= Multiplication-cycle yield
A <multiplication-cycle yield> specifies how successfully a cycle of a <chemical multiplication reaction> converts entering material into the intended output. It must state what happens to material that fails to complete the route. A stoichiometric multiplier alone is insufficient to determine the achieved gain.
Suppose successful processing produces $m$ copies of the measured species per entering copy, the success fraction is $p$, and a fraction $s$ of unsuccessful material survives unchanged. The net cycle multiplier is
$$
K=mp+s(1-p).
$$
Here $s=1$ represents survival of all failures and $s=0$ destruction of all failures. In an original example, $m=3$ and $p=0.8$ give $K=2.6$ when failures survive, but $K=2.4$ when they are lost. After five cycles the gains are approximately 118.8 and 79.6. The chemistry's failure pathway materially affects the answer.
For different cycle yields, total gain is $G=\prod_jK_j$. IUPAC's ideal direct-multiplication description gives $K^n$ for repeated identical cycles; the variable-yield expression follows by applying each cycle in sequence.<Classification and Use of Terms for Amplification and Related Reactions by D. T. Burns and others>[p.1332, Direct multiplication reaction.]{ref}
<Cycle-gain uncertainty> depends strongly on whether cycle errors are shared. Taking small perturbations,
$$
\delta\ln G\approx\sum_j\frac{\delta K_j}{K_j}.
$$
For $n$ independent multipliers with the same relative <standard uncertainty> $v$, the relative uncertainty of $G$ is approximately $\sqrt n\,v$. If one uncertain multiplier applies identically to every cycle, it is approximately $nv$. With $n=9$ and $v=0.01$, these models give 3% and 9%, respectively. Repeating a shared error is different from averaging independent errors.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5.2, correlated inputs; sensitivities here are derived for the stated gain model.]{ref}
Measure standards over the intended cycle range, with matched <chemical amplification blanks>. Reagent depletion, side reactions or inhibition can make the actual <multiplication-cycle yield> change with time or amount. A nominal gain calculated from ideal <stoichiometry> does not replace that experimental test.
Back to article page
