Article pubs.acs.org/JPCA
Rate Rules, Branching Ratios, and Pressure Dependence of the HO2 + Olefin Addition Channels Stephanie M. Villano, 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: In this work, we present high-pressure rate rules and branching ratios for the addition of HO2 to olefins through the concerted addition channel to form an alkyl peroxy radical (HO2 + olefin → RO2) and through the radical addition channel to form a β-hydroperoxy alkyl radical (HO2 + olefin → β-QOOH). These rate rules were developed by calculating rate constants for a series of addition reactions involving olefins with varying degrees of branching. The individual rate expressions were determined from electronic structure calculations performed at the CBS-QB3 level of theory combined with TST calculations. The calculated rate constants were found to be in good agreement with those reported in the literature. Next, we calculated apparent pressure- and temperature-dependent rate constants for HO2 addition to the terminal site of 1-butene using an energy-grained master equation (ME) approach and QRRK calculations with a modified strong collision (MSC) approximation. The two methods gave similar results for both reaction classes. We found that, for the radical addition reaction, the high-pressure limit for the stabilization channel is not reached until unusually high pressures (>1000 atm). Instead, this reaction leads to the direct formation of an oxirane + OH. In general, the results for the major channels are in reasonable agreement with prior theoretical and experimental data. Finally, to explicitly examine the effect of pressure, we compared concentration−time profiles for the reactions of HO2 plus butene in air that were obtained using both high-pressure and pressure-dependent mechanisms at 10 and 100 atm. These simulations showed that, contrary to general expectations, the manifestation of pressure falloff effects in kinetic modeling studies might be more prevalent at increasing pressures. This behavior is attributed to the reaction of β-QOOH with O2, the rate of which increases with increasing pressure of air. This bimolecular reaction competes with the unimolecular reactions of β-QOOH under conditions where falloff effects are important for that channel.
1. INTRODUCTION The reactions of alkyl radicals with O2 play a central role in the low-temperature oxidation of hydrocarbons, partially accounting for chain branching and the observed negative-temperaturecoefficient (NTC) behavior. This sequence of reactions, shown in Figure 1, primarily leads to the formation of two radicals: OH through cyclic ether formation and/or the addition of O2 to the hydroperoxy alkyl radical (QOOH) and HO2 through concerted elimination. The formation of OH accelerates ignition because its subsequent H-abstraction reactions are highly exothermic, thereby raising the temperature, which increases the rates of other reactions. In contrast, at low temperatures, HO2 is relatively stable and, therefore, generally present in high concentrations. If the concentration of olefins is also sufficiently high, then HO2 can add to these olefins to reenter the alkyl + O2 subset of reactions. Olefins are formed as intermediates during fuel oxidation from concerted elimination reactions and during fuel pyrolysis from the β-scission reactions of alkyl radicals, and they can also be present in the initial fuel blend (e.g., as a component in gasoline, diesel, and biodiesel). The occurrence of this addition reaction has some important implications. It has previously been shown that this reaction results in the formation of an oxirane (three-membered-ring cyclic ether) plus OH.1−6 Because this barrier is lower in energy than the HO2 plus olefin entrance channel, the rate for this low © 2013 American Chemical Society
energy exit channel could compete with the collisional stabilization rate even at very high pressures. Thus, this reaction converts a relatively unreactive HO2 radical into a reactive OH radical, which perhaps might serve to offset some of the inhibiting effect of the concerted elimination reaction. HO2 addition to an olefin can proceed by two distinct pathways: through the concerted addition reaction to form an alkyl peroxy radical (RO2) and through the radical addition channel to form a β-hydroperoxy alkyl radical (β-QOOH). The structures of these two transition states are shown schematically in Figure 2. (A three-dimensional representation is also shown in the Table of Contents figure.) The concerted addition reaction proceeds through a planar five-membered-ring transition state in which the HOO and CC bonds are partially broken and there is partial bond formation between the H atom and one side of the double bond and the OO group and the other side. The radical addition reaction proceeds through a transition state in which the CC bond is partially broken and there is partial bond formation between the OO group and one side of the double bond. These two pathways initially lead to the formation of chemically activated RO2* and Received: May 28, 2013 Revised: July 3, 2013 Published: July 5, 2013 6458
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other olefins were then referenced to the reaction with ethylene. These results showed that the activation energies for these reactions are sensitive to the structure of the olefin and display a strong correlation with the ionization energy of the olefin. Several studies have investigated the formation of HO2 plus olefins from the reactions of alkyl radicals with O2. From this direction, the HO2 plus olefin products are primarily formed from concerted elimination through the thermally activated and/or chemically activated routes,7,8 rather than from βscission of the β-QOOH radical. This is due to the high barrier for isomerization of RO2 to the β-QOOH radical, as well as the presence of other fast competing channels from the β-QOOH isomer (i.e., cyclic ether formation and second O2 addition). Although some of these studies were conducted prior to the discovery of the concerted elimination channel,9 the data are largely consistent with the current understanding of the alkyl + O2 subsystem. Walker and co-workers10,11 reported Arrhenius parameters for the formation of ethylene and propylene from the ethyl + O2 and i-propyl + O2 reactions, respectively, based on relative rate measurements. Wagner et al.12 investigated the ethyl + O2 and n-propyl + O2 reactions in a heated tubular reactor coupled to a photoionization mass spectrometer. Total rate constants and olefin branching fractions were reported as functions of both temperature and pressure. Kaiser 13 investigated the ethyl + O2 reaction over a wide range of conditions using gas chromatography to monitor the concentrations of stable species. These data were then fit to a simplified kinetic scheme to determine Arrhenius parameters for the thermally activated and chemically activated production of ethylene. Taatjes and co-workers7,8,14−18 measured the timedependent production of HO2 (and OH) radicals for several small alkyl + O2 systems as a function of temperature and pressure. These results were compared to the predictions of time-dependent master equation calculations based on the ab initio characterization of the underlying potential energy surface. Good agreement between the experimental data and the model predictions was obtained with only modest adjustments of the calculated barrier heights. The mentioned experiments have provided important insights into the concerted addition and radical addition pathways. However, none of them directly addressed the competition between these two pathways or the impact of HO2 addition on combustion kinetics. Theoretical studies can shed
Figure 1. Schematic diagram of the important reactions in the alkyl + O2/HO2 + olefin subset. A bullet (•) indicates a radical, and an asterisk (*) indicates a chemically activated species. Oxirane, oxetane, and oxolane refer to classes of three-, four-, and five-membered-ring ethers, respectively. Note that the β isomer of QOOH is the only QOOH species that can react to form HO2 + olefin products.
β-QOOH* adducts, which can either isomerize, react to form bimolecular products, or undergo collisional stabilization (see Figure 1). Subsequent thermal activation of the stabilized RO2 and β-QOOH species also results in isomerization or production of bimolecular species. Because of the competition between collisional stabilization and reaction of the activated complexes, the reaction rate and product branching ratios are functions of both temperature and pressure. Since the various products can be formed through more than one reaction pathway, it is difficult to obtain accurate rate parameters for individual pathways experimentally. As a result, only limited experimental data are available in the literature regarding these addition pathways. Walker and co-workers1−6 investigated the formation of oxirane + OH from the addition of HO2 to several olefins. These experiments were conducted at a nominal pressure of 60 Torr from 650 to 775 K. HO2 radicals were produced from the decomposition of tetramethylbutane in the presence of O2, and the end products were quantified using gas chromatography. Using a simplified kinetic scheme, Arrhenius parameters for the reaction of HO2 with ethylene were obtained from relative rate measurements using the HO2 self-reaction rate constant as a reference. Arrhenius parameters for the reactions of HO2 with
Figure 2. Radical addition (1a and 1b) and concerted addition (2a and 2b) reactions of CCCC + HO2. 6459
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light on both issues. Zádor et al.19 reported high-pressure rate constants for both of these addition channels, along with the competing allylic H-atom abstraction reaction, for seven HO2 + olefin reactions. Their rate constant calculations were based on QCISD(T) (quadratic configuration interaction with single and double excitations and triple excitations added perturbatively) calculations that were slightly adjusted to better reproduce the experimental data of Taatjes and co-workers. Temperature- and pressure-dependent rate constants were obtained using timedependent master equation calculations. These results indicate that the relative importance of the three reaction channels depends on the structure of the olefin. In general, for the investigated reactions, H-atom abstraction and radical addition dominate at high temperature, whereas concerted addition dominates at lower temperatures. The reaction of 2-butene is an exception because radical addition was found to dominate over the entire temperature range. The impact of HO2 addition on the ignition delay time of a cyclohexane/cyclohexene mixture was examined at 700 K and 10 bar. In this work, we present rate estimation rules for both the concerted addition and radical addition channels. This study is the third installment in our effort to develop a consistent set of rate rules for the reactions in the alkyl + O2 subset.20,21 These rules were derived by systematically calculating rate constants for HO2 addition to a series of small- to intermediate-sized olefins with varying degrees of branching. The individual rate constants are based on the results of CBS-QB3 electronic structure calculations combined with canonical transition state theory (TST) calculations. These results were compared to previous studies by Zádor et al.,19 Miyoshi,22 and Chen and Bozzelli.23 Next, we compared calculated pressure- and temperature-dependent rate constants for these channels using time-dependent master equation (ME) calculations to those using quantum Rice−Ramsperger−Kassel (QRRK) calculations with a modified strong collision (MSC) approximation. In general, the rate constants calculated by the two methods were found to be in good agreement with each other and with prior theoretical19 and experimental3 data. The rate coefficients for the various channels of the radical addition reaction were found to be very sensitive to pressure, and falloff effects were predicted over an extended range of pressures that are relevant to most combustion/ignition applications. To explicitly examine the effect of pressure, we compared concentration−time profiles for the reactions of HO2 plus butene in air obtained using both high-pressure and pressuredependent mechanisms.
low-frequency vibrational modes that resemble torsions around single bonds were treated as hindered internal rotors rather than harmonic oscillators. Hindrance potentials were calculated at the B3LYP/6-31G(d) level of theory through relaxed surface scans with a step size of 10°. Hindrance 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 Radom35 based on the equilibrium geometry of the species. For each internal rotor, the one-dimensional Schrö dinger equation was solved numerically using the eigenfunctions of the one-dimensional free-rotor 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 were required in this work, bond additivity corrections were not employed. Inspections of the hinderedrotor potentials helped ensure that the optimized geometry of a molecule corresponded 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 corresponded to the desired reaction coordinate. High-pressure rate coefficients were calculated using canonical transition state theory (TST) as k(T) = κ(T)(kBT/h)Vn−1 exp(−ΔG⧧/RT), where κ(T) is the tunneling correction factor, V is the standard molar volume [V = RT/(1 atm)], n is equal to 2 for bimolecular reactions, and ΔG⧧ is the Gibbs free energy difference between the transition state and the contribution from the reaction coordinate and the reactants. The remaining variables have their usual meanings. Tunneling correction factors were calculated with an asymmetric Eckart potential,36 which has been shown to predict correction factors with an accuracy comparable to that achieved with more sophisticated methods.37 Rate constants were calculated over the temperature range of 300−1500 K in 50 K increments and fit to the modified Arrhenius expressions k(T) = ATn exp(−E/RT), where A is the pre-exponential factor, n is the temperature coefficient, and E is related to the activation energy by Ea = E + nRT. The error in the calculated barrier heights is estimated to be ±1.0 kcal mol−1, and the error in the pre-exponential factor is estimated to be a factor of 2. This leads to an uncertainty of a factor of ±2.6 near 1000 K and a factor of ±3.4 near 500 K in the calculated rate constant. 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 served as the basis for the development of the rate rule for a given reaction class. Rate rules were determined by averaging the individual rate constants within a given subset at each temperature and then fitting these averages to modified Arrhenius expressions. We evaluated the pressure and temperature dependence of the 1- and 2-butene (CCCC and CCCC, respectively) plus HO2 reaction systems using two different methods. [Note that throughout the text, we commonly use a notation that omits the hydrogen atoms and uses a bullet (•) to indicate a radical site.] The first method employed the code MESMER, which solves the one-dimensional time-dependent master equation (ME) using an eigenvector/eigenvalue approach.38
2. METHODS Electronic structure calculations were performed using the CBS-QB3 composite method24 as implemented in the Gaussian 03 and 09 software packages.25,26 This method has been shown to predict heats of formation for a large test set of molecules24 with an accuracy of just over 1 kcal mol−1 and has been successfully applied in numerous kinetic studies,27−29 including those of small alkyl radical plus O2 reactions.22,30−34 Geometries, rotational constants, and harmonic frequencies were calculated at the B3LYP/6-311G(d,p) level of theory. The electronic energy was obtained by performing single-point energy calculations at the CCSD(T)/6-31+G(d′) and MP4(SDQ)/6-31G+(d,p) levels of theory and extrapolating to the MP2 complete basis set limit. Additional corrections for spin contamination and systematic errors further improved the energy. To improve the accuracy of ΔfH298, S298, and Cp values, 6460
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olefin + HO2
6461
C3COO• CCC(C2)OO• (CC)2C(C)OO• CCC(C2)OO• CCC(CC)(C)OO• CCCC(C2)OO• C2CC(C2)OO• CCC(C)C(C2)OO• C2CC(CC)(C)OO•
→ → →
→ → →
→ → →
C2CC(CC)OO• C2CC(C)OO• CCC(C)C(C)OO•
C2COO• CCC(C)OO• C2CC(C)OO• CCCC(C)OO•
→ → → →
→ → →
C2CCOO• CCC(C)COO• CCCC(C)COO•
→ → →
CCC(C)OO• CCC(CC)OO• CCCC(C)OO•
CCCOO• CCCCOO• C2CCCOO• CCCCCOO•
→ → → →
→ → →
CCOO•
RO2
→
→
−1
mol )
5.01 × 10−1 1.05 × 10−1 4.70 × 10−1 1.95 × 10−1 1.32 × 10−1 1.53 × 10−1 4.98 × 10−2 1.51 × 10−1 4.94 × 10−2 4.96 × 10−2 4.30 × 10−1 6.57 × 10−1 4.76 × 10−1 2.89 × 10−1 4.27 × 10−1 4.28 × 10−1 1.15 × 100 3.41 × 10−1 2.47 × 10−1 2.16 × 10−2 2.57 × 10−2 5.01 × 10−2 2.70 × 10−2 1.09 × 10−1 7.94 × 10−1 1.46 × 10−1 2.87 × 10−2 7.03 × 10−2 1.71 × 10−1 5.46 × 10−2 1.03 × 10−1 5.08 × 10−3 7.02 × 10−3 2.23 × 10−3 8.93 × 10−3
−1
3.27 3.46 3.27 3.37 3.44 3.42 3.53 3.41 3.53 3.52 3.29 3.23 3.27 3.33 3.31 3.32 3.26 3.31 3.35 3.68 3.63 3.55 3.69 3.42 3.20 3.40 3.53 3.55 3.42 3.63 3.45 3.85 3.82 3.95 3.76
n
−1
7.5 8.9 9.8 9.1 8.6 9.0 11.1 11.7 10.9 10.9 5.4 5.6 5.4 5.3 5.5 7.3 7.5 7.5 6.9 8.1 8.2 8.7 7.9 3.3 3.9 3.4 2.7 4.8 5.2 4.7 4.6 5.6 5.7 5.6 5.5
E (kcal mol )
modified Arrhenius parameters A (cm s
3
−23.5 −23.1 −22.8
−23.4 −23.7 −23.6
−25.2 −25.3 −25.5
−20.2 −20.2 −20.1
−21.4 −21.3 −21.4
−23.7 −24.1 −23.7 −23.9
−17.1 −17.1 −16.9
−19.0 −19.0 −19.2 −18.8
−21.7
ΔrxnH298 (kcal mol )
−1
−1
−39.0 −41.2 −41.2
−39.0 −41.9 −39.3
−40.5 −40.1 −42.3
−37.9 −37.6 −37.6
−35.1 −38.3 −36.9
−37.3 −37.5 −37.7 −37.5
−34.7 −35.4 −35.1
−35.5 −35.4 −34.7 −34.8
−32.9
ΔrxnS298 (cal mol
thermochemistry K )
−1
1.82 × 105 2.93 × 104 1.67 × 104 2.53 × 104 4.48 × 104 2.98 × 104 2.44 × 103 1.90 × 103 2.81 × 103 2.59 × 103 1.34 × 106 1.24 × 106 1.32 × 106 1.34 × 106 1.46 × 106 2.64 × 105 3.75 × 105 1.58 × 105 2.56 × 105 5.57 × 104 4.36 × 104 3.14 × 104 9.18 × 104 6.66 × 106 7.03 × 106 6.92 × 106 5.95 × 106 2.15 × 106 1.47 × 106 3.01 × 106 1.96 × 106 4.44 × 105 4.67 × 105 3.58 × 105 5.05 × 105
500 K
(1.06) (0.81) (1.14)
(0.68) (1.40) (0.91)
(1.06) (1.04) (0.90)
(0.79) (0.57) (1.65)
(1.43) (0.60) (0.98)
(0.93) (0.99) (1.00) (1.09)
(0.78) (1.16) (1.07)
(0.57) (0.87) (1.54) (1.02)
1000 K 7.47 × 107 2.82 × 107 2.19 × 107 2.59 × 107 3.59 × 107 2.95 × 107 7.40 × 106 7.12 × 106 7.77 × 106 7.35 × 106 2.02 × 108 1.94 × 108 1.96 × 108 1.91 × 108 2.27 × 108 1.02 × 108 1.56 × 108 6.68 × 107 8.47 × 107 4.15 × 107 3.33 × 107 2.91 × 107 6.22 × 107 3.78 × 108 4.53 × 108 4.11 × 108 2.72 × 108 2.77 × 108 2.18 × 108 3.93 × 108 2.21 × 108 1.05 × 108 1.16 × 108 9.17 × 107 1.09 × 108
kTST (kTST/krule)
(1.10) (0.87) (1.04)
(0.79) (1.42) (0.80)
(1.20) (1.09) (0.72)
(0.80) (0.70) (1.50)
(1.53) (0.65) (0.83)
(0.96) (0.97) (0.95) (1.12)
(0.96) (1.05) (1.00)
(0.78) (0.92) (1.27) (1.05)
a In this notation, hydrogen atoms are assumed, and a bullet (•) indicates a radical site. The letters identifying rate rules refer to the nature of the α- and β-carbon atoms (see Figure 2) as primary (p), secondary (s), or tertiary (t). bPre-exponential factor divided by 2 to reflect the symmetry of the olefin.
pp 1 CC + HO2b ps rate rule 2 CCC + HO2 3 CCCC + HO2 4 C2CCC + HO2 5 CCCCC + HO2 pt rate rule 6 C2CC + HO2 7 CCC(C)C + HO2 8 CCCC(C)C + HO2 sp rate rule 9 CCC + HO2 10 CCCC + HO2 11 C2CCC + HO2 12 CCCCC + HO2 ss rate rule 13 t-CCCC + HO2b 14 t-CCCCC + HO2 15 t-CCCCC + HO2 st rate rule 16 CCCCC2 + HO2 17 C2CCC + HO2 18 CCC(C)CC + HO2 tp rate rule 19 C2CC + HO2 20 CCC(C)C + HO2 21 (CC)2CC + HO2 ts rate rule 22 C2CCC + HO2 23 CCC(C)CC + HO2 24 C2CCCC + HO2 tt rate rule 25 C2CCC2 + HO2b 26 CCC(C)CC2 + HO2 27 CCC(C)CC2 + HO2
no.
concerted addition reactiona
Table 1. TST Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set Used to Derive Rate Rules for the Concerted Addition Reactions, along with Rate Estimation Rules and Ratios of the Individually Calculated TST Rate Constants to the Rate Rules
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olefin + HO2
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C3•COOH CCC(2C•)OOH (CC)2C(C•)OOH CC•C(C2)OOH CC•C(CC)(C)OOH CCC•C(C2)OOH C2C•C(C2)OOH CCC•(C)C(C2)OOH C2C•C(CC)(C)OOH
→ → →
→ → →
→ → →
C2C•C(CC)OOH C2C•C(C)OOH CCC•(C)C(C)OOH
C2•COOH CCC(C•)OOH C2CC(C•)OOH CCCC(C•)OOH
→ → → →
→ → →
C2C•COOH CCC•(C)COOH CCCC•(C)COOH
→ → →
CC•C(C)OOH CC•C(CC)OOH CCC•C(C)OOH
CC•COOH CCC•COOH C2CC•COOH CCCC•COOH
→ → → →
→ → →
C•COOH
β-QOOH
→
→
−1
mol )
3.56 × 101 7.91 × 102 2.56 × 103 3.01 × 102 1.08 × 103 4.42 × 102 1.35 × 103 1.64 × 104 2.99 × 103 1.12 × 103 1.06 × 101 2.34 × 101 2.09 × 101 2.51 × 101 1.95 × 101 4.62 × 101 3.87 × 102 1.71 × 102 4.95 × 101 1.86 × 102 3.27 × 101 3.09 × 103 3.14 × 102 3.37 × 10−1 8.19 × 100 9.49 × 10−1 8.81 × 10−4 1.72 × 10−1 1.31 × 100 1.07 × 10−1 5.86 × 10−1 1.69 × 100 1.96 × 101 4.97 × 10−1 3.16 × 10−1
A (cm s
−1
3.22 2.78 2.62 2.86 2.76 2.88 2.67 2.43 2.55 2.65 3.29 3.21 3.21 3.74 3.22 3.09 2.89 2.94 3.13 2.95 3.16 2.61 2.90 3.67 3.28 3.54 4.38 3.70 3.44 3.81 3.53 3.44 3.18 3.60 3.62
n
−1
11.1 9.5 9.7 9.5 9.4 9.4 7.9 8.3 7.8 8.0 9.1 9.1 9.2 8.8 9.1 7.2 7.5 7.4 7.3 5.4 5.2 5.8 5.8 7.2 7.9 7.3 6.1 4.7 5.2 4.5 5.0 2.7 3.0 2.5 2.9
E (kcal mol )
modified ME-derived parameters 3
−11.7 −11.1 −11.9
−8.8 −9.2 −9.0
−6.2 −6.8 −7.3
−9.0 −8.6 −8.6
−6.9 −6.9 −6.8
−5.5 −5.8 −5.3 −5.5
−6.0 −6.1 −6.0
−5.0 −5.1 −5.0 −5.1
−3.9
ΔrxnH298 (kcal mol )
−1
−1
−33.9 −35.3 −35.2
−33.0 −34.7 −32.6
−33.4 −35.8 −38.9
−30.4 −31.2 −31.5
−28.9 −29.8 −30.7
−31.4 −32.6 −32.5 −33.4
−29.5 −30.1 −30.1
−29.1 −29.9 −30.3 −29.7
−28.0
ΔrxnS298 (cal mol
thermochemistry K )
−1
2.51 × 105 1.82 × 106 1.73 × 106 1.05 × 106 2.50 × 106 2.01 × 106 7.49 × 106 1.37 × 107 8.89 × 106 4.82 × 106 8.79 × 105 1.10 × 106 9.41 × 105 4.51 × 105 1.03 × 106 1.03 × 107 1.28 × 107 9.20 × 106 8.76 × 106 6.99 × 107 6.02 × 107 9.80 × 107 6.53 × 107 2.05 × 106 2.14 × 106 2.21 × 106 1.31 × 106 1.56 × 107 1.44 × 107 2.22 × 107 1.32 × 107 2.15 × 108 3.60 × 108 2.10 × 108 1.06 × 108
500 K
(1.68) (0.98) (0.49)
(0.93) (1.43) (0.85)
(1.05) (1.08) (0.64)
(0.86) (1.41) (0.94)
(1.25) (0.90) (0.86)
(1.25) (1.08) (0.52) (1.17)
(1.83) (1.19) (0.65)
(0.96) (0.58) (1.38) (1.11)
6.12 × 108 1.48 × 109 1.38 × 109 9.29 × 108 1.90 × 109 1.71 × 109 2.59 × 109 4.86 × 109 2.68 × 109 1.71 × 109 8.26 × 108 1.00 × 109 8.68 × 108 5.05 × 108 9.27 × 108 3.33 × 109 4.07 × 109 2.87 × 109 3.05 × 109 8.31 × 109 7.21 × 109 1.10 × 1010 8.89 × 109 9.61 × 108 1.10 × 109 1.02 × 109 5.78 × 108 2.13 × 109 2.10 × 109 3.00 × 109 1.88 × 109 9.10 × 109 1.46 × 1010 9.07 × 109 5.55 × 109
1000 K
kTST (kTST/krule)
(1.61) (1.00) (0.61)
(0.99) (1.41) (0.88)
(1.14) (1.06) (0.60)
(0.87) (1.32) (1.07)
(1.22) (0.86) (0.92)
(1.22) (1.05) (0.61) (1.12)
(1.88) (1.04) (0.66)
(0.93) (0.63) (1.28) (1.16)
a In this notation, hydrogen atoms are assumed, and a bullet (•) indicates a radical site. The letters identifying rate rules refer to the nature of the α- and β-carbon atoms (see Figure 2) as primary (p), secondary (s), or tertiary (t). bPre-exponential factor divided by 2 to reflect the symmetry of the olefin.
pp 1 CC + HO2b ps rate rule 2 CCC + HO2 3 CCCC + HO2 4 C2CCC + HO2 5 CCCCC + HO2 pt rate rule 6 C2CC + HO2 7 CCC(C)C + HO2 8 CCCC(C)C + HO2 sp rate rule 9 CCC + HO2 10 CCCC + HO2 11 C2CCC + HO2 12 CCCCC + HO2 ss rate rule 13 t-CCCC + HO2b 14 t-CCCCC + HO2 15 t-CCCCC + HO2 st rate rule 16 CCCCC2 + HO2 17 C2CCC + HO2 18 CCC(C)CC + HO2 tp rate rule 19 C2CC + HO2 20 CCC(C)C + HO2 21 (CC)2CC + HO2 ts rate rule 22 C2CCC + HO2 23 CCC(C)CC + HO2 24 C2CCCC + HO2 tt rate rule 25 C2CCC2 + HO2b 26 CCC(C)CC2 + HO2 27 CCC(C)CC2 + HO2
no.
radical addition reactionsa
Table 2. TST Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set Used to Derive Rate Rules for the Radical Addition Reactions, along with Rate Estimation Rules and Ratios of the Individually Calculated TST Rate Constants to the Rate Rules
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Table 3. Arrhenius Fits (400−1100 K) of the High-Pressure Rate Rules and Average Heats of Reaction reaction typea αβ ppb ps pt sp ss st tp ts tt
concerted addition
radical addition
A (cm3 mol−1 s−1)
Ea (kcal mol−1)
ΔrxnH298 (kcal mol−1)
× × × × × × × × ×
11.8 13.7 15.8 9.6 11.5 13.0 7.9 9.3 10.7
−21.7 −19.0 −17.0 −23.8 −21.4 −20.2 −25.3 −23.6 −23.1
2.76 1.86 2.01 2.32 3.05 2.75 1.93 2.68 2.21
1010 1010 1010 1010 1010 1010 1010 1010 1010
A (cm3 mol−1 s−1)
Ea (kcal mol−1)
ΔrxnH298 (kcal mol−1)
× × × × × × × × ×
15.4 13.3 11.5 13.3 11.2 9.4 11.9 9.7 7.3
−3.9 −5.0 −6.0 −5.5 −8.7 −8.7 −6.8 −9.0 −11.5
1.34 7.69 9.23 5.92 8.47 9.94 3.17 2.89 3.77
1012 1011 1011 1011 1011 1011 1011 1011 1011
a Letters refer to the nature of the α- and β-carbon atoms as primary (p), secondary (s), or tertiary (t). bPre-exponential factor divided by 2 to reflect the symmetry of the olefin.
using the isothermal batch reactor module in the CHEMKINPRO software package.46
The microcanonical rate constants [k(E)] can be obtained either from Rice−Ramsperger−Kassel−Marcus (RRKM) calculations or from the inverse Laplace transform (ILT) of the high-pressure rate constant. The density and sum of states were calculated using exact counts. In this work, we employed the ILT method and treated the collisional energy transfer using an exponential down model. The phenomenological temperatureand pressure-dependent rate constants were extracted based on a procedure similar to that of Bartis and Widom.39 The second method employed quantum Rice−Ramsperger−Kassel (QRRK) calculations using the code ChemDis.40 In this method, the k(E) values were determined from the highpressure rate constants. The density of states for each isomer was determined from three representative frequencies and their degeneracies. The three frequencies were obtained from a fit of the corresponding heat capacity. Collisional energy transfer was treated using the modified strong collision (MSC) approximation,41 and a steady-state assumption was invoked for the energized isomers to obtain the apparent temperature- and pressure-dependent rate constants. Although this approach is clearly more approximate than the master equation approach, it does have a benefit in that it can be used to investigate the pressure dependence of larger-sized reactions for which highlevel ab initio calculations are often not practical. The required input parameters can easily be estimated using rate rules and group additivity. This work along with other previous studies42,43 provides a validation for this approach. For both sets of calculations, the average energy transferred per deactivating collision (ΔEdown) with N2 as the bath gas was assumed to be independent of temperature and was set at −300 cm−1. (For the MSC calculations, ΔEdown was converted to the corresponding average ΔEall value.) Although there is considerable uncertainty in the assignment of energy-transfer parameters, the value used in this study is typical of those utilized in the literature.19,44,45 The Lennard-Jones parameters used for N2 were σ = 3.6 Å and ε = 98 K, whereas those for the CCCCOO• and various hydroperoxy butyl radicals were set to σ = 5.4 Å and ε = 305 K. To explicitly examine the effect of pressure, concentration− time profiles for the reactions of CCCC and CCCC with HO2 radicals at 750 K and pressures of 10 and 100 atm were calculated with a high-pressure mechanism and compared to those predicted with a pressure-dependent mechanism. The initial mole fraction of HO2 was arbitrarily set to 0.001, and those of CCCC and CCCC were both set to 0.0005, with the remainder being air. These simulations were performed
3. RESULTS AND DISCUSSION 3.1. High-Pressure Rate Rules and Branching Ratios. High-pressure reaction rate coefficients for both the concerted addition and radical addition channels were calculated for a series of reactants. This series contained C2−C5 and several selected C6 and C7 olefins. The obtained rate coefficients are provided in modified Arrhenius form in Tables 1 and 2. Also included are the rate constants at 500 and 1000 K and the enthalpies and entropies of reaction. For symmetric olefins (reactions 1, 13, and 25), the pre-exponential factors are reported per addition site. For reactions involving olefins that exist as cis and trans isomers, only the trans olefin was considered. For both addition channels, the rate coefficients were grouped according to the nature of α- and β-carbon atoms in the transition state structure. The locations of the α- and βcarbon atoms are shown in Figure 2; the peroxy or hydroperoxy group is attached to the α-carbon, and the reactive neighboring carbon is defined as the β-carbon. Within the set of reactants considered, each of the two carbon atoms is either a primary (p), secondary (s), or tertiary (t) carbon. This leads to nine subclasses of reactions, namely, pp, ps, pt, sp, ss, st, tp, ts, and tt, where the first letter refers to the nature of the α-carbon site and the second to the nature of the β-carbon site. Aside from the addition of HO2 to ethylene, which is the only pp-type reaction, the combination of the different α- and β-carbon types leads to eight rate rules for each reaction channel. These rate rules, expressed on a per-site basis, are provided in Tables 1 and 2. Between 500 and 1000 K, these rules describe almost all of the investigated reactions within a factor of 2. In Table 3, we provide the rate rules in simple Arrhenius form for a narrower temperature range. This table is intended to show the trends in the pre-exponential factors and activation energies; however, because these reactions exhibit strong nonArrhenius behavior, we recommend the use of the rules in Tables 1 and 2. The pre-exponential factors for the concerted addition reactions group closely together. The pre-exponential factors for the radical addition reaction span a factor of 5. Of these, the tp, ts, and tt reaction subclasses have the lowest values. This is attributed to an unfavorable steric interaction between the substituted Cα atom and the OO group, which leads to a higher barrier for rotation about this bond. The same effect was not observed for the corresponding concerted addition reactions because the CαOO bond is not free to rotate (cf. Figure 2). The pre-exponential factors for the 6463
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from a primary β-carbon, to a secondary β-carbon, to a tertiary β-carbon. We have previously reported rate rules for the concerted elimination reaction20 of RO2 and the β-scission of β-QOOH to form HO2,21 which are the reverse reactions to the concerted addition and radical addition considered here. In the “reverse” directions, these reactions are endothermic, and the activation energies are surprisingly not sensitive to the overall thermochemistry of the reactions. These activation energies, which can be approximated from the differences in the activation energies for the forward addition reactions and the reaction enthalpies in Table 3, are essentially constant for all nine reaction classes. As a result, the rate constants group more closely together than those for the addition reactions. For the concerted elimination reaction, only two rules were assigned based on the degree of substitution of the olefin formed: kCE = 3.16 × 108 s−1 T1.20 exp(−29.4 kcal mol−1/RT) for the pp, ps, pt, sp, ss, and ts reaction types and kCE = 1.30 × 1011 s−1 T0.50 exp(−30.0 kcal mol−1/RT) for the st, ts, and tt reaction types. These two rate rules differ by a factor of 4 at temperatures below 1500 K. For the β-scission reaction, only one rate rule was assigned: kβS = 1.22 × 1011 s−1 T0.57 exp(−15.1 kcal mol−1/ RT) for all nine subclasses. These results are consistent with the Evans−Polanyi slopes of approximately 1 for the addition reactions. Considering the reactions in the exothermic direction, this suggests that the transition state structures are “late”. In other words, the transition state structure is much more sensitive to the structure of the product (RO2 or βQOOH) than to the structure of the reactants (HO2 + olefin). As the RO2 bond or the β-QOOH radical site changes from primary to secondary to tertiary, the stability of the product radical changes. Approximately the same magnitude of change also occurs in the corresponding transition states, and therefore, the changes in activation energies for the exothermic reactions are almost equivalent to the changes in the reaction enthalpies. HO2 addition can occur on either side of a double bond. If the olefin is not symmetric about the double bond, then this leads to two sets of products for each reaction channel. For a given olefin, the two potential reactions pair as pp + pp, ss + ss, and tt + tt if there is no difference in the degree of substitution on either side of the double bond and as ps + sp, pt + tp, and st + ts if the degree of substitution differs. For example, the reactions shown in Figure 2 correspond to the ps + sp category. Figure 4 shows the branching ratios for addition to the more substituted site of an olefin, defined as BR(x) = k(x)/[k(x) + k(y)], where x refers to the more substituted site. Again, using the reaction in Figure 2 as an illustrative example, the branching ratio is BR(sp) = k(sp)/[k(sp) + k(ps)]. For the concerted addition channel, HO2 addition to the more substituted site of the olefin is preferred. This is consistent with the fact that tertiary RO2 bonds are stronger than secondary RO2 bonds, which, in turn, are stronger than primary RO2 bonds. In contrast, for the radical addition reaction, HO2 is more likely to add to the less substituted side of the olefin. In this case, the barriers for the two competing reactions are similar, and the difference is mostly in the pre-exponential factors. The addition reaction to the less substituted site has a higher pre-exponential factor and is favored sterically. This trend is consistent with alkyl radical addition, which also favors the less substituted site of the olefin.47 Figure 5a compares the site-specific branching ratios for the radical addition reaction versus the concerted addition reaction [BRsite RA = kRA/(kRA + kCA)] for the nine subclasses of reactions.
concerted addition reactions were found to be more than one order of magnitude lower than those for radical addition. This difference is due to the different numbers of hindered rotors in the two transition states. The concerted addition reaction proceeds though a planar five-membered-ring transition state and has two fewer internal rotor modes than the corresponding transition state for radical addition. The activation energies for both reaction channels follow Evans−Polanyi relationships, as shown in Figure 3. In these
Figure 3. Evans−Polanyi relationship for the (a) concerted and (b) radical addition reactions. The notation in the legend identifies the nature of the α- and β-carbons in the alkyl peroxy or β-hydroperoxy alkyl radical products as primary (p), secondary (s), or tertiary (t). The first letter corresponds to the α-position, and the second letter to the β-position.
plots, the Arrhenius activation energies (Ea) were calculated from the n and E parameters of the modified Arrhenius fits in Tables 1 and 2 at 750 K. Notice that, although the two reaction channels have very different exothermicities, both the magnitudes and ranges of calculated activation energies for the two channels are comparable. Furthermore, even though these reactions are exothermic, the slopes of the Evans−Polanyi plots are close to 1. This means that, as the reaction enthalpy changes, almost the entire enthalpy difference is reflected in the change in activation energy. In both cases, for a given type of βcarbon, the barrier heights were found to increase in going from a tertiary α-carbon, to a secondary α-carbon, and even further in going to a primary α-carbon (see also Table 3). For the radical addition reaction, if the type of α-carbon is kept constant, the barrier heights also increase in going from a tertiary β-carbon, to a secondary β-carbon, to a primary βcarbon. In contrast, for the concerted addition reaction, for a constant type of α-carbon, the barrier heights increase in going 6464
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temperature decreases, the concerted additions that involve a primary β-carbon begin to become more important and dominate below 500 K. As shown in Table 3, for these three reaction classes (pp, sp, and tp), the barriers for concerted addition are lower than those for radical addition. These concerted addition reactions thus can compete against the radical addition reactions despite their lower pre-exponential factors. For the other six reaction classes, the barriers for concerted addition are higher than or approximately equal to those for radical addition, and addition reactions with the higher pre-exponential factors are favored. Whereas Figure 5a relates to site-specific branching ratios, Figure 5b presents the branching ratios for the total addition reactions to an olefin; hence, we consider that HO2 could add to either side of the double bond. Considering the six categories discussed thus far, this leads to the following reactions: pp + pp, ps + sp, ss + ss, pt + tp, st + ts, and tt + tt. For the reactions shown in Figure 2, the ps + sp class, the radical addition branching ratio is expressed as BRtotal RA = [kRA(ps) + kRA(sp)]/ [kRA(ps) + kRA(sp) + kCA(ps) + kCA(sp)]. Figure 5b reveals that the concerted addition channel is important only for α-olefins (terminal double bonds) at lower temperatures. Such reactions could be important in modeling ignition kinetics because the reactions of branched alkyl radicals with O2 have the potential to form more of these α-type olefins. The reason that the concerted addition reaction becomes important for these types of olefins at lower temperature is that the activation energies for the sp and tp concerted addition reactions are much lower than the activation energies for the sp, ps, tp, and pt radical addition reactions. Rate rules for both addition channels were previously calculated by Miyoshi22 at the CBS-QB3 level, but with a somewhat different way of postprocessing the ab initio data; these differences were discussed in an earlier work.20 A comparison of the rate rule for the sp subclass derived in this work to that of Miyoshi is provided in panels a and b of Figure 6 for the concerted and radical addition reactions, respectively. Also included are individual rate constants from Zádor et al.19 and Chen and Bozzelli.23 The rate constants from Zádor et al. are based on adjusted QCISD(T)/cc-pV∞Z//B3LYP/6-311+ +G(d,p) calculations. The rate constant from Chen and Bozzelli is for the radical addition channel and is based on CBS-q//B3LYP/6-31G(d) calculations. The various rate constants for the concerted addition channel are in remarkably good agreement. More spread is observed among the rate constants for the radical addition channel. The two CBS-QB3derived rate rules agree within the estimated error of the rate constant calculation. Chen and Bozzelli’s rate constant is approximately a factor of 3 lower than our rate rule over the entire temperature range. Of the two rate constants from Zádor et al., one is in good agreement with the rate rule reported here whereas the other is significantly lower. The difference in the latter value is mostly attributed to a higher activation energy. For the other values, the observed differences are mostly attributed to differences in the pre-exponential factors. This is likely related to subtle differences in the hindered-rotor treatment and/or in the optical isomers assignments. The spread in available radical addition rate constants leads to significantly different branching ratios in the literature for the two addition channels. 3.2. Pressure-Dependent Rate Constants. In an effort to characterize the pressure dependence of these two reaction channels, we calculated apparent temperature- and pressure-
Figure 4. Branching ratios for HO2 addition to the more substituted side of the olefin [BR = kx/(kx + ky), where x refers to the highersubstituted sites sp, tp, and ts] for the concerted addition (red) and radical addition (blue) reaction. The notation identifies the nature of the α- and β-carbons in the hydroperoxy alkyl or alkyl peroxy products as primary (p), secondary (s), or tertiary (t). The first letter corresponds to the α-position, and the second letter to the β-position.
Figure 5. Branching ratios for the radical addition reaction versus concerted addition. (a) Site-specific branching ratios for the nine subclasses: BRsite RA = kRA/(kRA + kCA). (b) Total branching ratios for the six types of olefins: BRtotal RA = [kRA(x) + kRA(y)]/[kRA(x) + kRA(y) + kCA(x) + kCA(y)], where x refers to one site on the olefin and y to the other site. The notation identifies the nature of the α- and β-carbons in the hydroperoxy alkyl or alkyl peroxy products as primary (p), secondary (s), or tertiary (t). The first letter corresponds to the αposition, and the second letter to the β-position.
The indices RA and CA here and in subsequent formulas refer to radical addition and concerted addition, respectively. At high temperatures, radical addition dominates in all subclasses. This is because the radical addition reaction has a higher preexponential factor than the concerted addition reaction. As the 6465
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CCCC• + O2 (P1). Whereas the barriers for isomerization through six- and seven-membered rings (W3 and W4) are considerably lower in energy than the concerted addition entrance barrier, the barriers for the subsequent oxetane (fourmembered-ring cyclic ether) plus OH and oxolane (fivemembered-ring cyclic ether) plus OH (P3 and P4) formation are only 0.4 and 2.1 kcal mol−1 lower in energy, respectively, than the entrance barrier. Moreover, both of these isomerization and subsequent cyclic ether formation reactions have significantly smaller pre-exponential factors compared to that for dissociation to form CCCC• + O2 (P1). Based on this analysis, one would expect concerted addition to yield CCCC• + O2 (P1) as bimolecular products and also perhaps stabilized CCCCOO• (W1). The radical addition reaction mainly produces oxirane + OH (P2). The pressure-dependent rate constants for the chemically activated reactions for HO2 addition to the terminal end of CCCC at 750 K are shown in Figure 8. Plots a and b correspond to addition through the concerted addition channel, whereas plots c and d are for the radical addition channel. Plots a and c were obtained using the ME approach as implemented in the program MESMER, and plots b and d were obtained with the QRRK/MSC approach. For the concerted addition channel, the two sets of rate constants for stabilization to the CCCCOO• adduct (W1) and for dissociation to form CCCC• + O2 (P1) are in very good agreement. Some small discrepancies are observed for the minor channels. The rate constant for isomerization to W3 is slightly lower in the QRRK/MSC calculation than in the ME calculation. The QRRK/MSC approach predicts larger contributions from the redissociation to the initial reactants [R(CA)] pathway, and the rate constants for cyclic ether + OH formation (P2−P4) display stronger falloff behavior as the pressure increases than the ME rate constants do. For the radical addition channel, the differences in the two methods are somewhat larger. The QRRK/MSC approach predicts that the rate constant for oxirane + OH formation (P2) begins to fall off from the low-pressure limit value at a slightly lower pressure than the ME calculation does. This is related to a slight increase in the stabilization channel (W2) relative to the ME result. Similarly, the QRRK-predicted rate constants for stabilization to CCCCOO• (W1) and dissociation back out the entrance channel (RA) and to form CCCC• + O2 (P1) are higher than found with the ME approach. The rate constants for the other two cyclic ether + OH formation (P3 and P4) channels show falloff behavior at lower pressures. Despite these differences, the general agreement between the two methods is rather good. For the major channels, oxirane + OH formation (P2), stabilization to CCC•COOH (W2), and dissociation to CCCC + HO2 [R(RA)], the difference in the two sets of rate constants is less than a factor of 2 over the entire pressure range. The two types of HO2 addition reactions display very different pressure dependences. In the case of the concerted addition channel, collisional stabilization dominates over most of the investigated pressure range, and the high-pressure limit is reached at approximately 10 atm. In contrast, for the radical addition channel, the rate constant for the formation of ethyl oxirane + OH (P2) is in the low-pressure limit (independent of pressure) until approximately 10 atm; collisional stabilization to the CCC•COOH radical (W2) is not competitive with that channel until about 100 atm, and the high-pressure limit for stabilization is not reached until pressures in excess of 1000
Figure 6. Comparisons of the rate rules derived in this work (red lines) with those of Miyoshi22 (blue lines) and the individual values of Zádor et al.19 (black lines) and Chen and Bozzelli23 (green line) for the (a) concerted and (b) radical addition channels of the sp reaction subclass.
dependent rate coefficients for the addition of HO2 to several olefins using a time-dependent ME method and/or a steadystate QRRK/MSC method. In both cases, the k(E) values were obtained from the corresponding high-pressure rate constants. The high-pressure rate constants for concerted and radical addition reactions were taken from Tables 1 and 2, and the remaining high-pressure rate constants were taken from Villano et al.20,21 In the first part of this section, we present the rate constants for the addition of HO2 to the terminal site of 1butene (CCCC). Next, we compare the pressure-dependent rate constants for the addition of HO2 to the central position of propene (CCC) to those calculated by Zádor et al.19 and to the rate constants for several other HO2 + olefin reactions to those measured by Walker and co-workers.3,10,11 The high-pressure rate constants for HO2 addition to the terminal end of CCCC can be calculated from the Arrhenius parameters in Table 4 (reactions 1, 2, 7, 9−11, 15, 18, and 20). This potential energy surface is shown in Figure 7. For the radical addition pathway, the entrance barrier (to form W2) is 10.5 kcal mol−1 relative to the CCCC + HO2 reactants (R). The barrier for isomerization through a five-membered ring to form CCCCOO• (W1) is 2.7 kcal mol−1 higher in energy than the entrance barrier. However, perhaps the most interesting feature of this pathway is that the exit channel to form oxirane + OH (P2) has a barrier that is 3.9 kcal mol−1 lower in energy than the entrance barrier. In contrast, the entrance barrier for concerted addition is 11.9 kcal mol−1 relative to the reactants (R) and is 4.3 kcal mol−1 lower in energy than that to form 6466
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Table 4. High-Pressure Mechanism for the Addition of HO2 to 1- and 2-Butenea kfor no.
reactionb
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
CCCC + HO2 ↔ CCCCOO• CCCC + HO2 ↔ CCC•COOH CCCC + HO2 ↔ CCC(C)OO• CCCC + HO2 ↔ CCC(C•)OOH CCCC + HO2 ↔ CCC(C)OO• CCCC + HO2 ↔ CC•C(C)OOH CCCCOO• ↔ CCCC• + O2 CCC(C)OO• ↔ CCC•C• + O2 CCCCOO• ↔ CCC•COOH CCCCOO• ↔ CC•CCOOH CCCCOO• ↔ C•CCCOOH CCC(C)OO• ↔ CCC(C•)OOH CCC(C)OO• ↔ CC•C(C)OOH CCC(C)OO• ↔ C•CC(C)OOH CCC•COOH → ethyl oxirane + OH CCC(C•)OOH → ethyl oxirane + OH CC•C(C)OOH → 2,3-dimethyl oxirane + OH CC•CCOOH → methyl oxetane + OH C•CC(C)OOH → methyl oxetane + OH C•CCCOOH → oxolane + OH • OOC(CC)COOH ↔ CCC•COOH + O2 • OOCC(CC)OOH ↔ CCC(C•OH + O2 • OOC(C)C(C)OOH ↔ CC•C(C)OOH + O2 • OOC(C)CCOOH ↔ CC•CCOOH + O2 • OOCCC(C)OOH ↔ C•CC(C)OOH + O2 • OOCCCCOOH ↔ C•CCCOOH + O2 • OOC(CC)COOH → HOOC(CC)CO + OH • OO CC(CC)OOH → HOOCC(CC)O + OH • OOC(C)C(C)OOH → HOOC(C)C(C)O + OH • OOC(C)CCOOH → HOOC(C)CCO + OH • OOCCC(C)OOH → HOOCCC(C)O + OH • OOCCCCOOH → HOOCCCCO + OH HOOC(CC)CO → CCCO + C•O + OH HOOCC(CC)O → CO + CCC•O + OH HOOC(C)C(C)O → CCO + CC•O + OH HOOC(C)CCO → CCO + C•CO + OH HOOCCC(C)O → CO + CC(C•)O + OH HOOCCCCO → CO + C•CCO + OH
krev
A (s−1 or cm3 s−1 mol−1)
Ea (kcal mol−1)
× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×
13.7 13.5 9.6 13.2 11.6 11.1 32.3 33.8 30.7 20.0 22.3 34.3 30.3 22.9 12.4 11.6 10.6 17.0 19.0 12.0 33.8 32.3 33.8 33.8 32.3 32.3 25.3 24.5 23.6 18.8 17.3 16.9 45.5 45.5 45.5 45.5 45.5 45.5
2.31 7.64 2.66 7.35 1.19 2.39 1.50 7.90 2.05 3.02 5.25 4.52 4.71 7.39 5.17 4.07 3.92 5.02 8.73 5.17 7.90 1.50 7.90 7.90 1.50 1.50 2.82 9.08 4.83 2.51 2.91 2.27 1.50 1.50 1.50 1.50 1.50 1.50
1010 1011 1010 1011 1011 1012 1014 1014 1012 1011 1010 1012 1012 1011 1012 1012 1012 1011 1011 1010 1014 1014 1014 1014 1014 1014 1012 1012 1012 1011 1011 1010 1016 1016 1016 1016 1016 1016
A (s−1 or cm3 s−1 mol−1)
Ea (kcal mol−1)
4.42 8.10 1.05 1.17 1.21 9.04 8.10 1.31 1.14 1.91 3.83 1.82 1.76 3.03
× × × × × × × × × × × × × ×
1012 1012 1013 1013 1013 1012 1011 1012 1011 1010 109 1011 1011 1010
30.8 16.5 31.6 16.5 30.9 15.8 −1.0 −1.5 16.7 6.5 5.7 15.4 15.7 5.8
1.31 8.10 1.31 1.31 8.10 8.10
× × × × × ×
1012 1011 1012 1012 1011 1011
−1.5 −1.0 −1.5 −1.5 −1.0 −1.0
a
Rate parameters for reactions 1−6 were taken from Tables 1 and 2, and those for reactions 7−20 were taken from Villano et al.,20,21 but were refit to a simple Arrhenius expression as this is the preferred format in this QRRK code. See text for the origin of the rate parameters for reactions 21−38. b In this notation, hydrogen atoms are assumed, and a bullet (•) indicates a radical site.
atm. These differences can be attributed to both the shallow βQOOH (W2) well, compared to the relatively deep concerted addition RO2 (W1) well, and the presence of the low-energy exit channel (P2) from W2 (cf. Figure 7). A consequence of the shallow β-QOOH well is that its density of states is low. This results in high k(E) values for all of its unimolecular processes, in particular, for the low-energy exit channel, and therefore, very high pressures are required for bimolecular stabilization to be competitive. Master equation approaches are particularly well-suited to describe such reaction systems as the energy transfer is treated in a physically correct way in the sense that, on average, only small amounts of energy are transferred per collision. Therefore, the populations of the energy states above and slightly below the barriers for product channels are well described. On the other hand, the MSC approximation considers that only a small fraction (β) of fully deactivating
collisions occur whereas the remaining collisions are completely elastic. This could lead to a poor description of the energy populations near the exit channels and, consequently, less reliable predictions of the actual rate constants. From this point of view, the rather good agreement (factors of
