Bjerrum radius

XX, YY, ZZ component of quadrupolar tensor q universal gas constant 8.31432 J/mol K radius of a curvature (Ch. 3) of a spherical particle or an atom (Ch. 15) of a capillary mid-point radius in Couette geometry Bjerrum radius... 

It is remarkable that an approach (Bjerrum s association theory ), which is based on ion pair formation, was very successful for very concentrated solutions. The Bjerrum radius... 

Distance of ions in an ionic lattice Rate of reaction specified by the subscript Bjerrum radius... 

Bjerrum considered the case of spherical ions in a solvent of dielectric constant e. The probability of finding two ions of opposite charge at a distance A from each other is calculated from the number of ions surrounding a central ion of opposite charge in a spherical shell of thickness dA and radius A. This probability, fV(A), is given by... 

Bjerrum concluded therefore that ion-pair formation occurs when an ion of one type of charge (e.g., a negative ion) enters a sphere of radius q drawn around a reference ion of the opposite charge (e.g., a positive ion). However, it is the ion size parameter that defines the distance of closest approach of a pair of ions. The Bjerrum hypothesis can therefore be stated as follows If a <, then ion-pair formation can occur if a> q, the ions remain free (Fig. 3.45). 

What is the total charge on an ionic atmosphere around an anion of valence z From the data in the text, examine logy vs. Vm, where tn is the molality of the solution, from 0 to 1 mol dm". The plots always pass through a minimum. Use the fully extended Debye-Hlickel theory, including the Bjerrum-Stokes and Robinson terms, to find the significance of the minimum at which the electrolyte concentration increases with the increase of the cation radius. 

On the basis of tois model, Lee and Wheaton arrived at an equation for in terms of q, the Bjerrum distance. The equation is several lines long and clearly only fit for use in appropriate software. The application of experimental data of to the equation allows one to find values of and the co-sphere radius R. These values... 

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. 

Born (1) and later Bjerrum (2) developed a theoretical approach to ion-solvent interactions based on a rather simple electrostatic model. Ions are considered as rigid spheres of radius r and charge z in a solvent continuum of dielectric constant e. Changes in enthalpy AH av) and in free energy AG av), respectively, associated with the transfer of the gaseous ions into the solvent are represented by the following equations ... 

Figure 11. Empirical — aG° data for 1 1 metal sulfate complexes (1 = 0) plotted against z z./(ym + r o J, where z+ and z. are the valence of cation and sulfate ion, Ym is the crystallographic radius in Angstroms of the cation in sixfold coordination (3S), and ysoj, = 3.05 A (59). The locus of — aG° values computed for the complexes by the simple electrostatic model is shown as a dashed line, and computed by the Fuoss and Bjerrum equations as lines labeled (F) and (B), respectively.

The suggestion was made by Bjerrum that all ions lying within a sphere of radius rmin. should be regarded as associated to form ion-pairs,... 

The value of the Bjerrum distance depends on the ionic charges, the nature of the solvent and the temperature but not on electrolyte concentration. For aqueous solutions of 1-1 electrolytes at 25°C, it is equal to 357 pm. Bjerrum proposed that all ions contained in a sphere with this radius are paired. This is a reasonable proposal for aqueous systems, since ions of typical size would be close to contact at such small separations. However, if the solvent has a lower dielectric permittivity, the distance over which ions are considered to be paired increases and the assumption that they are in contact is more diiScult to accept. For example, when the solvent s relative permittivity is 10, the Bjerrum distance r-Q increases to 2800 pm. 

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 Fuoss estimate of is based on a more reasonable model than that of Bjerrum and therefore is preferred. However, there are also problems with the Fuoss treatment in so far as it considers the solvent to be a dielectric continuum. Dielectric saturation effects are expected to be important, especially near the ions involved in ion pair formation. The second problem relates to the choice of the effective size for the ions. In the calculation made here the value of a for MgS04 was chosen to be much bigger than the crystallographic radius of Mg. This presumably is because the cation is strongly hydrated in aqueous solution. One is then faced with the question whether the ion pair involves contact of the two ions or whether it is better considered to be a species in which the two ions are separated by at least one water molecule. These questions can only be properly resolved using other experimental methods. 

This minimum value of r is frequently referred to as the Bjerrum distance and given, as shown, the symbol q. Bjerrum makes the arbitrary assumption that ions inside the sphere with radius q are associated, and that those outside this sphere are free. The value of q evidently depends upon the ratio of the electrostatic potential energy of an ion pair, t2/Dr, to the thermal kinetic energy of an ion, which is 3... 

This section discusses simulations in which the parameters are explicitly mapped to experimental systems. In particular, this mapping affects ion diameter cr, rod radius r0, Bjerrum length B, and line charge density A. In order to have a rod radius different from cr this requires the introduction of a new potential for ion-rod interactions, for which a modified truncated and shifted Lennard-Jones potential has been used ... 

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. 

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