Aliquot-aware dye inventory in titration

An aliquot-aware dye inventory accounts for dye removed in samples as well as dilution by titrant. In dipyrrin titration mass balance, an absorbance change can result from changing protonation, changing solution volume or loss of material from the vessel. Those effects require separate terms even when the underlying acid-base transformation is reversible.

Let a well-mixed vessel contain volume $V_j$, dye amount $n_j$ and total dye concentration $C_j=n_j/V_j$. Suppose an aliquot $v_j$ is removed first, then dye-free titrant volume $a_j$ is added, with no other gain or loss. The next inventory issoThe order is part of the protocol. Adding titrant before withdrawing the same numerical volume would remove dye from a different concentration.

For repeated sampling followed by dye-free replacement to restore a fixed volume $V$, let $r=v/V$. Then $n_k=n_0(1-r{)}^k$. Three $10\%$ withdrawals leave $0.9^3=0.729$ of the original dye, so $100\mathrm{nmol}$ becomes $72.9\mathrm{nmol}$. Subtracting three times the first aliquot's amount would incorrectly give $70\mathrm{nmol}$, because later aliquots contain less dye.

More generally, the removed amount is ${\sum }_j{v}_j{C}_j$, with each concentration evaluated at the actual sampling time. Under an additive Beer-Lambert law model,where the species fractions $f_{ij}$ sum to one on the chosen dye-unit basis. Correcting $V_j$ alone does not correct $n_j$. Returning to the initial protonation conditions need not restore the initial absorbance after sampling has removed material.

Primary acid titrations of fused dipyrrins show derivative-dependent spectral changes and multiple protonation stages.^[ref] The recurrence here is an explicit inventory model for interpreting such measurements, not a claim about aliquots in that paper. Adsorption, precipitation or chemical destruction would add separate loss terms; disagreement with the sampling-only prediction is a reason to test those possibilities.