The Bjerrum theory of ion association

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 ... 

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. 

J. Bjerrum (1926) first developed the theory of ion association. He introduced the concept of a certain critical distance between the cation and the anion at which the electrostatic attractive force is balanced by the mean force corresponding to thermal motion. The energy of the ion is at a minimum at this distance. The method of calculation is analogous to that of Debye and Hiickel in the theory of activity coefficients (see Section 1.3.1). The probability Pt dr has to be found for the ith ion species to be present in a volume element in the shape of a spherical shell with thickness dr at a sufficiently small distance r from the central ion (index k). 

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. 

Fuoss developed a new theory of ion association in 1958 [27] which overcame some of the difficulties associated with the Bjerrum approach. The cations in the solution were assumed to be conducting spheres of radius a and the anions to be point charges. The ions are assumed to be immersed in a dielectric continuum of permittivity Sj. Only oppositely charged ions separated by the distance a are assumed to form ion pairs. The resulting expression for the association constant is... 

The Bjerrum theory of electrostatic ion pairing, as applied to conductivity data, has been well substantiated by Justice and Justice [36]. The interpretation of such data that have traditionally been one of the main methods for studying ion association according to Martell and Motekaitis [37] is according to the following expression by Femandez-Prini and Justice [38] ... 

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... 

In solution theory the specialized distribution functions of this kind should appear in the theory of ion pairs in ionic solutions, and a form of the Bjerrum-Fuoss ionic association theory adapted to a discrete lattice is generally used for the treatment of the complexes in ionic crystals mentioned above. In fact, the above equation is not used in this treatment. Comparison of the two procedures is made in Section VI-D. 

The two curves in Fig. 2.14 are the relationships between log KA and log er for a 1 1 electrolyte. The solid curve was obtained by Bjerrum s theory [Eq. (2.17)] and the dotted curve by Fuoss theory [Eq. (2.19)], both assuming a=0.5 nm. The big difference between the two theories is that, according to Bjerrum s theory, ion association does not occur if r exceeds a certain value ( 50 in Fig. 2.14), although the value depends on the value of a. Both theories are not perfect and could be improved. In recent treatments of ion association, non-coulombic short-range interactions between ions are also taken into account [40]. By introducing non-coulombic interactions, W (r), Eq. (2.17) is modified to a form as in Eq. (2.20) ... 

If the association of ions to ion pairs is solely due to electrostatic forces, then there should be a correlation between the association constant KA and the dielectric constant of the solvent. The relation proposed by Bjerrum [35] has been found to describe satisfactorily ion association in solvents of low dielectric constants [36], In the case of solvents of moderate to high dielectric constants, the electrostatic theory of association leads to the equation [34,37]... 

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... 

Bjerrum s theory of ionic association gives rise to an expression for the fraction of ions in an ionic solution which are associated. Use the theory to calculate the degree of association of a 0.01 M MgClj solution in ethanol (e = 32). 

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 ... 

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... 

This expression for differs by a factor of 2 from the length below which ion association is assumed to take place in electrolyte solutions, according to Bjerrum s theory, see (1.5.2.30a). The reason for this factor of 2 is that for counterion association on a polyelectrolyte only the former loses its kinetic energy, whereas for association of two small ions this occurs for both. At low polylon charge, is of course simply given by... 

The major feature is a rapid decrease of A at low salt concentrations, followed by a minimum and pronounced increase. At the CP there is a substantial conductance. To interpret this behavior, we first note that the Debye- Hiickel (DH) theory itself predicts an instability regime at low T, but if compared with experiment C is far too low. Taking account for ion association considerably improves thew results. In the presence of ion association, a higher salt concentration is needed to achieve the concentration of free ions to drive phase separation, i.e. C is shifted to higher values. In particular, the Bjerrum model for ion pair association yields ... 

The concept of coordination in the second sphere was introduced by Werner. All authors agree that such outer-sphere association exists in solution, hut they disagree about the kind and the extent of this association. Some advocate a second-sphere coordination which is closely analogous to the inner-sphere coordination. The data which support this hypothesis are not very convincing and can be criticized in various ways. The present author finds that the electrostatic theories of N. Bjerrum, Fuoss, and Kraus, according to which the formation of the ion-associates is a result of coulombic attraction, both qualitatively and quantitatively, give the most trustworthy picture of the outer-sphere association. However, this does not exclude the fact that some preferred mutual orientation exists in the ion pairs. 

A concept of ion association in electrolyte solutions was introduced about eighty years ago by Bjerrum [1] in order to improve the Debye-Hiickel (DH) theory [2], In accordance with this concept an electrolyte solution is considered to be a mixture of free ions and ion clusters (usually ion pairs and some-... 

At the same time, theories of ionic association were worked out by Bjerrum and others (3,5,14,16,19). According to these, f-ree ions of opposite charge getting closer than a certain critical distance form separate associated entities. Thereby, the total number of moles of solute in the solution becomes lower than that expected on the basis of complete dissociation. These theories show that ion pairs can be formed, although to a small extent, even in aqueous 1 1 electrolytes where the critical distance is 3.57 at 25 ( ) for higher valent ions in solvents... 

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... 

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. 

---