Wilson energy parameters

Recently, a modification of the Wilson expressions for LCs was proposed by assuming that the interaction energy parameter between molecules of different t5rpes depends on the composition, and the following expressions were obtained ... 

The effectiveness of Wilson s model lies in the fact that only two parameters are required to describe the Gibbs energy at a given temperature. Its weakness lies in the fact that there is no clear molecular interpretation of these parameters. Wilson s approach works for a great variety of systems but when the departures from ideality are complex, more detailed models are required. Some extensions of Wilson s work have been discussed by Renon and Prausnitz [8] but they require introduction of more adjustable parameters. 

Here, a, a 2, and fl2i are the binary adjustable parameters estimated from experimental vapor-liquid equilibrium data. The adjustable energy parameters, a 2 and 21. are independent of composition and temperature. However, when the parameters are temperature-dependent, prediction ability of the NRTL model enhances. The Wilson, NRTL, and UNIQUAC equations are readily generalized to multicomponent mixtures. 

A variety of different approaches to the prediction of toxicity have been developed under the sponsorship of the Predictive Toxicology Evalnation project of the National Institnte of Environmental Health Sciences. The widespread application of compnta-tional techniqnes to stndies in biology, chemistry, and environmental sciences has led to a qnest for important, characteristic molecnlar parameters that may be directly derived from these compntational methods. Theoretical linear solvation energy relationships combine compntational molecular orbital parameters with the linear solvation energy relationship of Kamlet and Taft to characterize, nnderstand, and predict biological, chemical, and physical properties of chemical componnds (Eamini and Wilson, 1997). 

While there are reviews of the application of various quantum chemical parameters in QSARs (Karelson et al., 1996 Famini and Wilson, 2002), little attention has been paid so far to the dependence of descriptor values on the level of theory. This holds true in particular with respect to potential discrepancies between semiempirical and ab initio methods when calculating parameters such as frontier orbital energies and descriptors that characterize the molecular charge distribution. 

The electron gas model adequately describes the conduction of electrons in metals however, it has a problem, that is, the electrons with energy near the Fermi level have wavelength values comparable to the lattice parameters of the crystal. Consequently, strong diffraction effects must be present (see below the diffraction condition (Equation 1.47). A more realistic description of the state of the electrons inside solids is necessary. This more accurate description is carried out with the help of the Bloch and Wilson band model [18],... 

In these equations, y, and y2 are activity coefficients of components 1 and 2, respectively, GE is Gibbs molar excess free energy, A, 2 and A2j, are Van Laar parameters, G, 2 and G2j, are Wilson parameters, that is,... 

In principle, the parameters can be evaluated from minimal experimental data. If vapor-liquid equilibrium data at a series of compositions are available, the parameters in a given excess-free-energy model can be found by numerical regression techniques. The goodness of fit in each case depends on the suitability of the form of the equation. If a plot of GE/X X2RT versus X is nearly linear, use the Margules equation (see Section 3). If a plot of Xi X2RT/GE is linear, then use the Van Laar equation. If neither plot approaches linearity, apply the Wilson equation or some other model with more than two parameters. 

Theoretical Linear Solvation Energy Relationship (TLSER) With the LSER descriptors of Kamlet and Taft in mind, Famini and Wilson developed QM-derived parameters to model terms in Eq. [18] and dubbed these the TLSER descriptors. Descriptor calculations are done with the MNDO Hamiltonian in MOPAC and AMP AC. MNDO has greater systematic errors than do AMI and PM3, but the errors tend to cancel out better in MNDO-derived correlation equations. A program called MADCAP was developed to facilitate descriptor calculation from MOPAC output files. 

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... 

In usual applications, the parameters of the UNIQUAC and Wilson activity coefficient models are fitted to experimental phase equilibria data. However, in the development of these models, the adjustable parameters correspond to the difference of interaction energies between the like and the unlike species. 

Once the interaction energies were obtained, they were used to calculate the parameters in the UNIQUAC and Wilson models given by Eq. (24). To test the validity of the method, low-pressure vapor-liquid equilibrium (VLE) predictions were made for several binary aqueous systems. The calculations were done using the usual method assuming an ideal vapor phase (Sandler, 1999). Figures 7 and 8 show the low-pressure VLE diagrams for the binary aqueous mixtures of ethanol and acetone [see Sum and Sandler (1999a,b) for results for additional systems and values of the... 

Using Wilson s parameters for the carbon tetrachloride-acetonitrile system, estimate the Raoult law activity coefficients for each component in a equimolar solution. Then estimate the molar Gibbs energy of mixing. 

Fig. 1.11 Plot of the excess Gibbs energy of mixing for the carbon tetrachloride-acetonitrile system against the mole fraction of acetonitrile at 45°C. The points show the experimental results and the solid curve was calculated using equation (1.10.7) with the parameters given by Wilson [7] (see text).

This equation (Wilson, 1964) was developed on the basis of a liquid molecular model which assumes that interactions between molecules of two different components depend on the volume fraction of each component in the vicinity of a molecule of a given component. The volume fractions are determined from the probability of finding a molecule of one component or another in the vicinity of a molecule of the given component. The probabilities, and hence the volume fractions, are expressed through Boltzmann distribution functions of energy in terms of binary interaction parameters. The result for volume fractions is... 

The relationship between the potential function K(R) and the observable spectroscopic parameters is summarized in Figure 2. The harmonic vibration frequencies are obtained as the eigenvalues of a secular determinant involving the quadratic force constants and the atomic masses and molecular geometry (the F and G matrices of Wilson s well-known formalism) by a calculational procedure discussed in detail by Wilson, Decius, and Cross.1 The eigenvectors determine the normal coordinates Q in terms of which the kinetic and quadratic potential energy terms are both diagonal (R = LQ). The various anharmonidty constants and vibration/rotation interaction constants are obtained in terms of the... 

The activity coefficients are typically computed from a model for the excess Gibbs energy g, as described in the thermodynamics chapter (Chapter 4). The most popular are the Wilson, NRTL, and Uniquac models, described in detail in many places [15, 36 0]. They contain two or three adjustable (and possibly temperature-dependent) parameters per binary. One cannot predict which model will be best for a given system however, the Wilson equation is incapable of describing LEE. 

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