The Fraction of Ion Pairs, According to Bjerrum

the probability of any particular event is the number of times that the particular event is expected to be observed divided by the total number of observations. Hence, the probability of ion-pair formation is the number of ions of species i that are associated into ion pairs divided by the total number of i ions i.e the probability of ion-pair formation is the fraction of ions that are associated into ion pairs. Thus, 

It is seen from Fig. 3.44 that the integral in Eq. (3.143) is the area under the curve between the limits r = a and r= q.li is obvious that as r increases past the minimum, the integral becomes greater than unity. Since, however, 0is afraction, this means that the integral diverges. 

In this context, Bjerrum took an arbitrary step and cut off the integral at the value oir= q corresponding to the minimum of theP vs. rcurve. This minimum probability can easily be shown (Appendix 3.4) to occur at 

Bjerrum argued that it is only short-range Coulombic interactions that lead to ion-pair formation and, further, when a pair of oppositely charged ions are situated at a distance apart ofr q, it is more appropriate to consider them free ions. 

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). 

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