Past exam of the soft condensed matter course of the University of Cambridge / 2015 / Paper 1 / 1 / a / Solution

At fixed temperature and solvent permittivity, the Debye–Hückel screening length satisfies ${\lambda }_D^{-2}\propto {\sum }_i{c}_i{z}_i^2=2I$, where $I=\frac{1}{2}{\sum }_i{c}_i{z}_i^2$ is ionic strength. Fully dissociated sodium chloride has two monovalent ion species, giving $I_1=0.01$ M. Fully dissociated magnesium sulfate has two divalent species, givingThe multivalent-electrolyte screening-length ratio is thereforeThe squared valences are essential: counting only the total ion concentration misses a factor of four in the screening strength per formula unit. This is the comparable linear-screening estimate, assuming unchanged dielectric response and full dissociation. ion pairs and concentrated-electrolyte nonideality can make a real divalent-salt solution depart from it; no such correction is specified.