Multicomponent systems, pair

The most recendy developed model is called UNIQUAC (21). Comparisons of measured VLE and predicted values from the Van Laar, Wilson, NRTL, and UNIQUAC models, as well as an older model, are available (3,22). Thousands of comparisons have been made, and Reference 3, which covers the Dortmund Data Base, available for purchase and use with standard computers, should be consulted by anyone considering the measurement or prediction of VLE. The predictive VLE models can be accommodated to multicomponent systems through the use of certain combining rules. These rules require the determination of parameters for all possible binary pairs in the multicomponent mixture. It is possible to use more than one model in determining binary pair data for a given mixture (23). 

Function g contains pure-species parameters only, whereas function g incorporates two binary parameters for each pair of molecules. For a multicomponent system. 

Only the Wilson, NRTL, and UNIQUAC equations are suited to the treatment of multicomponent systems. For such systems, the parameters are determined for pairs of species exactly as for binary systems. 

Diffusion of ions can be observed in multicomponent systems where concentration gradients can arise. In individnal melts, self-diffnsion of ions can be studied with the aid of radiotracers. Whereas the mobilities of ions are lower in melts, the diffusion coefficients are of the same order of magnitude as in aqueous solutions (i.e., about 10 cmVs). Thus, for melts the Nemst relation (4.6) is not applicable. This can be explained in terms of an appreciable contribntion of ion pairs to diffusional transport since these pairs are nncharged, they do not carry cnrrent, so that values of ionic mobility calculated from diffusion coefficients will be high. 

A significant advantage of the Wilson equation is that it can be used to calculate the equilibrium compositions for multicomponent systems using only the Wilson coefficients obtained for the binary pairs that comprise the multicomponent mixture. The Wilson coefficients for several hundred binary systems are given in the DECHEMA vapour-liquid data collection, DECHEMA (1977), and by Hirata (1975). Hirata gives methods for calculating the Wilson coefficients from vapour liquid equilibrium experimental data. 

The extension of vector-algebraic techniques to multicomponent systems of higher dimensionality (degrees of freedom /> 2) can be carried out straightforwardly, even though one loses the convenience of mutually complementary pairs (X, X ) and orthogonal complementary conjugates (X, x ) that are a special feature of /= 2. In a space of / dimensions, a... 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

This important peculiarity, which allows one to determine the kinetic parameters of m-component copolymerization on the basis of the analysis of the experimental data obtained under the copolymerization of m(m — l)/2 monomer pairs vanishes if one uses other kinetic models instead of the terminal one. There are a number of models describing multicomponent systems which account for the influence of the penultimate unit [47], the formation of the binary [48] and triple [49] complexes and also for the depolymerization reactions [50], However, up to now, all such models have a limited range of application since the current experimental techniques do not allow one to determine correctly a great number of their kinetic parameters. 

As long ago as 1960, Tarasov et al. [121] presented some examples of the concrete three-component systems for which the existence of the azeotropic composition had already been predicted theoretically. The list of such systems was widened substantially after publication of the important paper [125], where a set of the known tabulated values of 653 pairs of reactivity ratios for a computer search of the possible multicomponent azeotropes was employed. For this aim one should, at first, reveal all the completely characterized multicomponent systems for which the values of reactivity ratios of all monomer pairs are tabulated. This problem can be formalized by reducing it to the search on the graph with 653 lines of a... 

The prediction of efficiencies for multicomponent systems is also discussed by Chen and Fair (1984b). For mixtures of dissimilar compounds the efficiency can be very different from that predicted for each binary pair, and laboratory or pilot-plant studies should be made to confirm any predictions. 

Thermodynamic effects of directional forces in liquid mixtures.— The theory applied to pure liquids in the last two sections can be generalized to liquid mixtures and can be used to discuss the effects of directional forces on the thermodynamic functions of mixing. Classical statistical mechanics leads to a complete expression for the free energy of a multicomponent system in terms of the intermolecular energies Ust for all pairs of components s and t. Each Ust can be expanded in the general manner (2.1), so that it is separated into a spherically symmetric part and various directional terms. 

Multicomponent Systems. Most efficiency models and test data are based on binary systems. For multicomponent systems it is often possible to use the key components as a pseudobinary system. Chan and Fair (1984b) found that selection of the two components should be based on the dominant pair (if not the key components). Rigorous treatments of multicomponent separations are given by Taylor and Krishna (1993). 

It is inconvenient for multicomponent systems because interactions between each pair of components must be known. 

Applications of polymer solution theory to the studies of the dissolution of humic acids and of their extraction from soils suffer most because interactions betvi een each pair of components must be known (criterion 4, p. 343). Unfractionated humic and fulvic acids and humic substances in the soil are parts of multicomponent systems, and interactions between the different components are unknown. 

The requirement [Sc] = [/] for a multicomponent system is a much more special case than for a corresponding binary system for it requires that all binary pair diffusivities in the multicomponent system be equal to one another and, furthermore, that r/D = 1, a situation realizable only for ideal mixtures made up of species of similar size and nature. 

The numbers of transfer units for each binary pair may be obtained as described in Section 12.1.5 or from experimental data and these binary numbers of transfer units used directly in the estimation of the matrices of numbers of transfer units for multicomponent systems as Example 12.2.3 demonstrates. 

In any event, we hope it is now well understood that mass transfer in multicomponent systems is described better by the full set of Maxwell-Stefan or generalized Fick s law equations than by a pseudobinary method. A pseudobinary method cannot be capable of superior predictions of efficiency. For a simpler method to provide consistently better predictions of efficiency than a more rigorous method could mean that an inappropriate model of point or tray efficiency is being employed. In addition, uncertainties in the estimation of the necessary transport and thermodynamic properties could easily mask more subtle diffusional interaction effects in the estimation of multicomponent tray efficiencies. It should also be borne in mind that a pseudobinary approach to the prediction of efficiency requires the a priori selection of the pair of components that are representative of the... 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

The generalization to multicomponent systems is quite straightforward. Instead of one pair correlation g(X, X"), we shall have pair correlation functions for each pair of species a/ . For instance, if A and B are spherical particles, then we have three different pair correlation functions g,, (R), gAB(R) = 8ba(R) and gBB(R)- We shall describe these in more detail in section 2.9. 

The chemical potential is the most important quantity in chemical thermodynamics and, in particular, in solution chemistry. There are several routes for obtaining a relationship between the chemical potential and the pair correlation function. Again we start with the expression for the chemical potential in a one-component system, and then generalized to multicomponent systems simply by inspection and analyzing the significance of the various terms. 

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