For over the decades, hydrogen-oxygen reaction, being important for practical implementation, has been in focus of attention as a model system that contains all key kinetic features of gas-phase ignition, combustion, and detonation. After works of N. N. Semenov and C. N. Hinshelwood, this reaction has been investigated for more than seventy years, and presently its mechanism is considered as the most studied one (in contrast with, for example, the mechanism of hydrocarbon oxidation). However, the problem of quantitative description of kinetics in this system remains as before unsolved both from standpoint of an agreement between results of different experiments [1] and from standpoint of an agreement between theoretical models and experiments (see [2] ).

¹The use of the formal kinetic description was and remains relevant in terms of its use in macro-kinetic applications, i.e., when it is enough to quantitatively correctly simulate the density and “heat release function” in the specific conditions of composition, temperature, and pressure.

²According to existing concepts (see, for example, [4] ), if molecules or radicals AB are 2-atomics (O₂, H₂, OH, ∙∙∙), such a reaction proceeds in two the following elementary steps: forming vibrationally excited AB(v) and its subsequent relaxation to equilibrium distribution on vibrational states (in this case, deactivation in collisions). If the AB are 3(or more)-atomics (HO₂, H₂O, H₂O₂, ∙∙∙) such a reaction proceeds in three following elementary steps: forming vibrationally excited AB(v), intramolecular energy redistribution, and relaxation. The vibrational relaxation times, t_vib, directly dependent on conditions; so the value of the rate constant of the vibrational-to-translational energy exchange process is proportional to pressure, p, and depends on the buffer gas type, M. If in the course of the reaction is maintained equilibrium distribution in vibrational states (“high pressure limit”), i.e., (t_ch is the chemical reaction characteristic time), the equilibrium approximation is correct, the reaction rate of recombination (dissociation) is determined by the rate of its first stage, not depends on p and M, the recombination rate constant is defined by the value k_¥ (the dissociation rate constant). In the opposite case, i.e., when, (“low pressure limit”) the reaction rate is determined by the rate of vibrational relaxation stage, the relationship k/k' = K(T) is invalid. Really in practice, an intermediate case is realized; the reaction rate constant and efficiencies of third-bodies, M, in the framework of the formal kinetics (on the basis of assumed mechanism), is searched empirically as a result of fitting to the experiment. Herewith, the relationship k/k' = K(T) is used that, at least, is inconsistent.

All kinetic calculations performed for the interpretation of experiments based on the assumption of availability of equilibrium in the internal and external degrees of freedom of molecules and radicals. Using this, assumption radically simplifies the task of describing the kinetic mechanism, because it eliminates the need to account for a dependence of the reaction cross sections on the energy of the internal degrees of freedom of molecules. In this case, the kinetic equations are equations only for the concentrations, reaction rate constants only depend on gas temperature, i.e., all the reactions are regarded as elementary (proceeding in one step) processes, the ratio between the rate constants of direct and reverse reactions, k/k¢ = K(T) (K is the equilibrium constant) is used. Here, following [3] , we will call such a kinetics description as the “formal kinetic description”¹. Meanwhile, the characteristic relaxation times of internal states, for example, the times of vibrational relaxation is comparable or even greater than the characteristic times of fast chemical reactions, such as, for example, practically barrierless bimolecular recombination reactions. In the terms of formal kinetics, a reaction of recombination (dissociation) is considered as termolecular (bimolecular) process with the participation of the third body M, i.e., A + B + M = AB + M reactions².

The inconsequence and formal character of this equilibrium kinetic conception, which has become traditional, is illustrated by Figure 1. It shows the k/k_max ratio, where k is the rate constant value for dissociation reactions of the most important species and k_max is their maximally possible value. Here, for the k values used the accepted and recommended (for an interpretation of experiments) dissociation rate constant values, or received, using the relationship k = k_rec∙K(T), from accepted and recommended recombination rate constant values, k_rec (which really can be considered equilibrium); see, for example, [1] [5] -[8] and references therein. It will be seen that the rate constant values for these reactions exceed the maximally possible ones (collision frequency multiplied by the Arrhenius factor) in several times (H₂, O₂) or orders of magnitude (OH, HO₂, H₂O, H₂O₂) that takes out theirs beyond bounds of a physical sense () converting into formal fitting parameters. It means that these rate constant values cannot be identified with the rate constants of elementary dissociation reactions because they are effective rate constants of complex overall (detailed mechanism of which is unknown) reactions. (The curve 7 corresponding to the “high pressure limit” for HO₂ + M ® H + O₂ + M reaction, as was to be expected, located in a region of values z £ 1).

As a result of accounting, in implicit form, vibrational relaxation time dependencies on conditions, one comes to different formal kinetic schemes (mechanisms), which along with the elementary (proceeding in one step) chemical reactions include formal overall reactions detailed mechanism of which is unknown, and their effective rate constants depend on conditions (composition, pressure, and other). (It is these circumstances that are the

main cause of origin of the terms “uncertainty” and “sensitivity” in relation to the reaction rate constants.) Meanwhile, “the problem of the reaction mechanism can be completely solved only on the basis of quantitative information about rate constants of elementary processes” [9] . Currently, the most authoritative source of information on this topic is the Potential Energy Surface (PES) ab initio analysis and the relevant dynamic calculations.

Comparative successes (from the standpoint of use in gas-dynamic and macro-kinetic applications) of formal kinetic schemes in terms of this equilibrium approach have been achieved owing to considerable variations of the rate coefficients for important processes while modeling different conditions (on temperature, pressure, and composition). So, the rate constant of the most important chain branching reaction H + O₂ « O + OH obtained by different authors for temperatures over 1000 K [1] considerably differs depending on experimental conditions; about half of obtained values exceed the theoretical estimate from above (see [2] ). To an even greater extent, this is applied to the inhibiting H + O₂ + M ® HO₂ + M reaction (see [1] [5] ).

Theoretical predictions and experimental data are often reconciled via violation of the relationship between the rate constants of the forward and backward reactions, by introduction of not existing reactions, and/or by use of implausible rate constants. For example, the use of H₂ + O₂ ® 2OH reaction with activation energy around or less than 24,200 K (underestimated more than by a factor 1.5; see, for example, [10] [11] ) converts this practically non-existing reaction (see [12] -[14] ) into the most important channel of initiation and subsequent formation of OH radical.

Inability to provide physically adequate (in the terms of elementary processes) explanation of the formation (already during the earliest steps of combustion process) of electron-excited radicals OH^* º OH(²S⁺) is another illustrative example of the inconsistency of this (traditional) equilibrium kinetic concept. At the same time, the emission OH^* ® OH + hν has long been used in research practice to determine the induction period. The formation of OH^* during the most early steps of combustion process, is supposed (see also [2] ) to occur in the following two stages: H + O₂ ® HO₂(v) and HO₂(v) + H₂ ® OH^* + H₂O, i.e., through formation of the vibrationally excited HO₂(v) radical. This is in accordance with the only quantitative acceptable overall reaction H + O₂ + H₂ ® H₂O + OH^* proposed back in the 60s [15] . (Note that during later stages of combustion, when the OH^* concentration attains its maximum, another reactions play an important role also [2] [16] .)

These and other facts³ indicate the need not only to make the kinetic scheme, more precise and detailed but to revise the existing kinetic concept which is based on the assumption of equilibrium in the internal (vibrational) degrees of freedom. In accordance with the conclusions of [2] , the above said refers primarily to the reactions involving the vibrationally excited HO₂(v) radical which is formed as a result of bimolecular recombination H + O₂ ® HO₂(v). These are the following main (in conventional chain branching mechanism of hydrogen oxidation) reactions: the above reaction H + O₂ ® O + OH and the inhibiting reaction H + O₂ + M ® HO₂ + M. (In terms of the existing kinetic concept, these and other overall reactions are interpreted as elementary ones depending only on gas temperature, T.) Meanwhile, in [2] , we suggested “the dependence of the apparent rate constant of the process H + O₂ ® O + OH on the concentration of the third body M = H₂”, and it was stated the need to consider this reaction as the “H + O₂ ® HO₂(v) ® O + OH with consideration for intramolecular energy redistribution in the HO₂(v) radical and its vibrational relaxation in collisions”. Such a consideration was performed in [16] , where it was shown that “the nonequilibrium character of the process with respect to vibrational degrees of freedom is responsible for the observed dependence of the overall reaction H + O₂ ® O + OH effective rate constant on experimental conditions”. In turn, the presence of vibrationally excited HO₂(v) radicals leads to the necessity of taking into account the vibrational nonequilibrium of other components in the reacting hydrogen- oxygen mixture.

The subject of this paper is a presentation and approbation of the theoretical model based on a systematic account of vibrational nonequilibrium of HO₂ radicals which are assumed to act as a key intermediate in the chain branching process and in the formation of electronically excited species. The paper in a certain sense is the continuation of our previous works [2] [16] . In terms of the proposed model, the chain branching H + O₂ ® O + OH and inhibiting H + O₂ + M ® HO₂ + M formal reactions are considered as the set of elementary processes of forming, intramolecular energy redistribution between modes [17] , relaxation, and monomolecular decay of the comparatively long-lived (see [18] -[21] )⁴ vibrationally excited HO₂(v) radical which moreover is capable to react and to exchange an energy with other components of the mixture. This significantly changes the physical understanding of the chemical mechanism in the reacting hydrogen-oxygen system.

This paper is organized as follows. Section 2 gives a general formulation of the kinetic equations; in the Appendix A, there are the forms of these equations for the particular cases implemented in practice. The kinetic scheme are adduced in the Section 3; in the Appendix B, there is a brief outline of the method of theoretical estimation for unknown characteristic times and/or rate constants of vibrational relaxation channels. Then Section 4 follows where the calculation results are presented and discussed. The final section (Section 5) concludes.

Equations of chemical and vibrational kinetics for general case of reacting multi-component gas mixture in frames of macroscopic (or hydrodynamic) description, i.e., in the form of equations for the concentrations of mixture components, n_i, and average energies of vibrational degrees of freedom (modes), e_k, were first published in [22] (see also [23] ). These equations were used for calculations of chemical and vibrational nonequilibrium multi-component gas flow through a nozzle while studying the combustion-driven CO₂-gas-dynamic laser work- ing media. In deriving these equations from the equations of balance of the vibrational level populations (or Master equations) [24] , the following simplifying assumptions were made: 1) chemical reactions do not disturb the Maxwell distribution; 2) rotational degrees of freedom are in equilibrium with the translational ones; 3) each type of molecular vibrations (mode) is modeled by a harmonic oscillator with vibrational temperature T_k as a measure of the average energy of that or another mode (such as the kth). Following [22] [23] in general (see also [16] [24] ), we have the equations for the molar component concentrations per unit mass, n_i, and the average numbers of vibrational quanta of kth mode per one molecule, e_k, as the time functions at given gas temperature, T, and pressure, p. (In terms of nonequilibrium statistical mechanics, these equations are the corresponding moments of the Master equations.)

For the molar concentrations of the components per unit mass of the gas mixture (n_i),

Stoichiometric coefficients, n_jr and, of the jth chemical component, Y_j, in the rth reaction correspond to the following record of the chemical reaction:

An influence of the vibrational relaxation processes on kinetics of chemical transformations is expressed via the dependence of reaction rate coefficients on the average energies of reagent vibrational modes, e_k, in accordance with the following relationship:

Here, k_r is the rate constant of the rth reaction in the forward (backward) direction. (or equilibrium value) is the rate constant of the rth reaction when all degrees of freedom are at thermodynamic equilibrium. Theoretically, is the value that one obtains from dynamical calculations after averaging corresponding cross sections over equilibrium distributions of reagents and/or as a result of the use of approximate analytical methods as for example different variants of the transition state theory (TST)⁵. Experimentally, in application to the dissociation-recombination reactions, the rate constant values determined in the high pressure limit (see footnote 2) must coincide with. The is the nonequilibrium factor, is the set of vibrational temperatures for molecules participating in the rth reaction as reactants; T_k is the effective vibrational temperature as the measure of average energy of kth mode;, q_k is the characteristic temperature of the kth mode (the size of quantum for vibrational transition 1 ® 0 in K).

where is the portion of activation energy, (the activation barrier height), of the rth reaction belonging to the vibrational degrees of freedom of the molecules participating in the rth reaction as reactants, d_ri is a com-

parative efficiency of ith vibrational mode (with vibrational temperature T_i) of the reactants. We

assume here.

This rate constant dependency on vibrational temperatures of appropriate modes in the form of Equations (3),

(4) with (b_ri are the expansion coefficients of the reaction coordinate in normal vibrations) was

deduced [25] [26] for a specific case of the polyatomic molecule dissociation reaction. However, the mentioned specificity is not a matter of principle for its deducing (because of the harmonic oscillator model use) and this dependency has significantly more general character. In the simplest case of the two-atomic molecule dissociation reaction (one mode), Equation (4) gives the known results [27] , obtained for T_k < T, realized in the flow behind the bow shock of reentering hypersonic vehicles at high altitudes.

As an estimate for the portion of activation energy related to the reactant molecular vibrations, we have:

i.e., the activation barrier height, , minus an equilibrium energy of the relative translational motion of the pair (according to the assumption 1) we assume here E_trans = 2T [28] ) and rotational energy of the molecules participating in the rth reaction as reactants (according to the assumption 2) we assume here E_rot = x_r∙T/2). Here, x_r is the number of rotational degrees of freedom of reactants. The height of activation barrier, , of the reverse reaction r' defines the value of the vibrational energy of the rth reaction products (see below).

Simple formula (5) plays the important role in this model of the interaction between vibrational relaxation processes and chemical reactions. According to (5), the height of activation barrier, , is the main criterion for a degree of influence of the reagent vibrational excitation on the rth reaction rate constant. The value given by Equation (5) is the more accurate, the more the height of the activation barrier (i.e., when). This is just the case when the effect of vibrational nonequilibrium may be the greatest.

For the average numbers of the vibrational quanta of the modes (ε_k),

The first term in the right parts of the vibrational kinetics Equations (6) describes the change of the average energy (in q_k) of the kth vibrational mode (mode k belongs to the molecule of kind j) as a result of its interaction with the other vibrational modes of the gas mixture, (de_k/dt)_vibr in Appendix A. Here,

is the equilibrium value of e_k; is the mole fraction

of ith component; is the rate constant of the qth channel of vibrational relaxation in the interaction between j and i molecules,

Z_ji is the number of collisions of one jth type molecule with ith type molecules per time unit (the collision frequency), is the probability of corresponding qth vibrational energy exchange process of the type

in which a collision of molecules of the jth and ith kinds causes the decrease in vibrational quantum numbers of modes m, v_m, by l_mq and the increase in vibrational quantum numbers of modes n, v_n, by l_nq. In terms of the harmonic oscillator model, the probability of such a process is unambiguously characterized by the probability of

⁶In the simplest case of the single-quanta vibrational-to-translational (VT) transfer, one has the relation

.

the corresponding transition between the lowest states, ⁶. Kinetic values of this type characterize a

process of vibrational energy exchange between modes which are considered as a whole [24] ; a practical utilization of such values means the replacement of Equation (9) with the following one:

The value in equation for ε_k is the change in the amount of vibrational quanta of the kth mode as a result of qth (q' th) process.

The second term in the right side of Equation (6) describes the influence of chemical reactions on the process of vibrational relaxation (mode k belongs to jth type molecule formed in the sth reaction), (dε_k/dt)_chem in Appendix A. Here, c_sk is the quantity (in θ_k) of vibrational quanta obtained by kth mode in average at one act of formation of the j-type molecule in sth reaction. The total set of c_sk values characterizes the distribution of numbers of vibrational quanta between modes of the products of sth reaction.

where, defined by the formula (5), is the portion of activation energy belonging to the vibrational degrees of freedom for the reaction s', i.e., in the direction of annihilation the molecule containing the kth mode. In terms of the proposed model, it is natural to assume the uniform distribution of between product modes

.

The vibrational kinetic equations presented in this section are correct as far as harmonic approximation is correct, i.e., when. However, the application of such equations to diatomic gases and their mixtures (see [29] and references therein) showed that the influence of anharmonicity of molecular vibrations on the relaxation rate may be taken into account (approximately, in average) in the framework of this model by introducing constant correction factors to the corresponding kinetic parameters (see below Appendix B).

Appearing in the Equations (4), (5), (11) values, d_rk, and h_rk, as a rule, are unknown; in each case to determine them, the solution of the corresponding dynamic problem is required. For complex many-electron systems, this is a very complicated task (even for modern computing facilities), and the PESs used in such calculations (quasi-classical or quantum) are often rather arbitrary. However, as the experience of the calculation of thermal rate constants by solving a dynamic problem shows, there is an important feature of this approach, which consists in the fact that the results of the calculations of the averaged reaction parameters are much less sensitive to the shape of the PES, than many of details of the collision mechanism; the value of the activation barrier height, , is the most important dynamic parameter.

The system of Equations (1), (6) was solved numerically for n_i and ε_k (see [16] ). As the initial condition at t = 0, the n_i and ε_k values corresponding to the gas mixture state before the shock front (at T ≈ 300 K) were taken. The values t^* (the maximum OH^* concentration moment) and t₅₀ (the moment at which the OH concentration reaches its half-maximum) were determined from the calculated time dependences n_i(t) for OH and OH^* º OH(²S⁺) and compared with directly measured in experiments (see below Section 4.2).

Necessary information for solving kinetic Equations (1), (6) is presented in the form of the kinetic scheme for chemical reactions and channels of vibrational relaxation. For description of the initiation and chain reactions in the H₂ + O₂ + Ar system, we have used the kinetic scheme involving chemical reactions with participation H₂, O₂, H₂O, HO₂, H, O, OH, H₂O₂, O₃ in the ground electronic state as well as º O₂(¹D), O^* º O(¹D), OH^* º OH(²S⁺), and relaxation channels for the vibrational modes H₂ (k = 1), HO₂(ν₁) º HO₂(100) (k = 2), OH (k = 3), O₂ (k = 4), (k = 5), HO₂(ν₂) º HO₂(010) (k = 6), HO₂(ν₃) º HO₂(001) (k = 7), H₂O(ν₁) º H₂O(100) (k = 8), H₂O(ν₂) º H₂O(010) (k = 9), and H₂O(ν₃) º H₂O(001) (k = 10) with characteristic temperatures, θ_k: θ₁ = 5989, θ₂ = 5325, θ₃ = 5140, θ₄ = 2250, θ₅ = 2170, θ₆ = 2059, θ₇ = 1577, θ₈ = 5266, θ₉ = 2297, and θ₁₀ = 5409 K. We neglect an influence of vibrational nonequilibrium in OH^*, O₃, and H₂O₂ species because of their low concentrations and/or high rates of vibrational relaxation.

The behaviour of the most important intermediate HO₂ radicals is defined by the following elementary processes:

1) Formation as a result of the bimolecular recombination and the fast intramolecular redistribution of H-O₂ bond energy (or energy randomization), H + O₂ ® HO₂(v);

2) Dissociation along H-O₂ bond, HO₂ « H + O₂;

3) Dissociation along O-OH bond, HO₂ « O + OH;

4) Vibrational-to-vibrational (VV'-) energy transfer, HO₂(100) + X(0) « HO₂(000) + X(1), X = H₂, O₂, H₂O(ν₁), ∙∙∙;

5) VV'-energy transfer, HO₂(010) + X(0) « HO₂(000) + X(1), X = H₂, O₂, H₂O(ν₁), ∙∙∙;

6) Vibrational-to-translational (VT-) energy transfer, HO₂(001) + M « HO₂(000) + M, M = H₂, O₂, H₂O, Ar;

7) Chemical reactions, HO₂ + Y_i « Y_j + Y_k, i, j, k = 1, 2, ∙∙∙.

The process of the HO₂(v) radical formation has been presented by the fast bimolecular recombination reaction 1). The fast energy randomization (or, intramolecular energy redistribution) of HO₂ bond energy at the modes 2, HO₂(ν₁), 6, HO₂(ν₂), and 7, HO₂(ν₃), in the reaction 1) is simulated by setting up h_I,2 = h_I,6 = h_I,7; see Equation (11). The reactivity of HO₂(v) and its subsequent evolution are determined by the processes of decay 2), 3), relaxation 4)-6), and the reactions with its participation 7).

The recent results on dynamics in the HO₂ system (see [17] and references therein) showed the existence of significant regularity embedded in the chaotic phase space. I.e., the energy redistribution between HO₂ vibrational modes is only partial because of an availability of “a bottleneck in the phase space which inhibits the intramolecular transfer of energy between the H-O and O-O bonds” [17] . In another words, the HO₂ system was found to be (in dynamic sense) an intermediate case between regular and chaotic behavior (“the RRKM assumption or the strong-coupling approximation” [30] ). Here, that latter is simulated (on the kinetic level) by the processes 1)-3); see below reactions (c1)-(c3) in the Table 1. To take into account the regular dynamic behavior on the kinetic level, we introduce in the kinetic scheme, in addition to the processes 1)-3), the one-stage process H + O₂ ® O + OH (reaction (c4) in the Table 1).

The calculated data presented in this section illustrate the efficiency of the above approach and its potential for elucidating the physical essence of high-temperature hydrogen oxidation (as a process essentially nonequilibrium with respect to vibrational degrees of freedom) and for quantitative interpretation of experimental data.

In calculations, the coordinate system associated with the gas flow behind the wave front (gas mixture is at rest, T = const, p = const, and t = 0 is the instant when the shock front passes) was used.

The theoretical model of chemical and vibrational kinetics of hydrogen oxidation based on consistent account for

T = 1178, K T = 1809, K

the vibrational nonequilibrium of HO₂ radical formed in bimolecular recombination H + O₂ ® HO₂ with the super-equilibrium store of vibrational energy has been elaborated and experienced. The chain branching H + O₂ = O + OH and inhibiting H + O₂ + M = HO₂ + M formal reactions are considered (in the terms of elementary processes) as a general multi-channel process of formation and intramolecular energy redistribution between modes, relaxation, and decay of the comparatively long-lived vibrationally excited HO₂ radical which is capable to react and exchange of energy with another components of the mixture. For description of reactions in the H₂ + O₂ + Ar system, the kinetic scheme, which takes into account chemical reactions with participation H₂, O₂, H₂O, HO₂, H, O, OH, H₂O₂, O₃ in the ground electronic state as well as O₂(¹D), O(¹D), OH(²S⁺), and relaxation channels of the vibrational modes H₂, HO₂(ν₁), OH, O₂, O₂(¹D), HO₂(ν₂), HO₂(ν₃), H₂O(ν₁), H₂O(ν₂), and H₂O(ν₃), have been used.

The calculated results were compared with the shock tube experimental data for strongly diluted H₂-O₂ mixtures at 1000 < T < 2500 K, 0.5 < p < 4 atm and two equivalence ratio Φ = 1, 2. As a general result of this comparison, one can state qualitative and quantitative agreement between calculated and experimental data. This approach is promising from the standpoint of reconciling the predictions of the theoretical model with experimental data obtained by different authors for various compositions and conditions using different methods.

According to the calculations, the H₂ + O₂ reaction is essentially nonequilibrium and the vibrational nonequilibrium of the HO₂ radical is an intrinsic feature of hydrogen oxidation. Taking into account the vibrational nonequilibrium reveals the very important effects running throughout all the process. Reaction rate constants depend on the reactant vibrational temperatures, and this dependence is characterized by a factor as large as several orders of magnitude. It is demonstrated that the vibrational nonequilibrium (and consequently its influence on the rate constants and the process on the whole) decreases with increasing the gas temperature T and pressure p remaining determinant up to temperatures T = 1500 K. At T < 1500 K, the nature of hydrogen-oxygen reaction is especially nonequilibrium, and the vibrational nonequilibrium of HO₂ radical is the essence of this process.

The suggested model of vibrationally nonequilibrium kinetics is able to provide physically adequate (in the terms of elementary processes) solution of the problem of the H₂ + O₂ reaction mechanism without using formal overall reactions with their rate constants depending on experimental conditions. As a final result of numerous calculations, we have one (fixed) chemical kinetic scheme (without any “uncertainty” and “sensitivity” of the equilibrium rate constants of elementary reactions).

The characteristic time of the V-RT energy transfer united for all three modes of the HO₂ radical in its collisions with H₂ molecules, , was the only adjustable parameter in performing the calculations. The limits of variation are characterized by the following relation:

.

This relation should be considered as the quantitative estimation for presently unknown value of at 2500 > T > 1000 K with the maximum uncertainty equal at T about 1000 K. This uncertainty characterizes the extent of our ignorance of reaction dynamic details and effects of translational and rotational nonequilibrium.

In the future, as the range of compositions and conditions on temperature and pressure will be extended, the kinetic constants listed in Tables 1-3 can be corrected. For optimization of the kinetic scheme, the results of ab initio analysis of PESs for H_mO_n molecular systems as well as the results of corresponding dynamic calculations must become of crucial importance as the independent non-empirical source of kinetic information.

With regard to the currently existing formal kinetic schemes (which along with the elementary chemical reactions include formal overall reactions), the following should be noted. The use of these schemes was and remains to be actual from the standpoint of their use in macro-kinetic applications such as combustion and detonation. Many of the existing kinetic schemes are quite acceptable for these purposes, although in need of substantial reductions and some adjustments in terms of accounting, in implicit form, for the important effect of vibrational nonequilibrium.

This work was supported by Russian Foundation for Basic Research (Grant No. 12-03-0052).

The change of the average energy, (de_k/dt)_vibr, of the kth vibrational mode as a result of qth relaxation channel has the following forms for different implemented types of energy exchange; see Equations (6), (7).

One-quantum VT energy transfer (Table 2, q = 1 - 10):

,

mode k belongs to a molecule of kind j, the molecule of kind i does not change its vibrational state, i.e., , ,.

One-quantum VV' energy exchange (Table 3, q = 11 - 32, 34 - 38, 40 - 47):

,

mode k belongs to a molecule of kind j, mode l belongs to a molecule of kind i, i.e., , , ,.

1 ® 2 quantum VV' energy exchange (Table 3, q = 33, 39):

,

mode k belongs to a molecule of kind j, modes l and p belong to a molecule of kind i, i.e., , , , 1, , ,.

In accordance with the SSH theory, the probability of qth complex vibrational transition from modes m to modes n during one collision of jth and ith molecules, see Equation (10), has the form:

Here, is the orientation factor (it was varied from 0.1 to 0.02 on the base of the simplest intuitive collision mechanism considerations); is the matrix element of 0 ® l_mq transition for harmonic oscillator (mode) of the mth type participating in the qth process; w_m is the frequency of the mth oscillator; is the collision adiabaticity parameter; is the characteristic time of interaction; F(x) is the adiabaticity factor taking into account the translational energy change during an exchange with vibrations,

Although the dependence (B.1), (B.2) has been obtained as a result of the linear dynamic problem solution (the semi-classical calculation in the first order approximation of perturbation theory for the model of an oscillator with an induced force) with using the exponential repulsive potential it can be approximately considered as universal one for a variety of colliding pairs since it is possible to take account of rotation and long-range attraction effects by means of adapting the parameters and.

The adiabaticity parameter,

and matrix elements,

are defined by the relationships well-known in the theory of vibrational relaxation (see, for example, [29] and references herein). Here, M_ji is the reduced mass of the j-i collision, m_m is the reduced mass of the mth oscillator, a is the parameter of the intermolecular interaction potential (take here 1/α = 0.2 × 10^{- 8 cm ), D}_m and b_m are the parameters of respective intramolecular potential (corresponding Morse potential).

When applying formulae (B.1)-(B.4) to HO₂ molecule, the latter was considered as the set of the following oscillators: the stretching along H×××O bond with the reduced mass 1.0693 and the frequency 3698 cm⁻¹; the angle vibration near equilibrium angle value ÐHOO = 104.3˚ with the reduced mass 1.1311 and the frequency 1430 cm⁻¹; the stretching along O∙∙∙O bond with the reduced mass 13.0737 and the frequency 1120 cm⁻¹

For the processes of VT- and non-resonance VV'-energy transfers with participation of oscillators involving light H atom, the relative translational motion is slow as compared to rotation. In the simplest intuitive model of VR-energy transfer [38] , the dynamic parameters of the theory of VT- and VV'-energy transfer (velocity and reduced mass) which characterize the relative translational motion are replaced by the corresponding parameters which characterize the rotational motion; within this approach, the role of translational motion is merely in drawing the molecules together. Consequently, while calculating the rate constants of these processes using relationships (B.1)-(B.4), we have, as SSHM estimation, instead of (B.3), the following relation (see also [39] ):

The main (Table 2 and Table 3) VT- and VV'-relaxation channels in the reacting mixture H₂ + O₂ + Ar at T ³ 1000 K, p ~ 1 atm were chosen as a result of comparative analysis of the rate constant values calculated for these conditions. This selection was based on the following estimations:

i.e., in the Table 2 and Table 3, it was taken into consideration the processes for which

Of non-single-quanta transitions, only the VV' transfer processes (v33), H₂O(001) + HO₂(000) « H₂O(000) + HO₂(011), and (v39), H₂O(100) + HO₂(000) « H₂O(000) + HO₂(011), (Table 3) answer the condition (B.7).