Redlich-Kister expansion

An equivalent power series with certain advantages is known as the Redlich/Kister expansion (Redlich, Kister, and Turnquist, Chem. Eng. Progr. Symp. Ser. No. 2, 48, pp. 49-61 [1952]) qe... 

To this point we have used no specific mixing rule to describe the interactions of monomers of surfactants 1 and 2 in the micellar pseudophase. We have assumed, however, that only one micellar pseudo-phase exists. For our calculations we have used the Redlich-Kister expansion for w(x) with up to two parameters (10,12). Moreover, we have not yet specified the form of the function y9(x), which can be varied for modeling specific counterion association behavior. For our calculations we have used the following linear function for /3(x) ... 

Using the Redlich-Kister expansion (16) an expression for Y /acjj was developed at Kigh temperatures using vapor-liquid equilibrium data for... 

The Redlich/Kister expansion, the Margules equations, and the van equations are all special cases of a very general treatment based on ratio functions, i.e., on equations for Ge given by ratios of polynomials. These... 

Reaction coordinate, 497-501 Redlich/Kister expansion, 377 Redlich/Kwong equation of state, 82-86, 487-489... 

Since x2 = 1 - x, for a binary system of species 1 and 2, x, can be taken as the single independent variable. An equivalent poweT series with certain advantages is known as the Redlich/Kister expansion t... 

Such data are also often represented by equations similar to those used for GE data, in particular by the Redlich/Kister expansion (Sec. 12.4). 

Redlich/Kister expansion (Redlich, Kister, and Turnquist, Chem. Eng. 

Vapor/liquid equilibrium (VLE) block diagrams for, 382-386, 396,490 conditions for stability in, 452-454 correlation through excess Gibbs energy, 351-357, 377-381 by Margules equation, 351-357 by NRTL equation, 380 by Redlich/Kister expansion, 377 by the UNIFAC method, 379, 457, 678-683... 

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. 

Figure 12.11 shows experimental heats of mixing Tl// (or excess enthalpies Ji ) for the ethanol/water system as a function of composition for several temperatures between 303.15 to 383.15 K (30 and 110°C). This figure illustrates much of the variety of behavior found for = AH and = AV data for binary liquid systems. Such data are also often represented by equations similar to those used for G data, in particular by the Redlich/Kister expansion. 

The molecular-level assumption underlying the Redlich-Kister expansion is that completely random mixtures are formed, that is, that the ratio of species 1 to species 2 molecules in the vicinity of any molecule is, on the average, the same as the ratio of their mole fractions. A different class of excess Gibbs energy models can be formulated by assuming that the ratio of species 1 to species 2 molecules surrounding any molecule also depends on the differences in size and energies of interaction of the chosen molecule with species 1 and species 2. Thus, around each molecule there is a local composition that is different from the bulk composition. From this picture, the several binary mixture models have been developed. 

There are several possible expressions that can be used for the Gibbs excess energy. One is the Redlich-Kister expansion... 

An alternative form is known as Redlich-Kister expansion ... 

The parameters (A, B, C, etc.) are independent of composition they do depend on T and P, though the P dependence is usually ignored for liquids. The Redlich-Kister expansion is fully equivalent to (5.6.2), but in (5.6.3) the magnitudes of the parameters (A, B, C, etc.) are unaffected when the component labels are interchanged however, if the labels are interchanged, the signs of the coefficients on the odd-order terms (B, D, P, etc.) also change. At present, values for these parameters cannot be computed from... 

As with any infinite series, the Redlich-Kister expansion can be used for calculations only after it has been truncated. Truncation at low order can account only for small deviations from a quadratic in for highly nonquadratic behavior, we must use a high-order expansion. However, high-order expansions are troublesome to use, not only because their algebraic forms are complicated, but also because the value for each parameter must be obtained from a fit to experimental data. These complications become problematic when the expansion is applied to mixtures containing more than two components, because ternary and higher-order coefficients appear. Each level of truncation produces a different form for the activity coefficients, but since this is an introductory discussion, we consider only the simple forms that result from truncations after the first and second terms. 

On truncating the Redlich-Kister expansion (5.6.3) after the first term, we are left with the parabolic form in (5.6.1). Traditionally, (5.6.1) has been called the two-suffix Mar-gules equation [11], but this name can be ambiguous and so we prefer to call it Porter s equation [12]. Applying (5.4.10) to (5.6.1) shows that the activity coefficients are also quadratic in the mole fractions, as shown in Figure 5.8,... 

We caution that a binary mixture may obey the Porter equation (5.6.1) but still not be a quadratic mixture that is, may be parabolic in composition but and may not be. An example is the hexane-cydohexane mixture shown in Figure 5.2. Such behavior occurs because as)unmetries in and approximately cancel when they combine via the Legendre transform (5.2.18) to form g. Such cancellations are the norm rather than the exception. To say this another way, the Redlich-Kister expansion for (5.6.3) is usually dominated by the first term, which is symmetric in Xj and Xi-However, in the analogous expansions for and s, asymmetric terms are frequently important. 

If we truncate the Redlich-Kister expansion (5.6.3) after the second term, we are left with... 

The expressions in 5.6.1-5.6.3 apply only to binary mixtures however, the Redlich-Kister expansion can be extended to multicomponent solutions. One multicomponent version of the Redlich-Kister expansion is... 

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... 

The Redlich-Kister expansion for the excess Gibbs energy provides no guidance about the temperature dependence of its parameters, and so temperature effects can only be obtained from experiment. In contrast, Wilson s equation is based on a theory that estimates the temperature dependence of the parameters. 

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