The Application of the Debye-Hückel Theory to Polyions. Part I. The Range of Applicability of the Debye Limiting Law

The Debye limiting law has been used to deduce an expression for the electrical work which must be done to build up a polyion from the corresponding single ions in extremely dilute solutions containing various (appreciable) concentrations of 1 — 1 electrolyte. The case of an infinite straight rigid polyion having charges at equal intervals along its length was treated in greater detail, and a corrected expression was obtained for the above-mentioned electrical work, which makes allowance for the finite size of the polyion charges and takes some account of the steric effects due to the remainder of the polyion molecule. The error due to the high electric potential ψ surrounding such a polyion which invalidates the Debye approximation of replacing sinh (εψ/kT) by εψ/kT was evaluated by numerical integration of the Poisson-Boltzmann equation. The integration was started at a considerable distance from the polyion on the assumption of cylindrical symmetry and continued at closer distances on the assumption of spherical symmetry surrounding each polyion charge. The results obtained indicated that the error involved in using the simple Debye limiting law formula would seldom be more than a few percent, provided the concentration of the 1 — 1 electrolyte was rather low (not more than about N/100).