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ION–ION INTERACTIONS 247

Fig. 3.15. The distance variation (in units) of the
charge dq enclosed in a dr-thick spherical shell,
showing that dq is a maximum at

Hence, the maximum value of the charge contained in a spherical shell (of
infinitesimal thickness dr) is attained when the spherical shell is at a distance
from the reference ion (Fig. 3.15). For this reason (but see also Section 3.3.9), is
known as the thickness, or radius, of the ionic cloud that surrounds a reference ion.
An elementary dimensional analysis [e.g., of Eq. (3.43)] will indeed reveal that
has the dimensions of length. Consequently, is sometimes referred to as the
Debye–Hückel length.
It may be recalled that is given [from Eq. (3.20)] by

As the concentration tends toward zero, the cloud tends to spread out increasingly (Fig.
3.16). Values of the thickness of the ion atmosphere for various concentrations of the
electrolyte are presented in Table 3.2.

3.3.9. Contribution of the Ionic Cloud to the Electrostatic Potential
at a Distance r from the Central Ion

An improved feel for the effects of ionic clouds emerges from considering the
following interesting problem. Imagine, in a thought experiment, that the charge on
the ionic cloud does not exist. There is only one charge now, that on the central ion.
What is the potential at distance r from the central ion? It is simply given by the familiar
formula for the potential at a distance r from a single charge, namely,