Redlich-Kister expression

Figure 3.11 Contributions to the molar excess Gibbs energy of mixing from the four first terms of the Redlich-Kister expression (eq. 3.76). For convenience Q - A1 - A2 - A3 1.

The coefficients y. of the Redlich-Kister expression (3.43) for the molar excess Gibbs energies and enthalpies of the miscible aqueous cosolvents on the list are shown in Table 3.13 adapted from [56]. It should be noted that whereas the G (Xj.) curves for many aqueous cosolvent systems are fairly symmetrical, the curves for some systems are quite skew, even changing sign from negative at water-rich compositions to positive beyond a certain... 

Again because of the similarity of the two components, consider as a first approximation the single-parameter Redlich-Kister expression, Eq. 13.14.2 ... 

S.Hermsen and Prausnitz, for example, used a two-parameter Redlich-Kister expression, and also took into account the vapor-phase nonideality. The obtained fit of the data is, of course, somewhat better. 

A similar derivation gives equation (5.34). To use equations (5.33) and (5.34) to calculate V and 1+, must be known as a function of xj or.vi. An expression often used is the Redlich-Kister equation, given by... 

Equations (7.93) and (7.94) are usually applied to mixtures of nonelectrolytes where Raoult s law standard states are chosen for both components. For these mixtures, Hi is often expressed as a function of mole fraction by the Redlich-Kister equation given by equation (5.40). That is... 

Here the first term represents the lattice stability components of the phase , the second term the Gibbs energy contribution arising from cluster calculations and the third term is the excess Gibbs energy expressed in the form of a standard Redlich-Kister polynomial (see Chapter 5). 

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

The excess Gibbs energy of the ternary mixture was expressed through the Wilson [38], NRTL [39] and Zielkiewicz [32] expressions. Because of the agreement between the latter two expressions, detailed results are presented only for the more simple NRTL expression. The parameters in the NRTL equation were found by htting x-P (the composition of liquid phase-pressure) experimental data [32]. The derivatives (9 i/9xi) c2 ( IX2/dx2)xi and (diX2/dxi)x2 in the ternary mixture were found by the analytical differentiation of the NRTL equation. The excess molar volume (V ) in the binary mixtures (i-j) was expressed via the Redlich-Kister equation... 

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

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... 

You may have noticed that the Margules model is equivalent to the two-parameter Redlich-Kister polynomial used in Examples 12.6 and 12.7. This maybe confirmed by setting A12 = Oo andA = Oo + Oi to the above equations (see Example 12.8. below). In the form given here, the expressions for the activity coefficients are symmetric in the two components such that each expression is obtained from each other by switching the subscripts 1 and 2. 

The bulk properties are conveniently described by Redlich-Kister-type expressions for binary mixtures of water (W) and a co-solvent (S) ... 

Many equations, either empirical or derived from models, are available to represent excess thermodynamic properties as a function of the liquid mole fraction and of a number of adjustable parameters for use in Equations 4.3 and 4.4 (Prausnitz, Lichtenthaler, and Gomes de Azevedo 1999). In the treatment of our experimental data, we have examined the capability of many of these equations to fit data with the smallest number of parameters and with the least standard deviation of the O.F. (Lepori et al. 1998). For the majority of systems, either binary or ternary, examined by us in the course of about two decades, the rational form (Myers and Scott 1963) of the Redlich-Kister (RK) expression (Redlich and Kister 1948), and the Wilson (Wilson 1964) equation in the extended form (Novtik, MatouS, and Pick 1987), have resulted in the most appropriate representations of and For ternary systems, the excess functions can be expressed as the sum of a contribution (subscript B), which depends only on the parameters of the three binary systems, and a ternary contribution (subscript T), which involves additional parameters. 

For the description of g /RT for all other binary systems a more flexible expression is required. The simplest way is the introduction of further adjustable parameters, as in the Redlich-Kister expansion [10] ... 

By varying the values of parameters in an expression for the excess molar Gibbs energy, we can model the onset of phase separation caused by a temperature change. Figure 11.5 shows the results of using the two-parameter Redlich-Kister series (Eq. 11.1.36). 

The derived bitraty excess properties were correlated with the Redlich-Kister eqrratiort, according to the expression ... 

The Redlich - Kister equation [48Red], a power series expansion, is used to express the excess Gibbs energy, for the interaction between the two elements i and j as follows ... 

Margules, and Redlich-Kister and are considered purely empirical and the local composition type, that includes the Wilson, NRTL, and UNIQUAC expressions and are considered semi-empirical. 

Redlich and Kister (RK) [1] employ a parabola 12X1X2 as the basic form for the of a binary solution. A power series in (Xj - JC2) is added to to express the asymmetry, if any, of the function. 

In spite of the simplicity of this relationship, the expressions for computing the activity coefficients for multicomponent mixtures are quite cumbersome. Since the equations of Wohl,76 Van Laar,68 Margules,43 Scatchard and Hamer,57 Redlich and Kister,52 and others are well documented in the literature, a restatement of these equations is not presented. Instead, a brief introduction to the newer methods including the Wilson equation, the NRTL, UNIQUAC, ASOG, and the UNIFAC methods is presented. 

Still with the aim of having mathematical expressions for the representation of the solution, Redlich and Kister offered a representation that provides an expansion of the excess Gibbs energy, a pure-substance reference in the same state of segregation as the solution (reference (I)), the equivalent of ihe Margules expansion for the activity coefficients. For a two-component solution, the pol5momial expansion up to order m is written ... 

Redlich and Kister have proposed the following polynomial expression to represent of a binary mixture ... 

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