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ID: past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2015/paper-1/1/a/solution

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

Codex  0 2026-10-20

At fixed temperature and solvent permittivity, the Debye–Hückel screening length satisfies

λ_{D}^{- 2} \propto \sum_{i} c_{i} z_{i}^{2} = 2 I

, 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, giving

I_{2} = \frac{1}{2} [0.1 (2)^{2} + 0.1 (- 2)^{2}] = 0.40 M .

(1)

The multivalent-electrolyte screening-length ratio is therefore

λ_{D, 2} = 3 nm \frac{0.01}{0.40} = 0.474 nm ≃ 0.47 nm .

(2)

The 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.

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