This
work presents a comparison between a numerical solution of
the Poisson–Boltzmann equation and the analytical solution
of its linearized version through the Debye–Hückel equations
considering both size-dissimilar and common ion diameters approaches.
In order to verify the limits in which the linearized Poisson–Boltzmann
equation is capable to satisfactorily reproduce the nonlinear version
of Poisson–Boltzmann, we calculate mean ionic activity coefficients
for different types of electrolytes as various temperatures. The divergence
between the linearized and full Poisson–Boltzmann
equations is higher for lower molalities, and both solutions tend
to converge toward higher molalities. For electrolytes of lower valencies
(1:1, 1:2, 2:1, and 1:3) and higher distances of closest approach,
the full version of the Debye–Hückel equation is capable
of representing the activity coefficients with a low divergence from
the nonlinear Poisson–Boltzmann. The size-dissimilar full version
of Debye–Hückel represents a clear improvement over
the extended version that uses only common ion diameters when compared
to the numerical solution of the Poisson–Boltzmann equation.
We have derived a salt-specific index (Θ) to gradually classify
electrolytes in order of increasing influence of nonlinear ion–ion
interactions, which differentiate the Debye–Hückel equations
from the nonlinear Poisson–Boltzmann equation.