Activity coefficient Margules

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

For those dilute mixtures where the solute and the solvent are chemically very different, the activity coefficient of the solute soon becomes a function of solute mole fraction even when that mole fraction is small. That is, if solute and solvent are strongly dissimilar, the relations valid for an infinitely dilute solution rapidly become poor approximations as the concentration of solute rises. In such cases, it is necessary to relax the assumption (made by Krichevsky and Kasarnovsky) that at constant temperature the activity coefficient of the solute is a function of pressure but not of solute mole fraction. For those moderately dilute mixtures where the solute-solute interactions are very much different from the solute-solvent interactions, we can write the constant-pressure activity coefficients as Margules expansions in the mole fractions for the solvent (component 1), we write at constant temperature and at reference pressure Pr ... 

When the Krichevsky-Kasarnowsky equation fails it may be because of either changing activity coefficient of the solute gas with composition, changing partial molal volume of the gas with pressure, or both. The Krichevsky-Ilinskaya equation takes into account the variation in the activity coefficient of the solute gas with mole fraction by means of a two-suffix Margules equation. 

The Margules and van Laar equations apply only at constant temperature and pressure, as they were derived from equation 11.21, which also has this restriction. The effect of pressure upon y values and the constants and 2i is usually negligible, especially at pressures far removed from the critical. Correlation procedures for activity coefficients have been developed by Balzhiser et al.(ll Frendenslund et alSls>, Praunsitz et alS19>, Reid et al. 2 ) van Ness and Abbott(21) and Walas 22 and actual experimental data may be obtained from the PPDS system of the National Engineering Laboratory, UK1-23). When the liquid and vapour compositions are the same, that is xA = ya, point xg in... 

Adopting a subregular Margules model for the NaAlSi30g-KAlSi308 (Ab-Or) binary mixture and assuming that the activity coefficient of the albite component is not affected by the presence of limited amounts of the third component in the mixture (i.e., CaAljSijOg), equation 5.260 may be transformed into... 

In the A-B binary solution the activity coefficients are given by the three-suffix Margules equations ... 

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

To see this point in detail, one can express the activity coefficients in Eq. 5.33 with a third-order Margules expansion, as in Eq. 5.18 20... 

These equations can be solved for the coefficients c, d 0, c, and d 0 in terms of the infinite-dilution activity coefficients (which are binary-system properties), but the solution will not be unique. Equation 5.48d, connecting the ternary second-order Margules expansion coefficients to the binary infinite-dilution activity coefficients, shows that a constant (say, c0) can be added to any c and c[Pg.202]

Grover, J., Chemical mixing in multicomponent solutions An introduction to the use of Margules and other thermodynamic excess functions to represent non-ideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. This review article provides a fine introduction to the thermodynamic theory of mixtures underlying the Margules expansion for adsorbate-species activity coefficients. 

Given in what follows are values of infinite-dilution activity coefficients and pure-spedes vapor pressures for binary systems at spedfied temperatures. For one of the systems, determine the Margules parameters, and then apply the Margules equation to a suffident number of VLE calculations to allow construction of a Pxy diagram for the given temperature. Base your calculations on the modified Raoult s-law expression, i.e., Eq. (11.74). 

Related Calculations. The constants for the binary Margules and Van Laar models for predicting activity coefficients (see Related Calculations under Example 3.4) are simply the natural logarithms of the infinite-dilution activity coefficients A t2 = I n y(XJ and /12,1 = I n y2XJ. 

For this system, / " may be assumed to be the same as the vapor pressure (for a discussion of the grounds for this assumption, see Example 3.6). Activity coefficients can be calculated using the Wilson, Margules, or Van Laar equations (see Example 3.4). 

Use the Margules formulation rx - exp( x2/2) for the activity coefficient of species 1 and relate the mole fraction to molarity through the expression n /Xi - 1000/14 derived in Section 2.10. (a) Specializing to the case of a binary solution, show that it is necessary to invoke an equation of the form (3.4.3) to fix the parameter /9, determine /9, and then write down an expression from which m may be found, (b) Let components 1 and 2 represent water and sucrose respectively determine /9 and m numerically. 

These results again follow in a straightforward manner as an inescapable consequence from very modest beginnings the Margules formulation for activity coefficients in binary solutions. However, as with any elementary approach involving the use of many simplifying assumptions, one cannot expect quantitative agreement with experiment. Rather, the... 

Data reduction may be based on Barker s method, i.e., minimizing the sum of squares of the residuals between the experimental values of P and the values predicted by this equation (see Ex. 12.1). Assume that the activity coefficients can be adequately represented by the Margules equation. 

We used infinite-dilution activity coefficients of 10 and 20 to create Fig. 6. Both are greater than 9, so we should expect the Margules equations to predict liquid/liquid behavior. Water and toluene have infinite-dilution activity coefficients in the thousands. They really dislike each other and break into relatively pure phases. If we examine the total Gibbs free energy curve, we gain the impression that the curve is totally convex-upward however, there is a slight downward move at the extremes because of the infinite downward slope of the mixing term at the extreme compositions. The two liquid phases are almost, but not quite pure. 

Equation 19 can, however, be integrated using any of the anal5rtical expressions available for the activity coefficient In such as the van Laar, Margules,... 

The analytical expressions obtained using the two-suffix Margules equationsfor the activity coefficients and eq 20 for the molar volume are given in Appendix 2. 

All of the necessary experimental data [Vf, H2,i, 7 2,3, and E (Margules parameter)] were taken from the original publications (indicated as footnotes to Table 1) or calculated using the data from Gmehling s vapor-liquid equilibrium data compilation. Figure 1 and Table 1 show that the present eq 25 is in much better agreement with experiment than Krichevsky s eq 1 and equations A2-3—5 from Appendix 2, which involve the Margules expression for the activity coefficient. The new eq 25 provides predictions that are comparable to those of an empirical correlation for aqueous mixtures of solvents, which involves three adjustable parameters. 

The activity coefficients of a solute in a mixed solvent could be also calculated by employing various well-known phase equilibria models, such as the Wilson, NRTL, Margules, etc., which using information for binary subsystems could predict the activity coefficients in ternary mixtures (Fan and Jafvert, 1997 Domanska, 1990). 

The paper is organized as follows first, the thermodynamic relations for the solubility of poorly soluble solids in pure and multicomponent mixed solvents are written. Second, an equation for the activity coefficient of a solute at infinite dilution in a binary nonideal mixed solvent [23) is employed to derive an expression for its solubility in terms of the properties of the mixed solvent. Third, various expressions for the activity coefficients of the cosolvents, such as Margules and Wilson equations [19), are inserted into the above equation for the solubility. The obtained equations are used to correlate the HOP solubilities in binary aqueous mixed solvents and the results are compared with experiment. Finally, the case of an ideal multicomponent solvent is considered and used for ternary and higher mixed solvents. 

The activity coefficients of the constituents of the binary solvent are expressed through the two-suffix Margules equations (19)... 

Van Ness and Abbott, Int. DATA Ser., Ser. A, Sel. Data Mixtures, 1978 67 (1978)] and excess enthalpy data [Morris et al.,/, Chem. Eng. Data 20 403-T05 (1975)] are available. The VLE data are well correlated by the Margules equations. As noted in connection with Eq. (4-270), parameters Ai and A i relate directly to infinite dilution values of the activity coefficients. Thus, we have from the VLE data at 323.15 K ... 

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