Effect of valence and size on counterion binding

Equations (4.5) and (4.8), which were developed for univalent ions, can be rewritten, thus  

The critical value for Q is 1/z. There is a proportional increase in the number of free counterions, f/z, as Q increases from zero, reaching a plateau when Q = z. Also, below this value the degree of dissociation,) , increases as the concentration decreases, and tends to unity as v tends to zero. When Q /z, p decreases with 0 and tends to /zQ as 0 tends to zero. The number of free ions caimot exceed njz Q. Note that this number is inversely proportional to the square of the valence. The condensation of ions is thus very sensitive to valence for multivalent counterions it takes place at a lower value of Q and the number of free ions is much smaller 

Imai (1961) has observed that multivalent counterions are more strongly bound than are monovalent ones. This phenomenon can be demonstrated theoretically by considering equilibrium conditions for two counterions with valencies z and (Zj z and degrees of dissociation and P -For a cylindrical model the equilibrium equations are 

As 0 0 then the solutions to these equations fall into four groups depending on the value of Q. 

The selective binding of cations is not as sensitive to size as to valence. The value of Q for the condensation of counterions of the same valence is unaffected. In the case of monovalent cations, the dissociation of all counterions is complete at infinite dilution, when 2 1 When Q the 

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