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 isHere $s=1$ 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 $G={\prod }_j{K}_j$. IUPAC's ideal direct-multiplication description gives $K^n$ for repeated identical cycles; the variable-yield expression follows by applying each cycle in sequence.^[ref]

Cycle-gain uncertainty depends strongly on whether cycle errors are shared. Taking small perturbations,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.^[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.