Hydrogen Oxidation: Kinetics of Free-Radical Nonbranched-Chain Addition of Hydrogen Atom to Oxygen Molecule

The kinetics of inhibition for nonbranched-chain processes of saturated free-radical addition to the C=C and C=O double bonds of alkene and formaldehyde molecules, respectively, by low-reactive free radicals that can experience delocalization of the unpaired p-electron was first considered in [1]. In these processes a low-reactive free radical is formed in the reaction competing with chain propagation reactions through a reactive free radical. In the present work the kinetics of inhibition by low-reactive hydrotetraoxyl free radical is considered for the nonbranched-chain process of the addition of a hydrogen atom to one of the two multiply bonded atoms of the oxygen molecule yielding a hydroperoxyl free radical. The hydroperoxyl free radical then abstracts the most labile atom from a molecule of the compound being oxidized or decomposes to turn into a molecule of an oxidation product. The only reaction that can compete with these two reactions at the chain evolution stage is the addition of the peroxyl radical to the oxygen molecule (provided that the oxygen concentration is sufficiently high). This reaction yields the secondary hydrotetraoxyl 1:2 adduct radical, which is the heaviest and the largest among the reactants. It is less reactive than the primary peroxyl 1:1 adduct radical and, as a consequence, does not participate in further chain propagation. At moderate temperatures, the reaction proceeds via a nonbranched-chain mechanism.

The aim of this study was the mathematical simulation of oxidation process autoinhibited by oxygen, when the dependence of the peroxide formation rate on the dissolved oxygen concentration has a maximum. The simulation was based on experimental data obtained for γ-radiation-induced addition reaction of hydrogen atom to the molecular oxygen for which the initiation rate V1 is known (taking into account that V = GP and V1 = G(H•)P, where P is the dose rate, and G(H•) is the initial yield of the chain-carrier hydrogen atom H• - initiation yield [2,3]).Based on the reaction scheme suggested for the kinetic description of the addition process to oxygen, the kinetic equations with one to three parameters to be determined directly were derived. Reducing the number of unknown parameters in a kinetic equation will allow one to decrease the narrowness of the correlation of these parameters and to avoid a sharp buildup of the statistical error in the nonlinear estimation of these parameters in the case of a limited number of experimental data points. The rate constant of the addition to oxygen, estimated as a kinetic parameter, can be compared to its reference value if the latter is known. This provides a clear criterion to validate the mathematical description against experimental data.

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 [4-6] (Figure 1) and in the liquid phase [7] (Figure 2, curve 2), considered in my earlier works [8-11], can also be explained in terms of the competition kinetics of free radical addition [12,13]. From figure 1 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) [4].In the familiar monograph “Chain Reactions” by Semenov [14], 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 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) [14]. Note that in the case of a gas-phase oxidation of hydrogen at low pressures of 25-77 P? and a temperature of 77 ? [5] 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 2) that indicates a chemical mechanism providing the appearance of a maximum (see reaction 4 of Scheme).

 

Figure 1: (1, 2) Quantum yields of (1, ?) hydrogen peroxide and (2, ?) water resulting from the photochemical oxidation of hydrogen in the hydrogen-oxygen system as a function of the oxygen concentration x (light wavelength of 171.9-172.5 nm, total pressure of 10⁵ Pa, room temperature [4]). (3, 4) Hydrogen peroxide formation rate V(?₂?₂) (dashed curves) as a function of the rate V(?₂) at which molecular oxygen is passed through a gas-discharge tube filled with (3, Δ) 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 [5]). The symbols represent experimental data.

Figure 2:(1, ?) Reconstruction of the functional dependence of the total hydrogen peroxide formation rate V_{3, 7} (?₂?₂) on the dissolved oxygen concentration x from empirical data (symbols) using equation (1a) and (4a) (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 [7] (S_Y = 1.13^' 10^-8). (2) The dashed curve described V₃(H₂O₂) as a function of the oxygen concentration x based on equation (1a) (model optimization with respect to α) and the experimental data of curve 2 (S_Y = 1.73^´ 10-8).

The chain evolution (propagation and inhibition) stage of the Scheme includes consecutive reaction pairs 2-3 and 3-3'; the parallel (competing) reaction pair 3-4; and consecutive-parallel reactions 2 and 4.

The hydroperoxyl free radical Ho^.₂ [23-25] resulting from reaction 2 possesses an increased energy due to the energy released the conversion of the ?=? multiple bond into the ??-?• 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 ??• that appears and disappears in consecutive parallel reactions 3 (first variant) and 3? possesses additional energy owing to the exothermicity of the first variant of reaction 3, whose heat is distributed between the two products. As a consequence, this radical has a sufficiently high reactivity not to accumulate in the system during these reactions, whose rates are equal (V3 = V3') under quasi-steady-state conditions, according to the above scheme. Parallel reactions 3 (second, parenthesized variant) and 3' regenerate hydrogen atoms. It is assumed [9,10] that the hydrotetraoxyl radical ??^.₄ (first reported in [26-28]) resulting from endothermic reaction 4, which is responsible for the peak in the experimental rate curve (Figure 1, curve 3), is closed into a five-membered [?????????]• cycle due to weak intramolecular hydrogen bonding [29,30]. This structure imparts additional stability to this radical and makes it least reactive.

The ??^.₄ radical was discovered by Staehelin et al., in a pulsed radiolysis study of ozone degradation in water; its UV spectrum with an absorption maximum at 260 nm (∈??^._4280nm = 320 ±15 m² mol^-1) was reported [31]. The spectrum of the radical is similar to that of ozone, but the molar absorption coefficient (∈??^._4λmax of the former is almost two times larger [31]. The assumption about the cyclic structure of the radical can stem from the fact that its mean lifetime in water at 294 K, which is (3.6 ± 0.4) × 10^-5 s (as estimated [11] from the value of 1/k for the monomolecular decay reaction ??^._4→Ho^.2+O₂ [31]), is 3.9 times longer than that of the linear radical [16,32] estimated in the same way [11] for the same conditions [33], (9.1 ± 0.9) × 10^-6 s. 

MP2/6-311++G** calculations using the Gaussian-98 program confirmed that the cyclic structure of ??^.₄[34] is energetically more favorable than the helical structure [16] (the difference in energy is 4.8-7.3 kJ mol^-1, depending on the computational method and the basis set).² For example, with the MP₂(full)/6-31G(d) method, the difference between the full energies of the cyclic and acyclic ??^.₄ conformers with their Zero-point Energies (ZPE) values taken into account (which reduces the energy difference by 1.1 kJ mol^-1) is -5.1 kJ mol^-1 and the entropy of the acyclic-to-cyclic ??^.₄ transition is ΔS⁰₂₉₈ = -1.6 kJ mol^-1 K^-1. Therefore, under standard conditions, ??^.₄ can exist in both forms, but the cyclic structure is obviously dominant (87%, K_eq = 6.5) [34].

Reaction 4 and, to a much lesser degree, reaction 6 inhibit the chain process, because they lead to inefficient consumption of its main participants _-Ho^.2 and ?^•.

When there is no excess hydrogen in the hydrogen-oxygen system, the homomolecular dimer O₄ [19-22,40,41], which exists at low concentrations (depending on the pressure and temperature) in equilibrium with O₂ [18], can directly capture the ?^• atom to yield the heteronuclear cluster³ ??^.₄. This ??^.₄ cluster is more stable than O₄ [18] and cannot abstract a hydrogen atom from the hydrogen molecule. Therefore, in this case, nonchain hydrogen oxidation will occur to give molecular oxidation products via the disproportionation of free radicals. 

The low-reactive hydrotetraoxyl radical ??^.₄ [31], which presumably has a high energy density [19], may be an intermediate in the efficient absorption and conversion of biologically hazardous UV radiation energy the Earth upper atmosphere. The potential energy surface for the atmospheric reaction HO^• + ?₃, in which the adduct ??^.₄ (²?) was considered as an intermediate, was calculated by the DMBE method [42]. From this standpoint, the following reactions are possible in the upper troposphere, as well as in the lower and middle stratosphere, where most of the ozone layer is situated (altitude of 16-30 km, temperature of 217-227 K, pressure of 1.0 ×10⁴-1.2×10³ Pa [43]; the corresponding ΔH⁰₂₉₈ reaction values are given in kJ mol^-1 [15]):

The ??^.₄ radical can disappear via disproportionation with a molecule, free radical, or atom in addition to dissociation. Note that emission from ?2(a¹Δ_g) and ?2(b¹∑⁺_g) is observed at altitudes of 30-80 and 40-130 km, respectively [44].

Staehelin et al., [31] pointed out that, in natural systems in which the concentrations of intermediates are often very low, kinetic chains in chain reactions can be very long in the absence of scavengers since the rates of the chain termination reactions decrease with decreasing concentrations of the intermediates according to a quadratic law, whereas the rates of the chain propagation reactions decrease according to a linear law.

The kinetic description of the noncatalytic oxidation of hydrogen, including in an inert medium [35], in terms of the simplified scheme of free-radical nonbranched-chain reactions (Scheme), which considers only quadratic-law chain 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 [45], 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 ⁴ for the ?rst stages of the process: k₆=√2K₅2K₇ [46] and, hence, V₁ = V₅ + 2V₆ + V₇ = (√2K₅ [?•] +√2K₇ [??^.₄])² which allow the exponent of the 2k₅[?•]² term in the d[?•]/dt = 0 equation to be reduced to 1 [1,46]. The kinetic equations were derived for the rates (mol dm-3 s-1) of the elementary reactions for the formation of molecular products of hydrogen oxidation. 

The kinetic equations were derived for the rates (mol dm^-3 s^-1) of the elementary reactions for the formation of molecular products of hydrogen oxidation.

The rate constant 2k₅ in the case of the pulsed radiolysis of ammonia-oxygen (+ argon) gaseous mixtures at a total pressure of 10⁵ Pa and a temperature of 349 K was calculated to be 1.6 × 10⁸ dm³ mol^-1 s^-1 [6] (a similar value of this constant for the gas phase was reported in an earlier publication [47]). Pagsberg et al., [6] found that the dependence of the yield of the intermediate ??^• on the oxygen concentration has a maximum close to 5 × 10^-4 mol dm^-3. In the computer simulation of the process, they considered the strongly exothermic reaction _Ho^.2 + N?₃ → ?₂? + •N???, which is similar to reaction 3 in Scheme, whereas the competing reaction 4 was not taken into account.

The ratio of the rates of the competing reactions is V₃/V₄ = αl/x and the chain length is ν = V₃/V₁. equation (1a) was obtained by substitution of the rate constant k₂ into equation (1) with its analytical expression (in order to reduce the number of unknown parameters that are to be measured directly). The optimum concentration x_m of oxygen, at which the rate of oxidation is maximum, can be calculated from equation (1a) or the analytical expression for k₂ if other parameters that appear in these expressions are known.

The rates of nonchain formation of molecular hydrogen, hydrogen peroxide, and water in reactions 5, 6, and 7 of quadratic chain termination are as follows:
It is important to note that, if in the Scheme chain initiation via reaction 1 is due to the interaction between molecular hydrogen and molecular oxygen yielding the hydroxyl radical ??^• instead of ?^• atoms and if this radical reacts with an oxygen molecule (reaction 4) to form the hydrotrioxyl radical ??^.₃(which was obtained in the gas phase by Neutralization Reionization (NR) Mass Spectrometry [32] and has a lifetime of >10^-6 s at 298 K) and chain termination takes place via reactions 5-7 involving the ??^•and ??^.₃ , radicals instead of ?^• and ??^.₄ , 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: V3'(?₂?) =V₁K₃l/(K₄x+√2K₅V₁