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238 CHAPTER 3
analysis only for systems in which the average electrostatic potential would be much
smaller than the thermal energy kT. Then:
Based on this assumption, one can expand the exponential of Eq. (3.10) in a Taylor
series, i.e.,
and neglect all except the first two terms. Thus, in Eq. (3.10),
The first term gives the charge on the electrolytic solution as a whole. This
is zero because the solution as a whole must be electrically neutral. The local excess
charge densities near ions cancel out because the excess positive charge density near
a negative ion is compensated for by an excess negative charge density near a positive
ion. Hence,
and one is left with
3.3.6. The Linearized Poisson–Boltzmann Equation
The stage is now set for the calculation of the potential and the charge density
in terms of known parameters of the solution.
