Bjerrum association constant

In Eq. (15) 2 qB/r is the coulombic part of the mean force potential, and Wjj is the noncoulombic part. The earlier association constants of Fuoss, Prue, and Bjerrum are special cases of this general chemical model [15]. The importance of noncoulombic interactions is proved [ 16] by ... 

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). 

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... 

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]... 

The best-developed way to measure the association of ions is through the measurement of electrical conductance of dilute solutions. As mentioned, this realization occurred in the nineteenth century to Arrhenius and Ostwald. An elaborate development of conductance equations suitable to a range of ion concentrations of millimolar and lower by many authors (see Refs. 5, 33 and 34 for critical reviews) has made the determination of association constants common. Unfortunately, in dealing with solutions this dilute, the presence of impurities becomes very difficult to control and experimenters should exercise due caution, since this has been the source of many incorrect results. For example, 20 ppm water corresponds to 1 mM water in PC solution, so the effect of even small contaminants can be profound, especially if they upset the acid-base chemistry of association. The interpretation of these conductance measurements leads, by least squares analysis of the measurements, to a determination of the equivalent conductance at infinite dilution, Ao, the association constant for a positively and negatively charged ion pair, KA, and a distance of close approach, d, using a conductance equation of choice. One alternative is to choose the Bjerrum parameter for the distance, which is defined by... 

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. 

These ideas based on Bjerrum s picture of ion-pair formation have received considerable experimental support. Thus, in Fig. 3.46, the association constant is seen to inaease markedly with deaease of dielectric constant. The dependence ofion-pair formation on the distance of closest approach is seen in Fig. 3.47. 

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... 

The need for an analytical expressions for the equation of state have led to a revival of the macroscopic electrostatic theory due to Debye, Hiickel and Bjerrum. DH theory becomes exact for large particles. In pilot work by Fisher and Levin (FL) [31], DH-Bj theory is extended by considering the interactions of the pairs with the free ions. Weiss and Schroer (WS) [32] have supplemented this theory accounting for dipole-dipole interactions between pairs and the e-dependence of the association constant. 

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 momentary association of simple ions is a well-known phenomenon that has been treated in a number of ways. For example, the ion association constant of Bjerrum has received much experimental support. However, the association of simple electrolytes is considered to be shortlived and has been included in the Debye-Hiickel electrostatic theory as correction constants to the concentration. On the contrary, the hydration of the ions may be long-lived. This may be accounted for by considering additionally the ionic interaction ... 

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. 

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. 

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). 

Examination of electrostatic principles allows some conclusions to be drawn regarding the effect of ion pairing on the selectivity of salt partitioning or, equivalently, on the driving force for cation exchange. As outlined in a standard text [234], treatments of Fuoss [235] or of Bjerrum [236] may be applied to estimate the ion-pair association constant /Ca.,soc- The Fuoss treatment assumes contact ion pairs and is conceptually simpler to use and apply. As the simplification will not affect the conclusions to be drawn here, it will be employed with the additional proviso that the effect of water in the solvent will be neglected for the moment. According to Fuoss, the ion-pair association constant at 298 K may be expressed in terms of the solvent dielectric constant 6 and the internuclear distance i m-x (in nm) between the cation and anion ... 

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... 

The association constants of table III can be compared with those from conductance measurements, and are found to be in perfojt agreement, e.g. K (MgS04/H20) = 160 dm mol . The agreement of the R -values of Table III for aqueous solutions with those of the ion-pair model, Eq. (20), ould be stressed as an important result. The calculated values, Rcaic. correspond to R = a + 2s (here s = don. dimension of OH) according to this model. The agreement of R with Bjerrum s distaiKe parameter q, which is often used as the upper limit of association and which depends only on the permittivity of the solvent [cf. Eq. (19b)], is less satisfactory. For aqueous solutions of 2,2-electrolytes at 25 °C q equals 1.43 nm, independent of the ionic radii. 

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

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. 

A criterion for the presence of associated ion pairs was suggested by Bjerrum. This at first appeared to be somewhat arbitrary. An investigation by Fuoss,2 however, threw light on the details of the problem and set up a criterion that was the same as that suggested by Bjerrum. According to this criterion, atomic ions and small molecular ions will not behave as strong electrolytes in any solvent that has a dielectric constant less than about 40. Furthermore, di-divalent solutes will not behave as strong electrolytes even in aqueous solution.2 Both these predictions are borne out by the experimental data. 

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