High-Pressure Rate Rules for Alkyl + O2 Reactions. 2. The

Article pubs.acs.org/JPCA

High-Pressure Rate Rules for Alkyl + O2 Reactions. 2. The Isomerization, Cyclic Ether Formation, and β-Scission Reactions of Hydroperoxy Alkyl Radicals Stephanie M. Villano, Lam K. Huynh,† Hans-Heinrich Carstensen,‡ and Anthony M. Dean* Chemical and Biological Engineering Department, Colorado School of Mines, Golden, Colorado 80301, United States S Supporting Information *

ABSTRACT: The unimolecular reactions of hydroperoxy alkyl radicals (QOOH) play a central role in the low-temperature oxidation of hydrocarbons as they compete with the addition of a second O2 molecule, which is known to provide chain-branching. In this work we present high-pressure rate estimation rules for the most important unimolecular reactions of the β-, γ-, and δ-QOOH radicals: isomerization to RO2, cyclic ether formation, and selected β-scission reactions. These rate rules are derived from high-pressure rate constants for a series of reactions of a given reaction class. The individual rate expressions are determined from CBS-QB3 electronic structure calculations combined with canonical transition state theory calculations. Next we use the rate rules, along with previously published rate estimation rules for the reactions of alkyl peroxy radicals (RO2), to investigate the potential impact of falloff effects in combustion/ignition kinetic modeling. Pressure effects are examined for the reaction of n-butyl radical with O2 by comparison of concentration versus time profiles that were obtained using two mechanisms at 10 atm: one that contains pressure-dependent rate constants that are obtained from a QRRK/MSC analysis and another that only contains high-pressure rate expressions. These simulations reveal that under most conditions relevant to combustion/ignition problems, the high-pressure rate rules can be used directly to describe the reactions of RO2 and QOOH. For the same conditions, we also address whether the various isomers equilibrate during reaction. These results indicate that equilibrium is established between the alkyl, RO2, and γ- and δ-QOOH radicals.

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INTRODUCTION Detailed kinetic models that describe the low-temperature oxidation of hydrocarbon fuels are a valuable design tool that can be used to improve the efficiency and emissions of internal combustion engines. Currently, multiple mechanisms are available in the literature that are designed to predict the ignition behavior of single and multicomponent hydrocarbon fuels,1−5 and efforts are underway to extend these models toward surrogate mixtures for gasoline,6 diesel,7 and jet fuel.8,9 Such kinetic models have provided valuable insight into the observed negative temperature coefficient (NTC) behavior,10 soot formation,11 and engine knock.12 However, as engine technologies further advance and new fuels emerge, there continues to be a need for more accurate predictions over a wider range of conditions and for more diverse fuel structures and formulations. These mechanism upgrades consist of, among others, improving the accuracy of existing thermodynamic and kinetic parameters, incorporation of missing reaction pathways, and perhaps the inclusion of pressure-dependent rate coefficients to account for the wide range of operating conditions found in combustion devices. Typical low-temperature hydrocarbon oxidation mechanisms contain hundreds of species and thousands of reactions for which often very little or no experimental data are available. As a result, many of the required thermodynamic and kinetic input parameters are estimated. Though thermodynamic properties © 2012 American Chemical Society

can often be reliably estimated using group additivity methods, assignment of the required rate coefficients is more challenging. Commonly, unknown rate expressions are approximated with rate estimation rules for a given reaction class. In the past these rate rules had to be derived from very limited sets of published rate coefficients, resulting in considerable uncertainty in many of these assignments. More recently, however, computational chemistry has matured to a degree that allows it to be used to reliably predict thermodynamic and kinetic data with an accuracy that is generally comparable to many experimental methods. This approach offers the flexibility to access each reaction channel independently and under conditions that may be difficult to explore experimentally. Thus, rate estimation rules can now be derived by systematic calculating rate constants for a given reaction class using a test set of reactants.13,14 This presents the opportunity to verify and potentially improve the accuracy of previous assignments. The low-temperature oxidation of hydrocarbon fuels is largely governed by the reactions of alkyl peroxy radicals (RO2) and their hydroperoxy alkyl radical isomers (QOOH), shown in Figure 1. Depending upon the temperature and pressure, this sequence of reactions can either accelerate or decelerate the Received: March 12, 2012 Revised: April 30, 2012 Published: May 1, 2012 5068

dx.doi.org/10.1021/jp3023887 | J. Phys. Chem. A 2012, 116, 5068−5089

The Journal of Physical Chemistry A

Article

reaction of n-butyl radical reacting with O2. A comparison of concentration versus time profiles obtained using either the pressure-dependent rate constants or the corresponding highpressure values reveals that under most conditions relevant to combustion/ignition problems, the high-pressure rate rules can be used directly. The modeling results using the high-pressure mechanism are further examined to determine whether the reactants and intermediates equilibrate during the low temperature oxidation reactions.

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METHODS Electronic structure calculations were performed using the CBS-QB3 composite method21 as implemented in the Gaussian 03 software package.22 The CBS-QB3 method has been shown to predict heats of formation for a large test set of molecules21 with an accuracy of just over 1 kcal mol−1 and has been successfully applied in numerous kinetic studies,13,14,23 including those of small alkyl radical plus O2 reactions.16,24−29 Geometries, rotational constants, and harmonic frequencies are calculated at the B3LYP/6-311G(d,p) level of theory. The electronic energy is obtained by performing single point energy calculations at the CCSD(T)/6-31+G(d′) and MP4(SDQ)/631G+(d,p) levels of theory and extrapolating the MP2 energy to the complete basis set limit. Additional corrections for spin contamination and systematic errors further improve the energy. To obtain accurate ΔfH298, S298, and Cp values, low frequency vibrational modes that resemble torsions around single bonds were treated as hindered internal rotors rather than as harmonic oscillators. The hindrance potentials were calculated at the B3LYP/6-31G(d) level of theory via relaxed surface scans obtained with a step size of 10 degrees. Hindered potentials with barriers below 12 kcal mol−1 were fit to truncated Fourier series expansions. Reduced moments of inertia for asymmetric internal rotors were calculated at the I(2,3) level as defined by East and Radom on the basis of the equilibrium geometry of the species.30 For each internal rotor, the 1D Schrödinger equation was numerically solved using the eigenfunctions of the 1D free rotor as basis functions and the energy eigenvalues were then used to numerically calculate its contributions to the thermodynamic functions. All other modes were treated as harmonic oscillators and the unprojected frequencies were scaled by a factor of 0.99. The electronic energy of each species was converted to its heat of formation using the atomization method. Because only relative energies are required in this work, no attempts were made to improve the heats of formation using, for example, bond additivity corrections. Inspection of the hindered rotor potentials helps to ensure that the optimized geometry of a molecule corresponds to the lowest energy minimum. A normal-mode analysis was performed to identify the nature of the species. Transition states were identified by having one imaginary frequency, which was animated to verify that it corresponds to the desired reaction coordinate. High-pressure rate coefficients were calculated using canonical transition state theory (TST): k(T) = κ(T)·kBT/h·exp(−ΔG⧧/RT), where κ(T) is the tunneling correction factor, and ΔG⧧ is the Gibbs free energy difference between the transition state minus the contribution from the transitional mode and the reactants. The remaining variables have their usual meaning. Tunneling correction factors were calculated with an asymmetric Eckart potential,31 which has been shown to predict these correction factors with an accuracy comparable to that achieved with more sophisticated methods.32 Rate

Figure 1. Schematic diagram for the important reactions in the alkyl + O2 subset. The reactions investigated here are indicated by the red arrows.

overall ignition kinetics. In the first paper of this series we presented high-pressure rate estimation rules for the unimolecular reactions of RO2.15 The current work extends this set of rules to include unimolecular pathways of several QOOH isomers. QOOH radicals are primarily formed from RO2 isomerization. If isomerization proceeds via a six- or sevenmembered cyclic transition state structure, the γ- and δ-QOOH isomers, respectively, are produced. In the case of smaller sized RO2 species, isomerization via a five-membered cyclic transition state structure to form the β-QOOH isomer is also potentially important. The β-QOOH radical may also be formed from the addition of HO2 to olefins. (In this notation, “β”, “γ”, and “δ” indicate the location of the radical site in QOOH as being two, three, and four carbon atoms away from the hydroperoxy group, respectively.) Once formed, the QOOH radicals may either isomerize (e.g., to re-form RO2), decompose to bimolecular products, or react with molecular oxygen to form hydroperoxy alkyl peroxy radicals (O2QOOH). Subsequent reactions of O2QOOH radicals are known to lead to chain branching. Thus, it is important to properly account for the unimolecular reactions of QOOH to describe the chain branching processes correctly. In this work we employ electronic structure calculations at the CBS-QB3 level of theory combined with transition state theory (TST) to calculate high-pressure rate coefficients for isomerization to RO2, cyclic ether formation, and several βscission channels for the β-, γ-, and δ-QOOH radicals. These reaction pathways were identified from a more thorough investigation of potential reactions of the β-, γ, and δC4H8OOH isomers as being the dominant channels. The individual rate expressions for a set of selected reactants form the basis to develop rate rules for a given reaction class. Although several previous studies have already provided highpressure rate constants for some of these reactions,16−20 the current study is aimed to provide a consistent set of rate rules (for both the reactions of RO2 and QOOH) using the same level of theory and TST calculation methodology. As a result, the product branching ratios are expected to be more consistent. Furthermore, in contrast to a previous study,16 each rate rule is derived by systematically calculating rate coefficients for a series of small to intermediate sized reactants that have the same reaction moiety but different substituents. This leads to a range of individual values, which gives an indication as to the accuracy of the rate rule. In this study, we also examine the effect of pressure on the combined RO2 and QOOH chemistry using Quantum−Rice−Ramsperger−Kassel/ modified strong collision calculations (QRRK/MSC) for the 5069



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not very sensitive to this parameter.15 The Lennard-Jones parameters used for N2 were σ = 3.6 Å and ε = 98 K, whereas the ones for the n-C4H9 radicals were assumed to be the same as those of n-butane, which are σ = 5.4 Å and ε = 305 K.37 The temperature-dependent rate coefficients obtained from this analysis were fit to modified Arrhenius expressions for a temperature range of 500−1000 K in 50 K increments and a fixed pressure of 10 atm. Concentration versus time profiles at 500, 750, and 1000 K and at 10 atm for the reaction of n-butyl radical with molecular oxygen were obtained with the isothermal batch reactor module as implemented in the Chemkin Pro software package.38 The initial mole fraction of the n-C4H9 radical was arbitrarily set to 0.2% with the remainder being air. Under these conditions, the n-C4H9 + O2 reaction is pseudo-first-order with respect to the n-C4H9 radical.

constants were calculated over a temperature range of 300− 1500 K in 50 K increments and fit to modified Arrhenius expressions: k(T) = A·Tn·exp(−E/RT), where A is the preexponential factor, n is the temperature coefficient, and E is related to the activation energy by Ea = E + nRT. We estimate that the error in the calculated barrier heights is ±1.0 kcal mol−1 and the error in the pre-exponential factor is a factor of 2. Thus, the uncertainty in the calculated rate coefficients is roughly a factor of 2.6 near 1000 K and a factor of 3.4 near 500 K.13 These uncertainties arise from errors in the ab initio method such as variations in optimized reactant and TS geometries as well as errors in the harmonic frequencies and hindered rotor calculations. The individual rate coefficients serve as the basis to develop the rate rule for a given reaction class. For the isomerization and β-scission reactions, rate rules were determined by averaging the individual rate coefficients at each temperature and then fitting these averages to modified Arrhenius expressions. For the cyclic ether and C−C fission reaction classes, the investigated reactions were found to span a larger range of reaction exothermicities and the activation energies of the corresponding rate constants also varied substantially. Thus, the individual rate constants could not be approximated by simple rules as discussed above. Instead, the rate rules for these reaction classes were based on Evans−Polanyi relationships. More specifically, the activation energies were determined from least-squares fits of the Evans−Polanyi plots and the preexponential factors were taken as the average values of the preexponential factors of the individual rate constants for the test set of reactants. To more accurately determine these parameters, the individual activation energies and preexponential factors for each reaction class were represented by simple Arrhenius fits (k(T) = A·exp(−Ea/RT)) and the temperature range was reduced to 500−1000 K. The error in the rate rules, due to the variation in the individual rate constants within a given training set, is calculated as the rootmean-square deviation (rmsd). We report these values at 500 and 1000 K in the text below. However, the total error in the rate rules is the propagation of the random error and the systematic error discussed above. Using our high-pressure rate rules for the unimolecular reactions of both RO215 and QOOH, we evaluated the pressure and temperature dependence of the n-C4H9 plus O2 reaction system with Quantum−Rice−Ramsperger−Kassel (QRRK) theory.33 In the first paper in this series,15 we used the QRRK/MSC method to investigate the reactions of larger sized species for which high level ab initio calculations were not practical. In this work we continue to use this method to facilitate comparisons between the two studies. Prior comparisons to experimental results have shown that this approach is able to predict observed falloff behaviors with an accuracy that is comparable to more rigorous methods.34,35 In our implementation of this approach, three representative frequencies and their degeneracies are used to calculate the densities of states. To be consistent with our previous investigation, these values were derived from heat capacity data that were estimated using Benson’s group additivity method as implemented in the software THERM.36 Collisional stabilization rate coefficients were obtained using the modified strong collision (MSC) approximation.33 The average energy transferred per collision (ΔEall) with N2 as bath gas was assumed to be independent of temperature and was set at −154 cm−1; however, previous results suggest that the calculations are

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RESULTS AND DISCUSSION I. Unimolecular Channels of C4H8OOH. The QOOH isomers can undergo a variety of reactions. In this section we compare the high-pressure rate constants for several unimolecular channels with the goal of determining which of the reaction pathways are most important. The following reactions are considered: isomerization to RO2, H atom migration along the carbon backbone, OH migration, cyclic ether formation, and several β-scission channels. For this portion of the discussion we specifically focus on the reactions of the β-, γ-, and δ-hydroperoxy butyl radical isomers (C4H8OOH). The hydroperoxy butyl radical was chosen because it is the smallest QOOH radical that can accommodate all three isomers. These rate constants are presented in Figure 2a−c, respectively. For comparison, the pseudo-first-order rate constants for the addition of O2 to hydroperoxy n-butyl radicals at 1 and 40 atm of air are indicated by the black short- and long-dashed lines, respectively. The bimolecular rate coefficients used to obtain these values were assumed to be equivalent to those for the reactions of β- and γ-hydroperoxy propyl radical isomers + O2, which were calculated using variational TST calculations by Klippenstein and co-workers.39 Note that in Figure 1 and throughout the remaining text and tables we write reactions using a notation where the hydrogen atoms are assumed and the radical site is indicated by a “•” symbol. Comparison of the rate constants for the various nC4H8OOH isomers reveals that, even though there are multiple reaction channels, only a few pathways are important. However, which pathways dominate and to what extent they can compete with bimolecular O2 addition varies for the different isomers. For the β-C4H8OOH (Figure 2a), cyclic ether formation is the fastest channel. This reaction results in the formation of an oxirane plus OH. The subsequent abstraction reactions of OH radicals are exothermic and can potentially result in a temperature increase that can accelerate the overall oxidation process. The second most prominent channel is β-scission of HO2. At lower temperatures, this reaction effectively inhibits the overall oxidation process, because the HO2 radical is relatively nonreactive when compared to the OH radical. These two channels are sufficiently fast such that they can compete with the bimolecular addition of O2, especially at higher temperatures. The rate constant for isomerization to the nC4H9O2 radical, the next fastest channel, is 3 orders of magnitude lower than that for cyclic ether formation, whereas the rate constants for all remaining reactions are found to be even smaller. 5070



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migration is 2 orders of magnitude smaller than that for cyclic ether formation. The rate constants for H-atom migration reactions were found to be negligibly small. For the δ-isomer, the rate constants for both OH migration and H-atom migration from the α-position are at least 1 order of magnitude smaller than that for cyclic ether formation. The larger rate coefficients for the cyclic ether formation, OH migration, and H-atom migration reactions for the δ-isomer, as compared to those for the γ-isomer, are due to the greater distance between the δ-isomer radical site and the hydroperoxy group. As a result, these reactions proceed through energetically less strained larger sized cyclic transition states. However, even though the rate constants for OH and H-atom migration are larger, these reactions are still too slow to compete with bimolecular O2 addition, even at a relatively low pressure of 1 atm. Despite the weak O−O bond of the hydroperoxy group, the OH migration reactions were found to be slower than many of the other competing channels. This observation is consistent with results from previous studies.24,40 Green et al.40 investigated the competition between OH migration and cyclic ether formation for several hydroperoxy alkyl radicals, also using the CBS-QB3 method. For each investigated reactant, the cyclic ether reaction was found to be significantly faster than the OH shift reaction. The authors note, however, that the calculations of the transition state for OH migration in the γand δ-QOOH radials suffer from spin contamination. This leads to larger uncertainties of the barrier heights even though the CBS-QB3 method contains a correction term for this. To address this concern, the authors re-evaluated the reactants, transition states, and products of all reactions at the B3LYP/ CBSB7 level of theory, which is less sensitive to spin contamination. Although, in general, the CBS-QB3 barrier heights were found to be lower than the DFT ones, similar conclusions could be drawn using the results of either method. Our OH-migration transition state calculations also suffer from spin contamination. However, the other competing channels are sufficiently fast such that the conclusion that OH migration can be neglected should hold even if the CBS-QB3 derived rate constants were systematically low. This conclusion is supported by several detailed kinetic modeling studies of small alkyl plus O2 systems,24,41 which are able to satisfactorily reproduce experimental data without the inclusion of this reaction channel. The rate constants for various H atom migration reactions along the carbon backbone were also found to be significantly smaller than those of other competing reactions. Because in this section we focus on the reactions of the C4H8OOH isomers, most of the investigated H-shift reactions proceed either through a three- or four-membered cyclic transition states and therefore have large barriers. For the δ-isomer, H-atom migration from the α-carbon proceeds though an energetically more favorable five-membered ring transition state. The initial α-C4H8OOH product is unstable and dissociates to butanal + OH. Even though this reaction has a lower reaction barrier, it is still slower than the competing RO2 isomerization and cyclic ether formation reactions. For larger QOOH radicals, there is the potential to transfer a H-atom via a six-membered cyclic transition state from further down the carbon chain. We evaluated several of these reactions and found that their rate constants are also smaller than those for RO2 isomerization and cyclic ether formation. In general, the computed rate expressions for this reaction class are comparable to those obtained for H-atom migration reactions of alkyl radicals.42

Figure 2. Arrhenius plots of the various reactions of the (a) β-, (b) γ-, and (c) δ-hydroperoxy butyl radicals. In the reaction notation, the hydrogen atoms are assumed and a “•” symbolizes a radical site. The red lines correspond to the RO2 isomerization reaction, the blue lines to the cyclic ether formation reaction, the green lines to β-scission reactions, the gray lines to OH migration reactions, and the orange lines to H-atom migration along the carbon backbone. The short and long dashed black lines are the pseudo-first-order rate constants for bimolecular O2 addition at 1 and 40 atm of air, respectively.

For the γ-and δ-isomers (Figure 2b,c) isomerization to nC4H9O2 radical is favored. Because the n-C4H9O2 radical may isomerize again to the various C4H8OOH isomers, there is the possibility that these isomers equilibrate. This issue will be further addressed at the end of the discussion section. For both the γ-and δ-isomers, the next most prominent unimolecular channel is cyclic ether formation. At higher temperatures the rate constant for β-scission of a C−C bond is just slightly lower than that of the cyclic ether formation. This channel is especially important for the γ-isomer because the resulting CH2•OOH radical is unstable and spontaneously dissociates to form formaldehyde and OH. Both cyclic ether formation and βscission are competitive with bimolecular O2 addition at higher temperatures. For the γ-isomer, the rate constant for OH 5071


β-QOOH primary rate rule C•COOH → CCOO•c C2•COOH → C2COO• CCC(C•)OOH → CCC(C)OO• C3•COOH → C3COO• CCCC(C•)OOH → CCCC(C)OO• CCC(2C•)OOH → CCC(2C)OO• C2CC(C•)OOH → C2CC(C)OO• β-QOOH secondary rate rule CC•COOH → CCCOO• CCC•COOH → CCCCOO• CC•C(C)OOH → CCC(C)OO• CCCC•COOH → CCCCCOO• CCC•C(C)OOH → CCCC(C)OO• CC•C(CC)OOH → CCC(CC)OO• CC•C(2C)OOH → CCC(2C)OO• C2CC•COOH → C2CCCOO• β-QOOH tertiary rate rule C2C•COOH → C2CCOO• CCC•(C)COOH → CCC(C)COO• C2C•C(C)OOH → C2CC(C)OO• γ-QOOH primary rate rule C•CCOOH → CCCOO• C•CC(C)OOH → CCC(C)OO• C2•CCOOH → C2CCOO• C•CC(CC)OOH → CCC(CC)OO• CCC(C•)COOH → CCC(C)COO• C•CC(2C)OOH → CCC(2C)OO• C2•CC(C)OOH → C2CC(C)OO• C3•CCOOH → C3CCOO• γ-QOOH secondary rate rule CC•CCOOH → CCCCOO• CCC•CCOOH → CCCCCOO• CC•CC(C)OOH → CCCC(C)OO• CC•C(C)COOH → CCC(C)COO• γ-QOOH tertiary rate rule C2C•CCOOH → C2CCCOO• CCC•(C)CCOOH → CCC(C)CCOO• C2C•C(C)COOH → C2CC(C)COO• C2C•CC(C)OOH → C2CCC(C)OO•

reaction QOOH → RO2

b

7

9.13 × 10 3.47 × 106 3.02 × 107 6.54 × 107 2.93 × 107 2.49 × 108 1.12 × 109 1.16 × 108 1.16 × 106 2.95 × 105 5.98 × 105 2.01 × 107 8.40 × 105 1.55 × 107 8.14 × 106 5.15 × 106 2.62 × 106 1.67 × 104 7.79 × 104 1.49 × 105 8.34 × 105 2.77 × 108 1.09 × 108 1.83 × 108 8.89 × 107 1.09 × 109 5.15 × 109 8.39 × 108 9.63 × 108 2.82 × 108 3.58 × 107 2.25 × 107 6.42 × 106 2.06 × 108 6.04 × 107 5.97 × 105 4.79 × 105 6.45 × 105 1.01 × 106 1.56 × 106

A (s−1) 0.99 1.34 1.11 1.05 1.11 0.79 0.66 0.96 1.50 1.65 1.61 1.20 1.54 1.20 1.26 1.26 1.45 2.01 1.87 1.76 1.43 0.62 0.74 0.67 0.74 0.46 0.23 0.47 0.47 0.65 0.86 0.90 1.09 0.67 0.77 1.32 1.34 1.34 1.18 1.21

n 13.8 15.0 14.2 14.1 13.7 14.0 14.1 13.5 13.9 14.9 14.5 14.2 14.5 14.0 14.0 13.7 14.5 13.5 14.5 14.3 13.4 4.8 5.5 5.0 4.7 4.9 5.5 5.4 4.6 4.4 5.0 5.4 4.9 5.2 4.9 4.4 4.8 4.6 4.7 4.1

E (kcal mol−1)

modified Arrh. parameters

−4.9 −5.9 −4.9 −7.1 −4.4 −4.3 −4.7 −6.4 −5.4 −6.2 −5.1 −6.1 −8.5 −6.0 −4.4 −5.2 −5.2 −6.3 −3.7 −5.1 −4.3 −5.4 −1.7 −3.3 −6.7 −5.1 −4.8 −4.0 −5.7 −4.3 −5.1 −4.0 −5.4 −7.3

−14.0 −14.0 −14.4 −13.8 −14.6 −14.4 −14.6 −14.2 −11.0 −11.0 −11.6 −16.4 −16.7 −16.6 −16.6 −15.9 −16.5 −17.1 −17.6 −13.2 −13.1 −13.4 −13.0 −11.1 −10.7 −10.3 −11.8

ΔrxnS298 (cal mol−1 K−1)

−17.7 −18.2 −18.3 −19.0 −18.4 −18.4 −18.3

ΔrxnH298 (kcal mol−1)

thermochemistry

× 104e (0.11) (0.52) (0.86) (0.89) (0.74) (1.39) (1.63) × 104 (0.27) (0.60) (2.04) (0.53) (1.97) (1.47) (1.34) (0.93) × 103 (0.74) (0.93) (1.68) × 108 (0.40) (0.77) (0.72) (1.23) (0.80) (0.67) (1.51) (1.79) × 107 (0.55) (0.87) (1.54) (1.06) × 107 (0.56) (0.98) (0.54) (1.66)

500 K 3.40 (±1.31) 3.82 × 103 1.76 × 104 2.94 × 104 3.03 × 104 2.50 × 104 4.73 × 104 5.55 × 104 1.03 (±0.66) 2.74 × 103 6.18 × 103 2.10 × 104 5.45 × 103 2.02 × 104 1.51 × 104 1.38 × 104 9.54 × 103 5.67 (±2.06) 3.74 × 103 4.72 × 103 8.52 × 103 1.07 (±0.54) 4.30 × 107 8.28 × 107 7.74 × 107 1.32 × 108 8.56 × 107 7.19 × 107 1.62 × 108 1.92 × 108 4.77 (±1.72) 2.61 × 107 4.13 × 107 7.35 × 107 5.04 × 107 2.70 (±1.24) 1.51 × 107 2.65 × 107 1.45 × 107 4.48 × 107

× 107 (0.25) (0.70) (1.05) (0.89) (0.73) (1.28) (1.38) × 107 (0.47) (0.85) (1.90) (0.72) (1.63) (1.28) (1.00) (1.20) × 107 (1.20) (1.25) (1.12) × 109 (0.62) (0.88) (0.76) (1.20) (0.87) (0.81) (1.28) (1.49) × 109 (0.69) (0.93) (1.44) (0.94) × 108 (0.71) (1.10) (0.57) (1.36)

1000 K 7.04 (±1.82) 1.86 × 107 4.93 × 107 7.41 × 107 6.28 × 107 5.11 × 107 9.01 × 107 9.71 × 107 3.24 (±1.50) 1.52 × 107 2.76 × 107 6.16 × 107 2.33 × 107 5.28 × 107 4.14 × 107 3.24 × 107 3.90 × 107 2.02 (±0.10) 2.05 × 107 2.15 × 107 1.92 × 107 1.81 (±0.66) 1.12 × 109 1.60 × 109 1.37 × 109 2.18 × 109 1.58 × 109 1.48 × 109 2.32 × 109 2.70 × 109 1.09 (±0.30) 7.53 × 108 1.02 × 109 1.57 × 109 1.03 × 109 6.02 (±1.92) 4.29 × 108 6.63 × 108 3.43 × 108 8.19 × 108

kTST (kTST/krule)

Table 1. Modified Arrhenius Fits, Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set of Reactions Used To Derive Rate Rules for the Isomerization of QOOH to RO2a

The Journal of Physical Chemistry A Article

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5073

2.22 × 107 6.53 × 106 9.85 × 106 2.61 × 108 2.44 × 108 1.69 × 105 3.76 × 105 4.30 × 105 7.70 × 105 1.59 × 106 1.08 × 106 6.92 × 105 6.66 × 106 1.38 × 107 4.90 × 105 4.40 × 104 2.04 × 105 6.92 × 103 1.61 × 103

δ-QOOH primary rate rule C•CCCOOH → CCCCOO• C2•CCCOOH → C2CCCOO• C•CC(C)COOH → CCC(C)COO• C•CCC(C)OOH → CCCC(C)OO• δ-QOOH secondary rate rule CC•CCCOOH → CCCCCOO• CCC•CCCOOH → CCCCCCOO• CC•C(C)CCOOH → CCC(C)CCOO• δ-QOOH tertiary rate rule C2C•CCCOOH → C2CCCCOO• CCC•(C)CCCOOH → CCC(C)CCCOO• C2C•CCC(C)OOH → C2CCCC(C)OO• ε-QOOH primary rate ruled C•CCCCCOOH → CCCCCOO• ε-QOOH secondary rate ruled CC•CCCCCOOH → CCCCCCOO• ε-QOOH tertiary rate ruled C2C•CCCCCOOH → C2CCCCCOO• 0.69 0.86 0.81 0.37 0.37 1.29 1.22 1.11 1.13 0.98 1.04 1.05 0.82 0.49 0.90 1.16 1.08 1.33 1.42

n 4.2 4.7 3.7 4.7 4.7 3.2 4.2 3.5 3.1 3.6 3.7 3.3 3.8 4.0 3.2 2.9 2.9 3.3 2.2

E (kcal mol−1)

−3.6 −2.9 −4.7 −3.2 −2.4

−10.9 −10.6 −11.1 −15.8 −12.8 −4.9

−4.7 −5.1 −4.3

−12.7 −13.4 −13.0

−11.2

−3.5 −4.6 −3.0 −4.7


−15.9 −16.6 −15.6 −16.4


thermochemistry 500 K 2.32 (±0.63) × 107 1.25 × 107 (0.54) 3.57 × 107 (1.54) 2.45 × 107 (1.05) 2.01 × 107 (0.87) 2.04 (±1.25) × 107 1.02 × 107 (0.50) 1.29 × 107 (0.63) 3.79 × 107 (1.86) 1.89 (±0.27) × 107 1.63 × 107 (0.87) 1.77 × 107 (0.94) 2.27 × 107 (1.20) 5.13 × 106 5.35 × 106 (1.04) 3.12 × 106 8.81 × 106 (2.82) 9.56 × 105 1.17 × 106 (1.23)

1000 K 3.14 (±0.63) × 108 2.40 × 108 (0.76) 4.09 × 108 (1.30) 3.29 × 108 (1.05) 2.82 × 108 (0.90) 2.50 (±1.01) × 108 2.01 × 108 (0.80) 1.60 × 108 (0.64) 3.91 × 108 (1.57) 2.26 (±0.35) × 108 2.20 × 108 (0.97) 1.88 × 108 (0.83) 2.72 × 108 (1.20) 5.32 × 107 4.99 × 107 (0.94) 3.07 × 107 7.97 × 107 (2.59) 1.28 × 107 9.69 × 106 (0.76)

kTST (kTST/krule)

The rate estimation rules are provided in bold. Also provided is the ratio of the TST rate constant and the rate rule in parentheses. bIn all reaction notations, the hydrogen atoms are assumed and a “•” symbolizes a radical site. cExcluded from the rate rule. dEstimated on the basis of the trend in the pre-exponential factors and activation energies for the reverse reaction of RO2 → QOOH and the trends in the equilibrium constant (see text). eThe given error is the root-mean-square deviation.

a

A (s−1)

modified Arrh. parameters

reactionb QOOH → RO2

Table 1. continued




Article

The quality of the rate rules developed in this work is demonstrated by the ratio of the individually calculated TST rate coefficients and the rate estimation rule (Table 1). The rate rules are able to predict more than 90% of these reactions within a factor of 2 at 500 K. Larger deviations are mainly seen for reactions involving either a C2 or C3 hydroperoxy alkyl radical. The rate constant for the reaction of the hydroperoxy ethyl radical is at 500 K a factor of 10 smaller than predicted by the rate rule and should not be estimated using this rule. The rate coefficients for the various C3 species are also found to be smaller than the corresponding rate rules, with the largest deviation being a factor of 3.75 at 500 K for the reaction of the β-hydroperoxy propyl radical. For these cases, the lower rate constants are the result of slightly higher reaction barriers. At this point it is unclear why the barriers for these smaller species are higher than those of larger QOOH species. However, a similar observation was made for reactions of small alkyl peroxy radicals.15 This illustrates the importance of varying the size of the alkyl substrate in the formulation of these rules. The isomerization reactions of C2−C4 hydroperoxy alkyl radicals to the corresponding alkyl peroxy radicals have been investigated previously. Table 2 compares the barrier heights (E0K + ZPE) of the current study with previously published results. Rienstra-Kiracofe et al. investigated the reactions of the hydroperoxy ethyl radical using CCSD and CCSD(T) methods with various basis sets.43 Their results show that the barrier for this reaction decreases with increasing basis set size. At their highest level of theory (CCSD(T)/TZ2Pf//CCSD(T)/TZ2P) the barrier was found to be 20.8 kcal mol−1. This value is slightly larger than our CBS-QB3 result as well as the G2 result.44 DeSain et al. investigated the reactions of the several C3−C4 hydroperoxy alkyl radical isomers using a combination of quadratic configuration interaction calculations and MP2 corrections.45 The results from the C3 reactants were used as input parameters for a master equation analysis. Good agreement between kinetic modeling and experimentally observed time-dependent OH and HO2 profiles was achieved with modest adjustments of the calculated barrier heights.46,47 With the exception of the isomerization reaction of the δhydroperoxy butyl radical, the CBS-QB3 barriers are slightly lower (by an average of 1.1 kcal mol−1) than those obtained of DeSain et al. However, the overall agreement between the two sets of data is reasonable and the differences are comparable to the combined errors in the ab initio methods. Rate constants for the isomerization of QOOH to RO2 have previously been reported at the CBS-QB3 level of theory.16,19,28 A comparison of the rate rule derived in this work (red solid line) to previously reported rate rules and other individual rate constants for primary γ-QOOH radicals is shown in Figure 4. Comparisons for the other QOOH isomers are provided in the Supporting Information. The previously reported rate rules by Miyoshi16 are subdivided according to the nature of the hydroperoxy group and according to the nature of the radical site (shown by the black short-dashed lines labeled “pp”, “sp”, and “tp”, where the first letter indicates the nature of the hydroperoxy group as primary, secondary, or tertiary and the second letter indicates the nature of the radical site as primary). Because we do not observe any dependence of the rate constants on the nature of the hydroperoxy group, we collapse these values into a single rule. Similarly, the rate constants obtained by Sharma et al.28 (blue long-dashed lines) do not vary systematically with respect to the hydroperoxy moiety. The available data span roughly an order of magnitude. Our

Only a small difference is observed for reactions that involve the transfer of hydrogen atom from an α- or β-carbon atom hydroperoxy alkyl radical. These were found to be slightly slower. The data for H-atom migrations from a β-carbon is presented in the Supporting Information. In summary, even though the QOOH isomers can undergo a variety of reactions, only the isomerization to RO2, cyclic ether formation, and some select β-scission reactions appear to be important. Depending upon the temperature and pressure, these unimolecular channels can potentially compete with bimolecular O2 addition. Thus, the remainder of this work will focus on these three reaction classes. II. Rate Rule Assignments. In this section, we present rate rules for the isomerization to RO2, cyclic ether formation, and β-scission reactions for the β-, γ-, and δ-QOOH radicals. The rate rules have been developed from sets of reactions that consist of C2−C5 QOOH reactants and several selected C6 and C7 reactants. Isomerization to RO2. We systematically evaluated the rate constants for the isomerization reactions of the β-, γ-, and δQOOH radicals to RO2. The results are provided in modified Arrhenius form in Table 1. The rate coefficients for the isomerization reactions of each QOOH isomer were found to group together. Within these groups there are differences based on the nature of the radical site (e.g., primary, secondary, or tertiary). For a given QOOH isomer, the reactions of primary radicals are slightly faster than the reactions of secondary radicals. Likewise, the reactions of secondary radicals are slightly faster than those of tertiary radicals. These differences follow the trend in the reaction endothermicities. The corresponding rate rules for all QOOH isomers are shown in Figure 3. At temperatures below 1000 K, the isomerization

Figure 3. Arrhenius plots of the rate rules for isomerization of β-, γ-, and δ-QOOH radicals to RO2 (shown by the green, red, and blue lines, respectively).

reactions of β-QOOH are significantly slower than those of the γ- and δ-isomers. This is attributed to the high ring strain associated with the five-membered cyclic transition state. This leads to a barrier that is approximately 10 kcal mol−1 higher in energy than that for a six-membered cyclic transition state. Isomerization of the δ-QOOH radical is slower than that of the γ-QOOH radical. Although the barrier for isomerization of the δ-QOOH radical is slightly higher (∼1 kcal mol−1) than that of the γ-QOOH radical, the difference in the rate coefficients can mainly be attributed to the smaller pre-exponential factor associated with the δ-QOOH radical reaction. This is due to the loss of one additional rotor in the seven- versus sixmembered cyclic transition state. 5074



Article

Table 2. Comparison of the Current CBS-QB3 Barrier Heights (E0K + ZPE) for RO2 Isomerization (TSRO2), Cyclic Ether Formation (TScy ether), and β-Scission of HO2 (TSβ‑Scission) to Selected Values in the Literature TSRO2 (kcal mol−1)

TScy ether (kcal mol−1)

TSβ‑scission (kcal mol−1)

19.0 20.8 20.1 17.9 18.8

13.8 13.9 16.4 11.9 15.3

16.2 15.1 18.9 15.9 18.3

CC•COOH

18.7 19.0

12.1 15.1

15.3 18.2

CCC•COOH

18.1 19.7 17.8 20.0 17.6 19.0 18.1

15.4 17.6 16.2 17.1 15.0 16.0 14.8

20.2

11.5 14.3 11.8 14.4 10.4 12.7 10.4 19.3 12.3 10.5 15.6 11.8

8.0 8.6 8.1 8.5 7.4 8.9 7.7 8.1 7.7 7.0

reactiona β-QOOH C•COOH

C2•COOH

•

CCC(C )OOH •

CC C(C)OOH •

C2C COOH

C3•COOH

γ-QOOH C•CCOOH •

CC CCOOH C•CC(C)OOH C2•CCOOH δ-QOOH C•CCCOOH a

18.9 17.4

method

ref

CBS-QB3 CCSD(T)/TZ2P modified G2 CBS-QB3 QCISD(T)/6-311++G(3df,2pd)// B3LYP/6-311G(d,p)+ΔMP2 CBS-QB3 QCISD(T)/6-311++G(3df,2pd)// B3LYP/6-311G(d,p)+ΔMP2 CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 CBS-q/MP2(full)/6-31G* QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 CBS-q/MP2(full)/6-31G* QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2

this work Rienstra-Kiracofe et al.43 Miller et al.44 this work DeSain et al.45

19.8 23.4 17.2 30.3 19.4 22.2 18.9 20.7

CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2 CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2

this work DeSain et this work DeSain et this work DeSain et this work DeSain et

12.8 14.0

CBS-QB3 QCISD(T)/6-311G(d,p)//B3LYP/6-31G*+ΔMP2

this work DeSain et al.45

21.2 15.1 16.1

this work DeSain et al.45 this work DeSain et this work DeSain et this work DeSain et this work Chen and DeSain et this work Chen and DeSain et

al.45 al.45 al.45 Bozzelli26 al.45 Bozzelli26 al.45

al.45 al.45 al.45 al.45

In this reaction notation, the hydrogen atoms are assumed and a “•” symbolizes a radical site.

rate rule and the single rate constant from Cord et al.19 (green dotted line) fall in the middle of the Miyoshi and Sharma et al. data. There could be several reasons for the wide range of available data. Causes for the large deviations may include the unintended uses of higher energy conformer reactants and/or transition states and differences in the methods of accounting for tunneling, optical isomers, and hindered rotors. Because the slopes of the various Arrhenius plots are similar in Figure 4, the differences are largely associated with the pre-exponential factors, which points to optical isomer assignments and variations in the hindered rotor treatments as the most likely reasons for the observed spread. Several different methods of dealing with internal rotation have been presented in the literature. Pfaender et al.48 has provided a useful review of the 1D-hindered rotor approximation, which is employed in this work, as well as a discussion on how the various methods of obtaining the hindrance potential and reduced moments of inertial impact the resulting partition function. As previously mentioned, the isomerization reaction of QOOH to RO2 is reversible and therefore there is the potential for these species to equilibrate. Using the above rate rules along with the previously reported rules for isomerization of RO2 to QOOH, one can calculate the equilibrium constants for this

reaction class. These values are presented in Figure 5 on a per hydrogen atom basis to account for the number of equivalent hydrogen atoms in the forward isomerization of RO2 to QOOH. Consistent with expectations, the normalized equilibrium constants group closely together according to the nature of the radical site in QOOH. This trend simply reflects the difference in bond strengths for primary, secondary, and tertiary C−H bonds. The equilibrium constants for primary and secondary β-QOOH radicals are slightly smaller than those for the corresponding γ- and δ-isomers. This reflects a slightly stronger C−H bond in the corresponding peroxide caused by the close proximity of the β-carbon to the hydroperoxy group.49 Whether or not equilibrium conditions are established for these reactions is investigated below. However, if this were the case, then under NTC conditions the concentration of tertiary QOOH radicals would be higher than that of secondary radicals and significantly higher than that of primary radicals. Thus, even though the rate constants for the second O2 addition are not expected to be very sensitive to the nature of the radical site (this expectation is based on the observations made for R + O2),50,51 the rates for these various reactions may differ significantly, reflecting the differing partially equilibrated concentrations. 5075



Article

type. The reactions of β-QOOH isomers result in the formation of oxiranes (three-membered ring ethers) plus OH, reactions of γ-QOOH isomers produce oxetanes (four-membered ring ethers) plus OH, whereas the reactions of δ-QOOH isomers form oxolanes (five-membered ring ethers) plus OH. Additionally, we have investigated a few selected reactions of ε-QOOH isomers, which lead to the formation of oxanes (sixmembered ring ethers) plus OH. The results are provided in modified Arrhenius form in Table 3. Also provided are the rate constants evaluated at 500 and 1000 K as well as the heats of reaction at 298 K. All investigated cyclic ether-forming reactions are exothermic. Within a given QOOH isomer subset, we observe a significant spread in the reaction exothermicities. Interestingly, the range in values is largest for the reactions of the γ- and ε-isomers. In each case, the least exothermic reactions correspond to formation the unsubstituted cyclic ether. As the degree of substitution on the ring increases, the reaction exothermicity increases. This implies that the heats of reaction depend on both the nature of the QOOH radical site and hydroperoxy group. Within each subset there is also a wide range in the rate coefficients; at 500 K these values span more than an order of magnitude. In general, the magnitude of these values scales with the reaction exothermicities. The largest rate coefficients correspond to the most exothermic reactions and inspection of the rate constants reveals that this is related to the barrier heights. Consistent with previous results,16,17 the activation energies for each QOOH isomer subset were found to follow an Evans− Polanyi relationship, shown in Figure 6. To facilitate this comparison, the activation energies were determined by fitting the calculated rate coefficients of each reaction to a simple Arrhenius equation over a narrower temperature range of 500− 1000 K. Note that the heats of reaction data points for β- and γisomers refer to the upper x-axis, whereas those for the δ- and ε-isomers relate to the lower x-axis. If the reaction energetics depended solely upon the nature of the radical site and on the hydroperoxy group, we would expect nine groups of data points for each Evans−Polanyi plot. In general, within each reaction subset, the activation energies decrease in going from a primary QOOH radical (triangles), to a secondary radical (circles), to a tertiary radical (squares). However, the data points do not distinctly group into nine sets. This suggests that other structural factors influence the energetics of these reactions. Another intriguing aspect is the ordering in the activation energies. Most notable, the activation energies for reactions of the β-isomers are substantially lower (∼6 kcal mol−1) than those of the γ-isomers. The activation energies for the δQOOH are lower (∼2 kcal mol−1) than those for the ε-isomers. This order is counterintuitive to a simple ring strain argument, which would predict that the activation energy decreases as the transition state ring size increases from a three- to sixmembered ring. Given the large range in reaction exothermicities of each reaction subclass, it is difficult to describe these rate coefficients with a single or small set of rules. Instead, the rate estimation rules for each QOOH isomer class are formulated on the basis of common pre-exponential factors and the observed trend in the activation energies. The pre-exponential factor is taken as the average of the A-factors for the rate constants of the individual reactants, which are determined from simple Arrhenius fits. The activation energies of the rate rules are determined from least-squares fits of the corresponding Evans− Polanyi plots. These rules are provided in Table 3. They predict

Figure 4. Comparisons of the rate rule derived in this work (red solid line) to previously reported rules by Miyoshi16 (black short-dashed lines) and several individual rate constants by Sharma et al.28 (blue long-dashed line) and Cord et al.19 (green dotted line) for the isomerization of primary γ-QOOH radicals. “pp”, “sp”, and “tp” indicate the nature of the hydroperoxy group as primary, secondary, or tertiary by the first letter and the nature of the radical site as primary by the second letter. Because we do not observe any dependence of the rate constants on the nature of the hydroperoxy group, we collapse these values into a single rule.

Figure 5. Comparison of the ratio of forward to reverse rate rules for the isomerization reactions of RO2 to the β-, γ-, and δ-QOOH radicals (shown by the green, red, and blue lines, respectively). These ratios are reported on a per hydrogen atom basis to account for the equivalent hydrogen atoms in RO2.

For the previously investigated reactions of RO2 to the β-, γ-, and δ-QOOH radicals we demonstrated that the observed trends in the various pre-exponential factors and activation energies can be used to estimate rate constants for isomerization to the ε-QOOH radical, which was not systematically investigated. Using these previously outlined trends along with the equilibrium constants, the rate rules for isomerization of the ε-QOOH radical can be estimated. These trend-based rate rules are provided in Table 1. To test how well their predictions compare to the actual calculated TST rate coefficients, we calculated rate coefficients for three selected ε-QOOH isomerization reactions. The estimated rate coefficients agree with the TST results within a factor of 3. This level of agreement seems satisfactory as this reaction is relatively unimportant because the rate of formation of the ε-isomer from RO2 is lower than that for the competing γ- and δ-isomer formation. Cyclic Ether Formation. We have calculated rate coefficients for cyclic ether formation reactions of each QOOH isomer 5076


β-QOOH rate rule C•COOH → cy(CCO) + OH C2•COOH → Ccy(CCO) + OH CCC(C•)OOH → CCcy(CCO) + OH C3•COOH → C2cy(CCO) +OH CCCC(C•)OOH → CCCcy(CCO) + OH CCC(2C•)OOH → (CC)(C)cy(CCO) + OH C2CC(C•)OOH → C2Ccy(CCO) + OH CC•COOH → Ccy(CCO) + OH CCC•COOH → CCcy(CCO) + OH CC•C(C)OOH → anti-Ccy(CC(C)O) + OH CC•C(C)OOH → syn-Ccy(CC(C)O) + OH CCCC•COOH → CCCcy(CCO) + OH CCC•C(C)OOH → anti-CCcy(CC(C)O) + OH CCC•C(C)OOH → syn-CCcy(CC(C)O) + OH CCC•(C)COOH → CCcy(C(C)CO) + OH CC•C(CC)OOH → anti-CCcy(CC(C)O) + OH CC•C(CC)OOH → syn-CCcy(CC(C)O) + OH CC•C(2C)OOH → C2cy(CC(C)O) + OH C2CC•COOH → C2Ccy(CCO) + OH CCCCC•COOH → CCCCcy(CCO) + OH C2C•COOH → C2cy(CCO) + OH C2C•C(C)OOH → C2cy(CC(C)O) + OH γ-QOOH rate rule C•CCOOH → cy(CCCO) + OH C•CC(C)OOH → Ccy(CCCO) + OH C2•CCOOH → Ccy(CCOC) + OH C•CC(CC)OOH → CCcy(CCCO) + OH CCC(C•)COOH → cy(CC(CC)CO) + OH C•CC(2C)OOH → C2cy(CCCO) C2•CC(C)OOH → anti-Ccy(CC(C)CO) + OH C2•CC(C)OOH → syn-Ccy(CC(C)CO) + OH C3•CCOOH → cy(CC(2C)CO) + OH CC•CCOOH → Ccy(CCCO) + OH CCC•CCOOH → CCcy(CCCO) + OH CC•CC(C)OOH → anti-Ccy(CCC(C)O) + OH CC•CC(C)OOH → syn-Ccy(CCC(C)O) + OH CC•C(C)COOH → anti-Ccy(CC(C)CO) + OH CC•C(C)COOH → syn-Ccy(CC(C)CO) + OH

reactionb QOOH → cyclic ether + OH

5077

p p p p p p p p p s s s s s s

p p p p p p p s s s s s s s s s s s s s t t

nature of radical sitec

p s p s p t s s p p p s s p p

p s s t s t s p p s s p s s p s s t p p p s

nature of OOHc A (s ) 3.55 × 1012 1.71 × 109 4.45 × 109 2.35 × 1010 2.47 × 1010 1.16 × 1011 1.05 × 1012 1.53 × 1010 1.21 × 109 1.71 × 109 4.78 × 109 1.01 × 1010 7.34 × 109 5.34 × 109 1.97 × 109 7.17 × 109 1.27 × 1010 4.67 × 109 4.27 × 1010 1.12 × 1010 6.46 × 109 2.93 × 109 5.36 × 1010 6.34 × 1011 2.64 × 109 2.44 × 109 6.79 × 109 2.03 × 1010 4.10 × 1010 1.03 × 1010 2.20 × 107 1.22 × 109 3.71 × 109 2.59 × 109 8.45 × 108 1.38 × 109 2.68 × 109 2.03 × 109 7.73 × 108

−1

0.00 1.01 0.86 0.68 0.64 0.43 0.15 0.64 1.05 1.06 0.79 0.67 0.93 0.82 0.92 0.88 0.74 0.80 0.52 0.82 0.94 0.99 0.54 0.00 0.71 0.78 0.59 0.46 0.36 0.55 1.58 0.70 0.70 0.69 0.90 0.70 0.62 0.65 0.72

n

−1

1.21(ΔrxnH298) + 32.6 12.7 10.8 10.8 9.4 10.7 9.6 10.5 11.3 10.9 9.6 10.5 10.9 9.3 9.9 9.6 9.5 10.0 8.8 10.9 10.9 9.9 8.9 0.76(ΔrxnH298) + 31.4 18.5 18.0 17.5 17.7 17.2 16.2 16.0 16.6 15.9 16.0 15.3 15.2 15.9 14.9 15.6

E (kcal mol )

modified Arrh. parameter

−16.0 −17.5 −16.8 −17.9 −17.1 −18.8 −19.3 −18.0 −18.3 −19.0 −19.3 −20.0 −20.5 −20.0 −18.7

−15.9 −16.8 −17.0 −17.8 −17.0 −17.7 −17.7 −17.2 −17.5 −18.1 −16.9 −17.4 −18.5 −17.2 −18.4 −18.4 −17.1 −18.2 −18.0 −17.4 −18.0 −18.4


21.1 21.8 22.2 23.5 24.1 23.1 23.9 25.0 24.8 20.0 20.6 19.5 20.3 21.1 22.2

21.6 24.2 25.8 25.6 25.9 27.5 25.3 21.9 22.6 22.6 24.1 22.5 24.0 24.0 21.9 23.1 23.1 24.5 23.0 22.2 21.7 22.7


thermochemisty

1.85 4.14 5.75 6.21 1.10 2.54 4.19 5.55 3.17 1.88 4.37 2.53 1.48 3.33 1.02

2.55 1.82 3.15 9.59 3.56 1.71 2.05 9.86 2.16 4.21 1.71 4.04 7.74 2.86 1.06 8.68 2.99 1.48 3.26 3.80 6.24 1.97 × × × × × × × × × × × × × × ×

× × × × × × × × × × × × × × × × × × × × × × 106 107 107 107 107 108 107 106 107 107 107 107 107 107 108 107 107 108 107 107 107 108 e 103 103 103 103 104 104 104 103 104 104 104 104 104 104 104

d

500 K

(0.76) (0.53) (1.22) (0.60) (1.82) (1.21) (1.36) (0.48) (2.16) (0.77) (1.36) (0.46) (0.19) (0.62) (0.50)

(0.46) (1.21) (1.53) (1.80) (1.80) (3.61) (0.45) (0.37) (0.58) (0.54) (0.97) (1.22) (0.63) (1.06) (0.97) (0.79) (1.24) (1.73) (0.45) (1.09) (0.96) (1.75)

3.31 6.07 5.77 6.42 8.27 1.28 3.86 3.78 1.54 9.77 1.84 8.55 6.73 9.59 4.32

3.08 7.50 1.14 1.72 1.05 2.34 6.46 5.88 1.07 9.17 5.28 1.89 1.46 7.81 2.46 1.77 7.90 1.81 1.36 1.75 1.84 2.48

× × × × × × × × × × × × × × ×

× × × × × × × × × × × × × × × × × × × × × ×

kTST (kTST/krule)

d 109 109 1010 1010 1010 1010 109 109 1010 109 109 1010 1010 109 1010 1010 109 1010 1010 1010 1010 1010 e 107 107 107 107 107 108 108 107 108 107 108 107 107 107 107

1000 K

(0.84) (0.86) (1.06) (0.79) (1.34) (1.11) (2.77) (0.44) (1.60) (0.78) (1.29) (0.46) (0.30) (0.52) (0.38)

(0.69) (1.03) (1.34) (1.25) (1.26) (1.81) (0.51) (0.60) (0.93) (0.55) (0.66) (1.75) (0.70) (0.80) (1.25) (0.90) (0.85) (1.03) (0.85) (1.57) (1.21) (1.24)

Table 3. Modified Arrhenius Fits, Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set of Reactions Used To Derive Rate Rules for the Cyclic Ether Formation Reactionsa



p s p s p s s s p p p p s p

p p p s s t p p p p p s

nature of OOHc

t t

nature of radical sitec 2.85 × 108 2.75 × 108 5.54 × 1010 1.72 × 108 2.48 × 109 5.20 × 108 4.82 × 107 2.42 × 107 3.83 × 108 1.04 × 1010 2.89 × 107 4.36 × 107 2.65 × 108 3.18 × 108 5.92 × 107 3.05 × 107

A (s−1) 0.92 0.96 0.00 0.76 0.35 0.67 0.94 1.01 0.62 0.00 0.76 0.60 0.52 0.44 0.68 0.83

n 13.4 12.6 1.14(ΔrxnH298) + 53.0 11.1 11.2 10.5 9.0 8.3 7.5 0.82(ΔrxnH298) + 45.2 11.3 10.5 10.5 10.5 12.7 8.8

E (kcal mol−1)

modified Arrh. parameter

17.7 18.6 17.7 19.3 18.7 16.5 17.2 14.5 11.2 11.5 11.1 12.7 12.0 10.4

−21.0 −22.8 −35.7 −36.3 −36.7 −37.6 −38.3 −38.9 −40.6 −40.6 −41.8 −41.5 −38.4 −42.7



thermochemisty

500 K 1.17 × 105 3.58 × 105 f 2.90 × 105 2.67 × 105 9.15 × 105 1.91 × 106 2.95 × 106 9.77 × 106 g 3.70 × 104 4.62 × 104 1.81 × 105 1.19 × 105 1.12 × 104 7.44 × 105 (0.53) (0.62) (0.92) (0.78) (0.95) (1.86)

(1.37) (0.63) (1.34) (0.99) (0.70) (1.20)

(0.98) (0.76)

1000 K

1.84 1.37 5.02 3.26 1.07 1.12

1.29 9.73 2.80 3.47 3.98 6.53

× × × × × ×

× × × × × ×

f 108 107 108 108 108 108 g 107 107 107 107 107 108

1.90 × 108 3.87 × 108

kTST (kTST/krule)

(0.68) (0.50) (1.11) (0.82) (0.97) (1.74)

(1.19) (0.64) (1.44) (1.06) (0.83) (0.97)

(0.69) (0.71)

a The rate estimation rules are provided in bold. Also provided is the ratio of the TST rate constant and the rate rule in parentheses. bIn the reaction notation, the hydrogen atoms are assumed and a “•” symbolizes a radical site. cThe letters “p”, “s”, and “t” stand for primary, secondary, and tertiary. dThe root-mean-square deviation is ±4.00 × 107 s−1 at 500 K and ±4.64 × 109 s−1 at 1000 K. eThe rootmean-square deviation is ±3.30 × 104 s−1 at 500 K and ±9.36 × 107 s−1 at 1000 K. fThe root-mean-square deviation is ±8.39 × 105 s−1 at 500 K and ±5.56 × 107 s−1 at 1000 K. gThe root-mean-square deviation is ±1.34 × 104 s−1 at 500 K and ±3.47 × 106 s−1 at 1000 K.

C2C•CCOOH → C2cy(CCCO) + OH C2C•CC(C)OOH → C2cy(CCC(C)O) + OH δ-QOOH rate rule C•CCCOOH → cy(CCCCO) + OH C•CCC(C)OOH → Ccy(CCCCO) + OH C2•CCCOOH → cy(CC(C)CCO) + OH CC•CCCOOH → Ccy(CCCCO) + OH CCC•CCCOOH → CCcy(CCCCO) + OH C2C•CCCOOH → C2cy(CCCCO) + OH ε-QOOH rate rule C•CCCCOOH → cy(CCCCCO) + OH C•C(C)CCCOOH → cy(CC(C)CCCO) + OH C•CC(C)CCOOH → cy(CCC(C)CCO) + OH C•CCC(C)COOH → cy(CC(C)CCCO) + OH C•CCCC(C)OOH → Ccy(CCCCCO) + OH CC•CCCCOOH → Ccy(CCCCCO) + OH

reactionb QOOH → cyclic ether + OH

Table 3. continued


5078



Article

Figure 7. Comparisons of selected rate constants derived in this work (red lines) to previously reported rules by Miyoshi16 (blue lines), Cord et al.19 (black), and Wijaya et al.17 (green lines) for the cyclic ether formation reactions of primary, secondary, and tertiary β-QOOH radicals.

Figure 6. Evans−Polanyi relationships for the cyclic ether formation reactions of the β-, γ-, δ-, and ε-QOOH radicals. The triangle symbols correspond to primary radicals, the circles to secondary radicals, and the squares to tertiary radicals. Note that the heats of reaction for the β- and γ-isomers are provided on the upper x-axis, whereas those of the δ- and ε-isomers are provided on the lower x-axis.

used the 03 package. It is suspected that there may have been some errors in the implementation of the CBS-QB3 method in the earlier versions of this code. In fact, Wijaya et al. did identify a problem in several of their calculations with the ΔE(Emp) empirical correction extrapolation procedure. For these cases, the barrier heights were overestimated by ∼2.5 kcal mol−1. Although this error was only observed in some cases, one may suspect that there were additional problems with the implementation of this method within the code at that time, because the results from this study and the two other more recent studies agree fairly well. β-Scission Reactions. We evaluated rate coefficients for two types of β-scission reactions. The first set of reactions involves HO2 elimination from β-QOOH radicals to form an olefin. The second set of reactions contains C−C bond fissions of the β-, γ-, and δ-QOOH radicals. These reactions can result in the formation of either an alkyl radical and olefinic peroxide or a smaller QOOH radical and an olefin. The results for the HO2 elimination reactions of β-QOOH radicals are summarized in Table 4. Due to the relatively weak C−OOH bond in QOOH, these reactions occur more quickly than those β-scission reactions that result in the cleavage of a C−C bond. Because both the C-OOH bond dissociation energy and the stability of the radical site are sensitive to the structure of the QOOH radical, one may expect that the rate constants for this reaction type depend on the nature of the peroxy moiety as well as whether the radical is classified as primary, secondary, or tertiary. However, our calculations reveal that the majority of the rate coefficients group together regardless of the structure of the reactant. Thus, we assign a single rate rule for this reaction type. Our rate rule is able to predict the rate coefficients of approximately 75% of the reactions to within a factor of 2. With the exception of the reaction of the hydroperoxy ethyl radical, all of the remaining rate constant estimates are within a factor of 2.5 of the directly calculated rate constants. HO2 elimination from several of the investigated reactants may form either a trans- or a cis-olefin. We investigated both product channels for several reactants and found that the rate constants for formation of the cis-product are anywhere from 3 to 6 times lower than those for transproduct formation. Because the cis-product channel is substantially slower, these reactions are not used to derive

the rate coefficients of approximately 80% of the reactions of the test set to within a factor of 2 at 500 K. All “outlier” reactions belong either to the β- or γ-isomers subsets. Of these, the rate coefficients of only two reactions deviate significantly from the rate rule predictions. At 500 K the reaction of the CCC(2C•)OOH radical is calculated to be 3.6 times faster than the rate rule suggests, but its rate constant is within a factor of 2 at 1000 K. The reaction of CC•CC(C)OOH is roughly 5 and 3 times slower than the rate rule estimates at 500 and 1000 K. At present, the reasons for these two large deviations are not known. A variety of different computational methods have been used to investigate this class of reaction.17,19,20,43,45 Some selected results are summarized in Table 2. Our calculated barrier for the reaction of the hydroperoxy ethyl radical is in good agreement with the CCSD(T)/TZ2P results of RienstraKiracofe et al.43 However, on average our barriers are 2.4 kcal mol−1 lower in energy than those of DeSain et al.,45 which are determined using a combination of quadratic configuration interaction (QCISD(T)) and MP2 calculations. (Their computed barrier of 30.3 kcal mol−1 for the γ-hydroperoxy butyl radical reaction seems unusually large and is excluded from this comparison.) It is suspected that the higher barriers of DeSain et al. might be due to the presence of spin contamination in the transition state structure. It has previously been shown that QCISD(T) method provides similar quality predictions as the CCSD(T) method,52 which is incorporated into the CBS-QB3 procedure. However, the CBS-QB3 method applies an empirical correction to account for the presence of spin contamination. Three other studies have employed the CBS-QB3 method.16,17,19 Figure 7 provides a comparison of the rate coefficients for the reactions of several β-isomers. Comparisons for the γ- and δ-isomers are provided in the Supporting Information. Our rate constants (red lines) are in good agreement with those of Miyoshi16 (blue lines) and Cord et al.19 (black lines), but they are systematically higher than those of Wijaya et al.17 (green lines). The latter study was done several years before the others and was performed using the Gaussian 98 software package, whereas all the other studies 5079


5080

p p s s p t p s s s p t p p s p s s s s s t t s s s s s

β-QOOH rate rule C•COOH → CC + HO2d C2•COOH → CCC + HO2 CC•COOH → CCC + HO2 CCC•COOH → CCCC + HO2 CCC(C•)OOH → CCCC + HO2 C2C•COOH → C2CC + HO2 C3•COOH → C2CC + HO2 CC•C(C)OOH → t-CCCC + HO2 CC•C(C)OOH → c-CCCC + HO2d CCCC•COOH → CCCCC + HO2 CCCC(C•)OOH → CCCCC + HO2 CCC•(C)COOH → CCC(C)C + HO2 CCC(2C•)OOH → CCC(C)C + HO2 C2CC(C•)OOH → C2CCC + HO2 C2CC•COOH → C2CCC + HO2 CCCCC(C•)OOH → CCCCCC + HO2 CCC•C(C)OOH → t-CCCCC + HO2 CCC•C(C)OOH → c-CCCCC + HO2d CC•C(CC)OOH → t-CCCCC + HO2 CC•C(CC)OOH → c-CCCCC + HO2d CC•C(2C)OOH → C2CCC + HO2 C2C•C(C)OOH → C2CCC + HO2 C2C•CC(C)OOH → C2CCCC + HO2 CCCCC•COOH → CCCCCC + HO2 CCCC(C•C)OOH → t-CCCCCC + HO2 CCCC(C•C)OOH → c-CCCCCC + HO2d CCCC•C(C)OOH → t-CCCCCC + HO2 CCCC•C(C)OOH → c-CCCCCC + HO2d p s p p s p t s s p s p t s p s s s s s t s s p s s s s

nature of OOH sitec 1.22 × 1011 2.59 × 109 7.93 × 1010 2.34 × 1010 6.03 × 109 2.67 × 1011 5.35 × 1010 1.60 × 1012 2.88 × 1011 3.85 × 1011 1.04 × 1010 2.21 × 1012 8.38 × 109 3.89 × 1013 6.83 × 109 6.24 × 1010 1.78 × 1012 5.84 × 1010 5.30 × 1010 9.92 × 1010 7.06 × 108 3.01 × 1011 9.79 × 1012 7.82 × 109 5.63 × 109 3.50 × 1011 1.35 × 109 2.65 × 1011 1.51 × 1011

A (s−1) 0.57 1.07 0.62 0.77 0.95 0.50 0.78 0.23 0.46 0.38 0.94 0.23 0.92 −0.15 0.92 0.72 0.24 0.66 0.62 0.52 1.05 0.35 0.04 0.90 0.97 0.37 0.98 0.38 0.46

n 15.1 15.7 15.5 15.3 15.2 15.8 14.9 15.2 15.2 16.3 15.1 15.6 14.5 15.5 15.1 15.0 15.7 15.0 16.1 15.1 15.2 15.2 15.4 14.9 15.0 14.9 15.0 14.8 15.9

E (kcal mol−1) 3.9 5.5 5.0 5.1 5.8 6.0 6.2 6.9 8.0 5.1 5.5 6.1 6.8 5.3 5.0 5.5 6.8 8.0 6.9 8.1 8.8 8.6 9.0 5.0 6.7 7.9 6.8 8.0

ΔrxnH298 (kcal mol−1) 28.0 31.4 29.1 29.9 32.6 29.5 33.4 28.9 29.4 29.7 33.4 30.1 35.8 32.5 30.3 32.9 30.7 30.4 29.8 29.5 33.0 31.2 30.4 29.2 30.4 30.4 31.8 31.8


thermochemisty 1000 K

3.11 (±1.37) × 109 1.57 × 109 (0.51) 2.35 × 109 (0.76) 2.13 × 109 (0.69) 2.01 × 109 (0.65) 2.92 × 109 (0.94) 6.68 × 109 (2.15) 3.75 × 109 (1.21) 3.12 × 109 (1.00) 1.44 × 109 3.35 × 109 (1.08) 4.12 × 109 (1.33) 3.19 × 109 (1.03) 5.55 × 109 (1.79) 1.96 × 109 (0.63) 4.82 × 109 (1.55) 3.58 × 109 (1.15) 2.92 × 109 (0.94) 1.17 × 109 1.75 × 109 (0.56) 4.80 × 108 1.59 × 109 (0.51) 5.58 × 109 (1.80) 2.13 × 109 (0.69) 2.37 × 109 (0.76) 2.57 × 109 (0.83) 6.44 × 108 2.07 × 109 (0.66) 1.20 × 109

500 K 1.05 (±0.61) × 106 e 2.78 × 105 (0.26) 6.19 × 105 (0.59) 5.67 × 105 (0.54) 4.97 × 105 (0.47) 7.18 × 105 (0.68) 2.18 × 106 (2.06) 1.54 × 106 (1.46) 1.06 × 106 (1.01) 3.03 × 105 8.79 × 105 (0.83) 1.37 × 106 (1.29) 1.13 × 106 (1.07) 2.55 × 106 (2.42) 5.18 × 105 (0.49) 1.56 × 106 (1.48) 1.14 × 106 (1.08) 9.75 × 105 (0.92) 2.34 × 105 6.04 × 105 (0.57) 1.11 × 105 6.10 × 105 (0.58) 2.38 × 106 (2.26) 6.41 × 105 (0.61) 6.33 × 105 (0.60) 1.11 × 106 (1.05) 1.73 × 105 9.32 × 105 (0.88) 2.92 × 105

kTST (kTST/krule)

a The rate estimation rules are provided in bold. Also provided is the ratio of the TST rate constants to the rate rule in parentheses. bIn reaction notation, the hydrogen atoms are assumed and a “•” symbolizes a radical site. cThe letters “p”, “s”, and “t” stand for to primary, secondary, and tertiary. dExcluded from the rate rule. eThe given error is the root-mean-square deviation.


reactionb

modified Arrh. Parameters

Table 4. Modified Arrhenius Fits, Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set of Reactions Used to Derive Rate Rules for the β-Scission Reactions the β-QOOH Radical to Form HO2 and an Olefina




Article

results for both types of C−C bond fissions are provided in modified Arrhenius form in Table 5 along with evaluated rate constants at 500 and 1000 K and the heats of reaction at 298 K. Inspection of the heats of reaction shows that there is considerable spread in these values. There is also considerable spread in the evaluated rate constants, especially at lower temperatures. Previous investigations of the C−C fission reactions of alkyl radicals show that the energetics of these reactions are sensitive to the nature of the reactant and product radicals as well as the olefin.53 On the basis of these observations, we expect that the corresponding β-scission reactions of the QOOH radicals are also sensitive to the nature of both, the reactant and products. Furthermore, the reaction energetics might depend upon the distance of the hydroperoxy group from the reacting C−C bond. To explore these effects, we plot the activation energies for this reaction class as a function of the corresponding heats of reaction in Figure 9. For this comparison, the activation energies were determined by fitting the calculated rate coefficients of each reaction to a simple Arrhenius equation over a narrower temperature range of 500−1000 K. Note that the heats of reaction data points for the reactions of the γisomer that form an unstable α-QOOH isomer (open blue circles) correspond to the lower x-axis. The heats of reaction data points for all other reactions (solid symbols) correspond to the upper x-axis. The activation energies for the reactions of the γ- and δ-QOOH radicals to form a smaller QOOH radical and an olefin correlate well with their corresponding heats of reaction. In general, the barriers for the reactions of the γisomer are only slightly lower than those of the δ-isomer, even though the overall heats of reaction differ by ∼35 kcal mol−1. This indicates that the barrier heights for the γ-QOOH reactions are influenced by the thermochemistry of the unstable α-QOOH radical, rather than by the final products. The barriers for the C−C fission reactions that form an alkyl radical plus olefin peroxide are generally slightly higher than those of the reactions that form a smaller QOOH and alkyl radical. The barrier heights do not appear to correlate with the proximity of the reactive C−C bond to the hydroperoxy group. Sabbe et al. have previously investigated the C−C bond fission reactions of a large variety of alkyl radicals using the CBS-QB3 method.53 In Figure 10 we plot in Evans−Polanyi format their results (shown by the open symbols) along with the results for the various C−C fission reactions of QOOH (shown by the solid symbols) investigated here. Because the overall heats of reaction for the γ-QOOH isomer to form a ketone or aldehyde plus OH plus olefin are exothermic, these data are excluded from this comparison. The Evans−Polanyi relationships for the QOOH C−C fission reactions are similar to that for the alkyl radical C−C fission reactions. Thus, the presence of the hydroperoxy group does not appear to significantly affect the reaction energetics of the C−C fission reactions, even when the reacting C−C bond is adjacent to the hydroperoxy moiety. Both sets of data show that, for a given heat of reaction, there is an appreciable spread in the calculated activation energies. At present the reason for this spread is unclear. The rate estimation rules for each subclass are formulated on the basis of a common pre-exponential factor and the observed trend in the activation energies. The pre-exponential factor is taken as the average of the A-factors of the rate constants of the individual reactants, which are determined from simple Arrhenius fits. The activation energies used in the rate rules

the rate rule. Furthermore, most kinetic mechanisms do not differentiate between cis- and trans-isomers of olefins. Table 2 compares the barrier heights calculated in this work for several β-scission of HO2 reactions to some selected values found in the literature. Rienstra-Kiracofe et al. have performed high level CCSD(T)/TZ2P calculations for the reactions of the hydroperoxy ethyl radical.43 Their barrier is 1.1 kcal mol−1 lower in energy than the one calculated in this work and 3.8 kcal mol−1 lower than that predicted by Miller et al.44 The majority of the available data for larger sized β-QOOHs originates from the QCISD(T)/MP2 calculations of DeSain et al.45 Once again we observe that our values are consistently lower than their values. The largest difference is 6.4 kcal mol−1 for the reaction of the β-hydroperoxy iso-butyl radical. For the remaining reactions, the average difference between the two sets of data is 1.7 kcal mol−1. Rate rules for the reactions of β-QOOH radicals have previously been reported at the CBS-QB3 level of theory by Miyoshi.16 Figure 8 compares the rate rule derived in this work

Figure 8. Comparisons of the rate rule derived in this work (red lines) to previously reported rules by Miyoshi16 (black lines) and individual rate constants by Chen et al.18 (green lines) for the β-scission reaction of the β-QOOH radical.

(red lines) to these previously reported values. Miyoshi provides nine rules (black lines) that group the rate coefficients according to the nature of the radical site and of hydroperoxy group. However, consistent with our observation these reactions group closely together and could effectively be collapsed into a single rule. The agreement between these two sets of CBS-QB3 results is quite good. Several individual rate constants for this reaction class have also been calculated using a modified CBS-q method (CBS-q//B3LYP/6-31G(d)) by Chen et al.18 (green lines). With the exception of the reaction of C3•COOH → i-C4H8 + HO2 (shown by the upper solid green line), these rate constants group closely together and are significantly less than the CBS-QB3 values. This is not too surprising given the different methods used. The second type of investigated β-scission reactions involves the cleavage of a C−C bond. For the β-QOOH radical, C−C bond fission results in the formation of an alkyl radical and an olefin peroxide. For the γ- and δ-isomers, C−C bond fission results in either the formation of an alkyl radical and an olefin peroxide or a smaller QOOH radical and an olefin. For δisomers, this latter pathway forms a β-QOOH radical. For γisomers it forms an unstable α-QOOH, which spontaneously dissociates to form an aldehyde or ketone plus OH radical. The 5081


5082

s p s p s p s s s p t p p p s p p s p p s t p p p s p p s p p p

p p p s s s t t p p p t s p p s p p s p s s s s p p t p p p s p

nature of OOH sitec 8.13 × 1013 8.56 × 1010 4.95 × 1010 4.87 × 1012 1.09 × 109 2.78 × 1011 9.68 × 109 4.55 × 1010 4.65 × 1010 2.11 × 109 1.64 × 107 1.17 × 1015 8.71 × 108 9.23 × 106 2.14 × 109 4.64 × 1013 2.41 × 1010 3.081 × 1013 2.89 × 109 3.08 × 108 1.23 × 109 3.00 × 108 1.18 × 109 2.23 × 109 1.96 × 109 9.10 × 108 3.64 × 109 3.53 × 109 6.27 × 108 1.3924 × 1014 2.33 × 1012 6.86 × 1013 5.78 × 1014 5.08 × 1011 2.23 × 1013

A (s−1) 0.0 0.85 0.83 0.32 1.43 0.70 1.18 0.97 0.92 1.34 2.00 −0.29 1.52 1.97 1.35 −0.03 1.02 0.0 1.3 1.5 1.3 1.6 1.2 1.2 1.2 1.4 1.3 1.2 1.4 0.0 0.6 0.1 −0.2 0.7 0.2

n 0.71(ΔrxnH298) + 14.5 30.7 27.9 29.6 29.5 32.5 29.0 32.1 33.5 30.3 28.3 29.5 29.4 28.4 27.1 27.0 28.2 0.53(ΔrxnH298) + 32.4 26.7 23.5 24.9 25.0 22.7 21.5 23.4 25.5 23.7 21.9 23.6 0.93(ΔrxnH298) + 7.2 29.0 29.0 27.3 28.7 28.4

E (kcal mol−1)

35.4 35.2 37.8 35.8 38.1

60.6 62.5 61.2 64.0 64.2 66.0 60.5 61.8 65.8 67.0 59.9

−10.6 −14.3 −10.3 −11.6 −13.9 −17.0 −11.0 −10.1 −13.0 −16.1 −10.2 24.0 24.0 22.0 24.3 22.4

32.3 36.1 36.6 30.9 31.7 32.1 30.0 38.3 31.6 37.4 38.1 30.6 31.5 39.3 39.5 34.2


23.0 22.7 21.8 24.3 25.5 22.7 26.0 30.1 22.6 21.9 20.3 24.4 23.6 22.6 20.2 23.1


thermochemisty

1.65 2.17 1.91 1.35 3.82

2.06 1.62 6.02 6.20 2.32 1.49 1.82 3.23 3.90 1.43 1.64

6.38 5.53 4.27 9.80 1.27 3.09 1.67 3.12 4.67 1.83 2.48 1.55 6.88 1.39 5.74 6.24

× × × × ×

× × × × × × × × × × ×

× × × × × × × × × × × × × × × ×

10−1 100 100 10−1 10−1 100 10−1 10−2 10−1 100 101 100 10−1 101 101 100 e 101 102 101 101 102 103 102 101 102 103 102 f 101 101 102 101 101

d

500 K

(0.93) (1.30) (1.64) (1.02) (0.51)

(0.36) (0.37) (1.18) (0.62) (0.66) (0.83) (2.50) (0.72) (1.85) (1.26) (3.48)

(0.23) (1.55) (0.62) (0.84) (0.26) (0.85) (0.50) (1.74) (0.12) (0.29) (1.23) (1.42) (0.37) (3.72) (2.67) (2.26)

5.23 5.02 1.57 4.27 7.52

3.49 6.17 4.15 5.28 4.92 1.70 5.32 3.14 1.46 1.93 6.20

5.80 1.23 1.55 7.61 2.68 1.50 3.41 1.25 5.04 1.11 5.61 1.18 4.48 2.92 4.61 1.88

× × × × ×

× × × × × × × × × × ×

× × × × × × × × × × × × × × × ×

kTST (kTST/krule)

d 106 107 107 106 106 107 106 106 106 107 107 107 106 107 107 107 e 107 107 107 107 107 108 107 107 108 108 107 f 107 107 108 107 107

100 K

(1.06) (1.04) (1.23) (1.00) (0.74)

(0.83) (0.53) (1.05) (0.95) (0.47) (0.72) (1.12) (0.85) (1.82) (1.03) (1.62)

(0.38) (0.72) (0.66) (0.78) (0.43) (0.88) (0.65) (1.03) (0.28) (0.49) (1.39) (1.25) (0.37) (1.68) (1.10) (1.25)

a

The rate estimation rules are provided in bold. Also provided is the ratio of the TST rate constants to the rate rule in parentheses. bIn reaction notation, the hydrogen atoms are assumed and a “•” symbolizes a radical site. cThe letters “p”, “s”, and “t” stand for primary, secondary, and tertiary. dThe root-mean-square deviation is ±9.60 × 100 s−1 at 500 K and ±7.84 × 106 s−1 at 1000 K. eThe rootmean-square deviation is ±1.75 × 102 s−1 at 500 K and ±3.73 × 107 s−1 at 1000 K. fThe root-mean-square deviation is ±3.72 × 101 s−1 at 500 K and ±1.80 × 107 s−1 at 1000 K.

β-, γ-, δ-QOOH rate rule (QOOH → peroxide + R•) C2•COOH → CCOOH + CH3 C2•CCOOH → CCCOOH + CH3 CCC(C•)OOH → CCOOH + CC• CCC•COOH → CCCOOH + CH3 CC•C(C)OOH → CCCOOH + CH3 C2CC•COOH → CCCCOOH + CH3 C2C•C(C)OOH → C2CCOOH + CH3 C2C•C(C)OOH → CCOOH + CC•C C2•CC(C)OOH → CCC(C)OOH + CH3 C3•CCOOH → CC(C)COOH + CH3 CCC(C2•)OOH → CC(C)OOH + CC• CCC•(C)COOH → CC(C)COOH + CH3 CC•C(C)COOH → CCCCOOH + CH3 CCC(C•)COOH → CCOOH + CC• C2CC(C•)OOH → CCOOH + CC•C CCCC•COOH → CCCCOOH + CC• γ-QOOH rate rule (γ-QOOH → olefin + RO + OH) C•CCOOH → CC + CH2O + OH C•CC(C)OOH → CCC + CH2O + OH CC•CCOOH → CCC + CH2O + OH C2•CCOOH → CCC + CH2O + OH CC•CC(C)OOH → CCC + CCO + OH CC•C(C2)OOH → CCC + C2CO + OH CC•C(C)COOH → t-CCCC + CH2O + OH CCC•CCOOH → CCCC + CH2O + OH C3•CCOOH → C2CC + CH2O + OH C2•CC(C)OOH → CCC + CCO + OH C2C•CCOOH → C2CC + CH2O + OH δ-QOOH rate rule (δ-QOOH → olefin + β-QOOH) C•CCCOOH → CC + C•COOH C•CCC(C)OOH → CC + C2•COOH C•CC(C)COOH → CC + CC•COOH CC•CCCOOH → CCC + C•COOH C2•CCCOOH → CCC + C•COOH

reactionb


modified Arrh. Parameters

Table 5. Modified Arrhenius Fits, Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set of Reactions Used to Derive Rate Rules for the β-Scission Reactions of the β-, γ-, and δ-QOOH Radicals That Result in the Cleavage of a C−C Bonda




Article

pressure rate rules for the important unimolecular reactions of the various QOOH radicals. Combined with our previously reported rate rules for the reactions of RO2,15 there is now an internally consistent set of high-pressure rate rules available for the alkyl + O2 submechanism. This provides us with the opportunity to explore two important features of this system. First, we examine the pressure dependence of the QOOH reactions in an effort to provide some guidance as to the conditions for which high-pressure rules can directly be applied without appreciable error. This discussion builds upon our earlier investigation of the pressure dependence of RO2 reactions.15 Below, we briefly summarize the important findings of that work and extend the analysis by including the reactions of QOOH. Second, we investigate whether the RO2 and the QOOH isomers equilibrate with each other and with R + O2 during the low temperature oxidation process, and if so, at which conditions this happens. In other words, we are interested to see if the reversible alkyl peroxy radical dissociation (RO2 ↔ R + O2) and isomerization (RO2 ↔ QOOH) reactions are fast compared to irreversible product channels. This aspect is important for kinetic modeling studies because if these species are partially equilibrated, then the accuracy of the thermodynamic database, which is generally used to derive reverse reaction rate constants, becomes more important. In the first paper of this series, we examined the pressure dependence of the dissociation, concerted elimination, and isomerization reactions of alkyl peroxy radicals (Figure 1) at three different temperatures and a wide range of pressures.15 Specifically, we obtained apparent pressure and temperature dependent rate coefficients for a representative small (n-C4H9), a medium (n-C8H17), and a large (n-C12H25) alkyl radical reacting with O2 using a QRRK/MSC analysis. However, we greatly simplified this reaction subset by neglecting the subsequent reactions of QOOH. Pressure effects were explicitly examined by comparing the product concentration versus time profiles at 0.1 and 10 atm obtained with two mechanisms: one that contained all apparent reactions from a QRRK analysis with their pressure-dependent rate constants and a second one that only included the elementary reactions with their corresponding high-pressure rate expressions. At low temperatures and pressures (500 K and 0.1 atm) the predictions of both mechanisms are similar in terms of the final product distributions, even though some concentration profiles differed at early reaction times. As the temperature was increased (750 and 1000 K), small differences in the product distributions were observed. These differences result from chemically activated pathways that are not present in the high-pressure mechanism. These chemically activated pathways are more prevalent in the reactions of smaller sized species, especially at higher temperatures. At 10 atm, however, the predictions of the two mechanisms were in good agreement for all three n-alkyl radicals at each investigated temperature. This implies that under these conditions (typical for many ignition problems) the contributions from the chemically activated reactions are insignificant. Thus, the high-pressure rate rules can be used directly to describe the reactions of RO2. Our initial analysis and the conclusions drawn from it were based on the RO2 reaction subset that treated QOOH radicals as final products. With these new rate rules in hand, we can extend the analysis by including the subsequent reactions of QOOH. The primary objective is to verify that our previous conclusionthat the high-pressure rate rules can be used

Figure 9. Evans−Polanyi relationships for the C−C fission reactions of the β-, γ-, and δ-QOOH radicals. The red squares correspond to C−C fission reactions that result in an alkyl radical plus olefin peroxide. The green circles correspond to C−C fission reactions of the δ-QOOH radical to form a β-QOOH radical and an olefin. The open blue circles correspond to C−C fission reactions of the γ-QOOH radical to form an aldehyde or ketone + OH + olefin. The solid-filled symbols refer to the upper x-axis, whereas the open symbols refer to the lower x-axis.

Figure 10. Evans−Polanyi relationships for the C−C fission reactions of the QOOH radicals studied here (shown by the solid symbols) to that of alkyl radicals determined by Sabbe et al.53 The red squares correspond to C−C fission reactions of the β-, γ-, and δ-QOOH radicals that result in an alkyl radical plus olefin peroxide. The green circles correspond to C−C fission reactions of the δ-QOOH radical that result in the formation of a β-QOOH radical and an olefin. The open diamonds, squares, and triangles correspond to the data in Tables 1−3, respectively, in the Sabbe et al. study.

are based on the least-squares fits of the corresponding Evans− Polanyi plots. These rules are provided in Table 5. For the C− C fissions that form a smaller QOOH radical and olefin, these rules can predict approximately 75% of the individually evaluated rate constants at 500 K to within a factor of 2. At 1000 K the rate rule can predict over 90% of the individual rate constants to within a factor of 2. For the C−C fission reactions that form an alkyl radical plus olefinic peroxide, the rate rule can predict the individual rate constants of 50% of the reactions at 500 K to within a factor of 2. At 1000 K about 75%of the rate constants are reproduced within a factor of 2. This level of agreement seems satisfactory because this reaction pathway is generally slow compared to other competing reactions under typical NTC temperatures. These channels only become important for the γ- and δ-isomer at high temperatures (Figure 2). III. Implications for Kinetic Modeling Studies. Thus far, the focus of this work has been on the development of high5083



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Table 6. Reactions Included in the High-Pressure Mechanism along with the Corresponding Rate Constants at 500, 750, and 1000 K and the Sensitivity Factors for the HO2 and OH Concentrations k (cm3 s−1 mol−1 or s−1) reaction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

a

CCCC•+ O2 → CCCCOO• CCCCOO•→ CCCC• + O2 CCCCOO• → CCCC + HO2 CCCCOO• → CCCCO + OH CCCCOO•→ CCC•COOH CCCCOO• → CC•CCOOH CCCCOO• → C•CCCOOH CCC•COOH → CCCCOO• CCC•COOH → CCcy(CCO) CCC•COOH → CCCC + HO2 CC•CCOOH → CCCCOO• CC•CCOOH → Ccy(CCCO) + OH CC•CCOOH → CCC + CO + OH C•CCCOOH → CCCCOO• C•CCCOOH → cy(CCCCO) + OH C•CCCOOH → CC + C•COOH

500 K 2.31 4.94 1.50 5.08 1.68 1.05 1.58 9.86 1.96 1.06 4.57 1.84 1.40

× × × × × × × × × × × × ×

1012 10−1 10−1 10−5 10−1 103 101 103 107 106 107 104 102

1.81 × 107 3.16 × 105 1.03 × 101

750 K 1.63 3.65 4.48 2.55 3.83 7.06 2.38 2.27 1.26 2.20 3.79 5.52 7.55

× × × × × × × × × × × × ×

1012 104 103 101 103 105 104 106 109 108 108 106 105

1.04 × 108 1.78 × 107 2.51 × 105

sensitivity factor for [HO2]b 1000 K

1.37 9.94 7.74 1.80 5.79 1.83 9.23 3.44 1.01 3.17 1.09 9.58 5.55

× × × × × × × × × × × × ×

1012 106 105 104 105 107 105 107 1010 109 109 107 107

2.48 × 108 1.33 × 108 3.93 × 107

sensitivity factor for [OH]b

500 K

750 K

1000 K

500 K

750 K

1000 K

0.800

0.716

0.654

−0.146

−0.171

−0.109 −0.414 −0.264

−0.034 −0.441 −0.135

0.006 −0.561 −0.095

0.023 0.083 0.053

0.032 0.189 0.055

−0.194 0.002 0.002 0.188 0.003

−0.051 0.051 0.414 −0.411 −0.003

−0.096 0.096 0.434 −0.381 −0.052

−0.115 0.115 0.493 −0.311 −0.181

0.010 −0.010 −0.083 0.082

0.026 −0.026 −0.186 0.164 0.023

0.035 −0.035 −0.165 0.104 0.060

0.260 −0.260

0.115 −0.113 −0.002

0.056 −0.043 −0.013

−0.052 0.052

−0.047 0.049 −0.002

−0.002 0.024 −0.022

In reaction notation, the hydrogen atoms are assumed and a “•” symbolizes a radical site. bThe blank spaces correspond to sensitivity factors with absolute values

High-Pressure Rate Rules for Alkyl + O2 Reactions. 2. The

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