The kinetics of inhibition for nonbranched-chain processes of saturated free-radical addition to the C=C and C=O double bonds of alkene and formaldehyde molecules, respectively, by low-reactive free radicals that can experience delocalization of the unpaired p-electron was first considered in . In these processes a low-reactive free radical is formed in the reaction competing with chain propagation reactions through a reactive free radical. In the present work the kinetics of inhibition by low-reactive hydrotetraoxyl free radical is considered for the nonbranched-chain process of the addition of a hydrogen atom to one of the two multiply bonded atoms of the oxygen molecule yielding a hydroperoxyl free radical. The hydroperoxyl free radical then abstracts the most labile atom from a molecule of the compound being oxidized or decomposes to turn into a molecule of an oxidation product. The only reaction that can compete with these two reactions at the chain evolution stage is the addition of the peroxyl radical to the oxygen molecule (provided that the oxygen concentration is sufficiently high). This reaction yields the secondary hydrotetraoxyl 1:2 adduct radical, which is the heaviest and the largest among the reactants. It is less reactive than the primary peroxyl 1:1 adduct radical and, as a consequence, does not participate in further chain propagation. At moderate temperatures, the reaction proceeds via a nonbranched-chain mechanism.The aim of this study was the mathematical simulation of oxidation process autoinhibited by oxygen, when the dependence of the peroxide formation rate on the dissolved oxygen concentration has a maximum. The simulation was based on experimental data obtained for γ-radiation-induced addition reaction of hydrogen atom to the molecular oxygen for which the initiation rate V1 is known (taking into account that V = GP and V1 = G(H•)P, where P is the dose rate, and G(H•) is the initial yield of the chain-carrier hydrogen atom H• - initiation yield [2,3]).Based on the reaction scheme suggested for the kinetic description of the addition process to oxygen, the kinetic equations with one to three parameters to be determined directly were derived. Reducing the number of unknown parameters in a kinetic equation will allow one to decrease the narrowness of the correlation of these parameters and to avoid a sharp buildup of the statistical error in the nonlinear estimation of these parameters in the case of a limited number of experimental data points. The rate constant of the addition to oxygen, estimated as a kinetic parameter, can be compared to its reference value if the latter is known. This provides a clear criterion to validate the mathematical description against experimental data.
Kinetics of Hydrogen Atom Addition to the Oxygen Molecule
A number of experimental findings concerning the autoinhibiting effect of an increasing oxygen concentration at modest temperatures on hydrogen oxidation both in the gas phase [4-6] (Figure 1) and in the liquid phase  (Figure 2, curve 2), considered in my earlier works [8-11], can also be explained in terms of the competition kinetics of free radical addition [12,13]. From figure 1 shows that the quantum yields of hydrogen peroxide and water (of products of photochemical oxidation of hydrogen at atmospheric pressure and room temperature) are maximum in the region of small concentrations of oxygen in the hydrogen-oxygen system (curves 1 and 2, respectively) .In the familiar monograph “Chain Reactions” by Semenov , it is noted that raising the oxygen concentration when it is already sufficient usually slows down the oxidation process by shortening the chains. The existence of the upper (second) ignition limit in oxidation is due to chain termination in the bulk through triple collisions between an active species of the chain reaction and two oxygen molecules (at sufficiently high oxygen partial pressures). In the gas phase at atmospheric pressure, the number of triple collisions is roughly estimated to be 103 times smaller than the number of binary collisions (and the probability of a reaction taking place depends on the specificity of the action of the third particle) . Note that in the case of a gas-phase oxidation of hydrogen at low pressures of 25-77 Pа and a temperature of 77 К  when triple collisions are unlikely, the dependence of the rate of hydrogen peroxide formation on oxygen concentration (the rate of passing of molecular oxygen via the reaction tube) also has a pronounced maximum (see curves 3 and 4 in Figure 2) that indicates a chemical mechanism providing the appearance of a maximum (see reaction 4 of Scheme).
The chain evolution (propagation and inhibition) stage of the Scheme includes consecutive reaction pairs 2-3 and 3-3'; the parallel (competing) reaction pair 3-4; and consecutive-parallel reactions 2 and 4.The hydroperoxyl free radical Ho.2
[23-25] resulting from reaction 2 possesses an increased energy due to the energy released the conversion of the О=О multiple bond into the НО-О• ordinary bond. Therefore, before its possible decomposition, it can interact with a hydrogen or oxygen molecule as the third body via parallel (competing) reactions 3 and 4, respectively. The hydroxyl radical НО• that appears and disappears in consecutive parallel reactions 3 (first variant) and 3 possesses additional energy owing to the exothermicity of the first variant of reaction 3, whose heat is distributed between the two products. As a consequence, this radical has a sufficiently high reactivity not to accumulate in the system during these reactions, whose rates are equal (V3 = V3') under quasi-steady-state conditions, according to the above scheme. Parallel reactions 3 (second, parenthesized variant) and 3' regenerate hydrogen atoms. It is assumed [9,10] that the hydrotetraoxyl radical НО.4
(first reported in [26-28]) resulting from endothermic reaction 4, which is responsible for the peak in the experimental rate curve (Figure 1, curve 3), is closed into a five-membered [ОО─Н···ОО]• cycle due to weak intramolecular hydrogen bonding [29,30]. This structure imparts additional stability to this radical and makes it least reactive.The НО.4
radical was discovered by Staehelin et al., in a pulsed radiolysis study of ozone degradation in water; its UV spectrum with an absorption maximum at 260 nm (∈НО.4280nm
= 320 ±15 m2
) was reported . The spectrum of the radical is similar to that of ozone, but the molar absorption coefficient (∈НО.4λmax
of the former is almost two times larger . The assumption about the cyclic structure of the radical can stem from the fact that its mean lifetime in water at 294 K, which is (3.6 ± 0.4) × 10-5
s (as estimated  from the value of 1/k for the monomolecular decay reaction НО.4→Ho.2
), is 3.9 times longer than that of the linear radical [16,32] estimated in the same way  for the same conditions , (9.1 ± 0.9) × 10-6
s. MP2/6-311++G** calculations using the Gaussian-98 program confirmed that the cyclic structure of НО.4
 is energetically more favorable than the helical structure  (the difference in energy is 4.8-7.3 kJ mol-1
, depending on the computational method and the basis set).2
For example, with the MP2
(full)/6-31G(d) method, the difference between the full energies of the cyclic and acyclic НО.4
conformers with their Zero-point Energies (ZPE) values taken into account (which reduces the energy difference by 1.1 kJ mol-1
) is -5.1 kJ mol-1
and the entropy of the acyclic-to-cyclic НО.4
transition is ΔS0298
= -1.6 kJ mol-1
. Therefore, under standard conditions, НО.4
can exist in both forms, but the cyclic structure is obviously dominant (87%, Keq
= 6.5) .Reaction 4 and, to a much lesser degree, reaction 6 inhibit the chain process, because they lead to inefficient consumption of its main participants -Ho.
2 and Н•
When there is no excess hydrogen in the hydrogen-oxygen system, the homomolecular dimer O4
[19-22,40,41], which exists at low concentrations (depending on the pressure and temperature) in equilibrium with O2
, can directly capture the Н•
atom to yield the heteronuclear cluster3
. This НО.4
cluster is more stable than O4
 and cannot abstract a hydrogen atom from the hydrogen molecule. Therefore, in this case, nonchain hydrogen oxidation will occur to give molecular oxidation products via the disproportionation of free radicals. The low-reactive hydrotetraoxyl radical НО.4
, which presumably has a high energy density , may be an intermediate in the efficient absorption and conversion of biologically hazardous UV radiation energy the Earth upper atmosphere. The potential energy surface for the atmospheric reaction HO•
, in which the adduct НО.4
А) was considered as an intermediate, was calculated by the DMBE method . From this standpoint, the following reactions are possible in the upper troposphere, as well as in the lower and middle stratosphere, where most of the ozone layer is situated (altitude of 16-30 km, temperature of 217-227 K, pressure of 1.0 ×104
Pa ; the corresponding ΔH0298
reaction values are given in kJ mol-1
radical can disappear via disproportionation with a molecule, free radical, or atom in addition to dissociation. Note that emission from О2(a1
) and О2(b1
) is observed at altitudes of 30-80 and 40-130 km, respectively .Staehelin et al.,  pointed out that, in natural systems in which the concentrations of intermediates are often very low, kinetic chains in chain reactions can be very long in the absence of scavengers since the rates of the chain termination reactions decrease with decreasing concentrations of the intermediates according to a quadratic law, whereas the rates of the chain propagation reactions decrease according to a linear law.The kinetic description of the noncatalytic oxidation of hydrogen, including in an inert medium , in terms of the simplified scheme of free-radical nonbranched-chain reactions (Scheme), which considers only quadratic-law chain to unsteady-state critical regimes, and at a substantial excess of the hydrogen concentration over the oxygen concentration was obtained by means of quasi-steady-state treatment, as in the previous studies on the kinetics of the branched-chain free-radical oxidation of hydrogen , even though the applicability of this method in the latter case under unsteady states conditions was insufficiently substantiated. The method was used with the following condition 4
for the ﬁrst stages of the process: k6
 and, hence, V1
which allow the exponent of the 2k5
term in the d[Н•]/dt = 0 equation to be reduced to 1 [1,46]. The kinetic equations were derived for the rates (mol dm-3 s-1) of the elementary reactions for the formation of molecular products of hydrogen oxidation. The kinetic equations were derived for the rates (mol dm-3
) of the elementary reactions for the formation of molecular products of hydrogen oxidation.
The rate constant 2k5
in the case of the pulsed radiolysis of ammonia-oxygen (+ argon) gaseous mixtures at a total pressure of 105
Pa and a temperature of 349 K was calculated to be 1.6 × 108
 (a similar value of this constant for the gas phase was reported in an earlier publication ). Pagsberg et al.,  found that the dependence of the yield of the intermediate НО•
on the oxygen concentration has a maximum close to 5 × 10-4
. In the computer simulation of the process, they considered the strongly exothermic reaction Ho.
2 + NН3
О + •NНОН, which is similar to reaction 3 in Scheme, whereas the competing reaction 4 was not taken into account.The ratio of the rates of the competing reactions is V3
= αl/x and the chain length is ν = V3
. equation (1a) was obtained by substitution of the rate constant k2
into equation (1) with its analytical expression (in order to reduce the number of unknown parameters that are to be measured directly). The optimum concentration xm
of oxygen, at which the rate of oxidation is maximum, can be calculated from equation (1a) or the analytical expression for k2
if other parameters that appear in these expressions are known.The rates of nonchain formation of molecular hydrogen, hydrogen peroxide, and water in reactions 5, 6, and 7 of quadratic chain termination are as follows:
It is important to note that, if in the Scheme chain initiation via reaction 1 is due to the interaction between molecular hydrogen and molecular oxygen yielding the hydroxyl radical НО•
instead of Н•
atoms and if this radical reacts with an oxygen molecule (reaction 4) to form the hydrotrioxyl radical НО.3
(which was obtained in the gas phase by Neutralization Reionization (NR) Mass Spectrometry  and has a lifetime of >10-
6 s at 298 K) and chain termination takes place via reactions 5-7 involving the НО•
, radicals instead of Н•
, respectively, the expressions for the water chain formation rates derived in the same way will appear as a rational function of the oxygen concentration x without a maximum: V3'(Н2
Thus, the addition reaction of the HO2•
radical that possesses an elevated energy in statu nascendi
with the oxygen molecule (at sufﬁciently high oxygen concentrations) to give the НО4•
radical was used for the ﬁrst time in the kinetic description of the initiated hydrogen oxidation at moderate temperature and pressure [9,10]. This reaction is endothermic and competes with the chain propagation reaction via the Н•
atom. The radical generated in the former reaction has a low reactivity and inhibits the chain reaction.6
The above data concerning the competition kinetics of the nonbranched-chain addition of hydrogen atoms to the multiple bonds of the oxygen molecules make it possible to describe, using rate equations (1a) and (4a), obtained by quasi-steady-state treatment, the peaking experimental dependences of the formation rates of molecular 1:1 adduct H2
on the concentration of the oxygen over the entire range of its variation in binary system (Figure 2). In such reaction systems consisting of saturated and unsaturated components [50,51], the unsaturated compound (in this case the O2
) is both a reactant and an autoinhibitor, specifically, a source of low-reactive free radicals (in this case the НО.4
radicals) shortening kinetic chains.The progressive inhibition of the nonbranched-chain processes, which takes place as the concentration of the unsaturated compound is raised (after the maximum process rate is reached), can be an element of the self-regulation of the natural processes that returns them to the stable steady state.Using mechanism of the nonbranched-chain free-radical hydrogen oxidation considered here, it has been demonstrated that, in the Earth’s upper atmosphere, the decomposition of O3
in its reaction with the НО•
radical can occur via the addition of the latter to the ozone molecule, yielding the radical, which is capable of efficiently absorbing UV radiation .
1. There were calculations for the two conformers (cis and trans) of the HO4• radical  using large scale ab initio methods and density functional techniques with extended basis sets. Both conformers have a nearly planar geometry with respect to the four oxygen atoms and present an unusually long central O-O bond. The most stable conformer of HO4• radical is the cis one, which is computed to be endothermic with respect to HO2• (X2A") + O2 (X3∑-g) at 0 K.
1. Taking into account the principle of detailed balance for the various pathways of formation of products, whose numbers in the elementary reaction should not exceed three for possible involvement in the triple collisions in the case of the reverse reaction, since the probability of simultaneous interaction of four particles is negligible.
2.The planar, six-atom, cyclic, hydrogen-bonded dimer (HO2•)2 was calculated using quantum chemical methods (B3LYP density functional theory) . The hydrogen bond energy is 47.7 and 49.4 kJ mol-1 at 298 K for the triplet and singlet states of the dimer, respectively.
3. It is impossible to make a sharp distinction between the two-step bimolecular interaction of three species via the equilibrium formation of the labile intermediate O4 and the elementary trimolecular reaction О2 + О2 + Н• → НО4•
4.For example, the ratio of the rate constants of the bimolecular disproportionation and dimerization of free radicals at room temperature is k(HO• + HO2•)/[2k(2HO•)2k(2HO2•)]0.5 = 2.8 in the atmosphere  and k(H• + HO•)/[2k(2H•)2k(2HO•)]0.5 = 1.5 in water . These values that are fairly close to unity.
5.This equation can be used to describe a wide range of nonbranched-chain reactions of addition of any free radicals to the C=C bonds of unsaturated hydrocarbons, alcohols, etc., which result in the formation of molecular 1:1 adducts in binary reaction systems of saturated and unsaturated components [1,9,10].
6.Note that in the case of similar nonbranched-chain (i.e., "slow") oxidation of RH hydrocarbons, the corresponding reactive RO2• and low-reactive RO4• radicals participate in the process . The only difference between the kinetic model of oxidation and the kinetic model of the chain addition of 1-hydroxyalkyl radicals to the free (unsolvated) form of formaldehyde in nonmethanolic alcohol-formaldehyde systems [1, 9, 10] is that in the former does not include the formation of the molecular 1:1 adduct via reaction 4.
i.According to Francisco and Williams , the enthalpy of formation (ΔНf˚298) in the gas phase of Н•, НО•, HO2•, HO4• (the latter without the possible intramolecular hydrogen bond taken into account), О3, Н2О , Н2О2, and Н2О4 is 218.0 ± 0.0, 39.0 ± 1.2, 12.6 ± 1.7, 122.6 ± 13.7, 143.1 ± 1.7, -241.8 ± 0.0, -136.0 ± 0, and -26.0 ± 9 kJ mol-1, respectively. Calculations for the HO4• radical with a helical structure were carried out using the G2(MP2) method . The stabilization energies of HO2•, HO4• and HO3• were calculated in the same work to be 64.5 ± 0.1, 69.5 ± 0.8, and 88.5 ± 0.8 kJ mol-1, respectively. The types of the O4 molecular dimers, their IR spectra, and higher oxygen oligomers were reported [17,18]. The structure and IR spectrum of the hypothetical cyclotetraoxygen molecule O4, a species with a high energy density, were calculated by the CCSD method, and its enthalpy of formation was estimated . The photochemical properties of O4 and the van der Waals nature of the О2-О2 bond were investigated [20,21]. The most stable geometry of the dimer is two O2 molecules parallel to one another. The O4 molecule was identified by NR Mass Spectrometry .