5) by the virial coefficients for the differential enthalpy. This equation-of-state for adsorption of single gases provides an accurate basis for predicting the thermodynamic properties and phase equilibria for adsorption from gaseous mixtures. 

Data are readily available for pure component and binary interaction second virial coefficients for a large number of components and binaries. Binary interaction coefficients are required for extending the equation to mixtures. The simplicity of the equation, the availability of coefficient data, and its ability to represent mixtures are some of the reasons the virial equation of state is a viable option for representing gases at densities up to about 70% of the critical density. It may be used for calculating vapor phase properties at these conditions but is not applicable to dense gases or liquids. 

Extending the same procedure for mixtures, say of two components, A and B will give us the second virial coefficient for a mixture. The first-order correction to the ideal-gas behavior of the mixture is... 

The factor in the square brackets can be viewed as an average virial coefficient for the mixture of two components. We now invert this relation by assuming an expansion of the total density p = pT in the form... 

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 result of a calculation can be quite sensitive to the values for the k. Although these quantities can be correlated at times against combinations of properties for pure species i and / (e.g., critical-volume ratios), they are best treated as purely empirical parameters, values of which are (ideally) backed out of good experimental mixture data for the type of property which is to be represented. Thus, if accurate calculation of low-to-moderate-pressure volumetric properties is required, then the kif could be estimated from available data on mixture second virial coefficients for the constituent binaries. Alternatively, if application to multicomponent VLE calculations is envisioned, then the ki would be best estimated from available VLE data on the constituent binaries. (It... 

In Equation 2, B is the second virial coefficient for the natural gas mixture and is a function only of temperatnre and composition... 

Finally, we emphasize that, even if we had several virial coefficients for a substance, the virial equations still only apply to gases and gas mixtures— both the density expansion and the pressure expansion fail to converge for liquids. Moreover, in practice we can find data or correlations for, at most, B and C, so the expansions should only be used for gases at low to moderate densities. 

Table 4.2 Values of second virial coefficients for methane(l)-sulfur hexafiuoride(2) mixtures. values from [21]. Values of B and dB / dT are for equimolar mixtures.

T = 300 K, the second term contributes about 20 J mol to G (0.5). The third term contributes about 4 J mol, while the last term contributes generally less than 0.1 J mol k To obtain G (0.5) to an accuracy of 1 Jmol for the above conditions, it is necessary to know the second virial coefficients for the pure components and the mixture to within 20cm mol. For non-polar + polar gas mixtures and polar + polar gas mixtures the usual methods for calculating Sab are far from reliable. For example, if the calculated values of Sab for acetone + nitromethane at 300 K listed by Reid and Sherwood were used in equations (2) and (3) to calculate G (0.5) an error of approximately 30 J mol would be introduced. In circumstances where the second virial coefficients are not known and the prediction methods are unreliable, it is advisable that the coefficients for the pure components and the mixture be measured. It should be sufficient for Rab to be determined from one measurement at y 0.5. 

Here ns is the amount of substance of stationary liquid, pi is the saturated vapour pressure of the solute at temperature r. Bag is the mixture virial coefficient for solute 4- carrier gas interaction, Bcc is the virial coefficient of the carrier gas, Fjj is the partial molar volume of the solute at infinite dilution in the solvent, is the molar volume of pure liquid A, and pi and po are the column inlet and outlet pressures. The chemical potential at infinite dilution can be calculated by measuring the retention volume of an infinitely small sample for various inlet and outlet pressures and extrapolation to zero pressure drop across the column. Everett and Stoddart proposed using equation (33) to determine the mixture second virial coefficients. The precision in Bag from this method is nearly equivalent to the best static methods. The assumptions required to derive the above equation have been examined by a number of authors. - ... 

In complete contrast the measurements by Sigmund et al. of the second and third virial coefficients of CF4 + SF showed that the Lorentz-Berthelot rules predict the experimentally determined interaction parameters for the unlike interactions within experimental error. Mixtures of spherically symmetric fluorocarbons thus closely resemble similar hydrocarbon mixtures in this respect. Lange and Stein reported measurements of the second and third virial coefficients for CF3H + CF4 mixtures. A distinct weakness in the unlike interactions was noted although no detailed calculations were made. 

More recently, Khoury and Robinson calculated virial coefficients for ethane + hydrogen sulphide mixtures from an effective Lennard-Jones 6—12 potential that includes a contribution from the dipole-induced-dipole interaction. The parameters were evaluated by fitting their measurements on CgHg and HgS to a 6—12 potential and to a Stockmayer potential respectively. Bradley and King have used several potential functions to analyze interactions of phen-anthrene with a number of small polyatomic molecules. 

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