Concept of a smeared out spherically symmetrical charge density

7 Concept of a smeared out spherically symmetrical charge density 

This is an absolutely crucial part of the model and has been dealt with by statistical mechanical averaging procedures. But only spherical symmetry has been assumed. And so for large non-spherical ions a modification to the smearing out procedure is needed, but see Sections 10.17.3 and 10.19. 

Any one given distribution of ions around a spherically symmetrical central y-ion need not necessarily be spherically symmetrical, but on average all possible arrangements will correspond to spherical symmetry. A charge density necessarily corresponds to an average distribution of ions, so conversion of the Poisson-Boltzmann equation to spherical symmetry is purely formal. 

But as has been hinted at above, there is one important limitation to this when considering large complex electrolytes such as are found in aqueous solutions of biological materials. Here the central ion is non-spherical. An ion which is not spherically symmetrical may impose a non-spherically symmetrical distribution of charge around it, and this ought to be taken care of, but is not, in the theory. The Debye-Hiickel theory can thus only be approximate for non-spherical ions. 

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