Page 308 - MODERN ELECTROCHEMISTRY

244 CHAPTER 3

Fig. 3.11. The distribution of excess
charge density around a central ion
can be pictured as a cloud, or atmos-
phere, of net charge around the cen-
tral ion.

charge density decays with distance r in an exponential way. The excess charge
residing on the ion cloud is opposite in sign to that of the central ion. Thus, a positively
charged reference ion has a negatively charged ion atmosphere, and vice versa (Fig.
3.12).
Up to now, the charge density at a given distance has been discussed. The total
excess charge contained in the ionic atmosphere that surrounds the central ion can,
however, easily be computed. Consider a spherical shell of thickness dr at a distance
r from the origin, i.e., from the center of the reference ion (Fig. 3.13). The charge dq
in this thin shell is equal to the charge density times the volume of the shell,
i.e.,

The total charge contained in the ion atmosphere is obtained by summing
the charges dq contained in all the infinitesimally thick spherical shells. In other words,
the total excess charge surrounding the reference ion is computed by integrating dq
(which is a function of the distance r from the central ion) from a lower limit
corresponding to the distance from the central ion at which the cloud is taken to
commence to the point where the cloud ends. Now, the ion atmosphere begins at the
surface of the ion, so the lower limit depends upon the model of the ion. The first model

Fig. 3.12. A positively charged ion has a nega-
tively charged ionic cloud, and vice versa.