= Initiator-limited antioxidant exhaustion
Initiator-limited antioxidant exhaustion describes whether a finite radical source can consume all the trapping capacity in an <inhibited autoxidation assay>. A missing end to an <autoxidation inhibition period> can result from too little integrated initiation, rather than an unlimited amount of <radical-trapping antioxidant>.
Let $K_0=nC_{A0}$ be the initial radical-trapping capacity, measured in concentration of trapped chain carriers. Assume quantitative trapping, no regeneration or competing loss, and an initiation flux $R_i(t)=R_0e^{-kt}$ with $R_0,k>0$. Before exhaustion the remaining capacity is
$$
K(t)=K_0-\frac{R_0}{k}(1-e^{-kt}).
$$
The source can deliver only $Q_\infty=R_0/k$. Consequently, $K_0<Q_\infty$ gives finite exhaustion time
$$
t_*=-\frac1k\log\left(1-\frac{K_0}{Q_\infty}\right).
$$
If $K_0=Q_\infty$, capacity approaches zero only as $t\to\infty$. If $K_0>Q_\infty$, a positive capacity $K_0-Q_\infty$ remains even at infinite time. These are different outcomes despite both lacking a finite exhaustion breakpoint.
For an original example, $R_0=0.20\,\mathrm{\mu M\,s^{-1}}$ and $k=0.010\,\mathrm{s^{-1}}$ give total supply $20\,\mathrm{\mu M}$. Initial capacities $10$, $20$ and $25\,\mathrm{\mu M}$ respectively give exhaustion at $69.3\,\mathrm s$, asymptotic exhaustion, and a residual $5\,\mathrm{\mu M}$. The initial-rate estimate $K_0/R_0$ would wrongly predict a finite $100\,\mathrm s$ exhaustion time in the middle case.
The logarithm explains a sensitive regime. For $\alpha=K_0/Q_\infty\ll1$,
$$
t_*=\frac{K_0}{R_0}\left(1+\frac\alpha2+O(\alpha^2)\right),
$$
but $t_*$ diverges as $\alpha\to1^-$. Small capacity or initiation-calibration changes near that threshold can cause large lag changes.
A primary visible-light <autoxidation> method distinguishes trapping kinetics from stoichiometry through a coupled oxidation readout.<A Continuous Visible Light Spectrophotometric Approach To Accurately Determine the Reactivity of Radical-Trapping Antioxidants by Evan A. Haidasz and others>[Indexed abstract, reaction monitoring and kinetic/stoichiometric interpretation]{ref} In the model here, a controlled second initiator addition can test whether capacity remains after the first source declines. A matched blank and independent <autoxidation initiation-rate calibration> are essential: extra initiation can change the optical reaction, and a persistent background source would eventually exhaust even capacity above $Q_\infty$. An absent breakpoint alone does not distinguish these mechanisms.
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