The Ionic Cloud around a Central Ion

as explained in Section 3.3.2, the principal objective of the Debye-Htickel theory is to calculate the time-averaged spatial distribution of the excess charge density around a reference ion. How is this objective attained  

The Poisson equation (3.4) relates the potential at r from the sample ion to the charge density at r, i.e.. 

From these two equations, one has the linear relation between excess charge density and potential, i.e.. 

Here then is the desired expression for the spatial distribution of the charge density with distance rfrom the central ion (Fig. 3.10). Since the excess charge density results from an unequal distribution of positive and negative ions, Eq. (3.35) also describes the distribution of ions around a reference or sample ion. 

To understand this distribution of ions, however, one must be sufficiently attuned to mathematical language to read the physical significance ofEq. (3.35). The physical ideas implicit in the distribution will therefore be stated in pictorial terms. One can say that the central reference ion is surrounded by a cloud, or atmosphere, of excess charge (Fig. 3.11). This ionic cloud extends into the solution (i.e., r increases), and the excess 

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Fig. 4.92. Debye s thought experiment to calculate the time for the ion atmosphere to relax (a) the ionic cloud around a central ion ...

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