Bjerrum’s equation

The second ionization constant K2 of a dicarboxylic acid should, from statistical considerations alone, equal Ki/4. However, experimentally, values of K2 for all dicarboxylic acids are less than Ki/4, but approach Ki/4 as the distance between ionized carboxyl group and incipient ionized carboxyl group increases. Presumably the negative charge of the monoanion alters the electrostatic field about the remaining carboxyl group and increases the work required to remove the unit positive charge of the proton. The same relationship between Ki and K2 should hold for a diol. Calculation of an approximate value for Kt/K2 should be possible with Bjerrum s equation (9)... 

This last equation, derived from Bronsted s and Bjerrum s equations, has been the subject of intensive investigation.- It predicts that, at low... 

Bjerrum s equation can be extended to include dipolar substituents if the dipole moment (fi) of the substituent is treated as a point dipole located at a distance r from the acidic proton (see Fig. 2), The pKa difference between the unsubstituted and substituted acids is then given by... 

The evidence for complexing of halide ions by SO2 stands in curious contrast to the remarkable degree of adherence of ionophores to Bjerrum s equation which is discussed above. If Bjerrum s equation is used as a measure, then it appears that a specifically bound molecule of SO2 has no influence on the electrostatic association of a halide ion, perhaps because the solvent molecule remains bound to the paired anion, but at a point remote from the location of the cation. The analogy to the pairing behavior of CIO4" and BFi is interesting. The use of Bjerrum s equation in more than a relative way is, however, not really justified. 

The differentials of Equations 5 and 6 are introduced in the Gibbs-Duhem equation, the terms mid In (mj7 ) are replaced by Bjerrum s terms d(mi0-)), and the integration is carried out. The resulting equation is... 

Fuoss (40) has improved Bjerrum s original treatment (37) of this situation and, although a number of other sophistications have been introduced, his formulation (41) is the one most used today. In fact rather fortuitously the relatively low dielectric constants of solvents employed in organic chemical reactions, particularly ionic polymerisations, are ideal media for the application of these theories. The analysis carried out by Fuoss leads not surprisingly to an equation... 

Blum s use of the MSA represents a significant advance, but it does not take into account either ionic association or Bjerrum s very reasonable idea (Section 3.8) about the removal of free water in the solution by means of hydration. Furthermore, Blum s equations do not explain the relation between conductance and concentration noted for many electrolytes, particularly at high concentrations, that is. 

The Fuoss Model. Fuoss ( ) observed that because the BJerrum d value would generally exceed the sum of the radii of the addends , Bjerrum s so-called "ion pairs" might not be in contact. Accordingly, he developed an equation based on the assumption that an "ion pair" existed only when oppositely charged ions were in contact. Based on this and other arguments (12, ), Fuoss derived the expression... 

So far only the meaning of an ion pair has been discussed, and this has to be related to an equilibrium constant defining ion association. Bjerrum s treatment relates to very dilute solutions, and calculates an explicit value for the association constant, which is therefore an ideal constant. Consequentiy, the Debye-Hiickel equation must be used to enable the calculation to apply at higher concentrations. 

Figure 3 shows the family tree of some association constants which can be found in the literature and indicates the presuppositions for deducing them from the initial equation. For example, Bjerrum s association constant and its appropriate activity coefficient are obtained from Eqs. (19) by setting R = q and = 0. As a further... 

Perhaps the greatest uncertainty in the evaluation is caused by the selection of the appropriate ionic radii, / . In accordance with the introduction of Bjerrum s association constant [Bj 26], Justice [Ju 71b, Ju 75a, Ju 75b] assumes the equation Ry = R = q From chemical considerations, Barthel [Ba 78a] considers it more correct to describe the radius Ry as the sum of the contact distance, a, of the ions and the size, s, of the solvent molecule Xy=a4-s. 

Thus, if Bjerrum s derivation is modified by replacing the intermediate S by the activated complex X, the logical dilemma disappears. Equation (6) is the correct transition-state formula to use, and the ionic-strength effects have been important in leading to this fundamental equation. 

An initial step in data analysis is to develop an equation that represents the experimental data reasonably. Although previous sections dealt with this issue, the approach assumed that certain species are formed. Two alternatives to this procedure are discussed here, both yielding the approximate stoichiometry of the complexes formed in the system. The most elementary is referred to as Job s method, while the ligand number method, developed by J. Bjerrum, is slightly more advanced. 

Bronsted s salt effect Bronsted-Bjerrum equation, -> charge transfer reaction... 

The essence of this treatment lies in considering a Bjerrum field effect term as a separate component of the Hammett equation. Wepster s paper occupies some 25 pages of the Journal of Organic Chemistry and contains extensive tabulation of experimental results (partly Wepster s own work, but mainly from the literature, with many references) and of the results of applying the new treatment. The experimental results are for the effects of both unipolar and dipolar substituents in a variety of reactions and they have been analysed in terms of the classical ideas developed by G. N. Lewis and Bjerrum and revived (now some 20 years ago) by Palm and his coworkers The outcome is that the Hammett equation needs only a simple extension to cover the effects of both dipoles and unipoles. The present author has previously written" a fairly lengthy summary of Wepster s paper, while recommending that seriously interested readers should consult the original paper. Here, however, only a brief summary will be attempted, with an indication of application to N2. ... 

N. Bjerrum found the results did not agree with the velocity equation for monomolecular reactions but the results were better represented by velocity equations for two consecutive, bimolecular reactions, on the assumption that the reaction involves the sequence of changes [Cr(H20)4Cl2]Cl->[Cr(H20)5Cl]Cl2 ->[Cr(H20)e]Cl3. If x, y, z respectively denote the concentrations of these three salts, then dxjdt=—kix, and dxldt=k. It was found that at 25°, A i=0-C0272 -J-O-0000162/s, and A 2=(3I/s+0-005/s2)10—7, where s denotes the cone, of free hydrochloric acid. For soln. with M mols of dark green chromic chloride, the... 

Several equations have been formulated by various authors as presented in Table 8.8 for curves of S /cf versus consistency limits for remolded clays. For stable clays, ( ) varies between 20° and 35°. Stable loose silts and sands typically have values of ( ) between 28° and 34°. In addition, a correlation between and plasticity index has been given by Bjerrum and Simons (1960). 

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