Everything in the parent note is built from the small pile of symbols below. We define each one in plain words, draw the picture it stands for, and say why the topic can't proceed without it — in an order where each rests on the one before.
Picture it (figure below) — a positive ion (a cation) is a ball missing electrons; a negative ion (an anion) is a ball with spare electrons. The red ball is Cu2+, the blue ball is S2−; the yellow arrows are the electric pull between opposite charges — that pull is the whole reason a metal cation and a sulphide anion can lock together into a solid CuS.
Picture it — free ions drifting apart in water suddenly pair up and sink. That sinking is the observation we read (a colour and a cloud). The parent note's colours — white AgCl, black CuS, brown Fe(OH)3 — are exactly these precipitates.
Picture it — a see-saw that has stopped moving because both sides push equally (not because nothing happens — both directions run, but cancel). K is the tilt setting at which it balances.
Picture it — a fixed box of water; [X] is how many dots of X are packed inside it. Big number = crowded; small number = nearly empty.
Recall A note for the careful reader: activities vs concentrations
Strictly, mass-action laws use activities (an "effective concentration" that corrects for ions bumping into each other in crowded solutions), not raw concentrations. At the modest concentrations of qualitative analysis we approximate activity by concentration — good enough to decide which group precipitates, but remember it is an approximation, not an exact law.
Picture it — a slider on a log scale (below): each step left divides by ten. The x-axis is the exponent n in 10n; 10−7 and 10−13 look "close" as labels but the green double-arrow marks that they are a million-fold apart — that gap is why the parent note stresses the two dissociation steps are "very different".
Now apply the law of mass action (§3) to a dissolving solid. Since the solid is left out, only the ion terms survive:
Picture it — a glass filling with water (below). The yellow dashed line (Ksp) is the rim height; the blue level (Q) is the current water level. In the left glass Q<Ksp (dissolved); in the right glass Q crosses the rim and spills (red = precipitate).
H2S releases its two H+ in two steps, each with its own acid constant Ka1 (first H+) and Ka2 (second H+). Their sizes (Ka1∼10−7, Ka2∼10−13) are the "tiny powers of ten" §5 warned us about; the second step is a million times more reluctant than the first — see Weak Acid Dissociation — H2S.
Picture it — the same log slider as §5, now reading acidity. Adding HCl shoves it toward more H+ (acidic, low pH); adding NH4OH shoves it the other way (basic, high pH). Every "acidic medium / basic medium" instruction in the parent note is just which way we shoved this slider.
Before the common-ion algebra we need one rule: how the two H2S steps merge into a single overall equilibrium.
How it works quantitatively — solve the overall equilibrium above for [S2−]:
[S2−]=[H+]2Ka1Ka2[H2S]
Adding HCl raises [H+]. Since Ka1Ka2 is fixed and, as argued below, [H2S] stays fixed too, the only way to keep the equation true is for [S2−] to shrink (it sits under [H+]2). That is the common-ion effect written as algebra.
Picture it — a crowded doorway: if H+ is already jammed in the hallway (from HCl), H2S can't push its own H+ out, so it barely dissociates → [S2−] stays tiny. This is exactly how HCl keeps Group IV from precipitating early. Full treatment: Common Ion Effect, driven by Le Chatelier Principle.
How to read this map: each box is a foundation from the sections above; an arrow A→B means "you need A before B makes sense." Follow the arrows top-to-bottom and you retrace the exact build order of this page, ending at the full separation scheme.
