Past exam of the quantum condensed matter course of the University of Cambridge / 2009 / Section A / A1 / c

In the planar Debye–Hückel approximation, a surface potential decays as $\psi (x)={\psi }_s{e}^{-\kappa x}$, with $\kappa \propto \sqrt{I}$ at fixed temperature and solvent permittivity; $I$ is the ionic strength. For a monovalent salt, reducing its concentration by ten reduces $\kappa$ by $\sqrt{10}$. Calibrating the first decay from the supplied potential ratio, and assuming unchanged ${\psi }_s$, givesHere the printed meV unit is interpreted as the electrostatic potential energy of a singly charged probe; the same ratio calculation applies to potentials in mV.

This numerical answer uses an effective exponential decay calibrated from the data. The inferred initial Debye length is $5/\log (35/6.6)=3.00\mathrm{nm}$, whereas the standard dilute aqueous-electrolyte expression at room temperature gives about $0.96\mathrm{nm}$ for $0.1\mathrm{M}$ monovalent salt. The numbers therefore are not consistent with that absolute ideal-electrolyte model. Moreover a spherical colloid has an additional radius-dependent geometric factor, and its surface potential need not remain constant when salt is changed. Without a surface boundary condition and geometry, the new potential is not uniquely specified beyond the stated planar, fixed-potential approximation.