Consider an original toy AB <salt> model with $K_{\mathrm{sp},c}=10^{-8}\,\mathrm M^2$ stipulated at fixed <temperature> and fixed <activity coefficient> product. This is a conditional concentration constant, not measured data for <calcium carbonate> or any identified material. A stock contains A <cations> and X <anions>; another contains Y <cations> and B <anions>. All four <ions> have unit <electric charge> magnitude. Only AB can form a <solid>; X and Y remain <spectator ions>. Assume full dissociation, no other <chemical reactions>, and a mixed <volume (thermodynamics)> equal to the sum of the initial <volumes>.

Mix two stocks, each of <volume (thermodynamics)> $V$,, each at <molar concentration> $C$ of its indicated <salt>. Before any <precipitation (chemistry)>, the mixed <volume (thermodynamics)> is $2V$, so
$$
c_{\mathrm A,0}=c_{\mathrm B,0}=\frac{CV}{2V}=\frac C2,\qquad Q_{c,0}=\frac{C^2}{4}.
$$
Multiplying the two stock <concentrations> would overestimate the mixed <ion product> by a factor of four. The saturation threshold is $C=2\sqrt{K_{\mathrm{sp},c}}=2\times10^{-4}\,\mathrm M$.

If $C=10^{-4}\,\mathrm M$, mixing gives $Q_{c,0}=2.5\times10^{-9}\,\mathrm M^2<K_{\mathrm{sp},c}$: no AB <precipitate> is favored. If $C=4\times10^{-4}\,\mathrm M$, mixing gives $Q_{c,0}=4\times10^{-8}\,\mathrm M^2>K_{\mathrm{sp},c}$. Suppose <nucleation> occurs and bulk AB <solid> remains at <chemical equilibrium>. Equal starting inventories and one-to-one <stoichiometry> require equal remaining <concentrations>, hence
$$
c_{\mathrm A,eq}=c_{\mathrm B,eq}=\sqrt{K_{\mathrm{sp},c}}=10^{-4}\,\mathrm M,\qquad
\frac{n_{\mathrm{AB},s}}{2V}=\frac C2-10^{-4}\,\mathrm M=10^{-4}\,\mathrm M.
$$
For two $10\,\mathrm{mL}$ portions in the second case, each reacting <ion> starts with $4\,\mathrm{\mu mol}$; the final $20\,\mathrm{mL}$ <chemical solution> retains $2\,\mathrm{\mu mol}$ of each, and $2\,\mathrm{\mu mol}$ of AB enters the <solid>. The respective X and Y inventories stay at $4\,\mathrm{\mu mol}$. Exactly half of each reacting inventory precipitates in this toy model.

\Image[/precipitation-mixing-saturation-threshold.png]
{title=Equal-volume dilution changes the <ion product> before <precipitation (chemistry)>}
{id=image-precipitation-mixing-saturation-threshold}
{description=Original conditional AB model with $K_{\mathrm{sp},c}=10^{-8}\,\mathrm M^2$. Left: the two reacting-ion <concentrations> occupy different stocks and each halves on mixing. Right: the initial mixed <ion product> relative to $K_{\mathrm{sp},c}$, with the two worked examples and saturation threshold. The curve is a model calculation, not measured cloudiness or a time trace.}

The <chemical equilibrium> calculation adds a phase constraint to the inventory balance. Write $x=n_{\mathrm{AB},s}/(2V)$ and $c=C/2-x$. In the equilibrated bulk model,
$$
x\ge0,\qquad c\ge0,\qquad c^2\le K_{\mathrm{sp},c},\qquad x\bigl(K_{\mathrm{sp},c}-c^2\bigr)=0.
$$
When $x>0$, the last condition requires saturation; when $x=0$, the inventory can leave an undersaturated <chemical solution>. Together these give $x=\max\{0,C/2-\sqrt{K_{\mathrm{sp},c}}\}$. This excludes a persistent <supersaturated solution> by its explicit <chemical equilibrium> assumption; it does not model a <nucleation> delay.


 Precipitation after equal-volume mixing (source code)