= Solution
The acid initially protonates 1 and switches the system into the lithium-bound reporter state, so \b[512 nm intensity rises rapidly]. The resulting carboxylate then decarboxylates irreversibly. Its <carbanion> removes the proton from 1, giving neutral waste 5 and restoring the original <zinc>/<lithium> distribution. Thus the 512 nm signal is a \b[transient pulse followed by recovery to its initial low baseline]; no repeated oscillations follow from one fuel addition.
For first-order <decarboxylation> the surviving fuel-derived carboxylate fraction is $e^{-kt}$ with $k=\ln2/(75\ \mathrm{min})$. If proton exchange and metal redistribution are fast, background is subtracted, and the population of the switched state is proportional to remaining protonated 1, the illustrative recovery is
$$
\boxed{I_{512}(t)=I_{\rm low}+(I_{\rm high}-I_{\rm low})2^{-t/(75\,\mathrm{min})}},\qquad t>0.
$$
Half the initial excess signal remains after 75 min and one quarter after 150 min in that limiting model. Actual binding equilibria can make the intensity nonlinear in remaining fuel; the supplied <half-life> fixes the fuel decay, not an exact reporter-intensity law. Monitor $^1$H <NMR> fuel consumption and waste formation alongside the optical trace to distinguish dissipation-driven recovery from <photobleaching> or simple dilution.
\Image[/past-exam-of-the-chemistry-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-3-switching.png]
{title=Original limiting-model pulse and recovery after one acid-fuel addition}
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