The Concept of Local Compositions

In the model of strictly-regular solutions, we have indicated a contradiction in the fact that the A-B bond involved a different amount of energy than the A-A and B-B bonds, but admitted that this difference did not prevent random distribution of the pairs A-B - in other words, the local composition of the liquid around a molecule of A or a molecule of B was the same as the overall composition. Now, we shall refine our models by examining local distributions different to the overall composition. 

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Renon used the concept of local composition to develop a non-random, two-liquid (NRTL) three parameter (al2, tp, ti() equations given below (Prausnitz et al., 1986). 

The long-range term has been satisfactorily described by the Debye-Huckel formula and is retained. The short-range contribution is modeled by utilizing the concept of local compositions in a manner similar to Renon and Prausnitz (20) but with additional assumptions appropriate for electrolyte systems. Preliminary results suggest the validity of the model since good fits to experimental data have been obtained for a wide range of binary and ternary systems with only binary parameters. 

Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are based on the concept of local composition. Within a liquid solution, local compositions, different from the overall mixture composition, are presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. The concept was introduced by G. M. Wilson in 1964 with the publication of a model of solution behavior since known as the Wilson equation. The success of this equation in the correlation of VLE data prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two Liquid) equation of Renon and Prausnitz and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz. A further significant development, based on the UNIQUAC equation, is the UNIFAC method,tt in which activity coefficients are calculated from contributions of the various groups making up the molecules of a solution. 

Local Composition. There have been many attempts to express the local composition in terms of the bulk composition and intermolecular interactions.2 °d3-i7 while Guggenheim was the first to introduce the concept of local composition, this idea became extensively used after it was apphed by Wilson to phase equihbria. A number of authors proposed various expressions for the local composition,2 ° 4-i7 among which we selected the following equations proposed in refs 7—9, because they were derived on the basis of some plausible theoretical considerations ... 

There are many other equations, which have been proposed, that do not result from Wohl s method. Two of the most popular equations are the Wilson and the universal quasi-chemical theory (UNIQUAC) by Abrams and Prausnitz.These equations are based on the concept of local composition models, which was proposed by Wilson in his paper. It is presumed in a solution that there are local compositions that differ... 

In the Wilson equation, the effects of difference both in molecular size and intermolecular forces are incorporated by an extension of the Flory-Huggins relation (5-8). Overall solution volume fractions ( i = XiViJvi) are replaced by local volume fractions, d> which are related to local molecule segregations caused by differing energies of interaction between pairs of molecules. The concept of local compositions that differ from overall compositions is shown schematically for an overall equimolar binary solution in Fig. 5.6, which is taken... 

Figure 5.6. The concept of local compositions. [From P. M. Cukor and J. M. Prausnitz, bill. Chetn. Eng. Symp. Ser. No. 32, Instn. Chem. Engrs., London, 3, 88 (1969).]...

As mentioned previously, one of the major advantages of FST is that it provides information on local density fluctuations. An alternative approach to this problem involves the concepts of local composition and preferential solvation (Ben-Naim 2006). To do this we first note that the average number of j particles in a region of spherical volume siurounding a central particle i is... 

It would be desirable to apply analytical expressions for the activity coefficient, which are not only able to describe the concentration dependence, but also the temperature dependence correctly. Presently, there is no approach completely fulfilling this task. But the newer approaches, as for example, the Wilson [13], NRTL (nonrandom two liquid theory) [14], and UNIQUAC (universal quasi-chemical theory) equation [15] allow for an improved description of the real behavior of multicomponent systems from the information of the binary systems. These approaches are based on the concept of local composition, introduced by Wilson [13]. This concept assumes that the local composition is different from the overall composition because of the interacting forces. For this approach, different boundary cases can be distinguished ... 

We find the concept of local composition in solid solutions, just as it is found for liquid solutions. Because of the existence of an exchange energy, we can imagine that if a molecule of A and a molecule of B experience a greater force of attraction than two molecules of A or two molecules of B, a molecule of A will tend to be surrounded by molecules of B and therefore the composition of the immediate environment of the molecule of A will not be the same as the overall composition of the solution the immediate vicinity is richer in molecules of B. The opposite effect is obtained if the molecules A and B attract each other less strongly than do two molecules of A or two molecules of B. This is the concept of local composition, which was first put forward by Wilson. 

There are several models of solutions which use this concept of overall composition - Wilson s, in particular. Given the restricted reach of these models, we have chosen to use the concept of local composition only in the context of the much more widely-applicable models UNIQUAC (see section 3.5) and UNIFAC (see section 3.6). 

Numerous models of ionic solutions have been put forward in the existing body of literature. The most important of these models, whieh is actually found to be included in all the others, is Debye and Htiekel s, which attributes the imperfection solely to the eleetrostatie forces between the ions but, in spite of this, is acceptable only for fiilly-dissoeiated strong electrolytes and very-dilute solutions. Then, we shall eite Pitzer s model (1973), which combines Debye and Hiickel s model with a virial-type expansion, and is therefore able to extend the range of concentrations examined. Beuner and Renon s model (1978) builds on Pitzer s, extending it to solutions containing neutral molecules such as SO2, NH3, CO2 or H2S. The most recent models take account of the concept of local composition (see section 3.2). In this category, we can cite the NRTL electrolyte model introduced by Chen (1979). Finally, other models have expressed the intermolecular interactions over short distances by a... 

Next, in Chapters, we present a number of solution models with microscopic definition, including random distribution models and models integrating the concepts of local composition and combinatorial excess entropy. 

The concept of local compositions revolutionized the whole approach to developing expressions for the excess Gibbs free energy and, consequently, the activity coefficient. Wilson used it in developing the equa-... 

Abrams and Prausnitz (1975) combined Guggenheim s quasi-chemical tiieory with the concept of local compositions to develop the Universal Quasi-Chemical (UNIQUAC) expression for the excess Gibbs free energy. [The equation can be also developed from the two-fluid theory (Maurer and Prausnitz, 1978).]... 

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