= Dipyrrin titration mass balance
\b[Dipyrrin titration mass balance] keeps track of the amount of dye while acid addition changes its <protonation> states and the solution volume. An <absorbance> decrease at one wavelength can result from redistribution or dilution, rather than destruction of <dipyrromethene> units. In a primary study of fused derivatives, acid titration produced several distinct spectral responses and, for some derivatives, more than one <protonation> step.<Naphthalimide-Fused Dipyrrins: Tunable Halochromic Switches and Photothermal NIR-II Dyes>[Figures 3–4 and captions; acid-induced absorption spectra of the 1a–1e and 2a–2e series]{ref}
For an original two-species model containing neutral $\mathrm{LH}$ and protonated $\mathrm{LH_2^+}$, conservation gives
$$
n_L=V([\mathrm{LH}]+[\mathrm{LH_2^+}]),\qquad C_L=\frac{n_L}{V}.
$$
The relevant <acid dissociation constant> supplies a second relation. In an ideal dilute aqueous model,
$$
K_a=\frac{[\mathrm H^+][\mathrm{LH}]}{[\mathrm{LH_2^+}]},\qquad
f_H=\frac{[\mathrm H^+]}{K_a+[\mathrm H^+]}.
$$
This chosen model does not assign an aqueous constant to the cited dyes in an organic solvent. If <pH> is not independently fixed, acid inventory and <electroneutrality> must also be solved; adding a known volume of acid does not prescribe the final free <hydrogen ion> concentration by itself.
The <Beer-Lambert law> predicts
$$
A=l\frac{n_L}{V}[\epsilon_H f_H+\epsilon_N(1-f_H)].
$$
For a stipulated 0.10 micromole of dye initially in 10 mL, adding 2.5 mL gives $C_L=8$ micromolar. At a measured condition with $f_H=1/2$, $l=1$ cm, $\epsilon_H=20000$ and $\epsilon_N=60000\,\mathrm{L\,mol^{-1}\,cm^{-1}}$, $A=0.32$. Omitting dilution predicts 0.40 despite exactly the same dye amount and species fraction.
<Aliquot-aware dye inventory in titration> adds another correction. If aliquots of volume $v_j$ are removed from well-mixed solutions having total dye concentrations $C_j$, the remaining amount is $n_0-\sum_j v_jC_j$, unless other gains or losses occur. <Adsorption>, precipitation or chemical transformation require additional inventories. For species containing two dye units, their concentration contributes twice to the conserved dye-unit sum.
A useful titration records additions, withdrawals and full spectra, then checks whether returning the acid-base conditions recovers the predicted inventory-corrected response. Conservation of dye units, charge balance and the spectral measurement equation impose different constraints; agreement with only one does not establish the others.
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