Bjerrum fraction associated

Bjerrum s theory of ionic association gives rise to an expression for the fraction of ions in an ionic solution which are associated. Use the theory to calculate the degree of association of a 0.01 M MgClj solution in ethanol (e = 32). 

Haymet and co-workers have calculated the mole fraction of dimers (associated ions) in electrolytic solutions, and some of their results are shown in Fig. 3.51. Use the equations of the Bjerrum theory applied to NajP04 and compare the results with those of the correlation function approach used by Haymet et al. The essential difference between the Haymet approach and that of Bjerrum is that... 

Consider KCl and take a to be the sum of the ionic radii. Use data from tables to get these. Thus, one can calculate d of the Bjerrum theory over a reasonable concentration range and, using appropriate tables, obtain the value of the fraction of associated ions. Now recalculate the values of log/. for KCl for 0.1 to 2 Af solutions from the full Debye-Hlickel theory involving allowance for ion size and hydration — but now also taking into account 0. In this approach, Cq(1 - 0) is the concentration of the ions that count in the expressions. (See Appendix 3.6.) Does this accounting for 0 improve the fit ... 

The values of Q(b) as defined above have been tabulated for various values of b from 1 to 80, and so by means of equation (76) it is possible to estimate the extent of association of a uni-univalent electrolyte consisting of ions of any required mean diameter a, at a concentration c in a medium of dielectric constant D, It will be seen from equation (76) that in general d increases as b increases, i.e., d increases as the mean diameter a of the ions and the dielectric constant of the solvent decrease. The values for the fraction of association of a uni-univalent electrolyte in water at 18 have been calculated by Bjerrum for various concentrations for four assumed ionic diameters the results are recorded in Table XXXVII. The extent of association is seen to increase markedly with decceasing ionic diameter and increasing concentration. The values are... 

The Bjerrum calculation gives the fraction of the ions associated to ion pairs, and this is calculated for very dilute solutions. From this a theoretical Kassoc at infinite dilution, i.e. for an ideal solution, can be found. 

In the inner region both ions are relatively close and ion i interacts with the unshielded potential of the central ion thus OJy = — (z e /e r) for symmetrical electrolytes. The fraction of ions forming ion pairs separated by distances between a and d are obtained by integration of eqn. 5.3.5 in the inner region. This fraction was termed associated by Bjerrum. In sufficiently dilute solutions the expression (5.3.2a) for Kj becomes... 

To compare both treatments it is useful to calculate the distribution of associated ions around the central one as predicted by Bjerrum s distribution function (5.3.5). We may inquire at which distance dfrom the central ion does a given fraction of associated ions exist, on the average. This fraction will be f(d) f l) and from eqn. 5.3.5... 

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