Excess Gibbs energy Redlich/Kister

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.

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). 

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 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... 

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. 

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. 

Many approximations used in modeling thermodynamic properties are based on the Taylor series. Examples are the virial expansions for the equation of state and the Redlich-Kister expansion of the excess Gibbs energy. Let/(x) and all its derivatives be continuous and single-valued on [a, b]. Then the Taylor series provides an approximation to f b) if we know/at a nearby point x = a and if we can evaluate derivatives of/ atx = a,... 

Smith and Brown have recently completed an extensive study of alcohol -I-alkane mixtures in which Barker s method is compared directly with chemical association models. They conclude that the equations of Barker s theory are more satisfactory than those of Kretschmer and Wiebe with concentration equilibrium constants and those of Redlich and Kister with mole fraction equilibrium constants. However, even Barker s equations are inadequate for predicting accurate excess Gibbs energies and enthalpies. Sosnkowska-Kehiaian, Hryniewicz, and Kehiaian accounted for their measurements of enthalpies of mixing of alkanes and other hydrocarbons with n-alkyl ethers using a zeroth approximation formula with one temperature-independent energy parameter. 

The Margules equation models the excess Gibbs free energy by a two-parameter Redlich-Kister polynomial. The excess Gibbs energy and the activity coefficients are given by the following equations ... 

With the help of the flexible Redlich-Kister expansion all kinds of concentration dependencies of g for binary systems can be described. The contribution of the different parameters to the value of the excess Gibbs energy is shown in Figure 5.16. However, both the Porter and the Redlich-Kister model can only be used for binary systems. Furthermore, the correct temperature dependence of the activity coefficients cannot be described using temperature-independent parameters. 

Figure 5.16 Contribution of the different parameters of the Redlich-Kister expansion to the value of the excess Gibbs energy.

Molar excess Gibbs energies of real liquid mixtures are often found to be unsymmetric functions. To represent them, a more general function is needed. A commonly used function for a binary mixture is the Redlich-Kister series given by... 

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... 

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 ... 

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 ... 

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

Use the data in Examnie 12.6 to fit the excess Gibbs free energy of the system ethanol/acetonitrile to a Redlich-Kister polynomial with two parameters, obtain equations for the activity coefficients. Use these equations to obtain the activity coefficients at infinite dilution and to construct the Pxy graph of this system at 40 "C. 

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). 

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