Bjerrum critical distance

NO3 in acetonitrile has been obtained by Janz and Muller. Associated structures of ions have been studied in nonaqueous solvents over a wide range of dielectric constants. LiCNS in solvents of low dielectric constant, such as ethers and thioethers, gives rise to several different types of ion aggregates. Many different types of contact ion pairs or agglomerates have been identified, and the role the solvent has in this association—whether the solvent separates the ions or not—has been determined. The Bjerrum critical distance, that is, the distance at which the ion is able to interact with other ions to form ion-pair stmctures (see Section 4.8.8), is of great use in these types of studies. Table 4.25 shows some values for 1 1, 2 2, and 3 3 electrolytes in different solvents. 

From (4-26), for a 1 1 electrolyte in water the Bjerrum critical distance is 3.6 X 10 cm. When it is realized that ions in water are normally highly solvated and that the sum of ionic crystal radii for typical anions and cations often approaches or exceeds 3.6 x 10 cm, it is reasonable to find that dissociation constants for ion pairs in water are large thus for sodium hydroxide the dissociation constant is about 5. On the other hand, for a 2 2 electrolyte in water the critical distance is 14.3 x 10 cm, and for a 1 1 electrolyte in ethanol, 11.5 x 10 cm. In these cases, even highly solvated ions can readily approach to the distance necessary to form an ion pair. For magnesium sulfate the dissociation constant in water is 6 x 10 , and for sodium sulfate, 0.2. 

However, it is still useful to see how the Bjerrum critical distance, q, varies with charge type (see Table 10.1). 

This expression was obtained by setting the Debye-Hiickel distance parameter a equal to the Bjerrum critical distance defined as z4.z t IlkTItT. The coefficient A then appears in the denominator as shown, giving a dependence on (eT) rather than on (eT)"... 

KCl and CsCl are associated (Bjerrum critical distance = 7.3 A) Me4NCl solvated by at least one complete solvation shell ... 

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). 

In order for a solvated ion to migrate under an electric field, it must be prevented from forming close ion pairs with its counterions by the solvating solvent. The effectiveness of the solvent molecule in shielding the interionic Coulombic attraction is closely related with its dielectric constant. The critical distance for the ion pair formation q is given by eq 4 according to Bjerrum s treatment, with the hypothesis that ion-pair formation occurs if the interionic distance is smaller than... 

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

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. 

Considerable effort has been made to develop a model for the parameter on the basis of statistical theories using simple electrostatic concepts. The first of these was proposed by Bjerrum [25]. It contains important ideas which are worth reviewing. He assumed that all oppositely charge ions within a certain distance of a central ion are paired. The major concept in this model is that there is a critical distance from the central ion over which ion association occurs. Obviously, it must be sufficiently small that the attractive Coulombic forces are stronger than thermal randomizing effects. Bjerrum assumed that at such short distances there is no ionic atmosphere between the central ion and a counter ion so that the electrostatic potential due to the central ion may be calculated directly from Coulomb s law. The value of this potential at a distance r is... 

As suggested by the above discussion, there are serious problems with the Bjerrum model. One of these relates to the fact that unreasonably large critical distances are involved in defining an ion pair in solutions of low permittivity. The second relates to the fact that the probability distribution is not normalized and continues to increase with increase in distance r. The latter problem is effectively avoided by considering only those values of P r) up to the minimum in the curve. 

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. 

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... 

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... 

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