 Formazan recovery identifiability

Formazan recovery identifiability asks whether measurements distinguish the amount of formazan formed from the fraction recovered in the optical reading. Better repeatability does not necessarily separate those quantities.

Let $n$ be total product formed, $r\in [0,1]$ the dissolved formazan recovery fraction, and $K=ϵ\ell /V$ a known optical conversion factor in a suitable fixed-path solution model. Then net absorbance isThe manufacturer's procedure separates formation of crystals, dissolution and measurement, motivating these distinct model quantities.^[ref] Knowing $B$ and $K$ determines the dissolved amount $d=B/K=rn$, but not the two parameters separately.

For any admissible scale $a>0$, replacing $(r,n)$ by $(ar,n/a)$ leaves $d$ unchanged, provided $ar\le 1$. For observed $d=6\mu \mathrm{mol}$, both $(r,n)=(1,6\mu \mathrm{mol})$ and $(1/2,12\mu \mathrm{mol})$ fit. In this conserved-product model the bound $r\le 1$ implies $n\ge d$, with no finite upper bound supplied by that measurement alone. This is an identifiability limitation, rather than merely a large measurement uncertainty.

Changing volumes or optical paths can provide several known factors $K_j$, but measurements $B_j=K_{j}rn$ still determine only the same product if $r$ and $n$ stay fixed. Such changes can test the optical law without resolving the underlying ambiguity.

An original matched-addition model illustrates what extra information could help. Suppose a known amount $\Delta n$ is added before the recovery step and experiences exactly the same unknown fraction $r$ as the native product. Converting both readings to dissolved amounts givesIf $d_0=6\mu \mathrm{mol}$ and adding $2\mu \mathrm{mol}$ increases the dissolved amount by $1\mu \mathrm{mol}$, the model gives $r=0.5$ and $n=12\mu \mathrm{mol}$. Adding already dissolved product after the recovery step would instead test later optical response and would not identify earlier losses.

The shared-recovery assumption must be justified: a dissolved spike can behave differently from product crystals formed inside a preparation. A second measurement helps only if its relation to the unknown quantities is independently established. Repeated measurement of the same product $rn$ cannot supply that missing relation.