= Oxygen-limited autoxidation
Oxygen-limited <autoxidation> occurs when the availability of molecular <oxygen> restricts the observed <reaction rate>. Oxygen can be consumed faster than it is replenished through <interphase mass transfer>, even when the gas above a liquid contains plenty of oxygen. A slow oxidation trace therefore does not by itself establish efficient <radical-trapping antioxidant> action.
A primary study measured dissolved oxygen availability during liquid-phase <hydrocarbon> <autoxidation>, rather than relying only on transport predictions. Its reported oxygen-transfer coefficients differed between benzene and indan under the stated conditions, illustrating why a gas atmosphere alone does not specify the liquid's oxygen supply.<In Situ Measurement of Liquid Phase Oxygen during Oxidation by Muhammad N. Siddiquee and Arno de Klerk>[Publisher abstract, in situ oxygen availability and solvent-specific mass-transfer coefficients]{ref}
An original well-mixed <mass balance> separates transfer from consumption. Let $C$ be dissolved oxygen <concentration>, $C^*$ its equilibrium value at the imposed gas composition, and $k_La$ the volumetric transfer coefficient. If a simplified local <rate law> consumes oxygen at $kC$, then
$$
\frac{dC}{dt}=k_La(C^*-C)-kC.
$$
With constant coefficients, the <steady state> is
$$
C_{\rm ss}=\frac{k_La}{k_La+k}C^*,\qquad
r_{\rm ss}=\frac{kk_La}{k_La+k}C^*.
$$
For $k\ll k_La$, the liquid stays near $C^*$ and the rate responds strongly to chemical kinetics. For $k\gg k_La$, the rate approaches the transfer ceiling $k_La C^*$. This first-order consumption model illustrates the transport mechanism; it is not a universal elementary <rate law> for radical-chain oxidation.
For $k_La=0.10\,\mathrm{s^{-1}}$ and $k=0.90\,\mathrm{s^{-1}}$, $C_{\rm ss}=0.10C^*$ and $r_{\rm ss}=0.090C^*\,\mathrm{s^{-1}}$. Doubling $k$ only raises the rate to about $0.0947C^*\,\mathrm{s^{-1}}$, a $5.3\%$ increase. A nearly flat response to a chemical perturbation can thus be a delivery limitation.
A discriminating experiment changes stirring or gas-liquid contact while matching <temperature>, substrate, initiator and optical settings, and records dissolved oxygen as well as the oxidation signal. Dependence on transfer conditions reveals a limitation that an optical trace alone can hide. In a closed vessel, finite total oxygen adds another constraint: the gas reservoir itself can become depleted, so $C^*$ cannot then be treated as constant.
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