Huckel model application

Some other kinds of models have shown parameters that seem to follow useful correlation relationships. Among these are the virial coefficient model of Bums (2), the interaction coefficient model of Helgeson, Kirkham, and Flowers (4), and the hydration theory model of Stokes and Robinson (1). The problem shared by all three of these models is that they employ individual ion size parameters in the Debye-Hiickel submodel. This led to restricted applicability to solutions of pure aqueous electrolytes, or thermodynamic inconsistencies in applications to electrolyte mixtures. Wolery and Jackson (in prep.) discuss empirical modification of the Debye-Huckel model to allow ion-size mixing without introducing thermodynamic inconsistencies. It appears worthwhile to examine what might be gained by modifying these other models. This paper looks at the hydration tlieory approach. 

In its original formulation, the Huckel model, as any 7c-electron scheme, is applicable... 

Gorin has extended this analysis to include (1) the effects of the finite size of the counterions in the double layer of spherical particles [137], and (2) the effects of geometry, i.e. for cylindrical particles [2]. The former is known as the Debye-Huckel-Henry-Gorin (DHHG) model. Stigter and coworkers [348,369-374] considered the electrophoretic mobility of polyelectrolytes with applications to the determination of the mobility of nucleic acids. 

One could use the Debye-Hiickel ionic-atmosphere model to study how ions of opposite charges attract each other, (a) Derive the radial distribution of cation ( +) and anion (nj concentration, respectively, around a central positive ion in a dilute aqueous solution of 1 1 electrolyte, (b) Plot these distributions and compare this model with Bjerrum s model ofion association. Comment on the applicability of this model in the study of ion association behavior, (c) Using the data in Table 3.2, compute the cation/anion concentrations at Debye-HUckel reciprocal lengths for NaCl concentrations of lO and 10 mol dm", respectively. Explain the applicability of the expressions derived. (Xu)... 

Thirdly, another corollary of the first limitation, is the inconsistency and inadequacy of activity coefficient equations. Some models use the extended Delbye-Huckel equation (EDH), others the extended Debye-Huckel with an additional linear term (B-dot, 78, 79) and others the Davies equation (some with the constant 0.2 and some with 0.3, M). The activity coefficients given in Table VIII for seawater show fair agreement because seawater ionic strength is not far from the range of applicability of the equations. However, the accumulation of errors from the consideration of several ions and complexes could lead to serious discrepancies. Another related problem is the calculation of activity coefficients for neutral complexes. Very little reliable information is available on the activity of neutral ion pairs and since these often comprise the dominant species in aqueous systems their activity coefficients can be an important source of uncertainty. 

As with the 2-0 ring systems discussed in the previous chapter, there have been very few reports of the application of theoretical methods to the systems under discussion here. Huckel molecular orbital calculations have been carried out on the imidazo[l,2-i>]pyrazole (35). This model predicted the IH tautomer to be more stable than the 5H and this is in agreement with the observation that the N-1 glycoside (38) is thermodynamically more stable than the N-5 isomer (39) (see Table 2 for structures) <84JOC3534). The same model also predicted a n electron density of greater than 1.1 on C-7, which is in agreement with the upheld chemical shifts ( ca. 65 ppm) observed for this carbon in the C NMR spectra of (35), (38), and (39). 

In support of this analysis, it was demonstrated that extended Huckel theory (EHT) correctly predicts a symmetrical transition state. Despite the substantial approximations of EHT, it does include overlap and thus closed-shell repulsion. In a clever "control experiment", a modified EHT code that does not include overlap produced an unsymmetrical transition state. It appears that the case of pericyclic transition states is one in which the approximations necessary to develop a rapid, semi-empirical computational model are too severe, and the semi-empirical methods are not applicable to such reactions. 

Attempts to improve the theory by solving the Poisson-Boltzmann equation present other difficulties first pointed out by Onsager (1933) one consequence of this is that the pair distribution functions g (r) and g (r) calculated for unsymmetrically charged electrolytes (e.g., LaCl or CaCl2) are not equal as they should be from their definitions. Recently Outhwaite (1975) and others have devised modifications to the Poisson-Boltzmann equation which make the equations self-consistent and more accurate, but the labor involved in solving them and their restriction to the primitive model electrolyte are drawbacks to the formulation of a comprehensive theory along these lines. The Poisson-Boltzmann equation, however, has found wide applicability in the theory of polyelectrolytes, colloids, and the electrical double-layer. Mou (1981) has derived a Debye-Huckel-like theory for a system of ions and point dipoles the results are similar but for the presence of a... 

Figure 4.3 shows a comparison of the variations of the mean activity coefficient of magnesium chloride as a function of the ionic strength. The points are obtained experimentally and the downward curve is obtained by application of Debye and HiickeTs law. The figure demonstrates that the model begins to deviate from the experimental values long before the ionic strength reaches one. In particular, for most electrolytes, the real curve exhibits an extremum which Debye and Huckel s model never shows. 

Uncertainty also arises in the accuracy of the measured chemical or physical data (field or experimental) that may be used as model input for a particular problem for example, measurements of pressure, temperature, alkalinity, pH, and Eh, petrographic descriptions, and mineral chemistries of phases all have uncertainty associated with them. Analytical incompleteness is also a concern if missing compositions must then be estimated. Additional uncertainty arises when formulae and supporting parameters are extrapolated beyond their range of applicability. The use of the Debye-Huckel expression to calculate ion activity coefficients at ionic strengths greater than 1 molal provides an example. 

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