Htickel Theory

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

The zeros in the 1.3 and 3,1 positions coirespond physically to the assumption that there is no interaction between rt electrons of atoms that are not neighbors, a standard assum rtion of Htickel theory,... 

For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients are approximately unity under most reasonable experimental conditions. For reactions involving only these species, differences between activity and concentration are negligible. Activity coefficients for ionic solutes, however, depend on the ionic composition of the solution. It is possible, using the extended Debye-Htickel theory, to calculate activity coefficients using equation 6.50... 

See any standard textbook on physical chemistry for more information on the Debye-Htickel theory and its application to solution equilibrium... 

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

Hermann RB. Structure-activity correlations in the cephalosporin C series using extended Htickel theory and CNDO/2.1 Antibiot 1973 26 223-7. 

Combining Eqs. (A.l) and (A.5), we find the basic differential equation ofDebye-Htickel theory ... 

Kirkwood, J. G. Poirier, J. C., The statistical mechanical basis of the Debye-Htickel theory of strong electrolytes, J. Phys. Chem. 1954, 86, 591-596... 

According to the Debye-Htickel theory, in the limit of the infinitely dilute solution, individual-ion activity coefficients are given by the equation... 

Recall from transition state theory that the rate of a reaction depends on kg (the catalytic rate constant at infinite dilution in the given solvent), the activity of the reactants, and the activity of the activated complex. If one or more of the reactants is a charged species, then the activity coefficient of any ion can be expressed in terms of the Debye-Htickel theory. The latter treats the behavior of dilute solutions of ions in terms of electrical charge, the distance of closest approach of another ion, ionic strength, absolute temperature, as well as other constants that are characteristic of each solvent. If any other factor alters the effect of ionic strength on reaction rates, then one must look beyond Debye-Hiickel theory for an appropriate treatment. 

The reactivity and regioselectivity of 1,3-dipolar cycloadditions have been discussed in terms of the frontier orbitals [271]. Most of the features may be understood on the basis of simple Htickel MO theory. The HOMO and LUMO n orbitals and n orbital energies for all 18 combinations of the parent dipoles are shown in Figure 12.8. The frontier orbitals of many of the 1,3-dipoles have previously been derived by CNDO/2 and extended Htickel theory [272]. The first six structures, all of 16-electron type, are shown in greater detail ... 

However, experimentally observed y =f c) functions usually first decrease, pass through a minimum, and then increase at high concentrations. In order to explain the increase of y with concentration, Stokes and Robinson modified the Debye-Htickel theory by introducing the effect of ion-solvent interaction. Thus, the modified theory is based on ion-ion and ion-solvent interactions. The modified theory is in good agreement with experimental results, up to an ionic strength of about 4, as shown in Figure 5.14. 

Reaction (15.37) is usually studied in dilute solution (ionic strength <0.1). If, as in our examples, the ligand is a nonelectrolyte, then it is a reasonable approximation to assume that tl 1. It is also not unreasonable to expect 7 mlj 7m v+ in these dilute solutions, since ions with the same charge behave in a somewhat similar manner, as suggested by the Debye-Htickel theory. Hence, /7 1 and K = Kc. Because we will not be overly concerned with quantitative results of high accuracy in this discussion, we will assume this approximation is sufficient and use K for Kc. It is not absolutely necessary that we do so, however, since corrections can be made for /7. 

In 1954 Sandorfyand Daudel15) published their C" approximation, a one-electron approximation which employs a linear combination of sp3 orbitals and borrows most of its simplifying features from the Htickel theory. The originality of this method lies essentially in the introduction of a resonance integral m/J between sp3 orbitals of the same carbon atom. Sandorfy 16> showed that the inductive effect due to a heteroatom can be reproduced by such a calculation. 

Likewise, yAB also cannot be measured experimentally, although, like Qa and Qb, 7a and yB can be measured, and at first sight the conversion given above may seem to give little improvement. However, for ion-ion and ion-molecule reactions, the Debye-Htickel theory, see Equation (7.8), can calculate the activity coefficient for any charged species and convert Equation (7.17) into a useful form. For other reactions the approach is only qualitative, but for them the effects of non-ideality are much smaller. 

The suspended particles are small compared with distances Z over which the potential

in potential from mean value, the ion concentrations deviate as in linearized Debye-Htickel theory ... 

In solvents of high dielectric constant such as water, the deviations from ideality caused by ion-ion interactions are reasonably small below concentrations of 0.1 ilf for 1 1 electrolytes and can be treated adequately by means of the Debye-Htickel theory. For polyvalent electrolytes or for higher concentrations of 1 1 electrolytes, or for either in solvents of lower dielectric constant, the situation is less fortunate. The deviations from ideality can become rather large, and there is no adequate theory for either correlating them or predicting them. 

Now, as explained in Section 3.3.2, the principal objective of the Debye-Htickel theory is to calculate the time-averaged spatial distribution of the excess charge density around a reference ion. How is this objective attained ... 

The ion size parameter a has done part of the job of extending the range of concentration in which the Debye-Htickel theory of ionic clouds agrees with experiment. Has it done the whole job One must start looking for discrepancies between theory and fact and for the less satisfactory features of the model. 

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