Kinetics of the Free-Radical Nonbranched-Chain Addition

The aim of this study was the conclusion of simple kinetic equations to describe ab initio initiated nonbranched-chain processes of the saturated free-radical addition to the double bonds of unsaturated molecules in the binary reaction systems of saturated and unsaturated components. In the processes of this kind the formation rate of the molecular addition products (1:1 adducts) as a function of concentration of the unsaturated component has a maximum. Five reaction schemes are suggested for this addition processes. The proposed schemes include the reaction competing with chain propagation reactions through a reactive free radical. The chain evolution stage in these schemes involves three or four types of free radicals. One of them is relatively low-reactive and inhibits the chain process by shortening of the kinetic chain length. Based on the suggested schemes, nine rate equations (containing one to three parameters to be determined directly) are deduced using quasi-steady-state treatment. These equations provide good fits for the nonmonotonic (peaking) dependences of the formation rates of the molecular products (1:1 adducts) on the concentration of the unsaturated component in binary systems consisting of a saturated component (hydrocarbon, alcohol, etc.) and an unsaturated component (alkene, allyl alcohol, formaldehyde, or dioxygen). The unsaturated compound in these systems is both a reactant and an autoinhibitor generating low-reactive free radicals. A similar kinetic description is applicable to the nonbranched-chain process of the free-radical hydrogen oxidation, in which the oxygen with the increase of its concentration begins to act as an oxidation autoingibitor (or an antioxidant). The energetics of the key radical-molecule reactions is considered.


Introduction
A free radical may be low-reactive if its unpaired p-electron may be delocalized, e.g., over conjugated bonds as in the case of allyl radical CH 2 =CHĊH 2 or along a double bond from carbon to the more electron-affine oxygen as in the case of formyl radical HĊ=O. Note that the activity of a free radical is also connected to the reaction heat in which it participates. In nonbranched-chain processes of reactive free radical (addend) addition to double bonds of molecules, the formation of rather low-reactive free radicals in reactions, which are parallel to or competing with propagation via a reactive radicals, lead to chain termination, because these low-reactive radicals do not participate in further chain propagation and because they decay when colliding with each other or with chain-carrier reactive radicals thus resulting in inefficient expenditure of the latter and process inhibition.
In similar processes involving the addend and inhibitor radicals in diffusion controlled bimolecular chain-termination reactions of three types, the dependences of the rate of molecular 1:1 adduct formation on the concentration of the unsaturated component (which is the source of low-reactive free radicals in a binary system of saturated and unsaturated components) have a maximum, usually in the region of small (optimal) concentrations. The progressive inhibition of nonbranched chain processes upon exceeding this optimal concentration may be an element of self-regulation of the natural processes returning them to a steady state condition.
Here, reactions of addition of reactive free radicals to multiple bonds of alkene, formaldehyde, and oxygen molecules to give 1:1 adduct radicals are taken as examples to consider the role of low-reactive free radicals as inhibitors of the nonbranched chain processes at moderate temperatures. In the case of oxidation, there are tetraoxyl 1:2 adduct radical arising upon addition of a peroxyl 1:1 adduct radical to molecular oxygen at high enough concentrations of the latter. The 1:1 adduct radical (which is the heaviest and the largest among the free radicals that result from the addition of one addend radical to the double bond of the molecule) may have an increased energy owing to the energy liberated in the transformation of a double bond into an ordinary bond (30-130 kJ mol -1 for the gas phase under standard conditions [1][2][3][4]). Therefore, it can decompose or react with one of the surrounding molecules in the place of its formation without diffusing in the solution and, hence, without participating in radical-radical chain termination reactions. Which of the two reactions of the adduct radical, the reaction with the saturated component or the reaction with the unsaturated component, dominates the kinetics of the process will depend on the reactivity and concentration ratios of the components in the binary system.
Earlier [5,6], there were attempts to describe such peaking dependences fragmentarily, assuming that the saturated or unsaturated component is in excess, in terms of the direct and inverse proportionalities, respectively, that result from the simplification of a particular case of the kinetic equation set up by the quasi-steady-state treatment of binary copolymerization involving fairly long chains [5]. This specific equation is based on an irrational function, whose plot is a monotonic curve representing the dependence of the product formation rate on the concentration of the unsaturated component. This curve comes out of the origin of coordinates, is convex upward, and has an asymptote parallel to the abscissa axis. Replacing the component concentrations with the corresponding mole fractions generates a peak in this irrational function and thereby makes it suitable to describe the experimental data [7]. However, this circumstance cannot serve as a sufficient validation criterion for the mechanism examined, because the new property imparted to the function by the above artificial transformation does not follow from the solution of the set of algebraic equations that are set up for the reaction scheme accepted for the process in a closed system and express the equality of the steady-state formation and disappearance rates of the reactive intermediates.
This publication presents a comprehensive review of the nonbranched-chain kinetic models developed for particular types of additions of saturated free radicals to multiple bonds [8][9][10][11][12][13][14]. It covers free radical additions to alkenes [10,11], their derivatives [8,9], formaldehyde (first compound in the aldehyde homological series) [8,9,12], and molecular oxygen [13,14] (which can add an unsaturated radical as well) yielding various 1:1 molecular adducts, whose formation rates as a function of the unsaturated compound concentration pass through a maximum (free radical chain additions to the С=N bond have not been studied adequately). In the kinetic description of these nontelomerization chain processes, the reaction between the 1:1 adduct radical and the unsaturated molecule, which is in competition with chain propagation through a reactive free radical ( • PCl 2 , С 2 Н 5 ĊНОН, etc.), is included for the first time in the chain propagation stage. This reaction yields a low-reactive radical (such as СН 2 =С(СН 3 )ĊН 2 or НĊ=О) and thus leads to chain termination because this radical does not continue the chain and thereby inhibits the chain process [8]. We will consider kinetic variants for the case of comparable component concentrations with an excess of the saturated component [10,11] and the case of an overwhelming excess of the saturated component over the unsaturated component [8,9,12]. Based on the reaction schemes suggested for the kinetic description of the addition process, we have derived kinetic equations with one to three parameters to be determined directly.
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 build-up of the statistical error in the nonlinear estimation of these parameters in the case of a limited number of experimental data points [15]. The rate constant of the addition of a free radical to the double bond of the unsaturated molecule, 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.
The kinetic equations were set up using the quasi-steady-state treatment. This method is the most suitable for processes that include eight to ten or more reactions and four to six different free radicals and are described by curves based on no more than three to seven experimental points. In order to reduce the exponent of the 2k 5 [ R ]/dt = 0 equation to unity [8], we used the following condition for the early stages of the process: k 6 = 7 5 2 2 k k [16] and, hence, V 1 = V 5 + 2V 6  R ] are the concentrations of the addend radical and the low-reactive (inhibitor) radical, respectively; V 1 is the initiation rate; V 5 , 2V 6 , and V 7 are the rates of the three types of diffusion-controlled quadratic-law chain termination reactions; 2k 5  The kinetic equations thus obtained fit the peaking rate curves well throughout the range of unsaturated component concentrations in the binary systems. Our mathematical simulation was based on experimental data obtained for γ-radiation-induced addition reactions for which the initiation rate V 1 is known.
The analysis of stable liquid-phase products was carried out by the gas chromatographic method.
accumulates an energy of 90-130 kJ mol -1 , which is released upon the transformation of the C=C bond to an ordinary bond (according to the data reported for the addition of nonbranched C 1 -C 4 alkyl radicals to propene and of similar C 1 and C 2 radicals to 1-butene in the gas phase under standard conditions [1][2][3][4]). Such adduct radicals, which do not decompose readily for structural reasons, can abstract the most labile atom from a neighbor molecule of the saturated or unsaturated component of the binary reaction system, thus turning into a 1:1 adduct molecule. The consecutive and parallel reactions involved in this free-radical nonbranched-chain addition process are presented below (Scheme 1). In the case of comparable component concentrations with a non-overwhelming excess of the saturated component, extra reaction (1b) (k 1b ≠ 0) is included in the initiation stage [10,11]. In the case of an overwhelming excess of the saturated component reaction (1b) is ignored (k 1b = 0) [8,9,12].
The initiation reaction 1 is either the decomposition of a chemical initiator [5,17,18] or a reaction induced by light [5,17,18] or ionizing radiation [19][20][21][22][23]. The overall rate of chain initiation (reactions 1, 1a, and 1b) is determined by the rate of the rate-limiting step (k 1b > k 1a ). The reaction between the free radical • 2 R , which results from reactions 1b and 4, and the saturated molecule R 1 А is energetically unfavorable because it implies the formation of the free radical • 1 R , which is less stable than the initial one. The addition reaction 2 may be accompanied by the abstraction reaction 2a. R , but to a much lesser extent. The rates of the formation (V, mol dm -3 s -1 ) of the 1:1 adducts R 3 A (via a chain mechanism) and R 3 B (via a nonchain mechanism) in reactions 3 and 4 are given by the equations where V 1 is the rate of the initiation reaction 1; l = [R 1 A] and x = [R 2 B] are the molar concentrations of the initial components, with l > x; k 2 is the rate constant of the addition of the • 1 R radical from the saturated component R 1 А to the unsaturated molecule R 2 В (reaction 2); and γ = k 1a /k 1b and α = k 3 /k 4 are the rate constant ratios for competing (parallel) reactions (α is the first chain-transfer constant for the free-radical telomerization process [5]). The rate ratio for the competing reactions is V 3 /V 4 = αl/x, and the chain length is v = V 3 /V 1 .
Earlier mathematical simulation [8] demonstrated that replacing the adduct radical R 3 with the radical R 2 [5] in the reaction between identical radicals and in the reaction involving R 1 gives rise to a peak in the curve of the 1:1 adduct formation rate as a function of the concentration of the unsaturated component. Reaction 1b, which is in competition with reaction 1a, is responsible for the maximum in the curve described by Eq. (2), and reaction 4, which is in competition with reaction (3), is responsible for the maximum in the curve defined by Eq. (1).
The number of unknown kinetic parameters to be determined directly (k 2 , α, and γ) can be reduced by introducing the condition γ ≅ α, which is suggested by the chemical analogy between the competing reactions pairs 1a-1b and 3-4. For example, the ratios of the rate constants of the reactions of • OН, СН 3 where l m and x m are the component concentrations l and x at the points of maximum of the function. Provided that V 1 is known, the only parameter in Eq. (3a) to be determined directly is α. If V 1 is known only for the saturated component where 1 -χ = l/(l + x) and χ = x/(l + x) are the mole fractions of the components R 1 A and R 2 В (0 < χ <1), respectively, and χ m is the χ value at the point of maximum.
The overall formation rate of the 1:1 adducts R 3 A and R 3 B is a sophisticated function of the formation and disappearance rates of the radicals The application of the above rate equations to particular single nonbranched-chain additions is illustrated in Figure 1. Curve 1 represents the results of simulation in terms of Eq. (3b) for the observed 1:1 adduct formation rate as a function of the mole fraction of the unsaturated component in the phosphorus trichloride-methylpropene 1 reaction system at 303 K [19]. In this simulation, the 60 Co γ-radiation dose rate was set at P = 0.01 Gy s -1 and the initiation yield was taken to be G( • PCl 2 ) = 2.8 particles per 100 eV (1.60 × 10 -17 J) of the energy absorbed by the solution [19]. The product of reaction 3 is Cl 2 PCH 2 C(Cl)(CH 3 )CH 3 (two isomers), V 1 = 4.65 × 10 -9 mol dm -3 s -1 at χ = 0, and 2k 5 = 3.2 × 10 8 dm 3 mol -1 s -1 . This leads to α = (2.5 ± 0.4) × 10 3 , and the rate constant of reaction 2 derived from this α value is k 2 = (1.1 ± 0.2) × 10 4 dm 3 mol -1 s -1 . Note that, if the R 2 -B bond dissociation energy for the unsaturated component of the binary system is approximately equal to or above, not below, the R 1 -A bond dissociation energy for the saturated component, then the rate of reaction 4 relative to the rate of the parallel reaction 3 (chain propagation through the reactive free radical 1 R • ) will be sufficiently high for adequate description of R 3 A and R 3 B adduct formation in terms of Eqs. (1)-(3b) only at high temperatures [20]. In the phosphorus trichloride-propene system, the difference between the R 2 -B (B = H) and R 1 -A (A = Hal) bond dissociation energies in the gas phase under standard conditions [1] is as small as 5 kJ mol -1 , while in the tetrachloromethane-methylpropene (or cyclohexene) and bromoethane-2-methyl-2-butene systems, this difference is 20.9 (37.7) and ~24 kJ mol -1 , respectively.

Excess of the Saturated Component
If the concentration of the saturated component exceeds the concentration of the unsaturated component in the binary system, reaction 1b can be neglected. If this is the case (k 1b = 0), then, in the numerators of the rate equations for reactions 3 and 4 (Eqs. (1) and (2)), γl/(γl + x) = 1 and the overall rate equation for the formation of the 1:1 adducts R 3 A and R 3 B will appear as where the parameters are designated in the same way as in Eqs.
The rate equations for the chain termination reactions 5-7 (Scheme 1, k 1b = 0) are identical to Eqs. Note that, if it is necessary to supplement Scheme 1 for k 1b = 0 with the formation of R 1 B via the possible nonchain reaction 2a (which is considered in the Section 2.1), the parameter k 2a should be included in the denominator of Eq. (4) to obtain The analytical expression for k 2 in the case of k 2a ≠ 0 is identical to the expression for k 2 for Eq. (4). The equation for the rate V 2a (R 1 B) can be derived by replacing k 2 with k 2a in the numerator of Eq. (4) containing k 2a in its denominator.

Addition to the C=O Bond of Formaldehyde
Free radicals add to the carbon atom at the double bond of the carbonyl group of dissolved free (unsolvated, monomer) formaldehyde. The concentration of free formaldehyde in the solution at room temperature is a fraction of a percent of the total formaldehyde concentration, which includes formaldehyde chemically bound to the solvent [27]. The concentration of free formaldehyde exponentially increases with increasing temperature [28]. The energy released as a result of this addition, when the C=O bond is converted into an ordinary bond, is 30 to 60 kJ mol -1 (according to the data on the addition of С 1 -С 4 alkyl radicals in the gas phase under standard conditions [1][2][3][4]). The resulting free 1:1 adduct radicals can both abstract hydrogen atoms from the nearest-neighbor molecules of the solvent or unsolvated formaldehyde and, due to its structure, decompose by a monomolecular mechanism including isomerization [9,12].

Addition of Free 1-Hydroxyalklyl Radicals with Two or More Carbon Atoms
Free 1-hydroxyalkyl radicals (which result from the abstraction of a hydrogen atom from the carbon atom bonded to the hydroxyl group in molecules of saturated aliphatic alcohols but methanol under the action of chemical initiators [29,30], light [17,31], or ionizing radiation [32,33]) add at the double bond of free formaldehyde dissolved in the alcohol, forming 1,2-alkanediols [8,9,12,[29][30][31][32][33][34][35][36], carbonyl compounds, and methanol [8,33] via the chaining mechanism. (The yields of the latter two products in the temperature range of 303 to 448 K are one order of magnitude lower.) In these processes, the determining role in the reactivity of the alcohols can be played by the desolvation of formaldehyde in alcoholformaldehyde solutions, which depends both on the temperature and on the polarity of the solvent [28,33]. For the γ-radiolysis of 1 (or 2)-propanol-formaldehyde system at a constant temperature, the dependences of the radiation-chemical yields of 1,2-alkanediols and carbonyl compounds as a function of the formaldehyde concentration show maxima and are symbatic [8,32]. For a constant total formaldehyde concentration of 1 mol dm -3 , the dependence of the 1,2-alkanediol yields as a function of temperature for 303-473 K shows a maximum, whereas the yields of carbonyl compounds and methanol increase monotonically [33] (along with the concentration of free formaldehyde [28]). In addition to the above products, the nonchain mechanism in the γ-radiolysis of the solutions of formaldehyde in ethanol and 1and 2-propanol gives ethanediol, carbon monoxide, and hydrogen in low radiation-chemical yields (which, however, exceed the yields of the same products in the γ-radiolysis of individual alcohols) [8,9,33]. The available experimental data can be described in terms of the following scheme of reactions: Scheme 2 Chain initiation 2CO + H 2 ). In these reactions, I is an initiator, e.g., a peroxide [29,30]; • 0 R , some reactive radical (initiator radical); R, an alkyl; ROH, a saturated aliphatic alcohol, either primary or secondary, beginning from ethanol; CH 2 O, the unsaturated moleculefree formaldehyde; • СН 2 ОН, the 1-hydroxymetyl fragment radical; • R (-H) OH, the reactive 1-hydroxyalkyl addend radical, beginning from 1-hydroxyethyl; R (-H) (ОH)СН 2 О • , the reactive hydroxyalkoxyl 1:1 adduct radical; • СНО, the low-reactive formyl radical (inhibitor radical); R 0 H, the molecular product; R (-H) (OH)СН 2 ОН, 1,2-alkanediol; R (-2H) HO, an aldehyde in the case of a primary alcohol and an R'R"CO ketone in the case of a secondary alcohol; R (-H) (ОH)R (-H) ОH, a vicinal alkanediol; R (-H) (OH)CHO, a hydroxyaldehyde. The chain evolution stage of Scheme 2 includes consecutive reaction pairs 2-3, 2-3a, and 3a-3b; parallel (competing) reaction pairs 3-3a, 3-3b, 3-4, and 3a-4; and consecutive-parallel reactions 2 and 4.
Scheme 2 does not include the same types of radical-molecule reactions as were considered in Section 2.1 for Scheme 1. In addition, it seems unlikely that free adduct radicals will add to formaldehyde at higher temperatures the reaction of adding is unlikely because this would result in an ether bond. The addition of hydroxymethyl radicals to formaldehyde, which is in competition with reaction 3b, is not included as well, because there is no chain formation of ethanediol at 303-448 K [33]. At the same time, small amounts of ethanediol can form via the dimerization of a small fraction of hydroxymethyl radicals, but this cannot have any appreciable effect on the overall process kinetics. The addition of free formyl radicals to formaldehyde cannot proceed at a significant rate, as is indicated by the fact that there is no chain formation of glycol aldehyde in the systems examined [33].
The mechanism of the decomposition of the free adduct radical via reaction 3a, which includes the formation of an intramolecular Н⋅⋅⋅О bond and isomerization, can be represented as follows [8,9,12]: The probability of the occurrence of reaction 3a should increase with increasing temperature. This is indicated by experimental data presented above [8,9,12]. The decomposition of the hydroxyalkoxyl radical. R (-H) (ОH)СН 2 О • (reaction 3a) is likely endothermic. The endothermic nature of reaction 3a is indirectly indicated by the fact that the decomposition of simple C 2 −C 4 alkoxyl radicals RО • in the gas phase is accompanied by heat absorption: [2−4]). Reaction 3b, subsequent to reaction 3a, is exothermic, and its heat for C 2 −C 3 alcohols in the gas phase is 298 Н ∆ = −40 to −60 kJ mol -1 [2][3][4]. As follows from the above scheme of the process, reactions 3a and 3b, in which the formation and consumption of the highly reactive free radical hydroxymethyl take place (at equal rates under steady-state conditions), can be represented as a single bimolecular reaction 3a,b occurring in a "cage" of solvent molecules.
The free formyl radical resulting from reaction 4, which is in competition with reactions 3 and 3a, is comparatively low-reactive because its spin density can be partially delocalized from the carbon atom via the double bond toward the oxygen atom, which possesses a higher electron affinity [1]. For example, in contrast to the methyl and alkoxyl π-radicals, the formyl σ-radical can be stabilized in glassy alcohols at 77 K [37]. In the gas phase, the dissociation energy of the C-H bond in formyl radicals is half that for acetyl radicals and is about 5 times lower than the dissociation energy of the С α -Н bond in saturated C 1 -C 3 alcohols [1].
As distinct from reactions 3 and 3a,b, reaction 4 leads to an inefficient consumption of hydroxyalkoxyl adduct radicals, without regenerating the initial 1-hydroxyalkyl addend radicals. Reaction 4 together with reaction 6 (mutual annihilation of free formyl and chain-carrier 1-hydroxyalkyl radicals) causes the inhibition of the nonbranched-chain process. For the disproportionation of the free radicals, the heats of reactions 5−7 for C 1 −C 3 alcohols in the gas phase vary in the range of The rates of the chain formation of 1,2-alkanediols in reaction 3 (and their nonchain formation in reaction 4), carbonyl compounds in reaction 3a, and methanol in reaction 3b are given by the following equations: where V 1 is the initiation rate, l is the molar concentration of the saturated alcohol at a given total concentration of formaldehyde 2 dissolved in it, x is the molar concentration of free formaldehyde (l >> x), k 2 is the rate constant of reaction 2 (addition of 1-hydroxyalkyl free radical to free formaldehyde), and α = k 3 /k 4 and β = k 3а /k 4 (mol dm -3 ) are the ratios of the rate constants of the competing (parallel) reactions. Estimates of 2k 5 were reported by Silaev et al. [39,40]. From the extremum condition for the reaction 3a rate function, derived the following analytical expression: The overall process rate is a complicated function of the formation and disappearance rates of the • R (-H) OH and • СНО free radicals: The ratios of the rates of the competing reactions are V 3 /V 4 = αl/x and V 3a /V 4 = β/x, and the chain length is ν = (V 3 + V 3a )/V 1 . The ratio of the rates of formation of 1,2-alkanediol and the carbonyl compound is a simple linear function of x: The equations for the rates of chain-termination reactions 5-7 are identical to Eqs. (12)- (14) (see below, Section 4.1).
Neutral formaldehyde solutions in alcohols at room temperature primarily consist of a mixture of formaldehyde polymer solvates reversibly bound to alcohols; these polymer solvates differ in molecular mass and have the general formula RO(CH 2 O) n H, where n = 1-4 [27]. The concentration of formaldehyde that occurs in solution as a free, unsolvated active species chemically unbound to the solvent (this species is capable of scavenging free radicals) at room temperature is lower than a percent of the total formaldehyde concentration [27]. The concentration x of the free formaldehyde species in solutions was determined by high-temperature UV spectrophotometry in the range 335-438 K at the total formaldehyde concentration c 0 (free and bound species including the concentration of polymer solvates) of 1.0-8.4 mol dm -3 in water, ethanediol, methanol, ethanol, 1-propanol, 2-propanol, and 2-methyl-2-propanol [28] (see Table 1 of the Appendix). This concentration increases with temperature according to an exponential law, and it can be as high as a few percent of the total concentration in solution under the test conditions, up to 19.3% in the case of 2-methyl-2-propanol at a total concentration of 1.0 mol dm -3 and a temperature of 398 K. The following empirical equation relating the concentration x (mol dm -3 ) of free formaldehyde to temperature T (K) and the total concentration c 0 in the solution (measured at room temperature), was developed by the treatment of 101 data points [28,39]: 2 The alcohol concentration in alcohol-formaldehyde solutions at any temperature can be estimated by the method suggested in [38,39]. The data necessary for estimating the concentration of free formaldehyde using the total formaldehyde concentration in the solution are reported by Silaev et al. [28,39].
where the coefficients a and b were calculated as the parameters of a straight-line equation by the least-squares technique from the dependence of lg x on 1/T at c 0 = 1.0 mol dm -3 for various solvents, and the coefficient h was obtained as the average value of the slopes of lg x as linear functions of lg c 0 at various series of fixed temperatures. The Table 2 summarizes these coefficients for each solvent. As regards the experimental data, the error in the calculations of the concentration x of free formaldehyde made by Eq. (7) in the specified temperature range was no higher than 25%. On the assumption that the dependence of the density of a given solution on the concentration of formaldehyde is similar to the analogous linear dependence found for aqueous formaldehyde solutions (0-14 mol dm -3 ; 291 K) [27], the concentrations l T (mol dm -3 ) of alcohols in alcoholformaldehyde solutions at a certain temperature can be estimated by the equation where c 0 is the total formaldehyde concentration (mol dm -3 ); M is the molecular mass (g mol -1 ) of the solvent; d and d T are the solvent densities (g cm -3 ) at room and given temperatures, respectively; the coefficients 8.4 × 10 -3 and 21.6 have the units of 10 3 g mol -1 and g mol -1 , respectively [38]. Earlier [28], it was found that the concentration x of the free formaldehyde species decreased with the solvent permittivity D 298 at a constant temperature. Water is an exception. Although water is more polar than alcohols, the concentration x of free formaldehyde in an aqueous solution is anomalously high and reaches the level of its concentration in 2-propanol, all other factors being the same (see Figure 2) [28,39]. This can be due to the specific instability of hydrated formaldehyde species and the ease of their conversion into free formaldehyde with increasing temperature.   4 and V3а on the concentration x of free formaldehyde (model optimization with respect to the parameters α, β and k2) from empirical data (symbols) for the 1-propanol-formaldehyde system at 413 K [8,9,41]: (1, ) Figure 3 illustrates the use of Eqs. (5) and (6) for describing the experimental dependences of the formation rates of 1,2-butanediol (curve 1) in reactions 3 and 4 and propanal (curve 2) in reaction 3a on the concentration of free formaldehyde in the 1-propanol-formaldehyde reacting system at a total formaldehyde concentration of 2.0 to 9.5 mol dm -3 and temperature of 413 K [8,9,41]. The concentration dependence of the propanal formation rate was described using the estimates of kinetic parameters obtained for the same dependence of the 1,2-butanediol formation rate. We considered these data more reliable for the reason that the carbonyl compounds forming in the alcohol-formaldehyde systems can react with the alcohol and this reaction depends considerably on the temperature and acidity of the medium [27]. The mathematical modeling of the process was carried out using a 137 Cs γ-radiation dose rate of P = 0.8 Gy s -1 [32,41], a total initiation yield of G(CH 3 СН 2 ĊНОН) = 9.0 particles per 100 eV [8,9] (V 1 = 4.07 × 10 -7 mol dm -3 s -1 ), and 2k 5 = 4.7 × 10 9 dm 3 mol -1 s -1 ). The following values of the parameters were obtained: α = 0.36 ± 0.07, β = 0.25 ± 0.05 mol dm -3 , and k 2 = (6.0 ± 1.4) × 10 3 dm 3 mol -1 s -1 .
Note that, as compared to the yields of 1,2-propanediol in the γ-radiolysis of the ethanol-formaldehyde system, the yields of 2,3-butanediol in the γ-radiolysis of the ethanolacetaldehyde system are one order of magnitude lower [41]. Using data from [8,9], it can be demonstrated that, at 433 K, the double bond of 2-propen-1-ol accepts the 1-hydroxyethyl radical 3.4 times more efficiently than the double bond of formaldehyde [42].

Addition of Hydroxymethyl Radicals
The addition of hydroxymethyl radicals to the carbon atom at the double bond of free formaldehyde molecules in methanol, initiated by the free-radical mechanism, results in the chain formation of ethanediol [34]. In this case, reaction 3a in Scheme 2 is the reverse of reaction 2, the 1-hydroxyalkyl radical • R (-H) OH is the hydroxymethyl radical • СН 2 ОН, so reaction 3b is eliminated (k 3b = 0), and reaction 5 yields an additional amount of ethanediol via the dimerization of chain-carrier hydroxymethyl radicals (their disproportionation can practically be ignored [43]). The scheme of these reactions is presented in [35].
The rate equation for ethanediol formation by the chain mechanism in reaction 3 and by the nonchain mechanism in reactions 4 and 5 in the methanol-formaldehyde system has a complicated form 3 as compared to Eq. (1) for the formation rate of the other 1,2-alkanediols [12]: where f = k 2 x 2 + (αl + β + x) 5 1 2k V .
If the rate of ethanediol formation by the dimerization mechanism in reaction 5 is ignored for the reason that it is small as compared to the total rate of ethanediol formation in reactions 3 and 4, Eq. (9) will be identical to Eq. (5). After the numerator and denominator on the right-hand side of Eq. (5) are divided by k -2 ≡ k 3a , one can replace k 2 in this equation with K 2 = k 2 /k -2 , which is the equilibrium constant for the reverse of reaction 2. Ignoring the reverse of reaction 2 (k 3a = 0, β = 0) makes Eq. (5) identical to Eq. (4) for Scheme 1 at k 3b = 0 (see the Section 2.1). In this case, the rate constant k 2 is effective.

Addition to Oxygen
The addition of a free radical or an atom to one of the two multiply bonded atoms of the oxygen molecule yields a peroxyl free radical and thus initiates 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 a secondary, tetraoxyalkyl, 1:2 adduct radical, which is the heaviest and the largest among the reactants. It is less reactive than the primary, 1:1 peroxyl adduct radical and, as a consequence, does not participate in further chain propagation. At moderate temperatures, the reaction proceeds via a nonbranched-chain mechanism.

Addition of Hydrocarbon Free Radicals
Usually, the convex curve of the hydrocarbon (RH) autooxidation rate as a function of the partial pressure of oxygen ascends up to some limit and then flattens out [6]. When this is the case, the oxidation kinetics is satisfactorily describable in terms of the conventional reaction scheme [2,5,6,16,44,45], which involves two types of free radicals. These are the hydrocarbon radical R • (addend radical) and the addition product 2 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 ( Figures. 4, 5) [46,47]. 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 4 RO • radical (1:2 adduct) inhibiting the chain process [13,14]. Scheme 3 Chain initiation 1. 2k RO 2 R + 2О 3 . The only difference between the kinetic model of oxidation represented by Scheme 3 and the kinetic model of the chain addition of 1-hydroxyalkyl radicals to the free (unsolvated) form of formaldehyde in nonmethanolic alcoholformaldehyde systems [8,9] (Scheme 2, Section 3.1) is that in the former does not include the formation of the molecular 1:1 adduct via reaction 4.
The decomposition of the initiator I in reaction 1 yields a reactive 0 R • radical, which turns into the ultimate product R 0 H 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  [6,44], 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 C 1 -C 4 radicals R, the pathway of reaction 3 leading to the alkyl hydroperoxide RO 2 H is endothermic (∆Н˚2 98 = 30-80 kJ mol -1 ) and the alternative pathway yielding the alcohol ROH is exothermic (∆Н˚2 98 = -120 to -190 kJ mol -1 ), while the parallel reaction 3a, which yields a carbonyl compound and the alkoxyl radical R′′О • or the hydroxyl radical • ОН, is exothermic in both cases (∆Н˚2 98 = -80 to -130 kJ mol -1 ), as also is reaction 3b (∆Н˚2 98 = -10 to -120 kJ mol -1 ), consecutive to reaction 3a, according to thermochemical data for the gas phase [2][3][4]. 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 4,5 4

RO
• , which has the largest weight and size. This radical is possibly stabilized by a weak intramolecular H···O hydrogen bond [54] shaping it into a six-membered cyclic structure 6 (seven-membered cyclic structure in the case of aromatic and certain branched acyclic hydrocarbons) [56,57]: Reaction 4 in the case of the methylperoxyl radical  [46] similar to the [ROO • ···(π-bond)RH] complex between the alkylperoxyl 1:1 adduct radical and an unsaturated hydrocarbon suggested in this work. The electronic structure of the π-complexes is considered elsewhere [48]. 5 Thermochemical 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 Н···О taken into account is ∆Нf˚298( 121.3 ± 15.3 kJ mol -1 ) and polyoxides (∆Нf˚298(СН3О4Н) = -21.0 ± 9 kJ mol -1 ) [49]. 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 [50][51][52]. The IR spectra of dimethyl tetraoxide with isotopically labeled groups in Ar-O2 matrices were also reported [53]. 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 [53]. 6 The R(-H)H···O(R)O3 ring consisting of the same six atoms (C, H, and 4O), presumably with a hydrogen bond [6], also forms in the transition state of the dimerization of primary and secondary alkylperoxyl radicals RО2 • via the Russell mechanism [5,55]. 7 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.
latter process is likely accompanied by chemiluminescence typical of hydrocarbon oxidation [52]. These reactions regenerate oxygen as O 2 molecules (including singlet oxygen 8 [52,59]) and, partially, as O 3 molecules and yield the carbonyl compound R (-2H) HO (possibly in the triplet excited state [52] [6]. The equations describing the formation rates of molecular products at the chain propagation and termination stages of the above reaction scheme, set up using the quasi-steady-state treatment, appear as follows: where Note that the alkylperoxyl radicals RО2 • are effective quenchers of singlet oxygen О2(a 1 ∆ g ) [58]. function, f = k 2 x 2 + (αl + β + x) 1 The ratios of the rates of the competing reactions are V 3 /V 4 = αl/x and V 3a /V 4 = β/x, and the chain length is ν = (V 3 + V 3a )/V 1 .
Eq. (11) is identical to Eq. (6). Eqs (10a) and (10a) were obtained by replacing the rate constant k 2 in Eqs. (10) and (11) with its analytical expression (for reducing the number of unknown parameters to be determined directly).
For αl >> β (V 3 >> V 3a ), 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. (10) and (10a) can be neglected (β = 0) and these equations become identical to Eqs.
In the alternative kinetic model of oxidation, whose chain termination stage involves, in place of R • (Scheme 3), 2

RO
• 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 ( In  [5], using the quasi-steady-state treatment in the long kinetic chain approximation, when it can be assumed that V 2 = V 3 , without using the substitution k 6 = 7 5 2 2 k k [5,6,16] (as distinct from this work), found that V 2 = V 3 is an irrational function of x: where a 1 , b 1 , c 1 , and d 1 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 2 RO • in three types of quadratic-law chain termination reactions (reactions 5-7) and are variants of the conventional model [2,5,6,16,44,45], 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 9 is [O 2 ] top ≤ x m . 9 The oxygen concentration attained in the liquid may be below the thermodynamically equilibrium oxygen concentration because of diffusion limitations hampering the establishment of the gas-liquid saturated solution Unlike the conventional model, the above kinetic model of free-radical nonbranched-chain oxidation, which includes the pairs of competing reactions 3-4 and 3a-4 (Scheme 3), allows us to describe the nonmonotonic (peaking) dependence of the oxidation rate on the oxygen concentration ( Figure 4). In this oxidation model, as the oxygen concentration in the binary system is increased, oxygen begins to act as an oxidation autoinhibitor or an antioxidant via the further oxidation of the alkylperoxyl 1:1 adduct radical shortening of the kinetic chains). The optimum oxygen concentration x m , at which the oxidation rate is the highest, can be calculated using kinetic equations (10a) and (11a) and Eq. (3a) with β = 0 or the corresponding analytical expression for k 2 . In the familiar monograph Chain Reactions by Semenov [60], 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) [60]. 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 [47] 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 5) that indicates a chemical mechanism providing the appearance of a maximum (see reaction 4 of Scheme 4). Curve 1 in Figure 4 illustrates the fit between Eq. (3a) 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 (V 3 >> V 3a and αl >> β) [46]. The oxygen concentration limit in o-xylene is reached at an oxygen concentration of [O 2 ] top > x m , which corresponds to the third experimental point [46]. The oxygen concentration was calculated from the oxygen solubility in liquid xylene at 373 K [61]. The following quantities were used in this mathematical description: 60 Co γ-radiation dose rate of P = 2.18 Gy s -1 and total initiation yield of G(o-СН 3 С 6 Н 4 ĊН 2 ) = 2.6 particles per 100 eV of the energy absorbed by the solution [46]; V 1 = 4.73 × 10 -7 mol dm -3 s -1 , and 2k 5 = 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, k 2 = (3.2 ± 0.8) × 10 5 dm 3 mol -1 s -1 . From the data presented in [62], it was estimated that k 4 = k 3 /α = (5.2 ± 1.2) × 10 2 dm 3 mol -1 s -1 . equilibrium under given experimental conditions (for example, when the gas is bubbled through the liquid) or because the Henry law is violated for the given gasliquid system under real conditions.

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 liquid phase [63] (Figure 4, curve 2) and in the gas phase [47,64,65] ( Figure 5), considered in our earlier work [13,56,57,66], can also be explained in terms of the competition kinetics of free radical addition [14,67]. From Figure 5 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) [64].  10 5 Pa, room temperature [64]). (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 [47]). The symbols represent experimental data.

Scheme 4
Nonbranched-chain oxidation of hydrogen and changes in enthalpy (∆Н˚2 98 , kJ mol -1 ) for elementary reactions 10 Chain initiation HO radical with a helical structure were carried out using the G2(MP2) method [68]. 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 [69,70]. 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 [71]. The photochemical properties of O4 and the van der Waals nature of the О2-О2 bond were investigated [72,73]. The most stable geometry of the dimer is two O2 molecules parallel to one another. The O4 molecule was identified by NR mass spectrometry [74] [75][76][77][78] 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 (V 3 = V 3′ ) under quasi-steady-state conditions, according to the above scheme. Parallel reactions 3 (second, parenthesized variant) and 3′ regenerate hydrogen atoms. It is assumed [56,57] that the hydrotetraoxyl radical 4 HO • (first reported in [79,80]) resulting from endothermic reaction 4, which is responsible for the peak in the experimental rate curve (Figure 4, curve 2), is closed into a five-membered [ОО─Н···ОО] • cycle due to weak intramolecular hydrogen bonding [54,81]. This structure imparts additional stability to this radical and makes it least reactive.
The 4 HO • radical was discovered by Staehelin et al. [82] in a pulsed radiolysis study of ozone degradation in water; its UV spectrum with an absorption maximum at 260 nm • radical [68,83] estimated in the same way [66] for the same conditions [84], (9.1 ± 0.9) × 10 -6 s. MP2/6-311++G** calculations using the Gaussian-98 program confirmed that the cyclic structure of 4 HO • [85] is energetically more favorable than the helical structure [68] (the difference in energy is 4.8-7.3 kJ mol -1 , depending on the computational method and the basis set). 11  • can exist in both forms, but the cyclic structure is obviously dominant (87%, K eq = 6.5) [85]. Reaction 4 and, to a much lesser degree, reaction 6 inhibit the chain process, because they lead to inefficient consumption of its main participants -2 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 [87]. 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 [2].  [88]. 12 Ozone does not interact with molecular hydrogen. At moderate temperatures, it decomposes fairly slowly, particularly in the presence of О 2 ( − Σ g X 3 ) [70]. 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 11 There were calculations for the two conformers (cis and trans) of the at 0 K. 12 The planar, six-atom, cyclic, hydrogen-bonded dimer 2 2 ) (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.

(Scheme 4).
When there is no excess hydrogen in the hydrogen-oxygen system and the homomolecular dimer O 4 [71][72][73][74]89,90], which exists at low concentrations (depending on the pressure and temperature) in equilibrium with O 2 [70], can directly capture the Н • atom to yield the heteronuclear cluster • 4 HO , 13 which is more stable than O 4 [70] and cannot abstract a hydrogen atom from the hydrogen molecule, nonchain hydrogen oxidation will occur to give molecular oxidation products via the disproportionation of free radicals.
The low-reactive hydrotetraoxyl radical 4
Staehelin et al. [82] 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 [87], in terms of the simplified scheme of free-radical nonbranched-chain reactions (Scheme 4), which considers only quadratic-law chain termination and ignores the surface effects [47], 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 13 It is impossible to make a sharp distinction between the two-step bimolecular quasi-steady-state treatment, as in the previous studies on the kinetics of the branched-chain free-radical oxidation of hydrogen [76], 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: 14 (6) and (7) -quadratic-law chain termination -are identical to Eqs. (13) and (14) provided that β = 0. In these equations, l and x are the molar concentrations of hydrogen and oxygen (l >> x), l m and x m are the respective concentrations at the maximum point of the function, V 1 is the rate of initiation (reaction 1), α = k 3 /k 4 , the rate constant is derived from the condition ∂V 3 /∂x = 0, and 2k 5 is the rate constant of reaction 5 (hydrogen atom recombination), which is considered as bimolecular within the given approximation. 15 In the case of nonchain hydrogen oxidation via the above addition reaction (Н • + О 4 add k  → 4 HO • ), the formation rates of the molecular oxidation products in reactions 6 and 7 (Scheme 4, k 2 = k 3 = k 4 = 0) are defined by modified Eqs. (13) and (14) in which β = 0, (αl + x) is replaced with 1, and k 2 is replaced with k add K eq (k add K eq is the effective rate constant of Н • addition to the О 4 dimer, К eq = k/k′ is the equilibrium  [94]. These values that are fairly close to unity. 15 This rate constant 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 [65] (a similar value of this constant for the gas phase was reported in an earlier publication [95]). Pagsberg et al. [65] 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 НО2 • + NН3 → Н2О + • NНОН, which is similar to reaction 3 in Scheme 4, whereas the competing reaction 4 was not taken into account.
was obtained in the gas phase by neutralization reionization (NR) mass spectrometry [83] and has a lifetime of >10 -6 s at 298 K) and chain termination takes place via reactions 5-7 involving the НО • and 3 HO • , radicals instead of Н • and 4 HO • , 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: Curve 2 in Figure 4 describes, in terms of the overall equation for the rates of reactions 3 and 7 (which was derived from Eqs. 3a and 14, respectively, the latter in the form of [96] in which k 2 is replaced with its analytical expression derived from Eq. (10) with β = 0 everywhere), the dependence of the hydrogen peroxide formation rate (minus the rate  [94]; V 1 = 4.15 × 10 -8 mol dm -3 s -1 ; 2k 5 = 2.0 × 10 10 dm 3 mol -1 s -1 [94]. As can be seen from Figure 4, the best description of the data with an increase in the oxygen concentration in water is attained when the rate V 7 of the formation of hydrogen peroxide via the nonchain mechanism in the chain termination reaction 7 (curve 1, α = (8.5 ± 2) × 10 -2 ) is taken into account in addition to the rate V 3 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: k 2 = 1.34 × 10 7 (vs. 2.0 × 10 10 [94]) 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 [66,97,98] 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 [94].

General Scheme of the Addition of Free Radicals to Molecules of Alkenes, Formaldehyde, and Oxygen
The general scheme of the nonbranched-chain addition of a free radical from a saturated compound to an alkene (and its functionalized derivative), formaldehyde, or dioxygen (which can add an unsaturated radical as well) in liquid homogeneous binary systems of these components includes the following reactions [57,97,98].
(a) k 2 x 2 << (αl + x) 1 5 2 V k (ascending branch of a peaked curve) and (b) k 2 x 2 >> (αl + x) 1 5 2 V k (descending branch) is transformed into simple functions (direct and inverse proportionality, respectively) of the concentration x of the unsaturated compound. These functions allow tentative estimates of the parameters k 2 and α to be derived from the experimental product formation rate V provided that V 1 and 2k 5 are known: where ϕ = 1 under conditions (a) and (b) and ϕ = 2 at the point of maximum (where k 2 x 2 ≅ (α + x) 1 5 2 V k ). Equations (10) and (11) under the condition k 2 x 2 >> (αl + β + x) 1 5 2 V k (descending branch of a peaked curve) can be transformed into Eqs. (17) and (18), respectively, which express the simple, inversely proportional dependences of reaction rates on x and provide tentative estimates of α and β: where ϕ = 2 at the point of maximum (where k 2 x 2 ≅ (αl + β + x) 1 5 2 V k ) and ϕ = 1 for the descending branch of the curve.

Equation (3) for V 3, 4 under condition (b) transforms into Eq. (17).
For radiation-chemical processes, the rates V in the kinetic equations should be replaced with radiation-chemical yields G using the necessary unit conversion factors and the relationships V = GP and V 1 = ε 1 G( 1 R • )P, where P is the dose rate, ε 1 is the electron fraction of the saturated component R 1 A in the reaction system [100], and G( 1 R • ) is the initial yield of the chain-carrier free radicals (addends) -initiation yield [39,94].

Conclusions
In summary, the material on the kinetics of nonbranched-chain addition of free saturated radicals to multiple bonds of alkene (and its derivative), formaldehyde, or oxygen molecules makes it possible to describe, using rate equations (1)-(6), (9)-(11) obtained by quasi-steady-state treatment, experimental dependences with a maximum of the formation rates of molecular 1:1 adducts on the concentration of an unsaturated compound over the entire region of its change in binary reaction systems consisting of saturated and unsaturated components ( Figures. 1, 3, 4).
The proposed addition mechanism involves the reaction of a free 1:1 adduct radical with an unsaturated molecule yielding a low-reactive free radical (the reaction 4 competing with the chain propagation reactions in Schemes 1-5). In such reaction systems, the unsaturated compound is both a reactant and an autoinhibitor, specifically, a source of low-reactive free 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.
A similar description is applicable to the nonbranched-chain free-radical hydrogen oxidation in water at 296 K [63] (Figure 4, curve 2). Using the hydrogen oxidation mechanism considered here, it has been demonstrated that, in the Earth's upper atmosphere, the decomposition of O 3 in its reaction with the НО • radical can occur via the addition of the latter to the ozone molecule, yielding the 4 HO • radical, which is capable of efficiently absorbing UV radiation [82].
The optimum concentration x m of unsaturated component in the binary system at which the process rate is maximal can be derived with the help of obtained kinetic equations (3a), (4a), (10a), and (11a) or from the corresponding analytical expressions for k 2 if other parameters are known. This opens a way to intensification of some technological processes that are based on the addition of free radicals to the double bonds of unsaturated molecules and occur via a nonbranched-chain mechanism through the formation of 1:1 adducts.