= 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 is
$$
n_{j+1}=n_j\left(1-\frac{v_j}{V_j}\right),\qquad
V_{j+1}=V_j-v_j+a_j,
$$
so
$$
C_{j+1}=C_j\frac{V_j-v_j}{V_j-v_j+a_j}.
$$
The 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 $\sum_jv_jC_j$, with each concentration evaluated at the actual sampling time. Under an additive <Beer-Lambert law> model,
$$
A_j=\ell\frac{n_j}{V_j}\sum_i\varepsilon_i f_{ij},
$$
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.<Naphthalimide-Fused Dipyrrins: Tunable Halochromic Switches and Photothermal NIR-II Dyes>[Figures 3–4 and captions, acid-induced spectral changes]{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.
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