Redlich-Kister polynomial

The third term of Eq. (14), Gxs, is the excess term of the free energy. Although several of the aluminum alloys considered here form ordered intermetallic compounds, a regular-solution type model was used to describe their excess free energy. Gxs is described by the following Redlich-Kister polynomial,... 

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). 

Models incorporating linear composition dependencies to O (the subregular solution model), as well as others allowing for complex composition dependencies, have been developed. The most commonly used model is by Austrian-born American immigrant Otto R. Redlich (1896-1978) and Albert Theodore Kister (d. 2002) of the Shell Development Company in 1948, which is now known as the Redlich-Kister polynomial (Redlich and Kister, 1948). The total Gibbs energy of a binary system, using the Redlich-Kister model is ... 

The descriptions of all ternary systems were combined in one dataset to simulate the phase equilibria in the quaternary Si-B-C-N system. The results of the thermodynamic calculations of individual systems are shown in the corresponding sections. Thermodynamic models used are the Redlich-Kister polynomial [39], extrapolations according to Muggianu et al. [40] and the compound energy formalism [41] to describe the solid solution phases )S-boron, SiBn, SiBg, SiBs and B4+ C. 

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. 

Solution We fit the ratio /RTto a two-parameter Redlich-Kister polynomial... 

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 ... 

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 terms and Li,ij represent the interaction parameters between the atoms on one sublattice for a given occupancy of the other, and can be described by a Redlich - Kister polynomial, as follows ... 

Guggenheim s polynomial expansion (equation 3.171 Guggenheim, 1937) and the two Redlich-Kister equations (3.172 and 3.173 Redlich and Kister, 1948) are of general applicability for any type of mixture ... 

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... 

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. 

The isothermal compressibilities have been calculated with eq 5, using for the isothermal compressibilities of the pure substances the data from refs 53—56 (only the value for 2-butanol was taken as that for isobutanol). The data have been fitted using the Redlich—Kister equation.The values of D have been obtained from the activity coefficients or total pressure data by the sliding polynomials method. To check the accuracy of our calculations, the D values have been... 

For more accurate calculations of the partial molar volume (or any other partial molar property), an analytical, rather than graphical, procedure is used. First, one fits the volume change on mixing, AmixY. with a polynomial in mole fraction, and then the necessary derivative is found analytically. Since AmixY must equal zero at X = 0 and xi = 1 (,X2 = 0), it is usually fit with a polynomial of the Redlich-Kister form ... 

Now, some considerations can be written about the Heric s and this last polynomial function. In fact, the interaction terms in Equations 7 and 13 could be formally derived from the more general Redlich-Kister one (39], being the adjustable coefficients derived from this relation truncated at different power terms. However, it should be noted that for these binary mixtures, Heric s equation provides better results if compared with the polynomial one, despite the number of adjustable parameters. 

The second polynomial is a Redlich-Kister type of equation which includes a skewing factor k, similar to Eq. (1.28), and this was used to fit measurements at 375 K and higher temperatures where the hydrogen bonding... 

The authors mention similarity between the Legendre polynomials and the Redhch-Kister expansion. This similarity is not that striking if we realize that the Legendre polynomials are orthogonal on interval ( — 1 1) while each even order term in the Redlich-Kister expansion is orthogonal to each odd order term on interval (0 1). Possibility of testing consistency of isobaric data is also mentioned. 

Excess molar volumes of v of 1,2-ethanediol + water were measured atp = 0.1 MPa with a faithful copy of the vibrating tube densimeter DMA 602 from Anton Paar. All binaiy mixtures were measured at the temperatures (308.2, 313.2, and 318.2) K. Values of V are negative for all the mixtures studied over the whole concentration range and for all temperatnres. Resnlts were correlated by polynomial equations of Redlich and Kister (1948). 

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

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