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ION–ION INTERACTIONS 245
Fig. 3.13. A spherical shell of thickness dr at a
distance r from the center of the reference ion.
chosen by Debye and Hückel was that of point-charge ions, in which case the lower
limit is r = 0. The upper limit for the integration is because the charge of the
ionic cloud decays exponentially into the solution and becomes zero only in the limit
Thus,
and by substituting for from Eq. (3.35), the result is
The integration can be done by parts (Appendix 3.2), leading to the result
which means that a central ion of charge is enveloped by a cloud containing a
total charge of (Fig. 3.14). Thus, the total charge on the surrounding volume is
just equal and opposite to that on the reference ion. This is of course precisely how
things should be so that there can be electroneutrality for the ionic solution taken as a
whole; a given ion, together with its cloud, has a zero net charge.
