Competition Kinetics of the Non-branched-Chain Processes of Free Radical Addition to the C=C, C=O, and O=O Bonds of Molecules

The aim of our study was the conclusion of the simple kinetic equations to describe ab initio the initiated non-branched-chain processes of saturated free radical addition to C=C, C=O, and O=O bonds of molecules in the binary 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 addition processes of free radicals to the multiple bonds of molecules. The schemes include reactions competing with chain propagation through a reactive free radical. The chain evolution stage in these schemes involves three or four types of free radicals, of which one 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 set up using quasi-steady-state treatment. These equations provide good fits for the non-monotonic (peaking) dependences of the formation rates of the molecular 1:1 adducts on the concentration of the unsaturated component in liquid homogeneous binary systems consisting of a saturated component (hydrocarbon, alcohol, etc.) and an unsaturated component (olefin, allyl alcohol, formaldehyde, or dioxygen). The unsaturated compound in these systems is both a reactant and an autoinhibitor generating low-reactive free radicals such as СН2=С(СН3)ĊН2, СН2=СНĊНОН, НĊ=O, o-СН3С6Н4СН2 4 O  , or 4 HO  . A similar kinetic description is applicable to non-branched-chain free-radical hydrogen oxidation. The energetics of the key radical-molecule reactions is considered.


Introduction
In a binary system consisting of a saturated component and an unsaturated one, the abstraction of the most labile atom from a saturated molecule by some initiator converts this molecule into a saturated free radical (addend) capable of adding to the double bond of an unsaturated molecule to yield a saturated 1:1 adduct radical. At a sufficiently high concentration of the unsaturated component in the system, this primary adduct radical can add to another unsaturated molecule under certain conditions to yield a secondary,1:2 adduct radical, and so on, resulting in telomerization. Under other conditions and at other relative reactivities of the components, the concentration of the saturated component can exceed the concentration of the unsaturated component so greatly that the most likely reaction for the primary adduct the molecule) may have an increased energy owing to the energy liberated in the transformation of a C=O, C=C, or O=O 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. In the processes of this kind, in which an addend radical and a low-reactivity, inhibiting radical are involved in three types of quadratic-law chain termination reactions, the formation rate of the 1:1 adduct as a function of the concentration of the unsaturated component has a maximum (which usually occurs at a low concentration of this component).
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 non-branched-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 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 non-telomerization 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   ] 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 and 2k 7 are the rate constants of the loss of identical free radicals via the reactions 1 R  + 1 R  and 2 R  + 2 R  , respectively; k 6 is the rate constant of the loss of different free radicals via the 1 R  + 2 R  reaction (see . 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.

Addition to the С=С Bond of Alkenes and Their Derivatives
When reacting with alkenes not inclined to free-radical polymerization, the free radicals originating from inefficient saturated telogens, such as alcohols [17] and amines [18], usually add to the least substituted carbon atom at the double bond, primarily yielding a free 1:1 adduct radical. This radical 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 nonoverwhelming 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].

Scheme 1
Chain initiation 1.  [20], alkyl [2,5], 1-hydroxyalkyl [5,6,17,[22][23][24], or a similar functionalized reactive addend radical [5]; 2 R  is an alkenyl radical (allyl or higher) [2,5,[17][18][19][20][21][22], 1-hydroxyalkenyl [5,17,18,23,24], or a similar functionalized low-reactive (inhibitor) radical [5,18]; 3 R  is a saturated reactive 1:1 adduct radical; R 0 A, R 0 B, and R 1 A are saturated molecules; R 2 B is an unsaturated molecule (olefin or its derivative); R 3 A and R 3 B are 1:1 adduct molecules; Prod designates the molecular products resulting from the dimerization or disproportionation of free radicals. The chain evolution (propagation and inhibition) stage of Scheme 1 include consecutive reactions 2 and 3, parallel (competing) reaction pairs 3 and 4, and consecutive-parallel reaction pair 2-4. 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. 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   12 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  = k 1a /k 1b and  = k 3 /k 4 are the rate constant ratios for competing (parallel) reactions ( 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 = l/x, and the chain length isv = 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) 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 1 R  and 2 R  : V(R 3 A, The application of the above rate equations to particular single nonbranched-chain additions is illustrated in Fig. 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, than 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)), l/(l + 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 12 3, 4 2 2 5 1 () where the parameters are designated in the same way as in 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 51 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 1-4). 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 alcohol-formaldehyde solutions, which depends both on the temperature and on the polarity of the solvent [28,33]. For the -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 -radiolysis of the solutions of formaldehyde in ethanol and 1-and 2-propanol gives ethanediol, carbon monoxide, and hydrogen in low radiation-chemical yields (which, however, exceed the yields of the same products in the -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 Chain propagation 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 radical (adduct radical), beginning from 1-hydroxyethyl; 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: ( ). Reaction 3b, subsequent to reaction 3a, is exothermic, and its heat for C 2 C 3 alcohols in the gas phase is [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 andβ = 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, 3a / 0 V х    , we 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  = (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. (9)-(11) (see below, Section 4.1). Figure 2 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 ).

Addition of the Hydroxymethyl Radical
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) 51 2kV .
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. (7) 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) in Scheme 1 (see the Section 2.2). 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 3 In an earlier publication [8], this equation does not take into account reaction 3a.
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 (Figs. 3, 4) [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 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 alcohol-formaldehyde 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 Addition to the C=C, C=O, and O=O Bonds of Molecules  [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 (ΔН˚2 98 = 30-80 kJ mol -1 ) and the alternative pathway yielding the alcohol ROH is exothermic (ΔН˚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 (ΔН˚2 98 = -80 to -130 kJ mol -1 ), as also is reaction 3b (ΔН˚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  , 4 It is hypothesized that raising the oxygen concentration in the o-xylene-oxygen system can lead to the formation of the RO • ···O 2  intermediate complex [46] 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 [48].
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]: . The latter process is likely accompanied by chemiluminescence typical of hydrocarbon oxidation [52]. These reactions regenerateoxygen as O 2 molecules (including singlet oxygen 7 [52,59]) and, partially, as O 3 molecules and yield the carbonyl compound R (-2H) HO (possibly in the triplet excited state [52]). Depending on the decomposition pathway, the other possible products are the alcohol ROH, the ether ROR, and the alkyl peroxide RO 2 R. 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 molecular oxygen 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  [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 50-52. The IR spectra of dimethyl tetraoxide with isotopically labeled groups in Ar-O 2 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)O 3 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 RО 2 • via the Russell mechanism [5,55]. 7 [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: V k x f where V 1 is the initiation rate, l =  9) is identical to Eq. (6). Eqs (8a) and (9a) were obtained by replacing the rate constant k 2 in Eqs. (8) and (9) 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. (8) and (8a) can be neglected (β = 0) and these equations become identical to Eqs. (3) and (3a) with the corresponding analytical expression for k 2 .
In the alternative kinetic model of oxidation, whose chain 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 8 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 (Fig. 3). 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  8 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 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 gas-liquid system under real conditions. chains). The optimum oxygen concentration x m , at which the oxidation rate is the highest, can be calculated using kinetic equations (8a) and (9a) 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). Reconstruction of the functional dependence of the total hydrogen peroxide formation rate V 3, 7 (Н 2 О 2 ) on the dissolved oxygen concentration x from empirical data (symbols) using Eqs. (3a) and (12) 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 [63] (S Y = 1.13  10 -8 ). The dashed curve described V 3 (H 2 O 2 ) as a function of the oxygen concentration x based on Eq. (3a) (model optimization with respect to α) and the experimental data of curve 2 (S Y = 1.73  10 -8 ) Curve 1 in Fig. 3 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 -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 .

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 [47,64,65] (Fig. 4) and in the liquid phase [63] (Fig. 3, curve 2), considered in our earlier work [66], can also be explained in terms of the competition kinetics of free radical addition [14,67].

Scheme 4
Nonbranched-chain oxidation of hydrogen and changes in enthalpy (ΔН˚2 98   The types of the O 4 molecular dimers, their IR spectra, and higher oxygen oligomers were reported [69,70]. The structure and IR spectrum of the hypothetical cyclotetraoxygen molecule O 4 , 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 O 4 and the van der Waals nature of the О 2 -О 2 bond were investigated [72,73]. The most stable geometry of the dimer is two O 2 molecules parallel to one another. The O 4 molecule was identified by NR mass spectrometry [74]. Chaintermination 5. 2Н • ( + М) 2  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 (Fig.  2, curve 3), 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 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]. 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 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  , 12 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. 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 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: 13 [94]. These values that are fairly close to unity. 14 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 НО 2 • + NН 3  Н 2 О + • NНОН, which is similar to reaction 3 inScheme 4, whereas the competing reaction 4 was not taken into account.
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 constant of the reversible reaction 2О 2 k k' The formation rates of the stable products of nonchain oxidation (k 3 = 0), provided that either reactions (2) and (4)   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 [64]). (3,4) Hydrogen peroxide formation rate V(Н 2 О 2 ) (dashed curves) as a function of the rate V(Н 2 О 2 ) 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 Note that, if in Scheme 4 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 [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: V 3 (Н 2 О) = 1 3' Curve 2 in Fig. 3 describes, in terms of the overall equation 32 3, 7 for the rates of reactions 3 and 7 (which was derived from Eqs. 3a and 12, respectively, the latter in the form of 42 71 m V V x f  (12a) [96] in which k 2 is replaced with its analytical expression derived from Eq. see below) [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 Fig. 3, 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:
For approximate estimation of the parameters of the kinetic equations (3), (4), (8), and (9), Eq. (4) under the conditions (a) k 2 x 2 << (αl + x) 5 1 2kV (ascending branch of a peaked curve) and (b) k 2 x 2 >> (αl + x) 5 1 2kV (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  = 1 under conditions (a) and (b) and  = 2 at the point of maximum (where k 2 x 2  (α + x) 5 1 2kV ). Equations (8) and (9) under the condition k 2 x 2 >> (αl + β + x) 5 1 2kV (descending branch of a peaked curve) can be transformed into Eqs. (15) and (16), respectively, which express the simple, inversely proportional dependences of reaction rates on x and provide tentative estimates of α and β: where  = 2 at the point of maximum (where k 2 x 2  (αl + β + x) 5 1 2kV ) and  = 1 for the descending branch of the curve.
Equation (3) for V 3, 4 under condition (b) transforms into Eq. (15). 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 =  1 G( 1 R  )P, where P is the dose rate,  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]. The optimum concentration of the unsaturated component in the system maximizing the process rate, x m , can be calculated using rate equations (3a), (4a), (8a), and (9a) or the corresponding analytical expressions for k 2 provided that 15 The stabilization energy of the low-reactive free radicals СН 2 =С(СН 3 )ĊН 2 , СН 2 =СНĊНОН, andНĊ=O in the standard state in the gas phase is -52.0, -42.1, and -24.3 kJ mol -1 , respectively [4,99]. the other parameters involved in these equations are known. This opens up the way to intensification of some technological processes that are based on the addition of free radicals to С=С, С=О, and О=О bonds and occur via a nonbranched-chain mechanism through the formation of a 1:1 adduct.

Conclusions
The above data concerning the competition kinetics of the nonbranched-chain addition of saturated free radicals to the multiple bonds of olefins (and its derivatives), formaldehyde, and oxygen molecules make it possible to describe, using rate equations (1)-(9a), obtained by quasi-steady-state treatment, the peaking experimental dependences of the formation rates of molecular 1:1 adducts on the concentration of the unsaturated compound over the entire range of its variation in binary systems consisting of saturated and unsaturated components (Figs. 1-3). 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 1-4). 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 nonbranchedchain free-radical hydrogen oxidation in water at 296 K [63] (Fig. 3, 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].