= Parallel first-order reaction mechanism
{title2=$\mathrm A\xrightarrow{k_P}\mathrm P,\quad\mathrm A\xrightarrow{k_Q}\mathrm Q$}
A <parallel first-order reaction mechanism> consumes the same <reactant> through competing routes. Extend the product-split idea in <catalytic selectivity> by following both <reaction products> over time. Stipulate irreversible <first-order reactions> $\mathrm A\xrightarrow{k_P}\mathrm P$ and $\mathrm A\xrightarrow{k_Q}\mathrm Q$ in a uniformly mixed <closed system> of fixed <volume> and <temperature>. Initially only A is present, with <concentration> $A_0$. With $K=k_P+k_Q$,
$$
[\mathrm A]=A_0e^{-Kt},\qquad
[\mathrm P]=A_0\frac{k_P}{K}(1-e^{-Kt}),\qquad
[\mathrm Q]=A_0\frac{k_Q}{K}(1-e^{-Kt}).
$$
Here $K$ abbreviates a sum of <rate constants>, not an <equilibrium constant>. The <rate equations> are $-d[\mathrm A]/dt=K[\mathrm A]$, $d[\mathrm P]/dt=k_P[\mathrm A]$, and $d[\mathrm Q]/dt=k_Q[\mathrm A]$. Integrating the last two after solving the first proves the expressions and their conserved total $[\mathrm A]+[\mathrm P]+[\mathrm Q]=A_0$.
Choose two alternative models: model U has $(k_P,k_Q)=(0.30,0.10)\,\mathrm{s^{-1}}$; model V has $(0.10,0.30)\,\mathrm{s^{-1}}$. Both have $K=0.40\,\mathrm{s^{-1}}$, so every measurement of A alone is identical, even with no <measurement error>. Yet their P-to-Q <concentration> ratios at any $t>0$ are $3$ and $1/3$, respectively. This is a precise failure of <identifiability> from that one observable. A calibrated measurement distinguishing P from Q resolves it within the stipulated two-route model.
\Image[/parallel-reaction-hidden-product-split.png]
{title=Identical <reactant> disappearance can hide opposite product splits in a <parallel first-order reaction mechanism>}
{id=image-parallel-reaction-hidden-product-split}
{description=Original chosen models U and V share $k_P+k_Q=0.40\,\mathrm{s^{-1}}$ and $A_0=1.00\,\mathrm{mmol\,L^{-1}}$, but exchange the two route <rate constants>. Left: indistinguishable A traces. Right: different predicted P and Q <concentrations> at $t=5.00\,\mathrm s$; each pair sums to the same total converted A. No curves or bars are experimental measurements.}
At $t=5.00\,\mathrm s$, both models give $[\mathrm A]=e^{-2}A_0\approx0.135335\,\mathrm{mmol\,L^{-1}}$. Model U gives $([\mathrm P],[\mathrm Q])\approx(0.648499,0.216166)\,\mathrm{mmol\,L^{-1}}$; model V exchanges these. Measuring only the summed <reaction product> <concentration> would also fail to distinguish them. Product-specific observations therefore answer a question that total conversion cannot. As with any <reaction mechanism>, fitting these observables still does not rule out a more elaborate untested route.
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