INTERACTIVE COMPANION · VOLUME 1, ISSUE 3 · SPRING 2025

The Solubility Bench

A live model accompanying “The Common Ion Effect Is Taught Backwards”

Model 1 solves the exact silver chloride system live, the same equation the article derives in §2, and draws it against the naive textbook rule so you can watch them agree and then stop agreeing. Model 2 turns a second ligand, ammonia, into a map of where the common-ion rule holds and where it reverses. Both run the identical arithmetic as the analysis script, in your browser, on every drag.

Model 1. Push chloride past the minimum

Added chloride 3.01e-3 M Start here, right on the minimum. Drag left toward 10^-6 M and solubility rises back toward the pure-water value. Drag all the way right, past 10^-1, and watch it rise again for a completely different reason. pH 7.0 Chloride is the conjugate base of a strong acid, so this barely moves the curve at all. Compare that with what pH does to calcium fluoride in §5 of the article. Ammonia (complexing agent) 0 M Ammonia gives silver a second way into solution that has nothing to do with chloride. Raise it and watch the minimum, and the orange marker in Model 2 below, slide to the right.

exact solubility: – naive rule (Ksp/C): – naive is off by: – minimum sits at: –

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Model 2. The regime map

Every point on this grid is its own exact solve of the two-ligand system, classified by whether raising chloride at that point would raise or lower solubility. The white marker tracks Model 1's sliders directly; move chloride or ammonia above and watch it cross the boundary.

this condition: – share of the map that reverses: –

What's going on here

Both panels solve \(K_{sp} = [\mathrm{Ag}^+][\mathrm{Cl}^-]\) together with four stepwise chloro-complex equilibria and two stepwise ammine equilibria, exactly as derived in the article, by bisection on \(\log[\mathrm{Cl}^-]\) for a target added-chloride concentration. Nothing here is a lookup table or a fit to the article's figures: every curve, every cell of the map, and every number in the readouts is this page's own root finder, run fresh in your browser.