Ion-association theory

In dealing with equilibria in natural waters, we wanted to give above all a feeling of the power of approach. In order not to overwhelm the reader with a large number of intricate details, we attempted to make nonideality corrections for electrolyte solutions simple and effective. The objective of this appendix is to review the various equilibria conventions usable for different natural water media—especially to compare the available conventions for describing (and measuring) pH and ionic equilibria in seawater—and to give an introduction to the ionic interaction theory, which is an expedient alternative and complementary approach to the ion association theory. 

A brief summary of the status of the thermodynamic properties for water-mineral reactions using the ion-association theory and revised data is ... 

The Bjerrum ion-association theory (q.v.) accounts for the results satisfactorily in some cases, but in others it is clear that additional factors must be taken into account. One of these is hydration—that is, the interaction of the free ions and ion-pairs, respectively, with the surrounding solvent. It has been shown that two types of ion-pair exist in, for instance, copper sulphate solutions ... 

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). 

Although the Bjerrum theory is thus not in general quantitatively applicable, the concept of ion association is very useful. It has assisted in an explanation of various phenomena observed in the study of homogeneous... 

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. 

Fig. 9. Logarithm of the cation activity coefficient versus the square root of the concentration for the system of manganese ions and cation vacancies in sodium chloride at 500°C. Filled-in circles represent the association theory with Rq = 2a, and open circles the association theory with R = 6/2. Crosses represent the present theory with cycle diagrams plus diagrams of two vertices, and triangles represent the same but with triangle diagram contributions added.

Fig. 10. The degree of association into nearest- and next-nearest-neighbour complexes, p, versus concentration, c, at 500°C for manganese ions and cation vacancies in sodium chloride. Filled circles represent the simple association theory, open circles the Lidiard association theory, and crosses the present theory using Eq. (173) when the first term only has been retained in the virial appearing in the equation for the defect distribution function (Eq. (168)). The point of highest concentration represented by a cross may be in error due to the neglect of higher terms in the virial series, and the dotted curve has not been extended to include it.

In support of the association theory, colloid chemists cited non-reproduceable cryoscopic molecular weight determinations (which were eventually shown to be caused by errors in technique) and claimed that the ordinary laws of chemistry were not applicable to matter in the colloid state. The latter claim was based, not completely without merit, on the ascerta-tion that the colloid particles are large aggregates of molecules, and thus not accessible to chemical reactants. After all many natural colloids were shown to form double electrical layers and adsorb ions, thus they were "autoregulative" by action of their "surface field" (29). Furthermore, colloidal solutions were known to have abnormally high solution viscosities and abnormally low osmotic pressures. 

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) ... 

The two mass action equilibria previously indicated have been used in conjunction with a modified form of the Shedlovsky conductance function to analyze the data in each of the cases listed in Table I. Where the data were precise enough, both K2 and K were calculated. As mentioned previously, the K s so evaluated are practically the same as those obtained for ion pairing in solutions of electrolytes in ammonia and amines. This is encouraging since it implies a fairly normal behavior (in the electrolyte sense) for dilute solutions of metals. Further support of the proposed mass action equilibria can be found in the conductance measurements of sodium in NH8 solutions with added salt. Bems, Lepoutre, Bockelman, and Patterson (4) assumed an additional equilibrium between sodium and chloride ions, associated to form NaCl, to compute the concentration of ionic species, monomers, and dimers when the common ion electrolyte is added. Calculated concentrations of conducting species are employed in the Onsager-Kim extension of the conductance theory for low-field conductance of a mixture of ions. Values of [Na]totai ranging from 5 X 10 4 to 6 X 10 2 and of the ratio of NaCl to [Na]totai ranging from zero to 28.5 are included in the calculations. 

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 mechanisms of reactions that occur in condensed phases involve the participation of solvent degrees of freedom. In some cases, such as in certain ion association reactions involving solvent-separated ion pairs, even the very existence of reactant or product states depends on the presence of the solvent. Traditionally the solvent is described in a continuum approximation by reaction-diffusion equations. Kapral s group is interested in microscopic theories that, by treating the solvent at a molecular level, allow one to investigate the origin and range of validity of conventional continuum theories and to understand in a detailed way how solvent motions influence reaction dynamics. 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

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]... 

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. 

Freedman TB, Cao XL, Young DA et al (2002) Density functional theory calculations of vibrational circular dichroism in transition metal complexes Identification of solution conformations and mode of chloride ion association for (+)-tris(ethylenediaminato)cobalt (III). J Phys Chem A 106 3560-3565... 

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