Model Bjerrum

Electrostatic interactions involving permanent charges (salt bridges). Accord ing to the Bjerrum model the binding constant between two ions A+ and B can be described in terms of the product of the ionic charges zA-zB and the mean effective distance between the ions. These parameters along with the dielectric permittivity (e) determine the magnitude of the Bjerrum function Q(b). The... 

The Bjerrum model for a water molecule has also been used in conjunction with a cell model (Angell, 1971) for water based on the... 

The Bjerrum Model. Bjerrum (see Robinson and Stokes (19)) defined an "ion pair" as existing when two ions of opposite charge approached such that the mutual potential energy between them equalled 2kT (k is the Boltzman constant). At 25 C, this means that an "ion pair" exists if the ion separation distance is equal... 

The Fuoss model yields more accurate AS° values for the sulfate complexes than the Bjerrum model. Most remarkable, however, is the even better overall accuracy of AS° calculated by the simple electrostatic model. 

Figure 12. Empirical and model-predicted aS° values for 1 1 metal sulfate complexes (1 = 0) plotted against z,z./(i y + (F) Fuoss and (B) Bjerrum models.

The major feature is a rapid decrease of A at low salt concentrations, followed by a minimum and pronounced increase. At the CP there is a substantial conductance. To interpret this behavior, we first note that the Debye- Hiickel (DH) theory itself predicts an instability regime at low T, but if compared with experiment C is far too low. Taking account for ion association considerably improves thew results. In the presence of ion association, a higher salt concentration is needed to achieve the concentration of free ions to drive phase separation, i.e. C is shifted to higher values. In particular, the Bjerrum model for ion pair association yields ... 

The stability constants of ion pairs (their log /Cassoc values) have been shown to be proportional to the electrostatic function ZMzJd, where z Z/. are the charge of metal cation and ligand, and d rM + ri, the sum of their crystal radii (cf. Fig. 3.5). Mathematical models for predicting ion pair stabilities generally assume this proportionality and include the simple electrostatic model, the Bjerrum model, and the Fuoss model (cf. Langmuir 1979). Such models can predict stabilities in fair agreement with empirical data for monovalent and divalent cation ion pairs. 

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. 

We shall now describe only two representative pair potentials one, based on the Bjerrum model, was the first to reproduce... 

Fig. 2.47 Schematic description of the various ingredients of the BNS potential function based on the Bjerrum model for water Use R) is the full potential function for the symmetric eclipse (SE) approach of two water molecules (plotted in the upper-right part of the figure). Also, the LJ and the HB parts of the potential for this specific orientation are shown. The function S R) is shown in the upper part of the figure.

We shall describe some results based on this pair potential in Sec. 2.7.10. We next turn to discussing a more general form of an effective pair potential that has some features in common with the Bjerrum model, but has some advantages for an analytical theory of liquid water. 

Uij = Rij/Rij. Similarly, Ga[(ljp u /) + 1] attains its maximum value whenever the direction of the acceptor arm Ij is in the direction -u//. Thus, the product of these three functions attains a value close to unity only if, simultaneously, Rij is about Rh, the direction of Hia is about that of Xij, and the direction of jp is about that of -u /. Such a configuration is said to be favorable for HB formation. Clearly, if all of the above three conditions are fulfilled, then the interaction energy is about sub- The sum of the various terms in the curly brackets of (2.7.8) arises from the total of eight possible favorable directions for HB formation (four when molecule i is a donor and four when molecule i is an acceptor). The variances a and a are considered as adjustable parameters. Note that of the eight terms in the curly brackets, only one may be appreciably different from zero at any given configuration X Xy. Clearly, the HB part of the potential in (2.7.8) does not suffer from a possible divergence as in the Bjerrum model, and clearly does not allow two bonds to be formed by a pair of water molecules (Fig. 2.48b). 

An approximate version of the Percus-Yevick ( Y) equation has been applied for the pair potential based on the Bjerrum model [Ben-Naim (1970)]. The pair correlation function is written in... 

Three well-established techniques in the theory of simple fluids have been adapted recently to the study of waterlike particles in three dimensions. The effective pair potential which has been tried in all of the methods is essentially the one based on the Bjerrum model described in Section 6.4. 

The spontaneous creation of ions in a liquid without an applied field proceeds from an equilibrium with two steps (Bjerrum model, 1926 Onsager, 1934 Eigen and De Maeyer, 1974),... 

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