Bjerrum s theory

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). 

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). 

According to Bjerrum s theory, the association constant is proportional to a b times a certain function of b, Q(b). The quantity a is the ionic distance parameter (i.e., the distance between the centers of the cation and the anion). 

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) ... 

Fig. 2.14 Comparison of the log /

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). 

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. 

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... 

In Bjerrum s theory (9), two ions of opposite charge constitute an ion pair if they are closer together than a certain critical distance ... 

Using Bjerrum s theory. Monk and co-workers 18y 21) calculated the radii of the ion pairs in question to have the expected order of magnitude 4-5 A. From the theory it can also be estimated that the constant for the formation of a triple ion [Coa6]X2 of the considered type and dimensions ... 

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. 

If the two values are similar, then it is likely that ion pairing is involved since Bjerrum s theory deals explicitly with short-range coulombic interactions which are predominantly dependent on the charges on the ions. 

Bjerrum s theory has been criticised because it involves the arbitrary cut-off at a critical distance q between ion pairs and free ions. It is felt that a more realistic situation would be one which would allow more of a fall-off between paired and free ions as the distance between them alters. 

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. 

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. 

The association constant arising from Bjerrum s theory is ... 

Once such functions are obtained, they can be used for the calculation of equilibrium constants of Eqs. (2.1), (2.4), and (2.5). As one can see, Eqs. (2.9) and (2.10) are quite similar to the functions widely used in Bjerrum s theory of stepwise complex formation in solutions (Leden s and Bjerrum s functions accordingly) [5,6]. Thus, the mathematical analysis and calculation of the constants become well-elaborated routine. Some examples of such calculations are given below. 

Bjerrum s theory for ion-pair formation relates the equilibrium constant for... 

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