Bjerrum relation

In order to achieve an adequately large KD, the adjustment of three variables is at our disposal, which are summarised in the Bjerrum-Fuoss equation (2) relating KD to D, the temperature T, and a quantity a which is determined by the distance of closest approach of the ions when paired ... 

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... 

Fig. 2.14 Comparison of the log /

Equation 33 is the familiar statistical relation between the equilibrium constants in a series of stepwise equilibria as derived by N. Bjerrum (II) for polyprotic acids and applied by J. Bjerrum (12) to complex ion equilibria. Substituting Equation 33 for K, into Equations 23 and 24 for 0f° and 0o° gives... 

Major proteins of globule and serum membranes are immunochemi-cally identical (Nielsen and Bjerrum 1977), and electrophoretic profiles of the proteins from either membrane are similar (Kitchen 1974). The major quantitative difference is the presence of higher amounts of a protein of Mr 85,000 in serum membrane fractions. In summary, the information suggests that milk serum membranes are related, but not identical, to milk lipid globule and plasma membranes. 

The effect of added salts on the rate constant of a given ionic reaction has been studied for many years. The Br nsted-Bjerrum treatment of these salt effects has been particularly successful, the rate constant being related to the ionic strength of the solution. The observed trends can be quantitatively accounted for using the DHLL or a related expression for the activity coefficients of reactants and transition state. This subject has been reviewed in detail (Perlmutter-Hayman, 1971). The ionic-strength principle appears satisfactory when the reaction involves ions of opposite charge but less so when it involves ions of the same charge. 

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]... 

Blum s use of the MSA represents a significant advance, but it does not take into account either ionic association or Bjerrum s very reasonable idea (Section 3.8) about the removal of free water in the solution by means of hydration. Furthermore, Blum s equations do not explain the relation between conductance and concentration noted for many electrolytes, particularly at high concentrations, that is. 

Avdeef (1998) has reported an automated potentiometric titration method for the determination of solubilities of drug substances containing ionizable groups, where a graphical procedure is used for the estimation of solubility constants based on Bjerrum difference plots. One useful relation derived in this work was ... 

Metal-enzyme complexes, a subgroup of metal-protein complexes, exhibit enzymatic activity consequent to readily dissociable combination with a variety of metal ions. Many of these studies have been performed with unpurified enzymes, and, even when pure enzymes were used, the stoichiometry of the interaction of the metal and enzyme has not been measured. Enhancement of enzymatic activity as a result of the addition of metal ions and its partial loss on their removal has been the chief criterion of assessment of physiological significance. Only in a few instances, e.g., enolase, has the stability and stoichiometry been studied in relation to function (Malmstrom, 1953, 1954). The study of metal complexes and particularly metal chelates (Bjerrum, 1941 Martell and Calvin, 1952 Calvin, 1954) has provided both new experimental and new theoretical backgrounds for the study of metals in relation to the specificity of enzyme action, metal-enzyme (Calvin, 1954), metal-substrate (Najjar, 1951), and metalloenzyme interaction, as well as metal-enzyme inhibition (James, 1953). 

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. 

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. 

Here, g0 is the polyion s charge density, and the potential of the finite line charge in the Debye-Hiickel approximation is denoted by linear charge density and related to the Bjerrum length ZB via i s) = ZB. One finds that the solution to Eq. 1 for the linear charge density is [84,83] ... 

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. 

The practical osmotic coefficient obtained directly from cryoscopic measurements gives a measure of the deviation from ideality for the solvent species. The corresponding measure of deviation from ideality for the solute species is the activity coefficient, y. y and (f> are related as a consequence of the Gibbs-Duhem equation. The following expression was first derived by Bjerrum ... 

One purpose of calculating y is to enable comparisons to be drawn between experimental data and theory. However, using Bjerrum s relation a theoretical expression for may be derived and compared directly with the experimental value. Taking this process a step further, a theoretical value of 0 may be calculated and compared directly with the experimentally determined value. 

Bjerrum, L. 1967. Engineering geology of Norwegian normally-consolidated marine clays as related to settlement of buildings. Geotechnique, 17(2) 81-118. 

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