Redlich-Kister series

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... 

For many binary liquid systems, the measured dependence of on composition is reproduced reasonably well by the two-parameter Redlich-Kister series... 

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). 

Figure 11.5 Binary liquid mixtures at 1 bar. The curves are calculated from the two-parameter Redlich-Kister series using the following parameter values.

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... 

Taking this process further, more complex composition dependencies to Q can be considered and it is straightforward to show that a general formula in terms of a power series should provide the capability to account for most types of composition dependence (Tomiska 1980). The most common method is based on the Redlich-Kister equation and Eq. (5.19) is expanded to become... 

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... 

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. 

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,... 

This four volumes series contains vapor-liquid equilibrium data (i.e., compositions of vapor and liquid phases as functions of temperature and pressure) covering the regions for which experimental measurements exi t. The systems are almost entirely organic. The data have been recalculated Into SI units and have been smoothed in many cases, often using the Redlich-Kister equation. There are detailed references to the source literature and there is an alphabetic index of the systems covered. The four volumes are ... 

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 ... 

Redlich and Kister suggested that solutions of different degrees of nonideality should be represented by a power series of the form... 

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. 

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