Systematic Computational Study on the Unimolecular Reactions of

ARTICLE pubs.acs.org/JPCA

Systematic Computational Study on the Unimolecular Reactions of Alkylperoxy (RO2), Hydroperoxyalkyl (QOOH), and Hydroperoxyalkylperoxy (O2QOOH) Radicals Akira Miyoshi* Department of Chemical Systems Engineering, School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

bS Supporting Information ABSTRACT: Unimolecular isomerization and decomposition reactions of alkylperoxy (RO2), hydroperoxyalkyl (QOOH), and hydroperoxyalkylperoxy (O2QOOH) radicals play important roles in the low-temperature oxidation of hydrocarbons. In this study, these reactions have been investigated by the CBS-QB3 quantum chemical method, and the variation of the rate parameters by the structural change of alkyl groups has been studied systematically for the rule-based construction of the low-temperature oxidation mechanisms of arbitrary noncyclic alkanes. The results can be well-interpreted in terms of the group additivity and the ring-strain effect of the cyclic transition states. To extract the important processes needed for the chemical kinetic modeling, the competing reaction channels were compared in detail by steady-state analysis with the high-pressure limiting rate constants. The importance of some reactions of O2QOOH radicals, which have not been considered in the previous modeling studies, such as the hydrogen exchange reactions between -OOH and -OO• groups and hydrogen shift reactions from non-OOH sites, is suggested.

1. INTRODUCTION The need for combustion modeling with chemical kinetic mechanisms is increasing, especially for the elucidation and control of spontaneous ignition of fuelair mixtures utilized in the HCCI (homogeneous-charge compression-ignition) technology1 and for the reduction of hazardous emission from combustion. On the other hand, a large change of the chemical constituents of practical fuel is expected in the near future due to the change of fossil fuel sources, from petroleum oil to new sources, such as oil sands, coal, and natural gas, and also due to the increasing demand for the use of carbon-neutral renewable fuels. More and more knowledge on the chemical kinetics of combustion is required to optimize the use of the variety of fuels for internal combustion engines. The autoiginition of the fuelair mixture is a combustion phenomenon representatively sensitive to the reaction kinetics, and it strongly depends on the chemical structure of the fuel molecule. The octane number is an index of the autoignitability of the fuels, which is determined by well-standardized procedures developed for the practical demand for qualifying the antiknock property of gasoline. It has been measured2 for a number of pure hydrocarbon compounds found as constituents of gasoline. Figure 1 shows the dependency of the research octane number (RON) on the change of the chemical structure of pure alkanes. The RON systematically decreases with the lengthening of the carbon chain while it increases by the substitution of a methyl group as a side chain. These clear tendencies of the octane numbers have been wellknown from more than 60 years ago, but have not been r 2011 American Chemical Society

quantitatively explained by means of chemical kinetic models, although our knowledge on the low-temperature oxidation of hydrocarbons has been significantly improved by a number of experimental,319 theoretical,2038 and modeling3947 studies. Though indirect, the pioneering works by Baldwin and coworkers,36 who investigated the oxidation mechanisms by adding hydrocarbons in the slowly reacting H2O2 mixture, revealed many important features of the low-temperature oxidation of hydrocarbons by product analysis. More recently, Taatjes and co-workers1118 performed time-resolved measurements of HO2 and OH, and, in conjunction with the theoretical works,25,26 the critical competition between the concerted HO2 elimination and the intramolecular hydrogen shift reaction of RO2 has been precisely investigated. The development of the quantum chemical methodology also significantly contributed to the understanding of the complex reaction systems. Especially, development of density functional theory48,49 (DFT) significantly improved the accuracy of the self-consistent field (SCF) level of calculations. The transition state for the concerted HO2 elimination from the ethylperoxy radical (C2H5OO•), which could only be located by the expensive coupled-cluster method,20 can now be easily found by DFT methods.21 Extensive studies on the C2H5 þ O2 system22,25,26,29 and several systematic studies on the reactions of RO2,35,37 QOOH,24,34 and O2QOOH 35,38 have Received: December 22, 2010 Revised: March 10, 2011 Published: March 29, 2011 3301

dx.doi.org/10.1021/jp112152n | J. Phys. Chem. A 2011, 115, 3301–3325

The Journal of Physical Chemistry A

Figure 1. Chemical-structure dependence of the research octane number. Reprinted with permission from ref 2. Copyright 1948 American Chemical Society.

been performed by DFT-based calculations with some post-SCF methods. The difficulty in constructing reliable kinetic models for lowtemperature oxidation of hydrocarbons lies in the huge number of reactions involved, for most of which experimental rate constants are unavailable. One solution for this problem is to estimate the rate constants by the rule-based method3941 based on the group additivity (GA) concept.50 Due to a number of reliable experimental thermodynamic data, the thermodynamic properties of the hydrocarbons and derived radicals can be estimated with acceptable accuracy by the GA method.51,52 However, nearly no direct experimental rate constant is available for some important types of reactions involved in the lowtemperature oxidation mechanisms. Recent development in the quantum chemistry and the computational technology enabled us to approach these reactions by quantum chemical calculations. However, our knowledge is still insufficient to construct the mechanisms for arbitrary alkanes by the rule-based method, since the rate constants have been evaluated for some limited structural variations, and the different theoretical methods used by the different researchers make it difficult to combine the available computational studies consistently. In the present study, for the purpose of evaluating the rate constants for the rule-based approach to the construction of the oxidation mechanisms for arbitrary noncyclic alkanes, unimolecular reactions of alkylperoxy (RO2), hydroperoxyalkyl (QOOH), and hydroperoxyalkylperoxy (O2QOOH) radicals have been investigated systematically for the possible combination of the alkyl substitutions by the same CBS-QB3 quantum chemical method. Although there are several possible problems in the rate constant evaluation for kinetic models, such as the complex falloff behavior of the pressure-dependent rate constants for the unimolecular, recombination, and chemically activated reactions, the present work is focused on the evaluation of the high-pressure limiting rate constants in order to provide a solid starting point for discussing problems in the kinetic models, including the pressure effects.

ARTICLE

2. COMPUTATIONAL METHOD The quantum chemical calculations were performed by using the Gaussian 03 program.53 Geometries, vibrational frequencies, and energies of the stationary points corresponding to the reactants, products, and transition states (TS) were calculated by the CBS-QB3 method.54,55 The vibrational frequency calculated by the B3LYP/CBSB7 method during the CBS-QB3 procedure was scaled by 0.99 for the zero point energy calculation and by 0.97 for the vibrational partition function calculations. The high-pressure limiting rate constants were calculated by the transition-state theory (TST) with one-dimensional semiclassical tunneling correction56 assuming the asymmetric Eckart potential by using the GPOP program.57 The internal rotors listed in Table 1 were treated as hindered rotors by using the PitzerGwinn approximation.58 The result of the analysis for the CH2 torsion of the n-propyl radical is shown in Figure 2. As shown in the inset of Figure 2a, the torsion angle, R, was defined as the dihedral angle between the C1C2C3 plane and the plane bisecting the C2C3H6 and C2C3H7 planes. The potential energies calculated by the CBS-QB3 method are shown by open circles in Figure 2a, and the solid curve shows the Fourier-series interpolation. The eigenstate energies of the hindered rotation obtained by solving the one-dimensional time-independent Schr€odinger equation by using the BEx1D program59 are shown by horizontal lines in Figure 2a with the vibrational quantum numbers (v) and rotational quantum numbers (J). A transition of the eigenstate character from vibration (v = 0 and 1) to rotation (J = 3 and higher) was found at around the energy of the barrier height (100 cm1). The partition function calculated directly from the eigenstate energies with proper nuclear statistical weights (1/4 for even-J states and 3/4 for odd-J states) is shown in Figure 2b by open circles (qexact). It should be noted that, here, all the partition functions were calculated consistently using the energy relative to the potential minimum. The harmonic oscillator approximation of the partition function, qHO, multiplied by the number of isomers, 2, considering the chiral structure at the potential minimum (indicated as 2qHO in Figure 2b), is only valid at low temperatures. For this motion, the free rotor approximation (qFR) is a fairly good approximation, and the PitzerGwinn approximation with the barrier height V0 = 100 cm1 (qPG) gave quantitative agreement with the exact partition function. For these types of internal rotors, which are nearly the free rotors, the partition function does not strongly depend on the estimation of the barrier height, V0, and, thus, all the radical-center CH2 rotors of the primary alkyl-type radicals, including the primary QOOH, were treated by the PitzerGwinn approximation with V0 = 100 cm1. Similarly, the internal rotors attached to the carbon radical centers and the rotors around the dissociating CC bonds of the β-CC fission transition states were treated with common estimated barrier heights listed in Table 1. The rotational conformers were included in the total partition functions via the rotational-conformer distribution partition functions, qRCD,60 εi qRCD ¼ ∑ gi exp ð1Þ kB T i where gi and εi denote the degeneracy and energy of the ith rotational conformer and kB is the Boltzmann constant. Here, it should be noted that the partition functions of most of the internal rotors consisting of heavy atoms, such as the CCOO torsion of peroxy radicals, can be well-approximated by the 3302

dx.doi.org/10.1021/jp112152n |J. Phys. Chem. A 2011, 115, 3301–3325


ARTICLE

Table 1. Estimated Barrier Heights for Hindered Rotations V0 a/cm1 CH2 rotor of a primary radical

H2C•R

CH3 rotor attached to a secondary or tertiary radical center

XYC•CH3

non-methyl rotor attached to a secondary or tertiary radical center

XYC•R •

rotor around the dissociating bond of a β-CC fission transition state a

100 80 400

CC 3 3 3 R

400

Barrier heights for the PitzerGwinn approximation.

Figure 2. Hindered rotor analysis for the CH2 torsion of n-propyl radical: (a) Potential energy curve and eigenstate energies; (b) partition function calculated from eigenstate energies, qexact, in comparison with harmonic oscillator (qHO), free rotor (qFR), and PitzerGwinn (qPG) approximations (see text for details).

harmonic oscillator partition function with the rotationalconformer distribution partition function (1). An example analysis of the CCOO torsion of the ethylperoxy radical is shown in Figure 3. The potential energy curve was estimated from the CBS-QBS potential energies and the second derivatives of the potential energy estimated by the B3LYP/CBSB7 frequency analysis at the four stationary points (Figure 3a). The definition of the torsion angle, R, is shown in the inset. The eigenstate energies calculated by the BEx1D program are shown by dotted horizontal lines in Figure 3a, and the partition functions directly calculated from the eigenstate energies are shown by open circles (qexact) in Figure 3b. The partition functions, qHO, qHO0 , etc., shown in the figure are defined as follows. qHO ¼ qðHO; gaucheÞ qHO 0 ¼ qðHO; transÞ expðΔE=kB TÞ qRCD ¼ 2 þ expðΔE=kB TÞ

ð2Þ ð3Þ ð4Þ

where q(HO,gauche) is the harmonic partition function of the lowest energy gauche isomer, q(HO,trans) is that for the trans isomer, and ΔE is the potential energy of the trans isomer relative to that of the gauche isomer. Again, all the partition functions were calculated consistently with the energy relative to the bottom of the potential energy of the gauche isomer. The results shown in Figure 3b indicate that the harmonic oscillator assumption is valid if the two isomers were treated with proper weights (2qHO þ qHO0 ). The slight deviation of qHOqRCD from qexact is due to the slightly smaller torsion harmonic frequency of the trans isomer than that of the gauche isomer. It should be noted that, in the qHOqRCD, all the vibrational frequencies of the trans isomer are approximated by those of the gauche isomer. The free rotor

Figure 3. Hindered rotor analysis for the CCOO torsion of ethylperoxy radical: (a) Potential energy and eigenenergies; (b) comparison of partition function calculated from eigenenergies (qexact) with free rotor (qFR) and harmonic oscillator (qHO) approximations (see text for details).

approximation (qFR) is a poor approximation for this case. The partition functions of these types of torsion motions were evaluated by the harmonic oscillator assumption with qRCD, that is, by qHOqRCD, in this study considering the cost efficiency. The energies of the all the rotational conformers were exhaustively examined by the CBS-QB3or B3LYP/ 6-31G(d) method for a limited number of C2C4 molecules. The conformer energy distributions for larger molecules were estimated on the basis of those for the C4 molecules since the maximum number of rotational conformers was as large as 36 = 729 for the molecules investigated in this study. The rate constants were evaluated for systematically chosen combinations of the methyl substitutions. For example, the intramolecular 1,5-H shift reactions of RO2 radicals (via sixmembered-ring TS) have been investigated for all nine combinations of the classes (primary, secondary, or tertiary) of the site where the -OO• group is attached and of the site from where the hydrogen atom is removed, listed as follows. •

OOCH2CH3 f HOOCH2 • CH2

•

OOCH2CðCH3 ÞH2 f HOOCH2• CðCH3 ÞH

ðpsÞ

OOCH2CðCH3 Þ2 H f HOOCH2• CðCH3 Þ2

ðptÞ

•

ðppÞ

•

OOCðCH3 ÞHCH3 f HOOCðCH3 ÞH• CH2

•

OOCðCH3 ÞHCðCH3 ÞH2 f HOOCðCH3 ÞH• CðCH3 ÞH

3303

ðspÞ

ðssÞ



ARTICLE

Figure 4. Averaged energy diagram for the unimolecular reactions of alkylperoxy (RO2 ) and hydroperoxyalkyl (QOOH) radicals. The indicated energies are relative to the alkylperoxy radicals and are the average over the possible sets of the methyl substitutions . The error bars denote the range of the variation by methyl substitutions. The chemical structures are drawn for 1-hexylperoxy (C6 H 13O 2 ) radical as representatives. •

OOCðCH3 ÞHCðCH3 Þ2 H f HOOCðCH3 ÞH• CðCH3 Þ2 •

OOCðCH3 Þ2CH3 f HOOCðCH3 Þ2• CH2

the rate constant for the “tp” class is one-third of the rate constant calculated for •OOC(CH3)2CH3 f HOOC(CH3)2•CH2. For some reactions involving cyclic transition states, the effects of the methyl substitution on other ringmember carbon were also investigated as described later.

ðstÞ ðtpÞ

•

OOCðCH3 Þ2CðCH3 ÞH2 f HOOCðCH3 Þ2• CðCH3 ÞH ðtsÞ

3. RESULTS FOR RO2 AND QOOH REACTIONS

•

OOCðCH3 Þ2CðCH3 Þ2 H f HOOCðCH3 Þ2• CðCH3 Þ2

3.1. Overview of the Unimolecular Reactions of RO2 Radicals. Figure 4 shows the generalized energy diagram for

ðttÞ

Here, “ps” denotes a class of alkyl substitution corresponding to the secondary hydrogen abstraction of a primary alkylperoxy radical. Hereinafter, unless otherwise noted, the rate constant or barrier height for a substitution class means that calculated for the minimum representative molecule; that is, the ps in the preceding examples means a quantity calculated for •OOCH2 C(CH3)H2 f HOOCH2•C(CH3)H, not for •OOCH2 C(CH2CH3)H2 f HOOCH2•C(CH2CH3)H. One important but a little complicated note is that the rate constant for the “sp” substitution class in the preceding example is half of the rate constant calculated for •OOC(CH3)HCH3 f HOOC(CH3)H•CH2 since the reactant molecule has two equivalent methyl groups. This is because, in the rule-based estimation of the rate constant, the group specific rate constant for sp is multiplied by the number of equivalent target sites (or by the number of equivalent hydrogen atoms when the ruled rate constant is evaluated per one target hydrogen atom). Similarly,

the unimolecular reactions of alkylperoxy (RO2) radicals and the subsequent reactions of the isomerization products, hydroperoxyalkyl (QOOH) radicals. The energies shown are the average over the systematically chosen methyl-substituted alkylperoxy radicals, and the error bars indicate the range of variation by methyl substitutions. The types of the reactions investigated in this studies are described as follows. The direct hydrogen abstraction reaction from alkyl radical by O2 (R1), the transition state for which is indicated by *m in Figure 4, has a significantly high barrier and will be unimportant in most of the circumstances.

It has been well-established11,26 that the formally “direct” reaction, which gives the same products, occurs on the potential 3304



ARTICLE

energy surface which forms peroxy radical (R2) followed by the concerted HO2 elimination (R3) through the transition state *e via chemically activated RO2.

reactions R8 from the carbon with the OOH group (*h2s, *h3s, ..., *h6s) may play some minor role.

This reaction is especially important for small alkyl radicals at low pressures.1118 For relatively large alkyl radicals, this reaction will occur via a two-step mechanism due to the rapid stabilization to RO2. Except for the back-dissociation (*d) to alkyl radical (R) þ O2, all of the other reaction channels of alkylperoxy radicals (P) are the intramolecular hydrogen abstraction reactions. The 1,3hydrogen abstraction from the R-position (hereinafter, the positions of the carbons in an RO2 radical are designated by R, β, γ, δ, ε, and ζ, respectively, from the carbon bonded to the OO• group) has a high barrier (*h1) and is unimportant. The energy of the five-membered-ring transition state (*h2) for the 1,4-hydrogen shift reaction from β-position (R4) is lower than that for the 1,3-shift (*h1), and those for six- (*h3), seven- (*h4), eight- (*h5), and nine-membered-ring (*h6) transition states for 1,5-, 1,6-, 1,7- and 1,8-hydrogen shift reactions, respectively, are even lower.

The R-hydroperoxyalkyl radicals (R-QOOH) produced by the 1,3-H shift reaction of RO2 (*h1) and by the H-shift reaction of other QOOH radicals (*h2s, *h3s, ..., *h6s) do not have minima on the B3LYP/CBSB7 potential energy surfaces and are expected to dissociate promptly to carbonyl compounds with OH as shown in (R8). The instability of the smallest R-QOOH, • CH2OOH, has been well-known from the experimental studies on the OD þ CH3OOH reaction,61 which promptly generates OH with CH2O. Also, the instability of R-QOOH species has been confirmed by theoretical studies.22,28,35 It should be noted that the calculations of some transition states involving the cleavage of OO bonds might suffer from the spin contamination. The ÆS2æ eigenvalues calculated in the MP2/CBSB3 step of the CBS-QB3 procedure and T1 diagnostic62 values in CCSD(T)/6-31þg(d0 ) calculations are shown in Table S1 of the Supporting Information for typical transition states investigated in the present study. Very large ÆS2æ eigenvalues (>1.5) were found for the transition states of the R þ O2 direct H-abstraction, the OH-transfer, and the H-shift reactions of β-QOOH, and the concerted CC/OO fission reaction (which will be described later) of γ-QOOH, but, fortunately, these reactions are relatively unimportant. Among the transition states for the major channels, those for the cyclic ether formation reactions of QOOH showed large ÆS2æ (∼1.27) and T1 (∼0.042). Considering the large spin-contamination correction in the CBS-QB3 method, which is 13 kJ mol1 for ÆS2æ = 1.27, the possible errors in the barrier heights may be as large as the magnitude of the correction, 1015 kJ mol1. The energies calculated in the present study are compared with the experimental values63 and previous theoretical works for the smallest C2H5 þ O2 system in Table S2 of the Supporting Information. The agreement was satisfactory considering the evaluated errors for the CBS-QB3 method,55 ≈6.1 kJ mol1 as the root-meansquare deviation. 3.2. Systematic Variation of the Barrier Heights. Figure 5 shows the effect of methyl substitutions on the threshold energies (barrier heights corrected for zero-point energy), E0, and the reaction energies at 0 K, ΔE, for the intramolecular hydrogen shift reactions of RO2 radicals. The definition of the numbers of substitution, m and n, is shown in the structure of the transition state on top of each plot, and the abscissa is an index defined as n m. The barrier heights and reaction energies for these types of reactions are essentially determined by the number of methyl substitutions on the target site, from where a hydrogen atom is abstracted, due to the well-known weakening of the CH bond dissociation energies by methyl substitutions, from primary to secondary to tertiary (pp f ps f pt, etc.). The results confirm the group additivity rule used in the construction of mechanisms. However, a small but clear non-next-neighbor effect was found for 1,7-H shift reactions (Figure 5d), for which, the methyl

The products of these types of reactions, hydroperoxyalkyl radicals (QOOH), play an essential role in the autoiginition of the hydrocarbonair mixture via the formation of O2QOOH, which eventually produces more than one OH radical; that is, it constitutes the chain-branching process. As expected from the group additivity, the energies of the β- (H2) to ζ-QOOH (H6) radicals are nearly the same (note that the energies shown are the average over primary, secondary, and tertiary QOOH radicals). As a result of extensive search for the transition states of possible subsequent reaction of QOOH radicals, four types of reactions were identified except for the back-isomerization reactions to RO2. The most important channel for β- (H2), γ- (H3), and δ-QOOH (H4) is the cyclic ether formation reactions (*h2r, *h3r, and *h4r).

The barrier height for this type of reaction is highest for γQOOH (*h3r). The β-fission reactions (*h2d, *h3d, ..., *h6d) are also important. The barrier is especially low for the β-QOOH (*h2d) radical since the weak CO bond is broken (R6) while the β-fission breaks a CC bond for other QOOH radicals.

The OH-transfer (R7) to the radical center (*h2t, *h3t, ..., *h6t) may play some role but is generally minor. Also the H-shift

3305



ARTICLE

Figure 5. Effect of methyl substitutions on the threshold energy (E0) and the reaction energy (ΔE) of the intramolecular hydrogen shift reactions of RO2 radicals: (a) 1,4-H shift, (b) 1,5-H shift, (c) 1,6-H shift, and (d) 1,7-H shift. The schematic structure of the transition state on top of the each plot shows the definition of m and n. The substitution classes “pp”, “ps”, and “pt”, etc., correspond to (m, n) = (0, 0), (0, 1), and (0, 2), etc. The solid circles and triangles denote ethyl and n-propyl substitutions, respectively, instead of methyl.

substitution on the carbon with an -OO• group also increases the barrier height slightly. This is probably due to the increase of the ring-strain energy, which will be discussed in detail later. Though limited, the effect of the size of the substituent was also investigated by substituting an ethyl or n-propyl group instead of a methyl group. The effect was concluded to be minor from the results shown by solid circles (ethyl) and triangles (n-propyl) in Figure 5. The substitution effect on the concerted HO2 elimination from RO2 shown in Figure 6 is totally different from that on the 1,5-H shift reaction (Figure 5a) despite the similarity of the transition-state structures, indicating the different electronic nature of this reaction from the H-shift reactions. The barrier height is nearly independent of the methyl substitution on either side, while the reaction energy decreases with the increase of the index, n m. Such a tendency on the methyl substitution rather resembles that of the HX (X = halogen atom) elimination reaction from haloalkanes.64 Figure 7 shows the barrier heights and reaction energies of the cyclic ether formation reactions of QOOH radicals. The tendency of the effect of the methyl substitution is essentially the same as those reported by Wijaya et al.34 and Chan et al.24 The

methyl substitution on the radical center carbon (for example, from pp to ps to pt in each figure) decreases both E0 and ΔE by similar amount except for the oxane formation reactions from ε-QOOH (Figure 7d), for which the ΔE shows different behavior. Since the methyl substitution scarcely affects the strength of the newly formed CO bond, the effect can be most probably ascribed to the relief of the ring strain of the transition states and products. The methyl substitution on the R-carbon to the OOH group also decreases the E0 and ΔE for the formation of threeand four-membered-ring cyclic ethers (*h2r and *h3r), while it increases the E0 for the formation of six-membered-ring ethers (*h5r). Since the strength of the OO bond broken in the reaction is nearly independent of the substitution,34 the effect should also be attributed to the change of the ring strain. Table 2 shows the change of the CCO bond angles of ethers by methyl substitutions. The decrease of the natural bond angle suggests the relaxation of the ring strain of products and transition states of the cyclic ether formation reactions by methyl substitutions. Figure 8 shows the CCO bond angles of the unsubstituted TS’s and cyclic ether products. The angle of the preserved CCO (θR) in the TS increases with the ring size and is nearly matched with 3306



Figure 6. Methyl-substitution dependence of the barrier heights (E0) and reaction energy (ΔE) for the concerted HO2 elimination reactions of RO2 radicals. The substitution classes “sp”, “ss”, and “st”, etc., correspond to (m, n) = (1, 0), (1, 1), and (1, 2), etc. The solid circles and triangles denote ethyl and n-propyl substitutions, respectively.

the unstrained angle of ethers (∼108.5) at ring size = 5. This well-explains the effect of methyl substitution on the R-carbon of QOOH in Figure 7; that is, the methyl substitution on the Rcarbon (from pp to sp to tp, etc.) relieves the ring strain of the three- or four-membered-ring TS (and thus decreases the E0) but has nearly no effect on the five-memeberd-ring TS, as shown in Figure 7c, and rather increases the ring strain of the sixmembered ring (Figure 7d). The CCO angle around the radical center carbon (θrc) of TS also increases with the ring size but does never exceed the natural angle, ∼108.5, up to ring size = 7. This is also consistent with the effect of methyl substitution on the radical-center carbon shown in Figure 7, as it always decreases the energy of the TS. It should be noted that the two types of methyl substitutions (to the R-carbon with -OOH and to the radical-center carbon) discussed here are equivalent on the stability of the cyclic ether products, and thus ΔE should be essentially symmetric for positive and negative indices (n m) in each figure except for the small difference in the stability of QOOH radicals. For these reactions, again, the effect of ethyl (solid circles) or propyl (solid triangles) substitution was confirmed to be essentially the same as that of methyl substitution. Since the substitution also decreases the CCC bond angles of alkanes as shown in Table 2, the methyl substitution on other inring carbons should also affect the barrier height. From the CCC bond angles shown in Figure 8b, the significant effect is expected for four- and five-membered-ring cyclic ether formation reactions since the CCC bond angles of the TS are significantly smaller than the unstrained angle of alkanes (∼113.3). For the oxetane formation reactions (*h3r), all of the combinations of methyl substitutions were investigated as shown in Figure 9a,b. The methyl substitution to the midcarbon was found to decrease both E0 and ΔE, by similar amounts to the R-substitution and radicalcenter substitution, as expected from the consideration on the ring strain. The effect was also investigated for the oxolane formation (*h4r), as shown in Figure 9c. The effect was also visible but, due to the reduced ring strain, the effect was smaller than that for the oxetane formation. Since similar effects may be also possible for the H-shift reactions of RO2, the CCC and CCO

ARTICLE

bond angles of the TS’s for the RO2 H-shift reactions are shown in Figure 10. The CCC bond angles are close to the unstrained angles especially for the six- and seven-membered-ring TS’s for 1,5- and 1,6-H shift reactions, and the methyl substitution on the center carbon of a CCC unit will not affect the ring-strain energy. This was further confirmed by the calculations shown in Figure 11, which shows the effect of in-ring quaternary carbon on the energetics of the 1,5 H-shift reactions. The existence of an in-ring quaternary carbon scarcely affects the threshold energy. The larger CCO angle of the seven- or eight-membered-ring TS than the unstrained one (Figure 10) well-explain the increase of the threshold energy by the methyl substitution on R-carbons, which can be seen in Figure 5c,d. Also the smaller CCO angle of the five-membered TS (Figure 10) suggests the decrease of the ringstrain energy by R-methyl substitutions, though the effect cannot be seen clearly in Figure 5a. Figure 12 shows the systematic variation of E0 and ΔE for βfission reactions of QOOH radicals. For β-QOOH radicals, as shown in Figure 12a, the threshold energy is nearly independent of the substitution while the reaction energy systematically changes with methyl substitutions. This means that the barrier heights for the reverse reactions significantly change by methyl substitutions, reflecting the stability of the product β-QOOH radicals. It should be noted that the reverse HO2-addition reactions to double bonds are important in the oxidation of alkenes and most of the oxiranes observed in earlier measurements3,4 were ascribed to the products of the addition reactions of HO2 or RO2 to the double bonds, followed by the cyclic ether formation (*h2r). The β-fission reactions (R9) of γQOOH shown in Figure 12b produce R-QOOH radicals which immediately dissociate to the carbonyl compound þ OH (R10), as discussed previously. •

CH2 CH2 CH2 OOH f C2 H4 þ ½• CH2 OOH simple CC fission ðh3dÞ •

½ CH2 OOH f CH2 O þ OH

ðR9Þ ðR10Þ

Due to the instability of the R-QOOH product, a transition state which directly correlates the γ-QOOH and three fragments (R11) was also found. •

CH2 CH2 CH2 OOH f C2 H4 þ CH2 O þ OH concerted CC=OO fission ðh3cÞ ðR11Þ

The coordinate of the imaginary frequency of the transition state or the intrinsic reaction coordinate involves the motion of simultaneous cleavage of CC and OO. This reaction will be called “concerted CC/OO fission reactions” hereinafter. Since the energies of the unstable R-QOOH radicals cannot be determined accurately, the reaction energy plotted in Figure 12b is the energy of the final three fragments. The threshold energy for the simple CC fission reactions shows the well-known tendency for the thermal decomposition of alkyl radicals, in which the methyl substitution on the carbon that becomes the radical center in the product significantly reduces ΔE and E0 by stabilizing the radical product. The tendency of the effect of the methyl substitution on the threshold energy for the concerted CC/OO fission reactions cannot be explained by simple electronic effect of the substitution. Figure 13 shows the substitution effect on the other two types of reactions of QOOH radicals. The energetics of the OH-transfer reactions of 3307



ARTICLE

Figure 7. Effect of methyl substitutions on the threshold energy (E0) and the reaction energy (ΔE) of the cyclic ether formation reactions of QOOH radicals: (a) β-QOOH, (b) γ-QOOH, (c) δ-QOOH, and (d) ε-QOOH. The solid circles and triangles denote ethyl and n-propyl substitutions, respectively.

Table 2. Methyl Substitution Effect on the Bond Angles

a

Bond angles of the geometry optimized by the B3LYP/CBSB7 method.

γ-QOOH (Figure 13a) is nearly independent of the substitution. The substitution effect on E0 of the H-shift reactions of ε-QOOH (Figure 13b) can be simply interpreted by the weakening of the CH bond at the R-position by substitution. ΔE shown in

Figure 13b is the energy of the final fragments since this reaction produces unstable R-QOOH radicals (R8). 3.3. High-Pressure Limiting Rate Constants. For all of the reactions investigated in the present study, high-pressure limiting rate constants were evaluated by the TST. The Arrhenius plots for three typical types of reactions are shown in Figure 14. For the same types of reactions, with the same ring size of the transition state, the preexponential factor does not change significantly by the methyl substitution and the variation of the rate constants mainly reflects the change of the barrier height. Since the variation of the threshold energy within the same type of the reaction has been described and discussed in the previous subsection, the rate constants for the competing different types of reactions are compared here. The TST rate constants for the competing reaction channels of RO2 radicals are compared in Figure 15. The rate constants plotted here are the geometric mean of the rate constants for the nine combinations of the substitutions, pp to tt, and, thus, in many cases, the value is close to ss substitution. For the relatively unimportant 1,8-H shift reaction (*h6), the rate coefficient was calculated for limited cases and the value plotted here is that for the ss substitution. Around 800 K, the ordering of the rate constants, from the largest 3308



ARTICLE

Figure 8. Ring-size dependence of the bond angles of TS’s and products of the QOOH cyclic ether formation reactions. Definitions of θR and θrc of TS are shown in the inset. The unstrained CCO (108.5) and CCC (113.3) bond angles of ethers and alkanes are shown by the horizontal lines for comparison.

to smaller, is as follows: 1,5-H shift (*h3), back-dissociation to R þ O2 (*d), 1,6-H shift (*h4), 1,7-H shift (*5), concerted HO2 elimination (*e), 1,4-H shift (*h2), 1,8-H shift (*h6), and 1,3-H shift (*h1). It should be noted that the relative importance among the competing channels differs from an alkylperoxy radical to another, considering the magnitude of the variation of the rate constant by substitution, such as those shown in Figure 14, and that some processes are impossible for some chemical structures. Moreover, the relative importance of the RO2 reactions cannot be determined by the rate constants only, since, for example, the small barriers between RO2 and QOOH suggest that the concentrations of RO2 and QOOH are sometimes in partial equilibrium and the H-shift reactions are not always the rate-determining steps. The comparison of RO2 reactions considering such situations will be given later in Discussion. The possible pressure dependence of the unimolecular reactions, including the chemical activation reactions and the interference between multiple channels,26,64 further makes the situation complex, but will not be discussed in this study as stated earlier. Parts ae of Figure 16 compare the mean rate constants for the reactions of β-, γ-, δ-, ε- and ζ-QOOH radicals, respectively. For these reactions, except for the back-isomerization reaction to RO2 (*h2, *h3, etc.), the reverse reactions are negligible since they produce fragmented products, and the relative significance among the competing reactions can be compared by the rate constants. The discussion that follows will be focused on the temperature around 800 K where the low-temperature oxidation is important. As shown in Figure 16a, the reaction of β-QOOH radicals is dominated by the rapid cyclic ether formation (*h2r) but the contribution of β-fission reaction (*h2d) may be also important considering the variation of the rate constants by substitution. The back-isomerization (*h2), OH-transfer (*h2t), and H shift (*h2s) reactions can be neglected in most of the cases. The second O2 addition reaction, for which the firstorder rate constant in 10 atm of air (that is, 2 atm of O2), as indicated by *O2(10), is only important at low temperature or at high pressures [note that, for example, the first-order rate constant at 100 atm is 1 order of magnitude larger than *O2(10)]. For γ-QOOH, as can be seen in Figure 16b, the fastest reaction is the back-isomerization to RO2 (*h3) and this

Figure 9. Effects of the in-ring quaternary and tertiary carbons on the energetics of the cyclic ether formation reactions of the γ-QOOH and δ-QOOH radicals: (a) in-ring tertiary effect for γ-QOOH, (b) inring quaternary effect for γ-QOOH, and (c) in-ring quaternary effect for δ-QOOH. The dotted lines indicate the energetics for the reactions without in-ring substitutions for comparison. The symbol “pTs” denotes that the classes of the carbon atoms are primary, tertiary, and secondary, from the R-carbon (with -OOH) to the radical-center carbon. 3309



ARTICLE

Figure 10. Ring-size dependence of the averaged CCC and CCO bond angles of the transition states for the H-shift reactions of unsubstituted RO2 radicals.

Figure 11. Change of the threshold energies (E0) and reaction energies (ΔE) of the 1,5-H shift reactions of RO2 radicals by the presence of a quaternary carbon in the ring. Dotted lines show the energetics for the reactions without in-ring quaternary carbon.

implies that the γ-QOOH radicals often exist in near partial equilibrium with RO2. The important exit channels are cyclic ether formation (*h3r) and β-fission (*h3d þ *h3c) reactions. The second O2 addition reaction to γ-QOOH is more important than the case of β-QOOH due to the relatively small rate constants for *h3r and *h3d þ *h3c. For δ-QOOH, at the temperature of low-temperature oxidation, ∼800 K, the ringclose (*h4r) and back-isomerization (*h4) reactions are in close competition, and the second O2 addition at 10 atm of air [*O2(10)] also competes with them. The other reactions of δQOOH are negligible. For ε- and ζ-QOOH, most of the exit channels, *hnr, *hnt, *hns, and *hnd, are in close competition, but all of them are unimportant compared to the second O2 addition under the pressure of 10 atm [*O2(10)] or higher. For ζ-QOOH, the six-membered-ring hydrogen shift reaction from the βposition to the OOH group may occur with the rate constant larger than that for *h6s. However, the rate constant is expected to be smaller than that for *h5s shown in Figure 16d considering the substitution effect of OOH, and, thus, it does not alter the essential picture. The hydrogen shift reactions of other QOOH radicals from non-OOH sites can be estimated to be unimportant

Figure 12. Effect of methyl substitutions on the threshold energy (E0) and the reaction energy (ΔE) of the β-fission reactions of QOOH radicals: (a) β-QOOH and (b) γ-QOOH.

since the rate constants for this type of reactions are largest for the six-membered-ring transition state65 and should be smaller than the *h5s shown in Figure 16d. The mean preexponential factors and activation energies for the reactions involving cyclic transition states are summarized in Figure 17. As have been discussed in many previous studies, the preexponential factors decrease with the ring size of the transition states approximately by a factor of 1/11 for cyclic ether formation (*hnr) and H-shift (*hns) reactions of QOOH and by a factor of 1/5 for OH-transfer reaction of QOOH (*hnt) and H-shift reactions of RO2 (*hn) per each increase of the ring size. The activation energies seem to reflect the ring-strain energy of the transition states and significantly increase at the ring size of 5 or smaller. The activation energies for cyclic ether formation reactions of QOOH (*hnr) do not seem to reflect the ring-strain energy directly since the activation energy for three-memberedring oxirane formation is apparently lower than that for fourmembered-ring oxetane formation.

4. RESULTS FOR O2QOOH REACTIONS 4.1. Investigations on the Smallest Typical Systems. Since a relatively small number of theoretical investigations have been 3310



ARTICLE

Figure 13. Effect of methyl substitutions on the threshold energy (E0) and the reaction energy (ΔE) of (a) OH-transfer reactions of γ-QOOH and (b) the H-shift reactions of ε-QOOH.

reported for these reactions,28,35,38 possible reaction channels were surveyed for the two smallest typical systems, O2CH2CH2OOH and O2CH2CH2CH2OOH. The results for the O2CH2CH2OOH system are summarized in Figure 18. The important channels were essentially the same as those reported by Bozzelli and Sheng.28 They are the 1,4-H shift reactions to produce OH þ hydroperoxyacetaldehyde (*h2) and the concerted HO2 elimination reaction (*e). The possibility of the reaction producing trioxolane þ OH (*r) was investigated, but it was concluded to be unimportant due to the high barrier. Also the OH-transfer (*t) and subsequent reactions were found to be unimportant. One implicative reaction found in this study is the H-atom exchange reaction between -OO• and the OOH group (*ho). Although this very reaction of • OOCH2CH2OOH does not have meaning since the reactant and product are identical, similar reactions of asymmetric O2QOOH radicals with respect to the -OO• and -OOH sites may play some role in the oxidation, which will be discusses in more detail later. The results for the O2CH2CH2CH2OOH system are summarized in Figure 19. As a conclusion, the 1,5-H shift reaction to produce hydroperoxypropanal þ OH (*h3) is the predominant reaction channel, and the essential parts agree well with the

Figure 14. Effect of the methyl substitutions on the high-pressure limiting rate constants for (a) 1,5-H shift reactions of RO2, (b) concerted HO2 elimination reaction of RO2, and (c) cyclic ether formation reactions of γ-QOOH.

recent study by Goldsmith et al.38 The reactions involving the hydrogen atom at the β-position are quite similar to those of RO2 radicals. Similarly to the O2CH2CH2OOH system, the ring formation (*r) and OH-transfer (*t) reactions were concluded to be minor channels. Again, the hydrogen exchange reaction between -OO• and -OOH groups (*ho) was found with a low barrier. 4.2. Systematic Variation of the Energetics and Rate Constants. Since the hydrogen shift reactions from nonOOH sites are expected to be similar to those of RO2 radicals and this was partly confirmed for the O2CH2CH2CH2OOH system, only the reactions specific to the O2QOOH radicals were investigated further for a set of methyl substitutions in 3311


The Journal of Physical Chemistry A this study. Figure 20 shows the energy diagrams of the reactions investigated in detail. The energies shown are the average and the error bars indicate the range of variation by methyl substitutions as in Figure 4. The variation of the threshold energy and reaction energy was investigated for a set of methyl substitutions, and the results are summarized in Figures 2123 for β-, γ-, and δ-O2QOOH, respectively.

Figure 15. Comparison of the mean high-pressure limiting rate constants for the unimolecular reactions of RO2 radicals. The reactions are designated by the symbols of the transition states in Figure 4 (*d, backdissociation to R þ O2; *h3, 1,5-H shift; *h4, 1,6-H shift; *e, concerted HO2 elimination; etc.).

ARTICLE

The methyl substitution effect was similar to that for the corresponding reaction of RO2 radicals except that the -OOH

Figure 17. Dependence of the Arrhenius preexponential factors (A) and activation energies (Ea ) on the ring size of the transition state (TS) for intramolecular reactions of RO 2, QOOH, and O 2QOOH which proceed via cyclic transitions states: *hn, H-shift reactions of RO2 ; *hnr, cyclic ether formation reactions of QOOH; *hnt, OHtransfer reaction of QOOH; *hns, H-shift reactions of QOOH; *hox, H-shift reactions of O 2 QOOH from OOH group described in section 4.

Figure 16. Comparison of the mean high-pressure limiting rate constants for the unimolecular reactions of (a) β-, (b) γ-, (c) δ-, (d) ε-, and (e) ζ-QOOH radicals. The reactions are designated by the symbols of the transition states in Figure 4. The reverse isomerization reactions to RO2 are indicated by *hn. The first-order rates for the second O2 addition reactions under 10 atm air (or 2 atm O2 ) are also shown for comparison [*O 2(10)]. 3312



Figure 18. Energy diagram for the unimolecular reactions of 2-hydroperoxyethylperoxy radical (HOOCH2CH2OO•).

ARTICLE

Figure 20. Averaged energy diagram for the unimolecular reaction of β-, γ-, and δ-O2QOOH. The energies are the average over the sets of methyl substitutions, and the error bars indicate the range of variation by methyl substitution. The minimum-size molecules are shown as representatives.

the -OOH group. The calculated high-pressure limiting rate constants for the 1,5-H shift reaction of γ-O2QOOH are shown in Figure 24. Comparison with those of 1,5-H shift reactions of RO2 (Figure 14a) shows that the rate constants are nearly equal to those of corresponding RO2 in which the OOH group is replaced by a methyl group. The H-exchange reactions between -OOH and -OO• groups do not show clear substitution effect on either side (Figures 21c, 22b, and 23b). Parts ad of Figure 25 compare the high-pressure limiting rate constants for the reactions of β-, γ, δ, and ε-O2QOOH, respectively. The rate constants for the reactions involving non-OOH sites are estimated to be equal to those of RO2 radicals. Below the discussion is focused on the temperature of low-temperature oxidation, 700900 K. β-O2QOOH. For β-O2QOOH, as shown in Figure 25a, one of the major reactions is the 1,5-H shift (*h3) reaction from a nonOOH site, if it exists, to produce β,γ-dihydroperoxyalkyl [P(OOH)2] radical (R12).

Figure 19. Energy diagram for the unimolecular reactions of 3-hydroperoxypropylperoxy radical (HOOCH2CH2CH2OO•).

substitution further decreases the barrier height for hydrogen abstraction reactions (Figures 21b, 22a, and 23a). The substitution effect of the -OOH group is similar to that of a methyl group; that is, the barrier height is nearly equal to that for the reaction of the RO2 radical with methyl group in place of

The estimated rate constants for the subsequent reactions of the β,γ-P(OOH)2 radicals are shown in Figure 26a. The β,γP(OOH)2 radical is expected to be predominantly consumed by the rapid cyclic ether formation (*h2r, R13).

This is in contrast to the γ-QOOH radicals produced by the 1, 5-H shift reaction of RO2, for which the back-isomerization (*h3) is important (Figure 16b), and they often exist in partial equilibrium with RO2. The existence of the additional -OOH 3313



ARTICLE

Figure 22. Effect of the methyl substitution on the energetics of γO2QOOH reactions: (a) 1,5-H shift reactions and (b) 1,7-H shift from OOH.

constitute the branching-chain process by producing the second OH radical (R14).

Figure 21. Effect of the methyl substitution on the energetics of βO2QOOH reactions: (a) concerted HO2 elimination reactions, (b) 1,4H shift reactions, and (c) 1,6-H shift from OOH.

group at the β-position opens up the fast oxirane formation channel (*h2r) for β,γ-P(OOH)2 , and the third O2 addition to this radical is unimportant up to ∼100 atm, shown as *O2 (100). The product of reaction R13, a compound with an oxirane ring and an OOH group, is expected to decompose similarly to the hydroperoxycarbonyl compound formed from γ-O2 QOOH, which will be described below, and

Another fast reaction of β-O2QOOH, the hydrogen exchange reactions between -OOH and -OO• groups (*ho-β), will be important in the case in which the rapid subsequent channel is accessible only from the product of isomerization. For example, the reactant of R15 has CH bonds only at the β-position to OO•, but the 1,4-H shift reaction (*h2-β or *h2) is slow, while the product of this reaction undergoes rapid 1,5-H shift reaction (*h3, R12).

For this case, the H-exchange reaction R15 promotes the oxidation process. The concerted HO2 elimination (*e-β) and 1,4-H shift reactions (*h2-β) are important for small radicals for which the other rapid reactions are geometrically prohibited. 3314



ARTICLE

Figure 25. Mean high-pressure limiting rate constants for the reactions of O2QOOH: (a) β-O2QOOH, (b) γ-O2QOOH, (c) δ-O2QOOH, and (d) ε-O2QOOH. Reactions are indicated by the symbols of the corresponding transition states in Figure 20. Rates of reactions involving non-OOH sites were estimated to be equal to those of RO2 reactions, and are indicated by the same symbols as in Figure 15. Figure 23. Effect of the methyl substitution on the energetics of δO2QOOH reactions: (a) 1,6-H shift reactions and (b) 1,8-H shift from OOH.

This is effectively an irreversible reaction since the product R-OOH radical rapidly decomposes28,35 into OH þ hydroperoxycarbonyl compound. The hydroperoxycarbonyl compound is a well-known intermediate in the low-temperature oxidation: its thermal decomposition produces the second OH radical (R17) and constitutes the major branching-chain process.

Figure 24. Rate constants for the 1,5-H shift reactions of γ-O2QOOH radicals.

γ-O2QOOH. For γ-O2QOOH, as shown in Figure 25b, the 1,5-H shift reaction from the carbon with the OOH group (*h3-γ) is dominant, if possible (R16).

The hydrogen exchange between -OOH and -OO• groups (*ho-γ) is also a rapid process and will be important if there is no hydrogen on the carbon with OOH; for example,

The *h3-γ reaction is impossible from the reactant of R18 since the γ-position is quaternary carbon, but it is possible from the 3315



ARTICLE

The β,δ-P(OOH)2 radical produced by R19 is expected to decompose predominantly and irreversibly to OH þ hydroperoxyoxirane compound (R20) as shown in Figure 26b.

δ- and ε-O2QOOH. The major reactions of δ-O2QOOH radicals are the 1,6-H shift reaction (*h4-δ) from the carbon with the OOH group (R21) and the 15-H shift reaction (*h3) similar to RO2 (R22) as shown in Figure 25c.

Due to the instability of the product R-OOH radical, reaction R21,R22R21 proceeds irreversibly. The product of R22 is also the βγ-P(OOH)2 radical, which was already discussed previously (Figure 26a) and decomposes to hydroperoxyoxirane þ OH (R23) nearly exclusively.

The situation is much the same for ε-O2QOOH, when the internal H-shift occurs from the δ-position (*h4, R24). The product of R24 β,δ-P(OOH)2 radical, rapidly and irreversibly decomposes to hydropeoxyoxirane þ OH (R25) as shown in Figure 26b.

However, for the fastest 1,5-H shift reaction (*h3, R26) the product is γ,γ-P(OOH)2 radical, and, as shown in Figure 26c, the subsequent cyclization reaction of to hydropeoxyoxetane (*h3r, R27) is relatively slow.

Figure 26. Estimated rate constants for the reactions of (a) β,γ-, (b) β,δ-, and (c) γ,γ- dihydroperoxyalkyl [P(OOH)2] radicals. Rate constants were estimated from those of corresponding reactions of QOOH radicals and are indicated by the same symbols as in Figure 16. The firstorder rates for the third O2 addition reactions under 10 and 100 atm air (or 2 and 20 atm O2) are also shown for comparison [*O2(10) and *O2(100)].

product, and, thus, this reaction leads to the promotion of the oxidation. For the special case in which H-abstraction from the γ-position is impossible even after the H-exchange reaction (*ho-γ), H-abstraction from the δ-position (R19) will be important.

For this case, the γ,γ-P(OOH)2 radical exists in partial equilibrium with ε-O2QOOH at low concentration of O2, and the third O2 addition reaction R28 may be important at high concentration of O2.

As shown in Figure 26, the third-O2 addition reactions are unimportant up to ∼100 atm for β,γ- and β,δ-P(OOH)2 but may be significant for γ,γ-P(OOH)2 at >10 atm. Since the γ,γP(OOH)2 radicals are formed from the relatively unimportant ε-O2QOOH radicals, the third-O2 addition reaction can be neglected for the small to medium size alkanes. For alkanes with 3316



ARTICLE

Table 3. High-Pressure Limiting Rate Constants for RO2 a R þ O2 forward rate constanta A/s1

substitutionb

reverse rate constanta

b

(Ea/R)/K

A/(cm3 molecule1 s1)

b

(Ea/R)/K

*d (decomposition to R þ O2) p

4.55 1025

3.593

18 430

1.14 107

1.627

100

s

25

2.15 10

3.209

18 930

5.79 1010

0.816

270

t

2.08 1026

3.094

1.62 1012

0.325

210

20 170

Rate constants are given by the modified Arrhenius expression, k = AT exp(Ea/RT). Substitutions “p”, “s”, and “t” mean the primary, secondary, and tertiary RO2.

a

b

b

Table 4. Rate Constants for R þ O2 Direct Hydrogen Abstraction Reaction rate constanta

rate constanta

substitutionb


(Ea/R)/K

substitutionb


(Ea/R)/K

pp

1.60 1011

8450

sp

1.20 1011

8080 c

ps

8.46 1012

7700

ss

7.61 1012

7500

pt

1.16 1012

7230

Rate constants are given by the Arrhenius expression k = A exp(Ea/RT). b Substitution “ps” means the secondary H-atom abstraction from a primary alkyl radical (R). Calculated only for the limited substitutions. c Rate constant for “sp” was evaluated as half of that for isopropyl þ O2 f propylene þ HO2. a

large chain, successive H-abstraction from “new” γ-positions may produce other types of P(OOH)2; for example,

for which the third-O2 addition may play some minor role in the low-temperature oxidation. In most of the previous modeling studies,3947 the O2QOOH radicals were assumed to decompose into hydroperoxycarbonyl compound þ OH exclusively via the internal H-abstraction from the -OOH sites. The detailed consideration above indicates that this is not the case, but the main story was not altered in the sense that most of the O2QOOH radicals decompose to OH þ hydroperoxycarbonyl compound or hydroperoxyoxirane which decomposes quite similarly to the hydroperoxycarbonyl, and the third-O2 addition reactions contribute little in many cases. The major difference from the previous modeling works is that the overall rate constants for the isomerization of O2QOOH are much larger than those for the H-shift reactions from the OOH sites only, since, in many cases, the isomerization occurs via the most preferable six-membered-ring transition states. The hydrogen exchange reactions between -OO• and -OOH groups further accelerate the overall process.

5. DISCUSSION 5.1. Evaluated Rate Parameters. The high-pressure limiting rate constants evaluated in the present study are listed in Tables 311. For all of the reactions investigated, the rate constants were calculated for the temperature range of 1000/ T = 0.82.0 with an interval of 0.1 and the Arrhenius expressions were derived by the linear regression. As described earlier, the rate constant for a class of substitution is that calculated for the minimum-sized representative reaction. The rate constants listed in Table 3 are the results of the variational transition-state theory

calculations on the CASPT2 potential energy curves with B3LYP/CBSB7 geometries and vibrational frequencies, details of which will be described elsewhere. The rate constants listed in Table 4 are those for the direct metathesis reactions via the transition state *m shown in Figure 4. These reactions were concluded to be unimportant since the rate constants are very small, which is less than 1% of the rate constant (∼1012 cm3 molecule1 s1) for the R þ O2 recombination reactions at 1000 K, as shown in Figure S1 of the Supporting Information. The formally direct reaction used in the modeling studies is the chemical activation reaction occurring via the alkylperoxy radical (P) and concerted HO2 elimination transition state (*e) in Figure 4 and will be important for relatively small alkyl radicals (R) at relatively low pressures. Rate constants for these chemical activation reactions were not evaluated in the present study. For some reactions, the rate constants for the reverse reactions are also listed in Tables 5 and 6. The reverse reactions of the concerted HO2 elimination (*e) from RO2 and β-CO fission (*h2d) of β-QOOH are listed mostly because they are important in the oxidation of alkenes. The rate constants for these reactions, HO2 addition to alkenes, are compared in Figure S2 in the Supporting Information. The rate constants are generally larger for the simple HO2 addition to form β-QOOH, but the contribution of the concerted HO2 addition to form RO2 cannot be neglected especially for the terminal alkenes as shown in Figure S2c. 5.2. Steady-Sate Analysis. As described earlier, the relative significance of the competing isomerization and decomposition reactions of RO2 radicals cannot be compared by the rate constants only, since the back-isomerization reactions are fast and QOOH radicals often exist in near partial equilibrium with RO2. For such a situation, the overall consumption of the RO2 radicals is not determined by the rate of isomerization but significantly depends on the subsequent reactions of the QOOH radicals including the second-O2 addition. To take into account this complex situation, a steady-state analysis was made by assuming 3317



ARTICLE

Table 5. High-Pressure Limiting Rate Constants for RO2 Reactions forward

reverse rate

rate constanta

constanta

Table 5. Continued

A/(cm substitution

A/s1

*e (concerted HO2 elimination) pp

5.47 10

ps pt

5.73 10 3.42 1012

sp

6.96 10

12

12

16 080 15 930 16 050 15 950

13

1.21 10

6150

14

6.41 10 4.95 1014

7370 8400

14

b

5.72 10

5170

13

ss

7.35 10

12

15 850

1.26 10

st

7.06 1012

15 780

3.91 1014

6820

14

c

tp

1.26 10

13

15 680

5.85 10

ts

1.77 1013

15 330

3.96 1014

tt

3.82 10

13

*h1 (1,3-H shift) pp

1.09 10

13

ss

5.36 10

12

substitution

15 860

6210 4280

A/s1

(Ea/R)/K

11 610

5.72 108

3030

15

2.90 10

1.94 1010

ps

1.19 10

9 700

3.43 108

2950

pt

5.73 109

8 430

1.20 108

2730

sp

4.53 1010

12 030

1.51 109

3080

ss

2.77 1010

10 400

5.36 108

2710

st

1.61 1010

8 900

2.06 108

2450

tp ts

2.81 1010 1.94 1010

12 530 10 740

7.49 108 4.91 108

3810 2750

1.02 1010

9 260

3.23 108

2500

pp

1.22 109

12 440

3.67 107

4030

ps

8

8.25 10

10 880

2.71 107

4030

sp

2.26 109

12 910

5.16 107

3620

ss

1.50 109

11 250

3.63 107

3640

10

*h6 (1,8-H shift)

5180

20 340

pp

tt

4940

d

a

19 550

forward

reverse

rate constanta

rate constanta

A/s1

(Ea/R)/K

*h5 (1,7-H shift)

(Ea/R)/K molecule1 s1) (Ea/R)/K

12


A/s1

substitution

3

forward rate constanta

(Ea/R)/K

A/s1

(Ea/R)/K

*h2 (1,4-H shift)

Rate constants are given by the Arrhenius expression, k = A exp(Ea/ RT). b Forward rate constant was evaluated as half of the rate constant for isopropylperoxy radical. c Forward rate constant was evaluated as one-third of the rate constant for tert-butylperoxy radical. d Calculated only for limited set of substitutions.

the steady state for QOOH and O2QOOH and by using the highpressure limiting rate constants listed in Tables 311. The reaction scheme considered for each isomerization channel is shown in Figure 27. For this scheme, the steady-state concentrations of O2QOOH and QOOH are expressed by

pp

1.26 1012

16 930

4.49 1010

7920

ps

1.02 10

12

15 130

1.73 1010

7920

pt sp

6.95 1011 1.54 1012

13 660 16 720

1.56 1010 5.81 1010

8330 7440 b

ss

8.50 10

11

15 310

1.37 10

st

5.41 1011

13 690

5.81 109

8480

tp

1.80 1012

16 810

8.59 1010

7270 c

ts

1.44 1012

14 510

2.55 1010

7270

tt

9.76 10

11

12 980

7.31 109

7520

and an effective first-order rate constant for the consumption of RO2 is given by

pp ps

3.07 1011 2.24 1011

11 560 9 900

1.21 1010 7.38 109

3080 3100

pt

1.31 1011

8 300

2.95 109

2880

sp

3.92 1011

11 360

6.23 1010

2560

ss

5.26 1011

9 970

6.17 1010

2770

st

2.42 1011

8 480

2.00 1010

2980

keff ¼ kðChBrÞ þ kðnbÞ ½QOOHss ½O2 QOOHss kðChBrÞ ¼ k4 ¼ rqk4 ½RO2 ½QOOHss ½QOOHss k2 ¼ rk2 kðnbÞ ¼ ½RO2

tp

5.05 1011

11 310

3.35 1010

2160

ts

3.48 1011

9 960

3.97 1010

2520

tt *h4 (1,6-H shift)

3.57 1011

8 500

8.50 109

2710

pp

4.36 1010

11 020

9.24 108

2700

ps

3.39 1010

9 220

1.29 109

2300

pt

1.53 1010

7 720

3.37 108

1980

sp

9.27 1010

11 210

2.85 109

2510

ss

7.93 1010

9 610

1.72 109

2000

st

3.28 1010

8 100

2.14 108

1290

tp ts

5.04 1010 5.78 1010

11 420 9 790

2.17 109 2.03 109

2450 1820

tt

3.27 1010

8 320

4.11 108

1260

10

8000

*h3 (1,5-H shift)

½O2 QOOHss k3 ½O2 ¼ k3 þ k4 ½QOOHss

ð5Þ

½QOOHss k1 ¼ ½RO2 k1 þ k2 þ k3 ½O2 k3 q

ð6Þ

q¼ r ¼

ð7Þ ð8Þ ð9Þ

where k(ChBr) and k(nb) denote the effective rate constants leading to chain branching and nonbranching processes, respectively. All of the reactions of RO2 shown in Figure 4 (*e, *h1, *h2, *h3, *h4, *h5, and *h6) except for the back-dissociation (*d) to R þ O2 were considered. As an approximation, some minor reactions of QOOH and O2QOOH were omitted from the analysis. Only the cyclic ether formation (*h2r) and CO fission (*h2d) were considered for the β-QOOH. Similarly, cyclic ether formation (*h3r) and β-fission reactions (*h3d þ *h3c) for γ-QOOH, cyclic ether formation (*h4r) for δ-QOOH, cyclic ether formation (*h5r) and 1,5-H shift (*h5s) for ε-QOOH, and cyclic ether formation (*h6r) and β-CC fission (*h6d) for ζ-QOOH were considered. The channels considered for O2QOOH are as follows: *h2-β, *e-β, and *h3 for β-O2QOOH, *h3-γ and *h4 for 3318



ARTICLE

Table 6. High-Pressure Limiting Rate Constants for β-QOOH Reactions forward rate constanta


A/s1

(Ea/R)/K


(Ea/R)/K

pp

9.80 1012

7 510

ps

6.22 1012

6 690

pt

1.07 1013

6 470

sp

7.43 1012

6 460

ss

3.02 1012

5 970

st

4.61 1012

5 870

tp ts

5.73 1012 7.70 1012

5 770 5 400

tt

4.59 1012

5 640

pp

8.80 1012

ps

6.25 10

12

8 660

5.48 1012

7750

8 660

4.11 1012

pt

7320

8.91 1012

8 650

5.74 1012

6330

sp

1.14 1013

8 610

2.50 1012

7110

ss st

6.68 1012 1.02 1013

8 460 9 100

7.08 1012 5.23 1012

6120 5370

tp

1.39 1013

8 080

1.35 1012

6210

ts

5.22 10

12

8 130

6.58 1013

4980

tt

9.16 1012

8 970

9.27 1014

3750

pp

6.99 1011

16 180

ss

7.60 1011

12 610

*h2s (1,2-H shift) b pp

2.31 1013

16 160

ps

3.36 1013

16 350

ss

1.95 1012

13 780

substitution *h2r (cyclic ether formation)

*h2d (β-CO fission)

*h2t (OH transfer) b

a

Rate constants are given by the Arrhenius expression, k = A exp(Ea/RT). b Calculated only for limited set of substitutions.

γ-O2QOOH, *h4-δ and *h3 for δ-O2QOOH, *h5-ε and *h4 for ε-O2QOOH and *h6-ζ for ζ-O2QOOH. As an exception, *e-β reaction which produces HO2 instead of OH was counted as a nonbranching reaction. The results are shown for some secondary alkylperoxy radicals in Figures 28 and 29. The contribution of each channel is indicated by the vertical width of the plot. The channels are indicated by the symbols of the transition states shown in Figure 4. For each isomerization channel, the fraction of the chain branching (indicated with “-ChBr”) and nonbranching (“-nb”) processes are shown separately. At 800 K, as shown in Figure 28a, 2-butylperoxy radical is dominantly consumed by the concerted HO2 elimination reaction (*e) up to ∼10 atm (air) with some minor contribution of nonbranching 1,4- (*h2-nb) and 1,5-H shift (*h3-nb) channels. However, at total pressures above ∼10 atm, the contribution of the chain-branching 1,5-H shift channel (*h3-ChBr) becomes larger. It should be remembered that the analysis is based on the high-pressure limiting rate constants, and the effect of the “pressure” in Figure 28 is essentially the effect of the partial pressure (or the concentration) of O2. The increase of *h3-ChBr at high pressure in Figure 28a is caused by the increase of the contribution of the second O2-addition to produce γ-O2QOOH in competition with the other reactions of γ-QOOH radical, β-fission (*h3dþ*h3c), cyclic ether formation (*h3r), and

back-isomerization (*h3) to RO2. Here, it is important to note, that the rate constant for the 1,5-H shift reaction (*h3) is more than 10 times larger than that for HO2 elimination (*e) at this temperature (800 K), but the dominant process is HO2 elimination up to ∼10 atm. This is because the rate constants for the subsequent reactions (*h3r and *h3d þ *h3c) of γ-QOOH is small and γ-QOOH is in near partial equilibrium with RO2. The relative fraction between *h3 and *e approaches the ratio of the rate constant (∼15:1) only at very high pressures (>1000 atm), where the rapid second-O2addition reaction makes the 1,5-H shift reaction effectively irreversible. The situation changes as the size of the alkylperoxy radical increases. Figure 28b shows the result for 2-pentylperoxy radical. For this case, the γ-QOOH radical is a secondary alkyl-type radical, while it is primary for 2-butyl. This increases the equilibrium constant between γ-QOOH and RO2 and also increases the rate constant for the cyclic ether formation reaction (*h3r). As a result, the contribution of the *h3 channel becomes significantly larger than the case of 2-butylperoxy even at low pressures. There is also a small contribution from the 1, 6-H shift reaction (*h4). The contribution of the chain-branching channel of 1,5-H shift (*h3-ChBr) becomes dominant above ∼10 atm. As a limiting long-chain radical, an analysis for 2-heptylperoxy radical, for which all the β, γ, δ, and ε positions 3319



ARTICLE

Table 7. High-Pressure Limiting Rate Constants for γQOOH Reactions

Table 7. Continued rate constanta

rate constanta substitution

1

A/s

substitution ts tt

(Ea/R)/K

A/s1

(Ea/R)/K

2.38 1011 6.99 1010

10 950 11 090

*h3s (1,3-H shift) c

*h3r (cyclic ether formation) pp ps

6.99 10 1.21 1012

10 140 8 970

pp

4.65 1011

18 610

ps

2.39 1011

18 750

pt

6.03 1011

7 690

ss

5.05 1011

17 840

sp

1.26 1012

9 220

ss

1.53 1012

8 370

st

3.75 1012

7 630

tp

1.06 1012

8 380

ts

6.26 1011

7 510

tt *h3r/T (formation of cyclic

6.95 1011

6 470

pTp

1.23 1012

9 670

pTs

3.00 1012

8 990

pTt

1.58 1012

7 190

sTp

8.63 1012

8 630

sTs

2.50 1012

8 340

sTt tTp

6.15 1012 3.18 1012

6 410 8 250

tTs

1.09 1012

6 370

tTt

9.38 1011

6 010

pQp

3.69 1011

8 070

pQs

1.08 1012

7 700

pQt sQp

9.04 1011 3.36 1011

6 970 7 020

sQs

1.10 1012

6 530

sQt

1.63 1012

6 500

tQp

1.26 1012

6 900

tQs

3.52 1011

6 150

tQt

4.33 1011

5 820

11

ethers with tertiary carbon in ring)

*h3r/Q (formation of cyclic ethers with quaternary carbon in ring)

*h3d (β-CC fission) þ*h3c (concerted CC/OO fission) b pp

1.95 1014

14 720

ps

1.57 1014

13 830

pt

9.09 1013

13 590

sp

4.40 1014

13 720

ss

2.59 1014

13 740

st

2.37 1014

13 620

tp

2.70 1014

12 110

ts tt

2.09 1014 1.44 1014

12 410 12 090

pp

6.30 1010

11 730

ps

5.37 1010

11 300

pt

2.73 1010

10 750

sp

2.96 1011

11 410

ss

3.42 1011

11 170

st tp

1.90 1011 1.30 1011

11 080 10 850

*h3t (OH transfer)

a

Rate constants are given by the Arrhenius expression, k = A exp(Ea/ RT). b The “concerted CC/OO fission” denotes the direct three-body dissociation reaction shown in R11 (see text for details). Since the βCC fission (R9) followed by the rapid OO bond cleavage (R10) gives the identical products as R11, sum of the rate constants for two reactions R9 and R11 are given here. c Calculated only for limited set of substitutions.

are secondary, is shown in Figure 28c. The 1,4- (*h2) and 1,7(*h5) H shift reactions are relatively unimportant due to the small rate of isomerization. At low partial pressure of O2, the relative contribution between 1,5- (*h3) and 1,6- (*h4) H shift reactions is determined by the rates of subsequent decomposition of γ-QOOH and δ-QOOH, not by the rates of isomerization which is much larger for 1,5-H shift than for 1,6-H shift. At large p(O2), the ratio between *h3 and *h4 is determined by the rates of isomerization due to the fast second-O2-addition reactions to QOOH. For this representative long-chain radical, the temperature dependence of the fractional contribution is also shown in Figure 29. The results of the steady-state analysis for more variety of alkylperoxy radicals are shown in Figures S3S14 of the Supporting Information. To compare the relative autoignitability of straight-chain alkanes, the sum of the k(ChBr) calculated by eq 8 for all of the isomerization channels are compared for some representative C3 to C7 secondary alkylperoxy radicals in Figure 30. The calculated ksum(ChBr) roughly represents the ignitability of alkanes (research octane numbers for propane (C3) to heptane (C7) are 112, 94, 62, 25, and 0). The slope of the plot is almost exactly unity at low pressures, indicating that the second-O2 addition is the rate-determining step for this chainbranching rate constant but significantly lowered at high pressures due to the switchover of the rate-determining steps. This gives a semiquantitative rationalization of the wellknown scaling rule for the ignition-delay times66 by Pn with n e 1, where P is the total pressure. Some more results for the other types of RO2 are shown in Figures S15 and S16 of the Supporting Information. The results of the steady-state analysis for several typical alkylperoxy radicals, shown in Figures 28 and 29, and Figures S3S14, can be summarized as follows: The 1,3-H shift (*h1) and 1,8-H shift (*h6) reactions were concluded to be negligible. The 1,5-H shift (*h3) and 1,6-H shift (*h4) reactions are dominant reactions when they are possible for relatively large alkyl radicals. The concerted HO2 elimination (*e) and 1,4-H shift (*h2) reactions are important for small or tertiary radicals. The contribution of 1,7-H (*h5) shift reaction is generally small but not negligible. 5.3. Rules for Rate Parameter Estimation. Tables 311 serve as the complete set of data necessary for the estimation of the rate constants of arbitrary noncyclic alkylperoxy radicals 3320



ARTICLE

Table 8. High-Pressure Limiting Rate Constants for δQOOH Reactions

Table 9. High-Pressure Limiting Rate Constants for εQOOH Reactions

rate constanta

rate constanta

A/s1

(Ea/R)/K

A/s1

(Ea/R)/K

pp

7.08 1010

6430

ps pt

11

pp

7.48 109

1.26 10 1.15 1011

6180

5270 4380

ps pt

5.99 109 2.81 109

5200 4830

sp

5.48 1010

6650

sp

5.43 109

7550

ss

10

7.45 10

5150

ss

2.28 109

6140

st

2.80 1010

3920

st

8.98 108

5750

tp

3.65 1010

6180

tp

4.50 109

9400

ts

3.40 1010

4710

ts

2.19 109

7430

tt

1.14 1010

3510

tt

2.48 109

5830

*h5s (1,5-H shift) pp

substitution

substitution

*h4r (cyclic ether formation)

*h5r (cyclic ether formation)

*h4r/Q (formation of cyclic ethers with quaternary carbon in ring)

1.41 1010

6330

pQSp

2.59 1011

5250

ps

7.43 109

6840

pSQp

9.49 1010

4990

pt

4.08 109

6850

pQSs

2.81 1011

4590

sp

4.59 109

5370

pSQs

5.35 1010

4170

ss

3.30 109

5430

pQSt

3.03 1011

4380

st

2.95 109

5520

pSQt

8.92 1010

3360

*h5d (β-CC fission) b

sQSp sSQp

1.02 1011 8.59 1010

5460 4840

pp ss

9.39 1013 6.34 1013

14970 14710

sQSs

1.71 1011

4390


sSQs

8.45 10

3970

pp

4.32 109

7110

sQSt

4.58 10

2830

b

ss

3.50 109

6130

sSQt

4.58 1010

2830 b

tQSp

7.87 1010

4810

tSQp

8.44 1010

4910

tQSs tSQs

5.55 1010 5.55 1010

3620 b 3620 b

tQSt

1.87 10

2430

tSQt

1.87 1010

2430 b

pp

8.03 1013

14860

ps

14

1.76 10

15210

sp

8.00 1013

15350

ss *h4t (OH transfer) c

7.85 1013

14920

pp

7.59 109

8180

ps

1.36 1010

8010

sp

1.72 1010

8320

ss

1.50 1010

7860

*h4d (β-CC fission)

10 10

10

b

a

Rate constants are given by the Arrhenius expression, k = A exp(Ea/ RT). b Calculated only for limited set of substitutions.

Table 10. High-Pressure Limiting Rate Constants for ζQOOH Reactions rate constanta

c

A/s1

(Ea/R)/K

pp

6.48 108

7500

ss

3.45 108

6740

pp

1.31 1014

15160

ps

1.52 1014

15390

sp

1.31 1014

15080

ss *h6s (1,6-H shift) b

1.49 1014

15340

substitution *h6r (cyclic ether formation) b

*h6d (β-CC fission) b

*h4s (1,4-H shift) c pp

3.08 1010

9910

pp

6.42 108

6000

ps sp

1.06 1011 3.91 1010

10300 8980

ss

5.94 108

5270

ss

4.38 1010

8940

pp

1.75 109

7760

ss

1.47 109

7040


a

Rate constants are given by the Arrhenius expression, k = A exp(Ea/ RT). b Estimated. c Calculated only for limited set of substitutions.

since they cover the complete set of substitution patterns, pp, ps, ..., tt, for the important classes of reactions. However, it will be also interesting and important to derive “rules” in terms of the changes of the Arrhenius parameters for a series of rate

a

Rate constants are given by the Arrhenius expression, k = A exp(Ea/ RT). b Calculated only for limited set of substitutions.

constants evaluated in the present study. The results are summarized in Table S3. For each class of reaction, substitution effect was evaluated as the geometric mean of the ratios of 3321



ARTICLE

Table 11. Continued

Table 11. High-Pressure Limiting Rate Constants for O2QOOH Reactions

rate constanta

rate constanta substitution

1

A/s

substitution

(Ea/R)/K

*e-β (concerted HO2 elimination involving β-COOH site) pp ps

3.79 1012 4.54 1012

15 720 16 480

sp

2.52 1013

16 320

ss

2.33 1013

15 790

tp

9.08 1012

15 000

ts

2.23 1013

15 620

pp

1.83 1011

13 340

ps

3.49 1011

12 160

sp

8.96 1011

15 120

ss

2.16 1012

12 940

tp ts

1.21 1012 1.30 1012

14 190 12 530

pp

8.69 108

6 410

ps pt

6.75 109 3.84 109

6 630 6 330

sp

9.34 109

7 110

ss

1.29 1010

6 590

st

9.06 109

5 840

tp

4.48 109

6 820

ts

8.78 109

7 290

tt

8.57 109

6 310

*h2-β (1,4-H shift from β-COOH site)

A/s1

(Ea/R)/K

*ho-δ (H shift from OOH at δ-position) pp

9.04 107

7 790

ps

4.22 108

7 470

pt

1.75 108

7 580

sp

2.34 108

8 010

ss

2.74 108

7 720

st

1.70 108

7 710

tp

1.69 108

7 650

ts tt

3.95 108 7.00 107

7 320 7 340

pp ps

1.19 1010 5.73 109

9 700 b 8 430 b

sp

2.77 1010

10 400 b

ss

*h5-ε (1,7-H shift from ε-COOH site)

*ho-β (H shift from OOH at β-position)

1.61 10

10

8 900 b

tp

1.94 10

10

10 740 b

ts

1.02 10

10

9 260 b

*ho-ε (H shift from OOH at ε-position) pp

9.04 106

8 050 c

ps

4.22 10

7

7 720 c

pt

1.75 10

7

7 830 c

sp

2.34 10

7

8 260 c

ss

2.74 10

7

7 980 c

st tp

7

1.70 10 1.69 107

7 960 c 7 900 c

ts

3.95 107

7 580 c

tt

7.00 10

7 590 c

6

a

*h3-γ (1,5-H shift from γ-COOH site) pp

9.35 1010

9 610

ps

1.07 1011

8 520

sp ss

1.67 1011 1.64 1011

9 330 8 290

tp

1.32 1011

9 460

ts

1.78 1011

8 510

Rate constants are given by the Arrhenius expression, k = A exp(Ea/ RT). b Estimated from the corresponding RO2 reactions. c Estimated.

Figure 27. Scheme considered for each isomerization channel in the steady-state analysis.

*ho-γ (H shift from OOH at γ-position) pp

1.21 109

7 080

ps

3.22 109

6 900

pt

1.95 10

9

6 400

sp

2.85 109

7 150

ss

5.54 109

6 980

st

5.09 109

6 490

tp

1.21 109

6 520

ts tt

3.47 109 3.56 109

6 330 7 140

pp

1.47 1010

8 600

ps

1.47 1010

7 620

sp

2.48 1010

9 000

ss

2.14 1010

8 050

tp ts

2.40 1010 2.10 1010

8 930 7 990

*h4-δ (1,6-H shift from δ-COOH site)

the preexponential factors and arithmetic mean of the difference of the activation energies. For example, for the effect of one methyl substitution on the target site of hydrogen abstraction, f_s(A) was calculated as the geometric mean of Aps/App, Ass/Asp, and Ats/Atp, and Δ_s(Ea/R) was the arithmetic mean of (Ea,ps/R) (Ea,pp/R), (Ea,ss/R) (Ea,sp/R), and (Ea,ts/R) (Ea,tp/R). Since clear systematic behavior was found for the E0 of the H-shift reactions (*h2, *h3, *h4, and *h5) of RO2 as shown in Figure 5, the rules for this type of reactions can be derived unambiguously. One methyl substitution on the target site decreases the Ea/R by ∼1690 K (ranging from 1470 to 1840 K) and two methyl substitutions by ∼3180 K (29803380 K). The effect of the second methyl substitution, ∼1490 K (=31801690), is slightly smaller than the first one but is approximately the same. As a consequence, the rule may be unified and simplified for all H-shift reactions of RO2, as each methyl substitution on the target site decreases Ea/R by ∼1590 K. 3322



ARTICLE

Figure 29. Steady-state contribution of the unimolecular processes of the 2-heptylperoxy radical as a function of temperature under 10 atm air (2 atm O2).

Figure 30. Sum of the effective rate constants for chain-branching channels, ksum(ChBr), of some secondary RO2 radicals as a function of partial pressure of O2 at 800 K.

substitution effect is not additive as can be found for tt substitution on the oxirane formation reaction shown in Figure 7a. For this case, it seems to be better to use the tabulated ktt for *h2r in Table 6, rather than that estimated from the rules in Table S3. Figure 28. Steady-state contribution of the unimolecular processes of some secondary RO2 radicals as a function of partial pressure of O2 at 800 K.

The effects on the A-factors are also similar for the substitution on the target site; that is, f_s(A) ≈ 0.78 and f_t(A) ≈ 0.45 for *h2 to *h5. Although these values do not coincide with the assumption that A-factors are proportional to the number of the hydrogen atoms on the site, f_s(A) = 2/3 and f_t(A) = 1/3, the assumption was found to be a fairly good approximation. It should be noted that the methyl substitution on the carbon with the -OO group of the RO2 radical also shows a nonnegligible effect especially for the eight-membered-ring 1,7-H shift reaction, for which methyl substitution increases the activation energy significantly probably due to the increase of the ring-strain energy as discussed in the previous section. The rules thus derived for the methyl (alkyl) substitution are intuitive and will be helpful for the estimation of the unknown reactions, for example, the reactions of -OH or -C(dO)OH substituted alkyl peroxy radicals, but, in some cases, the

6. CONCLUSIONS The reactions of RO2, QOOH, and O2QOOH important in the gas-phase low-temperature oxidation of alkanes have been investigated by the CBS-QB3 quantum chemical method. The variations of the barrier heights and rate constants by the methyl substitution were studied systematically so that the estimated rate constants can be used to construct the mechanisms for noncyclic alkanes with arbitrary structures. The methyl substitution effect can be well-interpreted by the group additivity and the ring strain effect. The rate constants for the competing processes of QOOH and O2QOOH radicals are compared to extract the important subsequent reactions of these radicals. The evaluated rate constants were also used to evaluate the effective rate of consumption of RO2 by an analysis assuming the steady-state for QOOH and O2QOOH concentrations in order to extract the essential processes needed for chemical kinetic modeling of the low-temperature oxidation. Importance of some reactions of O2QOOH which have not been considered in the modeling is suggested. 3323



’ ASSOCIATED CONTENT Table S1 listing the ÆS2æ eigenvalues and T1 diagnostics for typical transition-state calculations, Table S2 comparing the energies of the critical points on the C2H5 þ O2 potential energy surface calculated in the present study and in the previous theoretical investigations, figures comparing the rate constants for R þ O2 (Figure S1) and HO2 þ alkenes (Figure S2), Figures S3S16 showing the results of the steady-state analyses for some typical RO2 radicals, and Table S3 listing the rules for the estimation of the rate parameters. This material is available free of charge via the Internet at http://pubs. acs.org.

bS

Supporting Information.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by JX Nippon Oil & Energy Corp. ’ REFERENCES (1) Najt, P. M.; Foster, D. E. SAE Tech. Pap. Ser. 1983No. 830264. (2) Lovell, W. G. Ind. Eng. Chem. 1948, 40, 2388–2438. (3) Baldwin, R. R.; Walker, R. W.; Yorke, D. A. J. Chem. Soc., Faraday Trans. 1 1973, 69, 826–832. (4) Baldwin, R. R.; Walker, R. W.; Walker, R. W. J. Chem. Soc., Faraday Trans. 1 1980, 76, 825–837. (5) Baldwin, R. R.; Dean, C. E.; Walker, R. W. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1445–1455. (6) McAdam, K. G.; Walker, R. W. J. Chem. Soc., Faraday Trans. 2 1987, 83, 1509–1517. (7) Slagle, I. R.; Feng, Q.; Gutman, D. J. Phys. Chem. 1984, 88, 3648–3653. (8) Wagner, A. F.; Slagle, I. R.; Sarzynski, D.; Gutman, D. J. Phys. Chem. 1990, 94, 1853–1868. (9) Kaiser, E. W. J. Phys. Chem. 1995, 99, 707–711. (10) Kaiser, E. W. J. Phys. Chem. A 2002, 106, 1256–1265. (11) Clifford, E. P.; Farrell, J. T.; DeSain, J. D.; Taatjes, C. A. J. Phys. Chem. A 2000, 104, 11549–11560. (12) DeSain, J. D.; Clifford, E. P.; Taatjes, C. A. J. Phys. Chem. A 2001, 105, 3205–3213. (13) DeSain, J. D.; Taatjes, C. A. J. Phys. Chem. A 2001, 105, 6646–6654. (14) DeSain, J. D.; Taatjes, C. A.; Miller, J. A.; Klippenstein, S. J.; Hahn, D. K. Faraday Discuss. 2001, 119, 101–120. (15) DeSain, J. D.; Klippenstein, S. J.; Miller, J. A.; Taatjes, C. A. J. Phys. Chem. A 2003, 107, 4415–4427. (16) DeSain, J. D.; Klippenstein, S. J.; Taatjes, C. A. Phys. Chem. Chem. Phys. 2003, 5, 1584–1592. (17) Estupinan, E. G.; Klippenstein, S. J.; Taatjes, C. A. J. Phys. Chem. B 2005, 109, 8374–8387. (18) Taatjes, C. A. J. Phys. Chem. A 2006, 110, 4299–4312. (19) Petway, S. V.; Ismail, H.; Green, W. H.; Estupinan, E. G.; Jusinski, L. E.; Taatjes, C. A. J. Phys. Chem. A 2007, 111, 3891–3900. (20) Quelch, G. E.; Gallo, M. M.; Shen, M.; Xie, Y.; Shaefer, H. F., III; Moncrieff, D. J. Am. Chem. Soc. 1994, 116, 4953–4962. (21) Ignatyev, I. S.; Xie, Y.; Allen, W. D.; Schaefer, H. F., III. J. Chem. Phys. 1997, 107, 141–155. (22) Rienstra-Kiracofe, J. C.; Allen, W. D.; Schaefer, H. F., III. J. Phys. Chem. A 2000, 104, 9823–9840. (23) Wilke, J. J.; Allen, W. D.; Schaefer, H. F., III J. Chem. Phys. 2008, 128, No. 074308.

ARTICLE

(24) Chan, W. T.; Hamilton, I. P.; Pritchard, H. O. Phys. Chem. Chem. Phys. 1999, 1, 3715–3719. (25) Miller, J. A.; Klippenstein, S. J.; Robertson, S. Proc. Combust. Inst. 2000, 28, 1479–1486. (26) Miller, J. A.; Klippenstein, S. J. Int. J. Chem. Kinet. 2001, 33, 654–668. (27) Chen, C.-J.; Bozzelli, J. W. J. Phys. Chem. A 1999, 103, 9731–9769. (28) Bozzelli, J. W.; Sheng, C. J. Phys. Chem. A 2002, 106, 1113–1121. (29) Sheng, C. J.; Bozzelli, J. W.; Dean, A. M.; Chang, A. Y. J. Phys. Chem. A 2002, 106, 7276–7293. (30) Sun, H.; Bozzelli, J. W. J. Phys. Chem. A 2004, 108, 1694–1711. (31) Sun, H.; Bozzelli, J. W.; Law, C. K. J. Phys. Chem. A 2007, 111, 4974–4986. (32) Zhu, L.; Bozzelli, J. W.; Kardos, L. M. J. Phys. Chem. A 2007, 111, 6361–6377. (33) Asatryan, R.; Bozzelli, J. W. J. Phys. Chem. A 2010, 114, 7693–7708. (34) Wijaya, C. D.; Raman, S.; Green, W. J., Jr. J. Phys. Chem. A 2003, 107, 4908–4920. (35) Sharma, S.; Raman, S.; Green, W. H. J. Phys. Chem. A 2010, 114, 5689–5701. (36) Huynh, L. K.; Carstensen, H.-H.; Dean, A. M. J. Phys. Chem. A 2010, 114, 6594–6607. (37) Davis, A. C.; Francisco, J. S. J. Phys. Chem. A 2010, 114, 11492–11505. (38) Goldsmith, C. F.; Green, W. H.; Klippenstein, S. J. Presented at the 21st International Symposium on Gas Kinetics, Leuven, Belgium, Jul. 18-22, 2010. (39) Koert, D.; Pitz, W. J.; Bozzelli, J. W.; Cernansky, N. P. Proc. Combust. Inst. 1996, 26, 633–640. (40) Curran, H. J.; Gaffuri, P.; Pitz, W. J.; Westbrook, C. K. Combust. Flame 1998, 114, 149–177. (41) Curran, H. J.; Gaffuri, P.; Pitz, W. J.; Westbrook, C. K. Combust. Flame 2002, 129, 253–280. (42) Buda, F.; Bounaceur, R.; Warth, V.; Glaude, P. A.; Fournet, R.; Battin-Leclerc, F. Combust. Flame 2005, 142, 170–186. (43) Griffiths, J. F.; Hughes, K .J.; Schreiber, M.; Poppe, C. Combust. Flame 1994, 99, 533–540. (44) Ranzi, E.; Gaffuri, P.; Faravelli, T.; Dagaut, P. Combust. Flame 1995, 103, 91–106. (45) Muharam, Y.; Warnatz, J. Phys. Chem. Chem. Phys. 2007, 9, 4218–4229. (46) Westbrook, C. K.; Pitz, W. J.; Herbinet, O.; Curran, H. J.; Silke, E. J. Combust. Flame 2009, 156, 181–199. (47) Dagaut, P.; Hadj-Ali, K. Energy Fuels 2009, 23, 2389–2395. (48) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864–B871. (49) Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133–A1138. (50) Benson, S. W. Thermochemical Kinetics, 2nd ed.; John Wiley & Sons: New York, 1976. (51) Domalski, E. S.; Hearing, E. D. J. Phys. Chem. Ref. Data 1993, 22, 805–1159. (52) Ritter, E. R.; Bozzelli, J. W. Int. J. Chem. Kinet. 1992, 23, 767–778. (53) Frisch, M. J.; et al. Gaussian 03, Revision E.01; Gaussian: Wallingford, CT, 2004. (54) Montgomery, J. A., Jr.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 1999, 110, 2822–2827. (55) Montgomery, J. A., Jr.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 2000, 112, 6532–6542. (56) Garrett, B. C.; Truhlar, D. G. J. Phys. Chem. 1979, 83, 2921–2926. (57) Miyoshi, A. GPOP program suite, Revision 2010.02.14m3, available at http://www.frad.t.u-tokyo.ac.jp/∼miyoshi/gpop/. (58) Pitzer, K. S.; Gwinn, W. D. J. Chem. Phys. 1942, 10, 428–440. (59) Miyoshi, A. BEx1D program, Revision 2008.10.08, available at http://www.frad.t.u-tokyo.ac.jp/∼miyoshi/bex1d/. 3324



ARTICLE

(60) Miyoshi, A. Int. J. Chem. Kinet. 2010, 42, 273–288. (61) Vaghjiani, G. L.; Ravishankara, A. R. J. Phys. Chem. 1989, 93, 1948–1959. (62) Lee, T. J.; Taylor, P. R. Int. J. Quantum Chem., Quantum Chem. Symp. 1989, 23, 199–207. (63) Baulch, D. L.; Bowman, C. T.; Cobos, C. J.; Cox, R. A.; Just, Th.; Kerr, J. A.; Pilling, M. J.; Stocker, D.; Troe, J.; Tsang, W.; Walker, R. W.; Warnatz, J. J. Phys. Chem. Ref. Data 2005, 34, 757–1397. (64) Miyoshi, A.; Yamauchi, N.; Kosaka, K.; Koshi, M.; Matsui, H. J. Phys. Chem. A 1999, 103, 46–53. (65) Yamauchi, N.; Miyoshi, A.; Kosaka, K.; Koshi, M.; Matsui, H. J. Phys. Chem. A 1999, 103, 2723–2733. (66) Gauthier, B. M.; Davidson, D. F.; Hanson, R. K. Combust. Flame 2004, 139, 300–311.

3325


Systematic Computational Study on the Unimolecular Reactions of

Recommend Documents