During a redox titration the solution's electrode potential E climbs along a Nernstian S-curve as oxidant is added, jumping sharply near the equivalence volume Veq. For 1-electron couples:
V < Veq: E = E1° + 0.05916·log10(V / (Veq−V))
V > Veq: E = E2° + 0.05916·log10((V−Veq) / Veq)
V = Veq: E = (E1° + E2°) / 2
A redox indicator is itself a reversible couple, Inox + e⁻ ⇌ Inred, with its own standard potential E°ind and its own Nernst equation:
E = E°ind + 0.05916·log10([Inox]/[Inred])
fraction oxidized = 1 / (1 + 10^((E°ind − E)/0.05916))
The top panel plots the full titration curve. The bottom panel is a sorted population sweep: 400 indicator molecules, laid out in a grid ordered by their own transition threshold (lowest at top-left, highest at bottom-right — a rank plot rather than a spatial snapshot), flip from reduced to oxidized color as the solution potential sweeps past each one's personal threshold. Because the grid is sorted, the flipping molecules always form a clean diagonal-ish wavefront whose position along the grid directly visualizes the macroscopic sigmoid — the same shape as the "fraction oxidized" curve above, just decomposed molecule-by-molecule.
The indicator visibly changes color when the population sweep crosses the ~50% line, i.e. when the solution potential E passes through E°ind. That happens at whatever titrant volume makes E = E°ind — not necessarily at Veq. The steeper the potential jump and the closer E°ind sits to (E1°+E2°)/2, the smaller the resulting titration error.
- Volume slider / Auto-titrate — adds titrant; watch the point climb the S-curve and the sweep front cross the population grid.
- E°₁ / E°₂ sliders — reshape the S-curve, changing how sharp the equivalence-point jump is and where it sits.
- Indicator dropdown — swap indicators to see a poorly matched E°ind produce a large color-change error, even though the true equivalence point never moves.