Bjerrum probability distribution

FIGURE 2.9 Bjerrum probability distribution. Probability that a species of given charge (0 or -1) will be found at a distance r from a central ion of charge +1. A negative ion inside the distance q is considered paired with the central cation. If the distance of closest approach is a, the probability that a pair exists is proportional to the shaded area under the upper curve, aq, pairing does not occur. After Bjerrum (1926). 

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. 

A discussion of the statistical distribution of the population of the ion-pairs (9a) with respect to the inter ionic distance, based on the Bjerrum probability ratio and a continuum for the solvents reads ... 

Fuoss [26] revised the distribution function taking into account the fact that the probability should go to zero for large values of r. However, the modified function describing P(r) is essentially the same as that given by Bjerrum for small values of r. Therefore, the simpler theory is that usually used. 

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