—New reaction scheme is suggested for the initiated nonbranched-chain addition of free radicals to the multiple bond of the molecular oxygen. The scheme includes the addition reaction of the peroxyl free radical to the oxygen molecule to form the tetraoxyl free radical. This reaction competes with chain propagation reactions through a reactive free radical. The chain evolution stage in this scheme involves a few of free radicals, one of which (tetraoxyl) is relatively low-reactive and inhibits the chain process by shortening of the kinetic chain length. Based on the proposed scheme rate equations (containing one to three parameters to be determined directly) are deduced using quasi-steady-state treatment. The kinetic description with use the obtained rate equations is applied to the γ-induced nonbranched-chain processes of the free-radical oxidation of liquid o-xylene at 373 K and hydrogen dissolved in water containing different amounts of oxygen at 296 K. The ratios of rate constants of competing reactions and rate constants of addition reactions to the molecular oxygen are defined. In these processes the oxygen with the increase of its concentration begins to act as an oxidation autoinhibitor (or an antioxidant), and the rate of peroxide formation as a function of the dissolved oxygen concentration has a maximum. From the energetic standpoint possible nonchain pathways of the free-radical oxidation of hydrogen and the routes of ozone decay via the reaction with the hydroxyl free radical in the upper atmosphere were examined.