Bjerrum theory of ion pairing

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. 

Another example of specific ion adsorption was discussed in terms of the formation of interfacial ion pairs between ions in the aqueous and the organic phase. The contribution of specific ionic adsorption to the interfacial capacitance can be calculated using the Bjerrum theory of ion-pair formation. The results show that a phase boundary between two immiscible electrolyte solutions can be described as a mixed solvent region with varying penetration of ion pairs into it, depending on their ionic size. The capacitance increases with increasing ionic size in the order Ii+ < Na+ < K" " < Rb < Cs" ". Yufei et al. [22] found that significant specific ion adsorption occurs at the interface between two immiscible electrolytes... 

Bjerrum s theory of ion pairs qualitatively correctly explains a number of experimental data, but cannot be used to the full extent in quantitative calculations, particularly because of the provisional character of quantities a and (the integration limits). 

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. 

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 ... 

The theory of ion-pair formation in nonaqueous solutions has been substantially advanced by the work of Barthel, who demonstrated how important it is to take into account the non-Coulombic forces at small ionic distances in addition to the Coulom-bic ones used by Bjerrum. These non-Coulombic forces are represented by the mean force potential W (r) in the region a[Pg.551]

Studies on ion-pair formation initiated by N. BJerrum were extended theoretically by Onsager and Fuoss, and the theory of ion-pair formation almost completed in the 1960 s. Since they were in US A, the studies on ion-pair formation were more actively carried out in USA than other pans of the world. Since the study does not need expensive tools, the work could be done in Japan by using the Onsager-Fuoss theory. 

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. 

The first ideas concerning a role of pairwise electrostatic interaction between ions were advanced in 1924 by Vladimir K. Semenchenko. A quantitative theory of the formation of ion pairs was formulated in 1926 by Niels Bjerrum. 

Bjerrum s theory includes approximations that are not fully justified the ions are considered to be spheres, the dielectric constant in the vicinity of the ion is considered to be equal to that in the pure solvent, the possibility of interactions between ions other than pair formation (e.g. the formation of hydrogen bonds) is neglected and the effect of ion solvation during formation of ion pairs is not considered (the effect of the solvation on ion-pair structure is illustrated in Fig. 1.7). 

Fig. 3.54. Deviations from the Debye-Htickel limiting law (DHLL) for and of a 2 2 electrolyte for several theories. The ion-pairing cutoff distance d for the Bjerrum curve is 1.43 nm. / is the ionic strength. (Reprinted from J. C. Rasaiah, J. Chem. Phys. 56 3071, 1972.)...

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. 

In 1926, Bjerrum [137] used Debye-Hiickel theory to describe ion association and took into account the interaction of ions within a short range. He introduced an ion-pair concept, gave a definition of ion pairs as neutral species formed by electrostatic attraction between oppositely charged ions in solution, and showed how ion-pair formation was dependent on the ions size (radius of ions), solvent (dielectric constant), and temperature. 

Later Bjerrum s theory was supported by the work of Kraus [138], who showed importance of the dielectric constant, and Atherton [139], who demonstrated the existence of ion pairs using electron spin resonance spectroscopy. The formation of ion pairs may be studied by various methods conductance studies, UV-visible spectrometry, IR spectrophotometry, partition, distribution, or solvent extraction. The lifetime of ion pairs was determined to be at least 10 sec, which is equivalent to about 10 molecular vibrations, demonstrating that ion pairs can be considered as independent species [140]. Today, the ion-pair formation as independent species is widely accepted. 

The second golden time of solution chemistry staned by the work of Debye and Hiickel in 1923, which was based on statistical thermodynamics, one step further advanced from thermodynamics. This work was immediately followed by N. Bjerrum, whose idea of ion-pair formation was induced by the Debye-Hiickel theory. In these treatments they accepted the existence of ions and molecules with finite sizes as solutes, but solvent was regarded as a homogeneous continuum, and no molecular aspect was introduced in their concept. In this respect the second period can be said as the dawn of the molecular solution chemistry developed in the late 20th century. In this section, we have to mention the work by Bernal and Fowler (1933), which will be referred to again later. 

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 ... 

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... 

A modification of the Debye-Hiickel theory to include the possible formation of ion pairs was suggested as early as 1926 by Bjerrum.22 As has been shown in Chapter 7, the unmodified form of the Debye-Hiickel theory, leads, even with water solutions, to absurdities, for electrolytes with small ions, or salts of the higher valence types and the extended theory was necessary to account for these cases. 

Bjerrum s theory (Section 10.12) deals explicitly with formation of ion pairs, and can be used to calculate an expected value for the association constant for an equilibrium between two ions and an ion pair. This predicted value can be compared with the observed value. 

The dissociation constant in aprotic organic solvents can be derived from fundamental principles (Brandstorm, 1977), based on Bjerrum s theory for ion pairs, as a function of the dielectric constant of the solvent, temperature, and the distance between the ions in the ion pair. However, in most organic media, the dissociation constant of ion pairs is very small (on the order of 10 4-10 5), and hence, the free ion concentration is negligibly low. 

Various explanations have been given for deviations from the Debye-Hiickel-Onsager equations. A common type of behavior is for the negative slopes of the A versus /c plots to be greater than predicted by the equation that is, the experimental conductivities are lower than predicted by the theory. This has been explained in terms of ion pairing, a concept which was developed by the Danish physical chemist Niels Bjerrum (1879-1958) in 1926. Although most salts, such as sodium chloride, are present in the solid state and in solution as ions and not as covalent species, there is a tendency for them to come together from time to time to form ion pairs. 

The value of Kqx can be derived from Bjerrum s theory for ion pairs (Brandstrdm, 1977). In organic solvents with high dielectric constants such as acetone, the dissociation can be high enough for the anion formed to play an active role in the reaction. 

Soon after the appearance of the Debye-Htickel theory, it was found that the theory did not work well for many electrolytes. In 1926, Bjerrum suggested that electrostatic attraction between pairs of oppositely charged ions resulted in ion-pairs, which would account for the lower measured activity coefficients in these solutions. The problem then, as now, was how best to define and measure the extent of ion-pairing. How close must two ions be to become an ion-pair What is the difference between an ion-pair and a complex Is it really necessary to know this to use thermodynamics Helgeson (1981) notes that The distinction between ion association and short-range ionic interaction is nebulous at best. ... 

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] ... 

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