Solubility Due to Secondary Solvation

It has been stressed that solvation is a far-reaching phenomenon, although only the coordination number and the primary solvation number can be determined. However, there are effects of ions on the properties of solutions that lie outside the radius of the primary hydration sheath. These effects must now be accounted for, insofar as they relate to the solubility of a nonelectrolyte. Let the problem be tackled as though no primary solvation had withdrawn water from the solution. One can write 

A first-approximation calculation would be like this. The field X,of the ion at a 

a dipole (aligned parallel to the ionic field) is moved from infinity, where the field = 0, through a distance dr to a point where the corresponding field is dX, the elementary work done is -fi dX. Thus, the work to bring a mole of nonelectrolyte molecules from infinity to a distance r is 

This expression for the work of replacing a water molecule by a nonelectrolyte molecule at a distance r from an ion can now be introduced into Eq. (2.138) to give (in number of nonelectrolyte molecules per unit volume) 

The exponent of Eq. (2.144) is easily shown to be less than unity at 298 K for most ions. Thus, for distances outside the primary hydration shell of nearly all ions, the field X will be sufficiently small (because of the large dielectric constant of bulk water), so one can expand the exponential and retain only the first two terms, i.e., 

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