= Chemical multiplication reaction
A <chemical multiplication reaction> is a reaction sequence in which a species is converted through a cycle into more of that same species before measurement. Unlike a one-time conversion to a different reporter, the multiplied species can enter another cycle. IUPAC calls the case involving the species being determined a direct multiplication reaction.<Classification and Use of Terms for Amplification and Related Reactions by D. T. Burns and others>[p.1332, Direct multiplication reaction.]{ref}
Let $N_j$ be the amount of the multiplied species after cycle $j$. If every cycle has the same net multiplier $K$, then
$$
N_{j+1}=KN_j,\qquad N_n=K^nN_0.
$$
For an original ideal example, $K=3$ and four cycles convert $2\,\mathrm{pmol}$ into $162\,\mathrm{pmol}$. The gain is 81, not 12: repeating a threefold increase compounds it. The <stoichiometry> concerns a sequence supplied with reagents, rather than multiplication of isolated molecules without additional matter.
A <multiplication-cycle yield> changes the net multiplier. Suppose a fraction $p$ of the entering species completes a route producing three output molecules, while the remaining fraction survives unchanged. Then
$$
K_{\rm eff}=3p+(1-p)=1+2p.
$$
For stipulated $p=0.8$, four cycles give gain $2.6^4\approx45.7$, substantially below 81. If failed molecules are instead destroyed, the model would give $K_{\rm eff}=3p$: specifying the fate of failures is essential. For cycle-dependent multipliers, the gain is $\prod_{j=1}^nK_j$, not a fixed power by assumption.
The same compounding can affect an initial <chemical amplification blank>. Tenfold less contaminant at the start need not stay negligible after many cycles, and contamination introduced late experiences fewer cycles. Reagent depletion, side reactions and changing yield limit the range over which a constant-gain <measurement calibration> is credible. Cycle number and blank history belong in the experimental record.
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