Kinetic Equations for Radical-Chain Oxidation Involving Process-Inhibiting Alkyl(or Hydro)tetraoxyl Free Radical

: The derivation of kinetic equations for the oxidation processes by the free-radical non-branched chain mechanism is shown. This derivation is based on the proposed reaction scheme for the initiated addition of free radicals to the multiple bond of the molecular oxygen 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 – alkyl(or hydro)tetraoxyl – is relatively low-reactive and inhibits the chain process by shortening of the kinetic chain length. The rate equations (containing one to three parameters to be determined directly) are deduced using the quasi-steady-state treatment. These kinetic equations were used to describe the γ-induced non-branched chain processes of free-radical oxidation of liquid o -xylene at 373 K and hydrogen dissolved in water containing various 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 auto-inhibitor (or an antioxidant), and the rate of peroxide formation as a function of the dissolved oxygen concentration has a maximum. It is shown that a maximum in these curves arises from the competition between hydrocarbon (or hydrogen) molecules and dioxygen for reacting with the emerging peroxyl 1:1 adduct radical. From the energetic standpoint possible non-chain 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 (including the addition yielding the hydrotetraoxyl free radical, which can be an intermediate in the sequence of conversions of biologically hazardous UV radiation energy) were examined. The energetics of the key radical-molecule gas-phase reactions is considered.


INTRODUCTION
The kinetics of inhibition for non-branched 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 [1].In these processes a low-reactive free radical is formed in the reaction competing with chain propagation reactions through a reactive free radical. 1 In the present work the kinetics of inhibition by low-reactive tetraoxyl free radicals is considered for non-branched chain processes of the addition of a free radical or an atom to one of the two multiply bonded atoms of the oxygen molecule yielding a peroxyl free radical and thus initiating oxidation, which is the basic process of chemical evolution.The peroxyl 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 tetraoxyl 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 non-branched 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, and show the derivation of kinetic oxidation equations.The simulation was based on experimental data obtained for γ-radiation-induced addition reaction of hydrocarbon free radical R • (or hydrogen atom) to the molecular oxygen for which the initiation rate V1 is known (taking into account that V = GP and V1 = G(R • )P, where P is the dose rate,  is the electron fraction of the solvent RH in the binary system [2], and G(R • ) is the initial yield of the chain-carrier free radical R •initiation yield [3,4]).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.

II. ADDITION OF HYDROCARBON FREE RADICALS
Usually, the convex curve of the hydrocarbon auto-oxidation rate as a function of the partial pressure of molecular oxygen ascends up to some limit and then flattens out [5].When this is the case, the oxidation kinetics is satisfactorily describable in terms of the conventional reaction scheme 5-10, which involves two types of free radicals.These are the hydrocarbon radical R • (addend radical) and the addition product RO (1:1 adduct radical).However, the existing mechanisms are inapplicable to the cases in which the rate of initiated oxidation as a function of the oxygen concentration has a maximum (Figs. 1 and 2) [11,12].Such dependences can be described in terms of the competition kinetics of free-radical chain addition, whose reaction scheme involves not only the above two types of free radicals, but also the RO radical (1:2 adduct) inhibiting the chain process [13,14].
The decomposition of the initiator I in reaction 1 yields a reactive • 0 R radical, which turns into the ultimate product R0H via reaction 1a, generating an alkyl radical R • , which participates in chain propagation.In reaction 2, the addition of the free radical R • to the oxygen molecule yields a reactive alkylperoxyl 1:1 adduct radical • 2 RO [10], which possesses increased energy owing to the energy released upon the conversion of the O=O bond into the ordinary bond RО-О • (for addition in the gas phase under standard conditions, this energy is 115-130 kJ mol -1 for C1-C4 alkyl radicals [6,16,17] and 73 kJ mol -1 for the allyl radical [17]).Because of this, the adduct radical can decompose (reaction 3a) or react with some neighbor molecule (reaction 3 or 4) on the spot, without diffusing in the solution and, accordingly, without entering into any chain termination reaction.In reaction 3, the interaction between the radical adduct • 2 RO and the hydrocarbon molecule RH yields, via a chain mechanism, the alkyl hydroperoxide RO2H (this reaction regenerates the chain carrier R • and, under certain conditions, can be viewed as being reversible [6]) or the alcohol ROH (this is followed by the regeneration of R • via reaction 3b).The latter (alternative) pathway of reaction 3 consists of four steps, namely, the breaking of old bonds and the formation of two new bonds in the reacting structures.In reaction 3a, the isomerization and decomposition of the alkylperoxyl radical adduct  [5,9], yielding the carbonyl compound R(-Н)НО or R(-2Н)НО.Reaction 3b produces the alcohol ROH or water and regenerates the free radical R • (here, R and R are radicals having a smaller number of carbon atoms than R).As follows from the above scheme of the process, consecutive reactions 3a and 3b (whose rates are equal within the quasi-steady-state treatment), in which the highly reactive fragment, oxyl radical RО • (or • ОН) forms and then disappears, respectively, can be represented as a single, combined bimolecular reaction 3a,b occurring in a "cage" of solvent molecules.Likewise, the alternative (parenthesized) pathways of reactions 3 and 3b, which involve the alkoxyl radical RО • , can formally be treated as having equal rates.For simple alkyl C1-C4 radicals R, the pathway of reaction 3 leading to the alkyl hydroperoxide ), consecutive to reaction 3a, according to thermochemical data for the gas phase [6,17,18].In reaction 4, which is competing with (parallel to) reactions 3 and 3a (chain propagation through the reactive radical R • ), the resulting low-reactive radical that does not participate in further chain propagation and inhibits the chain process is supposed to be the alkyltetraoxyl 1:2 radical adduct 3,4   • 4 RO , which has the largest weight and 3 It is hypothesized that raising the oxygen concentration in the o-xylene-oxygen system can lead to the formation of an ROO • •••O2 intermediate complex [11] similar to the ROO • •••(-bond)RH complex between the alkylperoxyl 1:1 adduct radical and an unsaturated hydrocarbon suggested in this work.The electronic structure of the -complexes is considered elsewhere [19]. 4Thermochemical data are available for some polyoxyl free radicals (the enthalpy of formation of the methyltetraoxyl radical without the energy of the possible intramolecular hydrogen bond ) and polyoxides (ΔНf˚298(СН3О4Н) = -21.0 9 kJ mol -1 ) [20].These data were obtained using the group contribution approach.Some physicochemical and geometric parameters were calculated for the methyl hydrotetraoxide molecule as a model compound 21-23.The IR spectra of dimethyl tetraoxide with isotopically labeled groups in Ar-O2 matrices were also reported [24].For reliable determination of the number of oxygen atoms in an oxygen-containing species, it is necessary to use IR and EPR spectroscopy in combination with the isotope tracer method [24].
size.This radical is possibly stabilized by a weak intramolecular H•••O hydrogen bond [25] shaping it into a six-membered cyclic structure5 (seven-membered cyclic structure in the case of aromatic and certain branched acyclic hydrocarbons) [27,28]: Reaction 4 in the case of the methylperoxyl radical  RO radical.6 The latter process is likely accompanied by chemiluminescence typical of hydrocarbon oxidation [23].These reactions regenerate oxygen as O2 molecules (including singlet oxygen7 [23,30]) and, partially, as O3 molecules and yield the carbonyl compound R(-2H)HO (possibly in the triplet excited state [23]).Depending on the decomposition pathway, the other possible products are the alcohol ROH, the ether ROR, and the alkyl peroxide RO2R.It is likely that the isomerization and decomposition of the • 4 RO radical via reactions 6 and 7 can take place through the breaking of a C-C bond to yield carbonyl compounds, alcohols, ethers, and organic peroxides containing fewer carbon atoms than the initial hydrocarbon, as in the case of the alkylperoxyl radical • 2 RO in reaction 3a.At later stages of oxidation and at sufficiently high temperatures, the resulting aldehydes can be further oxidized into respective carboxylic acids.They can also react with transition state of the dimerization of alkylperoxyl radicals via the Russell mechanism [7,26]. 6Taking 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.This principle was not taken into account in the author's works on oxidation until 2012.
RO are effective quenchers of singlet oxygen О2( g a 

1
) [29].molecular oxygen so that a C-H bond in the aldehyde molecule breaks to yield two free radicals (

HO and • R(-H)O or • R(-2H)O
).This process, like possible ozone decomposition yielding an • O • atom or peroxide decomposition with O-O bond breaking, leads to degenerate chain branching [5].The kinetic equations describing the rates of formation of molecular products at the chain propagation and termination stages of the above reaction Scheme 1 were obtained using the quasi-steady-state treatment.To simplify the writing of equations in the derivation process, we introduce the following notations: • , and R4 = RO • (or • OH); α = k3 /k4 and β = k3a /k4 (mol dm -3 ).According to Scheme 1, V1 = V1a, V3a = V3b, and V2 = V3 +V3a +V4.In order to reduce the exponent of the 2k5[R • ] 2 term in the d[R • ]/dt = 0 equation to unity [1], were used the following condition for the early stages of the process: k6 = 57 22 kk [8] and, hence, 2kV and obtain the following expressions for free radicals.
The rate constant k2 of reaction 2 in the kinetic equations can be replaced with its analytical expression, which is obtained by solving the quadratic equation following from the rate function extremum condition The solution of this incomplete quadratic equation gives the value of the concentration xm at the maximum point: From here, the following kinetic equations of oxidation reaction rates were obtained.
( ) where V1 is the initiation rate, l = [RH] and x = [O2] are the molar concentrations of the starting components (l >> x), α = k3 /k4 and β = k3a /k4 (mol dm -3 ) are the ratios of the rate constants of the competing (parallel) reactions, is the rate constant of the addition of the alkyl radical R • to the oxygen molecule (reaction 2) as determined by solving the quadratic equation following from the rate function extremum condition x at the maximum point of the function, The ratios of the rates of the competing reactions are V3 /V4 = αl /x and V3a /V4 = β /x, and the chain length is  = (V3 + V3a)/V1.The overall rate of the process is a complicated function of the formation and disappearance rates of the free radicals R • and As distinct from the rates V4 (V4  V1), V5, and V7, the rates V2, V3, V3a, 3b , and 2V6 as a function of the concentration x have a maximum.
Eqs. (1a) and (2a) were obtained by replacing the rate constant k2 in Eqs. ( 1) and ( 2) with its analytical expression (for reducing the number of unknown parameters to be determined directly).For αl >> β (V3 >> V3a), when the total yield of alkyl hydroperoxides and alcohols having the same number of carbon atoms as the initial compound far exceeds the yield of carbonyl compounds, as in the case of the oxidation of some hydrocarbons, the parameter β in Eqs. ( 1) and (1a) can be neglected (β = 0): , and Equations ( 1) and ( 2) subject to the constraint k2x 2 >> (αl + (the descending branch of the peaking curve) can be transformed into Eqs.( 6) and ( 7), respectively, which express simple inverse proportionalities with respect to the oxygen concentration x and allow the parameters α and β to be tentatively estimated from the experimental product formation rate V provided that V1 is known: where  = 2 at the point of maximum (where ) 2 ( In the kinetic model of oxidation that does not include the competing reaction 4 (k4 = 0) and involves the radicals R • and , where a0, b0, and c0 are coefficients having no extremum.For a similar kinetic model in which reactions 3a,b and 4 appearing in the above scheme are missing (k3a = k4 = 0), Walling [7], using the quasi-steady-state treatment in the long kinetic chain approximation, when it can be assumed that V2 = V3, without using the substitution k6 = 7 5 2 2 k k [5,7,8] (as distinct from this work), found that V2 = V3 is an irrational function of x: where a1, b1, c1, and d1 are coefficients.Again, this function has no maximum with respect to the concentration of any of the two components.Thus, of the three kinetic models of oxidation mathematically analyzed above, which involve the radicals R • and in three types of quadratic-law chain termination reactions (reactions 5-7) and are variants of the conventional model [5][6][7][8][9][10], the last two lead to an oxidation rate versus oxygen concentration curve that emanates from the origin of coordinates, is convex upward, and has an asymptote parallel to the abscissa axis.Such monotonic dependences are observed when the oxygen solubility in the liquid is limited under given experimental conditions and the oxygen concentration attained 8 Unlike the conventional model, the above kinetic model of free-radical non-branched chain oxidation, which includes the pairs of competing reactions 3-4 and 3a-4 (Scheme 1), allows us to describe the non-monotonic (peaking) dependence of the oxidation rate on the oxygen concentration (Fig. 1).In this oxidation model, as the oxygen concentration in the binary system is increased, oxygen begins to act as an oxidation auto-inhibitor or an antioxidant via the further oxidation of the alkylperoxyl 1:1 adduct radical RO and R • and cause shortening of the kinetic chains).The optimum oxygen concentration xm, at which the oxidation rate is the highest, can be calculated using kinetic equations (1a) and (2a) or the corresponding analytical expression for k2.In the familiar monograph "Chain Reactions" by Semenov [31], 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 10 3 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 [12] 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 Fig. 2) that indicates a chemical mechanism providing the appearance of a maximum (see reaction 4 of Scheme 2).Curve 1 in Fig. 1 illustrates the fit between Eq. (1a) at αl >> β and experimental data for the radiation-induced oxidation of o-xylene in the liquid phase at 373 K in the case of 2-methylbenzyl hydroperoxide forming much more rapidly than o-tolualdehyde (V3 >> V3a and αl >> β, i.e., the parameter β can be neglected) [11].The oxygen concentration limit in o-xylene is reached at an oxygen concentration of [O2]top > xm, which corresponds to the third experimental point [11].The oxygen concentration was calculated from the oxygen solubility in liquid xylene at 373 K [32].The following quantities were used in this mathematical description: 60 Co -radiation dose rate of P = 2.18 Gy s -1 and total initiation yield of G(o-СН3С6Н4ĊН2) = 2.6 particles per 100 eV of the energy absorbed by the solution [11]; V1 = 4.73  10 -7 mol dm -3 s -1 , and 2k5 = 1.15  10 10 dm 3 mol -1 s -1 .The resulting value of the parameter α is (9.0 ± 1.8)  10 -3 ; hence, k2 = (3.2± 0.8)  10 5 dm 3 mol -1 s -1 .From the data presented in [33], it was estimated that k4 = k3/α = (5.2± 1.2)  10 2 dm 3 mol -1 s -1 .

Figure 1:
(1, ) Reconstruction of the functional dependence of the 2-methylbenzyl hydroperoxide formation rate V3(RO2H) on the dissolved oxygen concentration x from empirical data (points) using Eq.(1a) with β = 0 (model optimization with respect to the parameter α) for the o-xylene-oxygen system at 373 K [11] (standard deviation of SY = 5.37  10 -7 ).(2, ) Reconstruction of the functional dependence of the total hydrogen peroxide formation rate V3, 7(Н2О2) on the dissolved oxygen concentration x from empirical data (symbols) using Eqs.(1a) (5a) with β = 0 (model optimization with respect to the parameter α) for the γ-radiolysis of water saturated with hydrogen and containing different amounts of oxygen at 296 K [34] (SY = 1.13  10 -8 ).The dashed curve described V3(H2O2) as a function of the oxygen concentration x based on Eq. (1a) with β = 0 (model optimization with respect to α) and the experimental data of curve 2 (SY = 1.73  10 -8 ).

III. ADDITION OF THE HYDROGEN ATOM
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 [12,35,36] (Fig. 2) and in the liquid phase [34] (Fig. 1, curve 2), considered in our earlier works [13,27,28,37], can also be explained in terms of the competition kinetics of free radical addition [14,38].From Fig. 2 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) [35].(2, ○) water resulting from the photochemical oxidation of hydrogen in the hydrogen-oxygen system as a function of the oxygen concentration x (light wavelength of 171.9-172.5 nm, total pressure of 10 5 Pa, room temperature [35]).(3,4) Hydrogen peroxide formation rate V(Н2О2) (dashed curves) as a function of the rate V(О2) at which molecular oxygen is passed through a gas-discharge tube filled with (3, ) atomic and (4, ) molecular hydrogen.Atomic hydrogen was obtained from molecular hydrogen in the gas-discharge tube before the measurements (total pressure of 25-77 Pa, temperature of 77 K [12]).The symbols represent experimental data.HO [46][47][48][49][50] 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 9 According to Francisco and Williams [20], the enthalpy of formation (ΔНf˚298 ) in the gas phase of Н • , НО • , HO radical with a helical structure were carried out using the G2(MP2) method [39].The stabilization energies of HO 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 [40,41].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 [42].The photochemical properties of O4 and the van der Waals nature of the О2-О2 bond were investigated [43,44].The most stable geometry of the dimer is two O2 molecules parallel to one another.The O4 molecule was identified by NR mass spectrometry [45].
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 [27,28] that the hydrotetraoxyl radical HO (first reported in [51][52][53]) resulting from endothermic reaction 4, which is responsible for the peak in the experimental rate curve (Fig. 2, curve 3), is closed into a five-membered [ОО─Н•••ОО] • cycle due to weak intramolecular hydrogen bonding [25,54].This structure imparts additional stability to this radical and makes it least reactive.The HO 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 [37] from the value of 1/k for the monomolecular decay reaction ]), is 3.9 times longer than that of the linear HO radical [39,56] estimated in the same way [37] for the same conditions [57], (9.1 ± 0.9) × 10 -6 s.MP2/6-311++G** calculations using the Gaussian-98 program confirmed that the cyclic structure of HO [58] is energetically more favorable than the helical structure [39] (the difference in energy is 4.8-7.3kJ mol -1 , depending on the computational method and the basis set). 10For example, with the MP2(full)/6-31G(d) method, the difference between the full energies of the cyclic and 10 There were calculations for the two conformers (cis and trans) of the HO can exist in both forms, but the cyclic structure is obviously dominant (87%, Keq = 6.5) [58].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 and Н • .The hydrogen molecule that results from reaction 5 in the gas bulk possesses an excess energy, and, to acquire stability within the approximation used in this work, it should have time for deactivation via collision with a particle M capable of accepting the excess energy [60].To simplify the form of the kinetic equations, it was assumed that the rate of the bimolecular deactivation of the molecule substantially exceeds the rate of its monomolecular decomposition, which is the reverse of reaction 5 [6].41, which are deactivated by collisions, and in the form of O3) and yield hydrogen peroxide or water via a non-chain mechanism, presumably through the intermediate formation of the unstable hydrogen tetraoxide molecule H2O4 [61,62]. 11Ozone does not interact with molecular hydrogen.At moderate temperatures, it decomposes fairly slowly, particularly in the presence of О2( 3 g Χ −  ) 41.The reaction of ozone with Н • atoms, which is not impossible, results in their replacement with НО • radicals.The relative contributions from reactions 6 and 7 to the process kinetics can be roughly estimated from the corresponding enthalpy increments (Scheme 2).When there is no excess hydrogen in the hydrogen-oxygen system and the homomolecular dimer O4 [42-45, 63, 64], which exists at low concentrations (depending on the pressure and temperature) in equilibrium with O2 [41], can directly capture the Н • atom to yield the heteronuclear 11 The planar, six-atom, cyclic, hydrogen-bonded dimer 22 HO () • was calculated using quantum chemical methods (B3LYP density functional theory) [88].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.cluster 12 which is more stable than O4 [41] and cannot abstract a hydrogen atom from the hydrogen molecule, non-chain hydrogen oxidation will occur to give molecular oxidation products via the disproportionation of free radicals.The low-reactive hydrotetraoxyl radical • 4 HO [55], which presumably has a high energy density [42], 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 • + О3, in which the adduct • 4 HO ( 2 А) was considered as an intermediate, was calculated by the double many-body expansion (DMBE) method [65].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  ) is observed at altitudes of 30-80 and 40-130 km, respectively [67].Staehelin et al. [55] 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 non-catalytic oxidation of hydrogen, including in an inert medium [60], in terms of the simplified scheme of free-radical non-branched chain reactions (Scheme 2), which considers only quadratic-law chain termination and ignores the surface effects [12], at moderate temperatures and pressures, in the absence of transitions 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 [47], 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: 13 k6 = 57 22 kk [1,8].
The equation V3(H2O2; H2O) = V3(H2O) for the rate of the chain formation of hydrogen peroxide in the reaction (3) and water in reactions 3 and 3 with V3, 3(H2O) = 2V3 is identical to Eq. (1, 1a) with the corresponding analytical expression for k2 provided that β = 0 everywhere.The ratio of the rates of the competing reactions is V3 /V4 = αl /x, and the chain length is  = V3 /V1.The equations for the rates of non-chain formation of hydrogen peroxide and water via reactions 6 and 7quadratic-law chain terminationare identical to Eqs. ( 4) and ( 5) assuming that β = 0.In these equations, l and x are the molar concentrations of hydrogen and oxygen (l >> x), lm and xm are the respective concentrations at the maximum point of the function, V1 is the rate of initiation (reaction 1), α = k3/k4, the rate constant is derived from the condition ∂V3/∂x = 0, and 2k5 is the rate constant of reaction 5 (hydrogen atom recombination), which is considered as bimolecular within the given approximation.
The rate constant 2k5 in the case of the pulsed radiolysis of ammonia-oxygen (+ argon) gaseous mixtures at a total pressure of 10 5 Pa and a temperature of 349 K was calculated to be 1.6 × 10 8 dm 3 mol -1 s -1 [36] (a similar value of this constant for the gas phase was reported an earlier publication [68]).Pagsberg et al. [36] found that the dependence of the yield of the intermediate НО • on the oxygen concentration has a maximum close to 5 × 10 -4 mol dm -3 .In the computer simulation of the process, they considered the strongly exothermic reaction HO + NН3 → Н2О + • NНОН, which is similar to reaction 3 in Scheme 2, whereas the competing reaction 4 was not taken into account.The rate of the non-branched-chain free-radical oxidation of hydrogen is a complex function of the rates of formation and disappearance of Н • atoms and • 4 HO radicals: V1 + V3, 3' -V4 -V5 + V7.Unlike the dependences of the rates V4 13 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 [66] and k(H • + HO • )/2k(2H • )2k(2HO • ) 0.5 = 1.5 in water [4], the value that is fairly close to unity.
(V4  V1), V5 and V7, the dependences of the rates V2, V3, 3', and 2V6 on the oxygen concentration x show a maximum.Equation ( 1) with β = 0 under the conditions k2x 2 >> (αl + x) 51 2kV and k2x 2 << (αl + x) 51 2kV , αl >> x, corresponding to the descending and ascending branches of the curve with a maximum, can be transformed into simple equations which allow preliminary to estimate the parameters α and k2 and express inversely (see Eq. ( 6)) and directly proportional functions of concentration x : In the case of non-chain hydrogen oxidation via the above addition reaction (Н HO ), the formation rates of the molecular oxidation products in reactions 6 and 7 (Scheme 2, k2 = k3 = k4 = 0) are defined by modified Eqs. ( 4) and (5) in which β = 0, (αl + x) is replaced with 1, and k2 is replaced with kaddKeq (kaddKeq is the effective rate constant of Н • addition to the О4 dimer, Кeq = k/k is the equilibrium constant of the reversible reaction 2О2 k k О4 with k >> kaddН • ).The formation rates of the stable products of non-chain oxidation (k3 = 0), provided that either reactions 2 and 4 or reaction 2 alone (k4 = 0) occurs (Scheme 2; in the latter case, reactions 6 and 7 involve the  4) and ( 5) with β = 0, (αl + x) replaced with 1, and х 2 replaced with х.Note that, if in Scheme 2 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 • 3 HO (which was obtained in the gas phase by neutralization reionization (NR) mass spectrometry [56] and has a lifetime of >10 -6 s at 298 K) and chain termination takes place via reactions 5-7 involving the НО • and 10 -8 mol dm -3 s -1 of the primary formation of hydrogen peroxide after completion of the reactions in spurs) on the concentration of dissolved oxygen during the -radiolysis of water saturated with hydrogen (at the initial concentration 7  10 -4 mol dm -3 ) at 296 K [34].These data were calculated in the present work from the initial slopes of hydrogen peroxide buildup versus dose curves for a 60 Co -radiation dose rate of Р = 0.67 Gy s -1 and absorbed doses of D  22.5-304.0Gy.The following values of the primary radiation-chemical yield G (species per 100 eV of energy absorbed) for water -radiolysis products in the bulk of solution at pH 4-9 and room temperature were used (taking into account that V = GP and V1 = GHP):

IV. CONCLUSIONS
The above data concerning the competition kinetics of the non-branched chain addition of hydrocarbon free radicals and hydrogen atoms to the multiple bonds of the oxygen molecules make it possible to describe, using rate equations (1a) and (5a) obtained by quasi-steady-state treatment (the equation derivation is shown on p. 4), the peaking experimental dependences of the formation rates of molecular 1:1 adducts (in this case the o-СН3С6Н4CН2O2H or H2O2) on the concentration of the oxygen over the entire range of its variation in binary systems (Fig. 1).In such reaction systems consisting of saturated and unsaturated components [1, 13-15, 27, 28, 37, 38, 69-72], the unsaturated compound (in this case the O2) is both a reactant and an auto-inhibitor, specifically, a source of low-reactive free radicals (in this case the The progressive inhibition of the non-branched-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 non-branched-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 HO radical, which is capable of efficiently absorbing UV radiation [55].The kinetic equations obtained on the basis of the suggested reaction schemes involving pairs of competitive reactions at the chain evolution stage make it possible to directly determine the rate constant k2 of the addition reaction 2 without using a separate labor-intensive method of competing reactions as often happens in such cases.

•
the gas phase, with heat absorption equal to 110.0 ± 18.6 kJ mol -1 [20] (without the energy of the possible formation of a hydrogen bond taken into account).The exothermic reactions 6 and 7, in which the radical R • or • 4 RO undergoes disproportionation, include the isomerization and decomposition of the• 4 lm and xm are the values of l and

2 RO
 = 1 for the descending branch of the curve.In the alternative kinetic model of oxidation, whose chain termination stage involves, in place of R • (Scheme 1),• radicals reacting with one another and with • 4 RO radicals, the dependences of the chain formation rates of the products on the oxygen concentration x derived by the same method have no maximum:

• 2 RO (the latter instead of • 4 RO
in Scheme 1) in reactions 5-7, the reaction rate functions V3 and V3a obtained in the same way are fractional rational functions in the form of )

• 2 RO 4 RO
into the low-reactive 1:2 adduct radical • (reactions 4 and 6 lead to inefficient consumption of the free radicals • 2

• 4 HO
radical was discovered by Staehelin et al.[55] in a pulsed radiolysis study of ozone degradation in water; its UV spectrum with an absorption maximum at 260 nm ( two times larger[55].The assumption about the cyclic structure of the • 4

• 4 HO 4 HO 2 HO 4 HO 4 HO transition is 298 S
radical [86] 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 • radical is the cis one, which is computed to be endothermic with respect to• 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 •  = −1.6 kJ mol -1 K -1 .Therefore, under standard conditions,• 4 Reactions 6 and 7 (taking into account the principle of detailed balance for the various pathways) regenerate hydrogen and oxygen (in the form of О2(

4 HO
10 4 -1.2  10 3 Pa [66]; the corresponding 298 Н  reaction values are given in kJ mol -1 [20]): 8. Н2О(vapor) + h → Н • + HO • 66; radical can disappear via disproportionation with a molecule, free radical, or atom in addition to dissociation.Note that emission from О2( 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(Н2О) 7 (which was derived from Eqs. (1a) and (5), respectively, with β = 0, Eq. (5) in the form[69] of 42 71 m V V x f = (5a) in which k2 is replaced with its  derived from Eq. (1)with β = 0 everywhere), the dependence of the hydrogen peroxide formation rate (minus the rate 75 and GH = 0.6 (initiation yield)[4]; V1 = 4.15  10 -8 mol dm -3 s -1 ; 2k5 = 2.0 × 10 10 dm 3 mol -1 s -1 4.As can be seen from Fig.1, the best description of the data with an increase in the oxygen concentration in water is attained when the rate V7 of the formation of hydrogen peroxide via the non-chain mechanism in the chain termination reaction 7 (curve 1, α = (8.5  2)  10 -2 ) is taken into account in addition to the rate V3 of the chain formation of this product via the propagation reaction 3 (dashed curve 2, α = 0.11  0.026).The rate constant of addition reaction 2 determined from α is substantially underestimated: k2 = 1.34  10 7 (vs.2.0  10 10 [4]) dm 3 mol -1 s -1 .The difference can be due to the fact that the radiation-chemical specifics of the process were not considered in the kinetic description of the experimental data.These include oxygen consumption via reactions that are not involved in the hydrogen oxidation Scheme 2 and reverse reactions resulting in the decomposition of hydrogen peroxide by intermediate products of water radiolysis ( − aq e , Н • , НО • ), with the major role played by the hydrated electron[4].