The ionic atmosphere in more detail

In the theory, all ions are considered to be spherically symmetrical with the charge on the ion taken to be at the centre of the ion. One ion is taken to be a central reference ion with all the other ions which make up the ionic atmosphere arranged around it. There is no preferred direction for this arrangement and this implies that the ionic atmosphere is spherically symmetrical around the central reference ion. Polar coordinates are therefore used and the centre of the central reference ion is taken as the origin of the coordinate system. It is also the position of the charge on the central reference ion. 

Crystallographic measurements give sizes for unsolvated ions, and these vary considerably from ion to ion. But, in solution, most ions will be solvated and to differing extents. This is a problem which is not explicitly tackled by the simple Debye-Huckel theory. 

There are two related potentials pertinent to these distances which are of fundamental importance in the Debye-Hiickel theory  

It is vital to realise that these two potentials have the same magnitude, as shown by the following standard argument of electrostatics. 

Mathematically the ionic atmosphere can be treated as though it were a sphere of radius, a, with all the ions of the ionic atmosphere distributed on the surface of this sphere. Electrostatic theory states that inside such a sphere the potential due to the ionic atmosphere is constant at aU points within the sphere and is equal to the value of this potential at the surface of the sphere. It follows that  

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