Margules equations, binary

Broul et al. (13) and Hala (14) developed a correlation scheme for systems containing two solvents and one salt, which they applied to several salt concentrations, not just to the saturation level as in the studies mentioned above. They utilized the binary VLE data for the three binaries (solvent 1-salt solvent 2-salt and solvent 1-solvent 2) along with the ternary data to correlate successfully the ternary results. They employed the Margules equation (15) with the addition of a term to account for the coulombic interactions. 

Because of the limitations of the Margules equation—especially in predicting multicomponent VLE data—the Wilson, NRTL, and LEMF (16) equations are employed in this study. The experimental data on the systems presented in Table I were used in this work. These are the only systems for which both binary and ternary data could be found in the literature. As a matter of fact, uncertainties do exist about the accuracy of the two HgC systems. The maximum boiling... 

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

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). 

On the basis of the Duhem-Margules equation, prove that if one component of a binary mixture exhibits positive (negative) deviations from Raoult s Law, the second must do likewise. (See S. Glasstone, "Thermodynamics for Chemists", D. Van Nostrand, New York, 1947, Chapter 14.)... 

The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a general treatment based on rational functions, i.e., on equations for G /x X2RT given by ratios of polynomials. They provide great flexibility in the fitting of VLE data for binary systems. However, they have scant theoretical foundation, and therefore fail to admit a rational basis for extension to multicomponent systems. Moreover, they do not incorporate an explicit temperature dependence for the parameters, though this can be supplied on an ad hoc basis. 

A binary liquid system exhibits LLE at 298.15 K (25°C). Determine from each of the following sets of miscibilitv data estimates for parameters A12 and Aji in the Margules equation at 298.15 K(25°C) ... 

Domanska, U., 1990. Solubility of acetyl-substituted naphthols in binary solvent mixtures. Huid Phase Equilibria. 55, 125-145. Fan, C.H., Jafvert, C.T., 1997. Margules equations applied to PAH solubilities in alcohol-water mixtures. Environ. Sci. Technol. 31, 3516-3522. 

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

Sc. Liquid and Vapor Compositions.—Some general rules concerning the relative compositions of liquid and vapor in equilibrium, which are applicable to systems of all types, may be derived from the Duhem-Margules equation, using the form of (35.1). Since the increase in the mole fraction of one component of a binary mixture must be equal to the decrease for the other component, dNi is equal to — c/N2, as seen in 34b hence equation (35.1) may be written as... 

The liquid phase is a binary mixture so we may use the Margules equation (Table D.2) to estimate the activity coefficients. The interaction parameters are taken from Gmehling and Onken (1977ff, Vol. 1/3 + 4, p. 197)... 

At 120°C, both components in this binary indicate positive deviation in one composition range and negative deviation in another (Figure 1.13). The system also indicates a slight maximum pressure azeotrope near a water mole fraction of 0.95 (Figures 1.13 and 1.15). Additionally, a maximum is observed for the activity coefficient of water (Figure 1.14, curve 1) and a minimum for the activity coefficient of 3-hydroxy-2-butanone (curve 2). These phenomena are best represented by the Margules equation. 

Since we have the concentrations of the coexisting equilibrium liquid phase we can determine two binary parameters. Also, since we are interested in two different temperatures (LLE at 20°C and VLE at 73.4° C) we want an activity coefficient model with some built in temperature dependence (otherwise, we will get LLE with the same compositions at all temperatures.) Consequently, I will use the two constant Margules equation... 

In both solid and gaseous solutions, virial equation-based Raoultian coefficients have often been proposed. For example, the Margules equations, often used in binary and sometimes in ternary solid solutions and which have a virial equation basis, were proposed originally for gaseous solutions. However, there is no satisfactory general model for Raoultian coefficients in multi-component solid solutions, and the tendency in modeling has been to treat these solutions as ideal (i.e., to use the mole fraction of a solid solution component as its activity see Equation (3.13)). 

FIGURE 1.4-2 Composition dependence of g ( gKlxlx2RT) and of activiiy coefficients for binary mixtures. as represented by the two-parameter Margules equation (solid curves) and by the van Laar equation (dashed curves). For all cases. In y = 0.5 different cases correspond n> different values for In y, ... 

Flgare 1.5-2 shows exparimental and correlated binary VLE data for three dioxane-n-alkane systems at 80°C.m The pressure levels are modest (0.2-1.4 amt) liquid-phase nonidealities are sufficiently large to promote a2eotropy in all threa cases. Equations (1.5-12)—(1.5-15) were used for the data reduction, with experimental values for the Pf1 and virial coefficients were estimated from the correlation of Tsono-poulos.7 Activity coefficients were assumed to be represented by the three-parameter Margules equation, aed (he products of the data rednction were seis of valnes for parameters Al2, Ait. and D in Eqs. (1.4-10) and (1.4-11). The parameters so determined produce the correlations of the data shown by the solid curves in Fig. 1.5-2. For all threa systems, the data are represented to within their exparimental uncertainty. 

Fig. 11.11. Raoultian activities of H2O and CO2 in the binary solution at 600°Cand 2 kb. Data from Bowers and Helgeson (1983). The curved lines are fit to the data with Margules equations, discussed in Chapter 15. The inset refers to a discussion of standard states in Chapter 12.

Next we can finally see how the activity coefficient relates to the Margules equations for this case. Recall from Chapter 9 that the partial molar quantity of one component in a binary solution can be obtained graphically from the tangent (as with the chemical potentials /j,b and /ta in the coexisting solutions of Figure 15.3a). From equation (9.6), the partial molar free energy or chemical potential of component A in a solution of A and B is given by... 

The Margules equations such as those in Table 15.1 can be fitted by standard least-squares regression analysis to data for real solutions. For example, if data for the total free energy of a binary asymmetric solution is available over a range of compositions at different T and P, you could fit equation (15.41) for Greai (or the equation for in Table 15.1) and obtain Wq, and ITcj as regression parameters. The same could be done with the equations for excess enthalpy, entropy, and so on. This permits construction of phase diagrams and determination of thermodynamic properties based... 

A number of other models have been used in conjunction with Equation 7 to calculate the binary phase diagrams. Among these are the quasichemical equation (35,44), a truncated Margules equation (45,46), Darken s formalism (47,48), and various forms of the chemical theory, in which associated liquid species are postulated and some assumptions are made about the physical interactions between the species (49-51). Several of these studies have considered the liquid phase properties as well as the liquidus in the parameter estimation (45,46,51). 

So if we have experimental data for both Y Y2 / then (5.6.14) and (5.6.15) provide a straightforward way to obtain values for the Margules parameters. If a binary mixture happens to have Yi° = Y2 that A = A2, then the Margules equations collapse to the Porter equations. 

The Margules equations apply to many binary mixtures, including those that display positive deviations from ideality, mixtures that exhibit negative deviations from... 

For mixtures that do not obey the Porter or Margules equations, additional high-order terms must be kept in the Redlich-Kister expansion hence, more parameters must be evaluated from experimental data. Alternatively, if we want to keep only two parameters, then we must abandon the Redlich-Kister expansion for some more complicated representation of g. Many functional forms have been proposed [1, 2], but here we restrict our attention to a useful expression proposed by Wilson in 1964 [14] and now identified as one of the class of "local-composition" models [2], For binary mixtures Wilson s equation takes the form... 

Consider a binary mixture that obeys the Margules equations. What conditions, if any, must the parameters Aj and A2 obey if vs. passes through an extremum at some composition 0 < Xj < 1 ... 

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