Excess modelling

For molecules which differ in size or shape interactions between the surface of the molecules, different Gibbs excess models, such as NRTL [34] or UNIQUAC [35], are recommended, respectively. The predictive group contribution method UNIFAC [36] will fail if several polar groups compose a solvent or solute molecule. As a... 

The calculation of the equilibrium composition for a reactive liquid mixture requires specific excess models for the activity coefficient of every species [13]. Note that for ideal solutions the activity coefficient has a value of unity, 7 = 1. 

Figure 11. The dependence of surfactant adsorption on temperature, measured in Berea sandstone or silica sand. Adsorption levels were obtained using the surface excess model (1—10).

To classify geometric features, we select two thresholds for FVF. The values above 75% of iy-FVF are called high or excessive. Models with such iy-FVF do not necessarily exaggerate the actual fibre packing. The 75% boundary shows the models created at their geometrical limits. Likewise, moderate and low iy-FVF does not mean a better quality or a higher precision of the models. Relatively low o-FVF of... 

Therefore, the cluster of topics around the phase behavior of large molecules and charged species is one of the absolutely central themes in biothermodynamics. It forms an essential basis for instance, for all possible forms of bioseparation processes (Table 2). In some of these areas, a huge body of research is currently active. Basically three approaches can be distinguished. These are (1) the extension of existing methods and excess models (NRTL, UNI QUAG etc.) to aqueous, electrolyte systems containing biomolecules [5,6], (2) osmotic virial and closely related models based on the consideration of attractive and repulsive interactions between solutes via potentials of... 

The most straightforward (and the most developed) approach to multicomponent adsorption is in further development of the thermodynamics of a surface phase, similar to the bulk-phase thermodynamics. In this way, the Gibbs surface thermodynamics should be completed by an equation of state or by an excess model for a proper thermodynamie potential. An extended review of the fundamentals and the history of the development of this approaeh may be found in Refs. 8, 9, and 78. The approach has become espeeially popular and widely used for praetieal modeling of multicomponent adsorption after the works of de Boer [79] and, especially, Myers and Prausnitz [80]. The latter authors made the natural step of introducing the activity coefficients y of the components in an adsorbed phase. In terms of these coefficients, the chemical potentials of the adsorbate may be expressed as... 

It is important to note that whereas the BCS theory only applies to special low Tc materials, the golden-excess model covers all superconductors. The disadvantage of currently known HTSS is the brittleness of the ceramic materials that complicates the manufacture of flexible wires made up of well-aligned crystals. Raising the critical temperature, T, of cheap industrially important metals by isotopic enrichment offers a more manageable alternative technology. 

Since the accuracy of experimental data is frequently not high, and since experimental data are hardly ever plentiful, it is important to reduce the available data with care using a suitable statistical method and using a model for the excess Gibbs energy which contains only a minimum of binary parameters. Rarely are experimental data of sufficient quality and quantity to justify more than three binary parameters and, all too often, the data justify no more than two such parameters. When data sources (5) or (6) or (7) are used alone, it is not possible to use a three- (or more)-parameter model without making additional arbitrary assumptions. For typical engineering calculations, therefore, it is desirable to use a two-parameter model such as UNIQUAC. 

Null (1970) discusses some alternate models for the excess Gibbs energy which appear to be well suited for systems whose activity coefficients show extrema. 

In most cases only a single tie line is required. When several are available, the choice of which one to use is somewhat arbitrary. However, our experience has shown that tie lines which are near the middle of the two-phase region are most useful for estimating the parameters. Tie lines close to the plait point are less useful, since no common models for the excess Gibbs energy can adequately describe the flat region near the... 

A liquid-phase model for the excess Gibbs energy provides... 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

A quite different means for the experimental determination of surface excess quantities is ellipsometry. The technique is discussed in Section IV-3D, and it is sufficient to note here that the method allows the calculation of the thickness of an adsorbed film from the ellipticity produced in light reflected from the film covered surface. If this thickness, t, is known, F may be calculated from the relationship F = t/V, where V is the molecular volume. This last may be estimated either from molecular models or from the bulk liquid density. 

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

Face-centered cubic crystals of rare gases are a useful model system due to the simplicity of their interactions. Lattice sites are occupied by atoms interacting via a simple van der Waals potential with no orientation effects. The principal problem is to calculate the net energy of interaction across a plane, such as the one indicated by the dotted line in Fig. VII-4. In other words, as was the case with diamond, the surface energy at 0 K is essentially the excess potential energy of the molecules near the surface. 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Schematic model of the solid-solution interface at a particle of AgCI in a solution containing excess AgNOa.

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