Kinetic Equations of Free-Radical Nonbranched-Chain Processes of Addition to Alkenes, Formaldehyde and Oxygen

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 no monotonic (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 Nonbranchedchain 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 lowreactive radicals do not participate in further chain propagation and because they de-cay when colliding with each other or with chaincarrier 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 mo-lecular 1:1 adduct formation on the concentration of the un-saturated 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 non-branched chain processes upon exceeding this optimal con-centration 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 non-branched 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 (30-130kJ 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 de-scription of these nontelomerization chain processes, the re-action 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 CНОН, 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 )CН 2 or НC=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 qua-si-steadystate 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 2

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 90-130kJmol -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] Ib K ≠ is included in the initiation stage [10,11]. In the case of an overwhelming excess of the saturated component reaction (1b) is 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 • , whichis less stable than the initial one. The addition reaction 2 may be accompanied by the abstraction reaction 2a. V /V2 = k /k is independent of the concentration of the unsaturated component R 2 B in the system. The inhibition of the non branched-chain addition process is due to reaction 4, in which the adduct radical 3 R • is spent in an inefficient way, since this reaction, unlike reaction 3, does not The inhibiting effect is also due to the loss of chain carriers 1 R • through their collisions with low-reactive unsaturated radicals 2 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 non chain 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 1 a/k 1 b and α = k 3 /k 4 are the rate constant ratios for competing (parallel) reactions (α is the first chaintransfer constant for the free-radical telomerization process [5]).
The rate ratio for the competing reactions is V 3 /V 4 = al/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 О • , • СН 3 , 3 NO • , and with 2 4 H PO • methanol to the rate constants of the reactions of the same radicals with ethanol in aqueous solution at room temperature are 0.4-0.5 [25,26]. For the same purpose, the rate constant of reaction 2 in the kinetic equation can be replaced with its analytical expression (1 ) (1:1 ) 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 χm 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-methylpropene1 reaction system at 303K [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 100eV (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  temperatures [20]. In the phosphorus trichloride-propene system, the difference be-tween 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 5kJ mol -1 , while in the tetrachloro-methanemethylpropene (or cyclohexene) and bromoeth-ane-2-methyl-2-butene systems, this difference is 20.9 (37.7) and ~24 kJ mol -1 , respectively. γ γ + = and the overall rate equation for the formation of the 1:1 adducts R 3 A and R 3 B will appear as ( )

Excess of the Saturated Component
where the parameters are designated in the same way as in Eqs.

Addition to the C=O Bond of Formal-dehyde
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 formalde-hyde 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 addi-tion, when the C=O bond is converted into an ordinary bond, is 30 to 60Jmol -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 nearestneighbor molecules of the solvent or unsolvated formaldehyde and, due to its structure, decompose by a monomolecular mechanism including isom-erization [9,12].

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 448K 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 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 1mol dm -3 , the dependence of the 1,2-alkanediol yields as a function of temperature for 303-473K shows a maximum, whereas the yields of carbonyl compounds and methanol in-crease 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 1-and 2-propanol gives ethane-diol, 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: 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 molecule -free formaldehyde; • СН 2 ОН, the 1-hydroxymetyl fragment radical; • R (-H) OH, the reactive 1-hydroxyalkyl addend radical, beginning from 1-hydroxyethyl; and consecutive-parallel reactions 2 and 4.
Scheme 2 does not include the same types of radi-cal-molecule reactions as were considered in Section 2.1 for Scheme 1. In addition, it seems unlikely that free adduct rad-icals will add to formaldehyde at higher temperatures the re-action of adding is unlikely because this would result in an ether bond. The addition of hydroxymethyl radicals to for-maldehyde, which is in competition with reaction 3b, is not included as well, because there is no chain formation of ethanediol at 303-448K [33]. At the same time, smallamounts 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 repre-sented as follows [8,9,12]: (Picture 1)

Picture 1
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 ac-companiedby heat absorption: . Reaction 3b, subsequent to reaction 3a, is exothermic, and its heat for C 2 −C 3 alcohols in the gas phase is = 298 ∆  −40 to −60kJmol -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 delocal-ized 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 77K [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 Сα-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 radi-cals. Reaction 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 formal-dehyde 2 dissolved in it, x is the molar concentration of free formaldehyde (l >> x), k2 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 (moldm -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, 3 / 0 a V õ ∂ ∂ = 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 n H, where n = 1-4 [27]. The concentration of for-maldehyde 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 hightemperature UV spectro-photometry in the range 335-438 K at the total formaldehyde concentration c0 (free and bound species including the con-centration of polymer solvates) of 1.0-8.4moldm -3 in water, ethanediol, methanol, ethanol, 1-propanol, 2-propanol, and 2-methyl-2-propanol [28] (see Table of    where the coefficients a and b were calculated as the parame- Eq. (7) in the specified tem-perature 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 for-maldehyde solutions (0-14mol dm -3 ; 291 K) [27], the con-centrations lT (mol dm -3 ) of alcohols in alcohol-formaldehyde solutions at a certain temperature can be estimated by the equation where c 0 is the total formaldehyde concentration (moldm -3 ); M is the molecular mass (gmol -1 ) of the solvent; d and dT are the solvent densities (gcm -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. Alt-hough 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 spe-cies and the ease of their conversion into free formaldehyde with increasing temperature.  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].

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 form3 as compared to Eq. (1) for the formation rate of the other 1,2-alkanediols [12]: (4) for Scheme 1 at k 3b = 0 (see the Section 2.1). In this case, the rate constant k 2 is effec-tive.

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 per-oxyl 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 com-pound 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 aconsequence, does not participate in further chain propagation. At moderate temperatures, the reaction proceeds via a non-branched-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 hy-drocarbon radical R • (addend radical) and the addition product of free-radical chain ad-dition, whose reaction scheme involves not only the above two types of free radicals, but also the radical (1:2 adduct) inhibiting the chain process [13,14].   [8,9] (Scheme 2, Section 3.1) is that in the former does not include the formation of the mo-lecular 1:1 adduct via reaction 4.
The decomposition of the initiator I in reaction 1 yields a reactive radical 0 R • , 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 alkylperoxyl 1:1 adduct  [6,44], yielding the carbonyl com-pound 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 quasisteady-state treatment), in which the highly reactive fragment, oxyl radical R′′О • (or • ОН) forms and then disap-pears, 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. to -130kJ mol -1 ), as also is reaction 3b ( 298 Í • ∆ = -10 to -120kJmol -1 ), consecutive to reaction 3a, according to thermo chemical 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 sixmembered cyclic structure 6 (sev-en-membered cyclic structure in the case of aromatic and certain branched acyclic hydrocarbons) [56,57]: (Picture 2)

Picture 2
Reaction 4 in the case of the methylperoxyl radical 3 2 CH O • adding to the oxygen molecule to yield the methyltetraoxyl radical 3 4 CH O • takes place in the gas phase, with heat ab-sorption equal to 110.0±18.6kJ mol -1 [49] (without the energy of the possible formation of a hydrogen bond taken into account). The exothermic reactions 6 and 7, in which the radical R • or 4 RO • undergoes disproportionation, include the isomerization and decomposition of the radical 7 . The latter process is likely accompanied by chemiluminescence typical of hydrocarbon oxidation [52]. These reactions regen-erate 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]). Depending on the decomposition pathway, the other possible products are the alcohol ROH, the ether ROR, and the alkylperoxide RO 2 R. It is likely that the isomerization 4 RO • and decomposition of the 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 so that a C-H bond in the aldehyde molecule breaks to yield two free radicals breaking, leads to degenerate chain branching [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 V  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: in which reactions 3a,b and 4 appearing in the above scheme are missing (k 3a =k 4 =0), Walling [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 6 5 7 2 2 K K K = [5,6,16] (as distinct from this work), found that V 2 = V 3 is an irrational function of x: 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.  (1, 2) Quantum yields of (1, •) hydrogen peroxide and (2, ○) water resulting from the photochemical oxidation of hydrogen in the hydro-gen-oxygen system as a function of the oxygen concentration x (light wave-length of 171.9-172.5nm, total pressure of 105Pa, room temperature [64]). (3,4) Hydrogen peroxide formation rate V(Н 2 О 2 ) (dashed curves) as a function of the rate V(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 gasdischarge tube before the measurements (total pressure of 25-77Pa, temperature of 77K [47]). The symbols represent experimental data.
Unlike the conventional model, the above kinetic model of freeradical nonbranched-chain oxidation, which includes the pairs of competing reactions 3-4 and 3a-4 (Scheme 3), allows us to describe the non monotonic (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 auto inhibitor or an antioxidant via the further oxidation of the alkylperoxyl 1:1 adduct radical 2 RO • into the low-reactive 1:2 adduct radical 4 RO • (reactions 4 and 6 lead to inefficient consumption of the free radicals 2 RO • and R • and cause 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 al-ready sufficient usually slows down the oxidation process by shortening the chains.
The existence of the upper (second) ignition limit in oxidation is due to chain termination in the bulk through triple collisions between an active species of the chain reaction and two oxygen molecules (at sufficiently high oxygen partial pressures). In the gas phase at atmospheric pressure, the number of triple collisions is roughly estimated to be 103 times smaller than the number of binary collisions (and the probability of a reaction taking place depends on the specificity of the action of the third particle) [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 in-dicates a chemical mechanism providing the appearance of a maximum (see reaction 4 of Scheme 4).

Addition of the Hydrogen Atom
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].

Scheme 4
Nonbranched-chain oxidation of hydrogen and changes in enthalpy 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 (first reported in [79,80]) re-sulting 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 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 260nm 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 bimo-lecular deactivation of the molecule substantially exceeds the rate of its monomolecular decomposition, which is the reverse of reaction 5 [2].    [70] and cannot abstract a hydrogen atom from the hydrogen molecule, nonchain hydrogen oxida-tion will occur to give molecular oxidation products via the disproportionation of free radicals.
The low-reactive hydrotetraoxyl radical 4 HO • [82], which presumably has a high-energy density [71], may be an inter-   The kinetic description of the noncatalytic oxidation of hy-drogen, including in an inert medium [87], in terms of the simplified scheme of free-radical non branched-chain reactions (Scheme 4), which considers only quadratic-law chain termi-nation and ignores the surface effects [47], at moderate tem-peratures and pressures, in the absence of transitions to un-steady-state critical regimes, and at a substantial excess of the hydrogen concentration over the oxygen concentration was obtained by means of quasisteady-state treatment, as in the previous studies on the kinetics of the branched-chain free-radical oxidation of hydrogen [76], even though the ap-plicability 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 concentra-tions of hydrogen and oxygen (l >> x), l m and x m are the re-spective concentrations at the maximum point of the function, V1 is the rate of initiation (reaction 1), α = k 3 /k 4 , the rate con-stant is derived from the condition ∂V3/∂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 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 con-stant of 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 k2 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

Molecules of Alkenes, Formaldehyde, and Oxygen
The general scheme of the non branched-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].
Reaction 1b, which competes with reaction 1a, gives rise to a maximum in the dependence described by Eq. (2), whereas reaction 4 or 4a, competing with reactions 3 and 3a,b, is responsible for the maxima in the dependences defined by Eqs. (1), (3)- (6) or (10) and (11). The low-reactive radicals  ,where P is the dose rate, e1 is the electron fraction of the saturated component R 1 A in the reaction system [100], and 1 ( ) G 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 non-branchedchain 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 formationrates of molecular 1:1 adducts on the concentration of an un-saturated compound over the entire region of its change in binary reaction systems consisting of saturated and unsatu-rated components (Figures 1, 3, 4).
The proposed addition mechanism involves the reaction of a 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 radi-cals shortening kinetic chains. The progressive inhibition of the nonbranchedchain 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 296K [63] (Figure 4, curve 2). Using the hydrogen oxidation mechanism considered here, it has been demonstrated that, in the Earth's upper at-mosphere, 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 radical 4 HO • , which is capable of ef-ficiently 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 expres-sions 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 un-saturated molecules and occur via a nonbranched-chain mechanism through the formation of 1:1 adducts.