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No. 028 October 2025.
Materials science, homework one

Coulomb energy of one Na⁺–Cl⁻ pair: −8.15 × 10⁻¹⁹ J

Two ions carrying one electron’s charge each, 0.283 nm apart: the sodium radius, 0.102 nm, plus the chloride radius, 0.181 nm. The attraction is Coulomb’s law with nothing added.

EA = −A / r,A = e² / 4πε0 ≈ 2.307 × 10⁻²⁸ J m

EA ≈ −2.307 × 10⁻²⁸ / 2.83 × 10⁻¹⁰ ≈ −8.15 × 10⁻¹⁹ J per pair

Multiplied by Avogadro’s number, that is −491 kJ for a mole of pairs.

0-4-8-12-160.20.40.60.80.283 nm, −8.15 × 10⁻¹⁹ JE, 10⁻¹⁹ Jr, nm
0-4-8-12-160.20.40.60.80.283 nm, −8.15 × 10⁻¹⁹ JE, 10⁻¹⁹ Jr, nm
The attraction alone, −A/r, with the pair’s separation marked. Computed on this page from the assignment’s constants.

what I don’t understand yet

How one pair’s −491 kJ/mol becomes a crystal’s lattice energy. Every ion feels every other one, and the Madelung constant is the sum of that; the repulsion that stops the collapse comes after it.