Bjerrum theory

These large increases in rate might be attributed to the operation of a neutral salt effect, and, in fact, a plot of log k versus the square root of the ionic strength, fi, is linear. However, the reactants, in this case, are neutral molecules, not ions in the low dielectric constant solvent, chloroform, ionic species would be largely associated, and the Bronsted-Bjerrum theory of salt effects51 52, which is valid only for dilute-solution reactions between ions at small n (below 0.01 M for 1 1 electrolytes), does not properly apply. 

Although the Bjerrum theory is thus not in general quantitatively applicable, the concept of ion association is very useful. It has assisted in an explanation of various phenomena observed in the study of homogeneous... 

Once the value of maB has been established, the evaluation of the remaining terms in Eq. (150) for the activity coefficient follows quite simply. The fourth term on the right-hand side clearly involves interactions of oppositely charged defects of the sort considered in the Lidiard-Bjerrum theory. It can be written for the systems under consideration as... 

E. Munksgaard, 1949). Independently, Kemble worked along these lines at Harvard, so that the theory often is called the Kemble-Bjerrum theory for molecules (Assmus, 1990 109) see E. C. Kemble, R. T. Birge, W. F. 

There is some arbitrariness in the definition of the ion pair, and hence the association constant. Often a structural definition of the ion pairs is preferred—for example, by adopting a cutoff distance such as rc = 2a [141, 207] or similar choices [208, 209]. In contrast, Bjerrum (Bj) theory [140] uses an energetic criterion by defining ions as being associated, when their interaction energy is twice the thermal energy kBT. Bjerrum theory yields... 

Figure 10. Degree of dissociation a of the RPM as a function of the total ion density along the critical isotherms of Debye-Httckel-Ebeling (or Bjerrum) theory (DHEb), Fisher-Levin theory (FL), and Weiss-Schroer theory (WS) [138]. The asterisks show the critical points of the three models.

Solvent effects in electrochemistry are relevant to those solvents that permit at least some ionic dissociation of electrolytes, hence conductivities and electrode reactions. Certain electrolytes, such as tetraalkylammonium salts with large hydrophobic anions, can be dissolved in non-polar solvents, but they are hardly dissociated to ions in the solution. In solvents with relative permittivities (see Table 3.5) s < 10 little ionic dissociation takes place and ions tend to pair to neutral species, whereas in solvents with 8 > 30 little ion pairing occurs, and electrolytes, at least those with univalent cations and anions, are dissociated to a large or full extent. The Bjerrum theory of ion association, that considers the solvent surrounding an ion as a continuum characterized by its relative permittivity, can be invoked for this purpose. It considers ions to be paired and not contributing to conductivity and to effects of charges on thermodynamic properties even when separated by one or several solvent molecules, provided that the mutual electrostatic interaction energy is < 2 kBT. For ions with a diameter of a nm, the parameter b is of prime importance ... 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Dissociation of a salt in a solvent can similarly be treated taking into account ion pair formation. An ion association constant associated with the equilibrium established between ion pairs and dissociated ions is derived in the -> Bjerrum theory of ion pairs. 

Diamond was the first to focus on the concept of hydrophobic association and demonstrated that, at variance with the Bjerrum theory, ion-pairing of univalent organic electrolytes in water is possible [12]. He capitalized on the hydrophobic hydration concept [11,12] typical of large organic ions (yide supra) that increase the water structure via the formation of ice-like cages, thereby decreasing the system... 

Remember that the equations for the Bjerrum theory as presented here are correct only for electrolytes yielding ions of the same valence z-, i.e., only for symmetrical 1 1- or 2 2-valent electrolytes. 

What is the significance of these results on dimer and trimer formation for ionic solution theory In the post-Debye and HUckel world, particularly between about 1950 and 1980 (applications of the Mayer theory), some theorists made calculations in which it was assumed that aU electrolytes were completely dissociated at least up to 3 mol dm. The present work shows that the degree of association, even for 1 1 salts, is -10% at only 0.1 mol dm . One sees that these results are higher than those of the primitive Bjerrum theory. 

Haymet and co-workers have calculated the mole fraction of dimers (associated ions) in electrolytic solutions, and some of their results are shown in Fig. 3.51. Use the equations of the Bjerrum theory applied to NajP04 and compare the results with those of the correlation function approach used by Haymet et al. The essential difference between the Haymet approach and that of Bjerrum is that... 

Consider KCl and take a to be the sum of the ionic radii. Use data from tables to get these. Thus, one can calculate d of the Bjerrum theory over a reasonable concentration range and, using appropriate tables, obtain the value of the fraction of associated ions. Now recalculate the values of log/. for KCl for 0.1 to 2 Af solutions from the full Debye-Hlickel theory involving allowance for ion size and hydration — but now also taking into account 0. In this approach, Cq(1 - 0) is the concentration of the ions that count in the expressions. (See Appendix 3.6.) Does this accounting for 0 improve the fit ... 

The text gives the results of molecular dynamic calculations of the fraction of ion pairs as a function of concentration for univalent ions. Compare these values with those calculated by the Bjerrum theory. 

Jannik Bjerrum Theory of the reversible step reactions... 

It may be concluded that electrostatic models may be successfully applied only so far as interactions between weakly coordinating or noncoordinating species (such as tetraalkylammonium ions) are concerned. This is illustrated by Table X which shows that variations of association constants for tetrabutylammonium iodide as a function of dielectric constant roughly correspond to the trends predicted by the Bjerrum theory. When iodide, which is a comparatively weak base, is... 

Bjerrum theory (ion association) 1.5,2d black body (radiation) 1.7.22 blob 5.11... 

In solvents of low dielectric constant, where the Coulomb interactions are particularly strong, electrical conductance and dielectric spectra suggest that the ion distribution involves dipolar ion pairs, which then interact with the free ions and with other dipolar pairs. The ion pairs cause an increase of the dielectric constant, which in turn stabilizes the free ions, thus leading to redissociation at high salt concentrations. Extending the approach of Debye-Hiickel and Bjerrum, theory accounts for ion pairing, ion-ion pair and ion pair-ion pair interactions and rationalizes the basic features of the ion distribution in accordance with experiments and MC-simulations. 

When treating the association of counterions one may also apply the association statistics (AS) model which is equivalent to the Bjerrum theory for ion pairing in an electrolyte solution [29,30]. However, in the case of surface association spaee is available only on the side of the liquid. Another difference is due to the critical distance which depends on direction and is a function of the surface potential. This theory explains why two ionic species may associate at the surface despite the fact that they do not undergo ion pairing in the bulk of solution. According to the Bjerrum theory, ions of large effective size cannot approach the critical distance and such an electrolyte is completely dissociated. At the surface the critical distance extends by increasing surface potential and once the surface potential exceeds the critical value, at which the critical distance matches the minimum separation, association at the interface proceeds. 

Be very careful here The total potential at any distance, r, from the origin can be identified with the potential at any distance, r from the central z e ion due to that ion itself, only if nonideality is ignored. This means that corrections for non-ideality must be superimposed onto the Bjerrum theory after the association constant has been derived (see Section 10.12.4). 

In the Bjerrum theory, the potential which appears in the potential energy of interaction between the two ions of the ion pair is thus an approximate potential in so far as the term due to the ionic atmosphere is ignored. 

There have been attempts to modify Bjerrum s treatment to remove this arbitrariness, but none has been used universally to any great extent in the interpretation of experimental data. Nevertheless, despite this artificiality, the Bjerrum theory coupled with the Debye-Hiickel theory has proved a very useful and relatively successfiil tool in discussing electrolyte solutions. This success is especially noteworthy when the Bjerrum-Debye-Huckel theory is compared with the alternative approach of Guggenheim s numerical integration which gives similar results (see Section 10.13.1). Table 10.2 gives values of K ssoc for various charge types and for various values of a and q. 

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