High-Pressure Rate Rules for Alkyl + O2 Reactions ... - ACS Publications

ARTICLE pubs.acs.org/JPCA

High-Pressure Rate Rules for Alkyl + O2 Reactions. 1. The Dissociation, Concerted Elimination, and Isomerization Channels of the Alkyl Peroxy Radical Stephanie M. Villano,† Lam K. Huynh,‡ Hans-Heinrich Carstensen,† and Anthony M. Dean*,† † ‡

Chemical and Biological Engineering Department, Colorado School of Mines, Golden, Colorado 80401, United States International University, Vietnam National University
HCMC and Institute for Computational Science and Technology at HCMC, Vietnam

bS Supporting Information ABSTRACT: The reactions of alkyl peroxy radicals (RO2) play a central role in the low-temperature oxidation of hydrocarbons. In this work, we present high-pressure rate estimation rules for the dissociation, concerted elimination, and isomerization reactions of RO2. These rate rules are derived from a systematic investigation of sets of reactions within a given reaction class using electronic structure calculations performed at the CBS-QB3 level of theory. The rate constants for the dissociation reactions are obtained from calculated equilibrium constants and a literature review of experimental rate constants for the reverse association reactions. For the concerted elimination and isomerization channels, rate constants are calculated using canonical transition state theory. To determine if the high-pressure rate expressions from this work can directly be used in ignition models, we use the QRRK/ MSC method to calculate apparent pressure and temperature dependent rate constants for representative reactions of small, medium, and large alkyl radicals with O2. A comparison of concentration versus time profiles obtained using either the pressure dependent rate constants or the corresponding high-pressure values reveals that under most conditions relevant to combustion/ ignition problems, the high-pressure rate rules can be used directly to describe the reactions of RO2.

’ INTRODUCTION Decades of research have provided us with a solid understanding of the basic mechanism of hydrocarbon low-temperature oxidation. As a result, kinetic models are able to accurately predict the ignition behavior of a wide range of hydrocarbon fuels. Such models, when used in computational fluid dynamics (CFD) codes to simulate heat transfer, spray formation, vaporization, and mixing, provide powerful design tools to improve fuel efficiency and emissions in modern internal combustion engines and other applications. Despite the undisputable successes of these models, there continues to be a need for further improvements, either to expand the accuracy of predictions to a wider range of conditions or to extend the models to more complex fuels. This need is motivated by the demand for increased engine performance and efficiency, lower emissions, and the development of sustainable fuels. Current efforts are focused on extending these models to multicomponent fuels and new fuel types such as biofuels and synthetic fuels that may have significantly different ignition properties from traditional fuels. These extensions include incorporation of new mechanistic insights such as the concerted elimination pathway and special reactions of olefins, and the implementation of pressure dependence to account for the diversity of operating conditions found in typical combustion processes. Theoretical studies will play an increasingly important role in such mechanism upgrades and extensions. r 2011 American Chemical Society

Computational chemistry has matured to a point that its predictions nowadays often match the quality of experiments, whenever they are available, and it offers the opportunity to perform calculations for conditions that are difficult to explore experimentally. Furthermore, each individual reaction channel can be accessed independently, and all reactions can be consistently described using the same level of theory. However, due to the large size (typically hundreds of species and thousands of reactions) of modern kinetic mechanisms, it is not feasible to calculate rate expressions for each reaction separately with sophisticated electronic structure methods. Instead, large mechanisms are typically constructed in a systematic way on the basis of rate estimation rules for the various reaction classes and of group additivity concepts to estimate thermodynamic properties. Because in the past the rate rules had to be assigned on the basis of limited sets of published rate constants, which often lead to large uncertainties in the assignments, we propose to validate and if necessary to replace those rules with new ones based on calculated averaged rate constants taken from a set of training reactions. This provides the main motivation of the current study. Received: August 17, 2011 Revised: October 14, 2011 Published: October 17, 2011 13425

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collision analysis for representative small, medium, and large sized alkyl peroxy radicals over a wide range of temperatures and pressures. These results indicate that, under most conditions relevant to combustion/ignition problems, the high-pressure rate rules can be used directly to describe the reactions of RO2.

Figure 1. Potential energy diagram describing the unimolecular reaction channels of RO2 radicals: the concerted elimination (


), isomerization (
), and dissociation to reactants.

A crucial subset of reactions in hydrocarbon low temperature oxidation involves the reactions of the alkyl peroxy radicals: the dissociation (RO2 f R + O2), concerted elimination (RO2 f olefin + HO2), and isomerization to form the hydroperoxy alkyl radical (RO2 f QOOH). The competition between these channels is shown in Figure 1. At lower temperatures, the RO2 adduct preferably reacts by concerted elimination and/or isomerization. The concerted elimination channel is believed to inhibit ignition because it produces the relatively unreactive HO2 radical, whereas the isomerization channel may lead to chain branching through the subsequent reactions of the hydroperoxy alkyl radical. As the temperature increases, dissociation of the alkyl peroxy radical becomes more important; this reversibility is linked to the characteristic negative temperature coefficient (NTC) behavior. Several theoretical studies have evaluated high-pressure rate constants for individual RO2 reactions.1
4 However, comparison of the results shows a considerable spread in the derived values, which might be caused by different calculation methods employed. As a result, it is difficult to use these values to construct a consistent set of rate rules. An improvement would be to perform calculations for all reaction pathways at the same level of theory, which should be sophisticated enough to provide reliable barriers (activation energies) for all channels. Furthermore, because the reactions are generally pressure dependent, it is important to determine under which conditions (sets of pressures and temperatures) these high-pressure rate rules are valid and when falloff effects must be accounted for. In this study, we present electronic structure calculations at the CBS-QB3 level of theory combined with transition state theory to calculate high-pressure rate constants, which then serve as the basis to develop rate rules for the dissociation, concerted elimination, and isomerization reactions of alkyl peroxy radicals. (Subsequent papers in progress will focus on reactions of hydroperoxy alkyl radicals and the second O2 addition reactions.) In contrast to a previous study,1 the proposed internally consistent sets of rate estimation rules are derived by systematically calculating rate constants for a series of small to intermediate sized alkyl peroxy species that have the same reactive moiety but different substituents. Provided that the same reaction types also dominate the ignition chemistry of larger alkyl peroxy radicals, these rate rules should be sufficient to describe the initial oxidation of any open-chain alkyl peroxy radical. Next, we explore the role of pressure effects on the RO2 chemistry using a Quantum
Rice
Ramsperger
Kassel (QRRK)/modified strong

’ METHODS Electronic structure calculations were performed using the CBS-QB3 composite method5 as implemented in the Gaussian 03 software package.6 CBS-QB3 calculations have been shown to predict heats of formation for a large test set of molecules with an accuracy of just over 1 kcal mol
1.5 Furthermore, this method has been successfully applied in numerous kinetic studies7
9 including those of small alkyl radical plus O2 reactions.1
3,10
13 Geometries, rotational constants, and harmonic frequencies (scaled by a factor of 0.99) are calculated at the B3LYP/ 6-311 G(d,p) level of theory. The final electronic energy is obtained by performing a series of single point energy calculations at the CCSD(T)/6-31+G(d0 ) and MP4(SDQ)/6-31G+(d,p) levels of theory and then extrapolating to the MP2 complete basis set limit. Additional corrections for spin contamination and systematic errors further improve the final energy. To obtain accurate values for ΔfH298, S298, and Cp, low frequency vibrational modes that resemble torsions around single bonds were treated as hindered internal rotors rather than as harmonic oscillators. The required hindrance potentials were calculated at the B3LYP/ 6-31G(d) level of theory by optimizing the remaining degrees of freedom while scanning the specified dihedral angle in increments of 10°. Only hindered rotations with barriers below 12 kcal mol
1 were considered. The hindered potentials of these modes were fitted to truncated Fourier series expansions. Reduced moments of inertia for asymmetric internal rotors were calculated at the I(2,3) level based on the equilibrium geometry as defined by East and Radom.14 The 1-D Schr€odinger equation was numerically solved for each internal rotor using the eigenfunctions of the 1-D free rotor as basis functions. The energy eigenvalues are then used to numerically calculate their contributions to thermodynamic functions. All other modes are treated as harmonic oscillators, and the unprojected frequencies are used in the calculations of thermodynamic properties. 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. The reported results relate to the lowest energy conformer of a given species. This was verified by inspection of the hindered rotor potentials to ensure that the starting geometry is 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 pathway. For the concerted elimination and isomerization channels, high-pressure rate coefficients were calculated using canonical transition state theory (TST): k(T) = k(T) 3 kBT/h 3 exp(
ΔGq/ RT), where k(T) is a tunneling correction factor, and ΔGq is the Gibbs free energy difference between the transition state minus the contribution from the reaction coordinate and the reactants. The remaining variables have their usual meaning. Tunneling correction factors were calculated with an asymmetric Eckart potential. 15 Recent calculations on the unimolecular reactions of the propyl peroxy radical suggest that tunneling 13426

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The Journal of Physical Chemistry A corrections obtained by the Eckart method are comparable to more sophisticated tunneling treatments.16 Rate constants were calculated over a temperature range of 300
1500 K in 50 K increments and fitted to a modified Arrhenius expression: k(T) = nH 3 AH 3 Tn 3 exp(
E/RT), where nH is the number of equivalent hydrogen atoms, AH is the pre-exponential factor expressed on a per hydrogen atom basis, n is the temperature coefficient, and E is related to the activation energy (by Ea = E + nRT). Because the dissociation of RO2 has no barrier above that of the reaction endothermicity, these rate constants were derived from the calculated equilibrium constants and association rate constants, which are based on a review of the available data in the literature. Rate estimation rules were determined by averaging the individual rate constants for a given reaction class at each temperature and fitting these averages to a modified Arrhenius expression. On the basis of prior experience, the uncertainty in the calculated rate constants is estimated to be approximately a factor of 2 (near 1000 K).9 This uncertainty arises 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 rotors calculations. We expect a similar uncertainty for the rate rules. All transition states investigated in this study have a cyclic structure. For the concerted elimination and 1
3 H isomerization (abstraction from the α-carbon) reactions, these transition states have a planar ring structure. Isomerization reactions that proceed through five-, six-, or seven-membered rings have a nonplanar transition state ring structure. For the latter two reactions, the chair conformers of the transition states were found to be lower in energy than the corresponding boat conformers. The nonplanar ring structures lead to the occurrence of a pseudochiral OOH center. As a result, we corrected the entropies of these transition states by adding R 3 ln(2), which increases the rate constants by a factor of 2. If the cyclic transition state structure contains substituents, additional chiral centers may be present and this was accounted for in the same way. Because this treatment assumes that these isomers are energetically equivalent, the rate constants represent an upper bound. As an example, consider the isomerization of s-butyl peroxy radical to β-hydroperoxy butyl radical [CCC(C)OO• f CC•C(C)OOH; in this and following structural representations, most H atoms are omitted for clarity]. Using the approach discussed above, we obtain a rate constant that is approximately 1.5 times the correct rate constant calculated as the Boltzmann weighted sum of the individual rate constants for each isomer. Thus, approximating the rate constant using this simplified method only introduces a small error. Using the high-pressure rate rules derived in this study, we evaluated the pressure and temperature dependence for the reactions of n-C4H9, n-C8H17, and n-C12H25 radicals with O2 using Quantum
Rice
Ramsperger
Kassel (QRRK) theory.17 In our implementation of this approach, three representative frequencies and their degeneracies are used to calculate the density of states. These values were derived from heat capacity data that were estimated using Benson’s group additivity method as implemented in the software THERM.18 Collisional stabilization rate constants were obtained using the modified strong collision (MSC) approximation.17 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. Rate constants for the n-C4H9 + O2 were also calculated using a ΔEall of
80 cm
1 to demonstrate the relative insensitivity of the predictions with respect to this parameter. The Lennard-Jones parameters used for N2 were σ = 3.6 Å and ε = 98 K, while the

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ones for the n-alkyl radicals were estimated on the basis of known values for series of normal alkanes (n-C4H9, σ = 5.4 Å and ε = 305 K; n-C8H17, σ = 7.0 Å and ε = 360 K; and n-C12H25, σ = 7.8 Å and ε = 365 K).19 The QRRK/MSC method was chosen because it can easily be used with rate rules. Besides the high-pressure rate constants, this method only requires minimal information about the reactants and the various isomers, which can easily be estimated, while no details about the transition states are needed. (Use of the RRKM approach requires detailed information about the transition state geometry and frequencies. Such detailed information is usually not available for larger species, because high level ab initio calculations are often not practical.) Despite its simplicity, 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.20,21 Because this work focuses on the reactions of RO2, we neglected the subsequent reactions of QOOH in our pressure dependent analysis. Consequently, the pressure dependent rate constants that we obtained are not suitable for direct use in mechanisms, but allow us to compare trends for the various reactions. The pressure and temperature dependent rate constants obtained from this analysis were fitted to a modified Arrhenius expression from 500 to 1000 K in 50 K increments at a given pressure. These results were then used to obtain concentration versus time profiles at 500, 750, and 1000 K and at 0.1 and 10 atm using the isothermal batch reactor module as implemented in the Chemkin Pro software package.22

’ RESULTS AND DISCUSSION I. RO2 Thermochemistry. The R
O2 bond dissociation energies (BDE) were calculated for a series of C1
C6 alkyl peroxy radicals. The average values are listed in Table 1, and the individual values are given in Table 3. The R
O2 BDE increases as the degree of substitution around the α-carbon increases (methyl < primary < secondary < tertiary). This trend is opposite to the stability ordering of alkyl radicals and, hence, the corresponding R
H bond dissociation energies in alkanes. This counterintuitive R
O2 BDE ordering has previously been observed,23,24 and the origin has been discussed in terms of hyperconjugation effects.25 The BDE of methyl peroxy radical is found to be 33.1 kcal mol
1, while on average we find that the BDE for a primary R
O2 bond is 35.6 kcal mol
1, that of a secondary R
O2 bond is 37.4 kcal mol
1, and that of a tertiary bond is 38.7 kcal mol
1. The BDE of the neo-pentyl peroxy radical is found to be slightly higher (36.8 kcal mol
1) than those of the other primary R
O2 (Table 3). At present, it is unclear if this higher value is a reflection of an unusually high stability of the neo-pentyl radical, when compared to other primary alkyl radicals, or simply due to the inability of the calculation method to accurately describe this molecule. The BDEs of C1
C4 alkyl peroxy radicals have previously been determined using isodesmic work reactions.23,26 Isodesmic reactions are commonly employed to account for systematic errors in electronic structure calculations. However, the implementation of this method, which attempts to formulate reactions with similar bonding environments in the reactants and products, is limited by the availability of suitable reference compounds whose heats of formation are well established. In general, the BDEs reported in this work are slightly higher than those determined through isodesmic reactions (by ∼1 kcal mol
1); however, the differences between the various bonding types are in excellent 13427

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Table 1. Average Calculated Equilibrium Constants and Bond Dissociation Energies (ΔHrxn) for the Reaction of RO2 f R + O2 ΔHrxn(298) (kcal mol
1)

Keq = cTx exp(
d/RT)a d

Keq(298 K)

Keq(750 K)

Keq(1000 K)

(kcal mol
1)

(mol cm
3)

(mol cm
3)

(mol cm
3)


22


8


6

c RO2

(mol cm
3)

methyl

1.35  10

4

primary

7.51  10

6

secondary

2.65  10

10

x 0.65

33.3

1.29  10

3.66  10

atomizationb

8.10  10

isodesmic

33.1

experimental 32.7 ( 0.9e

c

32.0

31.3 ( 1.2f 1.37

35.9


23


8

1.43  10

2.98  10


6

8.30  10

35.6

35.5 ( 2.0e

c

34.5

35.8 ( 2.3f 2.30

38.2


24


8

5.19  10

4.77  10


5

1.49  10

37.4

36.4

36.6 tertiary

2.13  1013

3.17

39.8

1.97  10
24

4.13  10
8

1.32  10
5

38.7

37.1 ( 2.3c

c d

36.5 ( 0.9e 37.5 ( 2.5f

37.8c 37.5 d

38.5 ( 1.0g a g

The individual values are fit from 300 to 1500 K in 50 K increments. This work. Reference 23. Reference 4. Reference 24. References 27 and 29. References 28 and 29. b

c

d

e

f

Table 2. Selected Published Room-Temperature Rate Constants for the Association Reaction of R + O2a R

klit (1012 cm3 mol
1 s
1)

krule (1012 cm3 mol
1 s
1)

reference (method)

4.2b

primary CC•

CCC•

CCCC•

4.8 ( 0.6 4.7+2.7
1.7

54 (review) 55 (review)

4.9 ( 0.20

33 (relative rate measurement; 3
1500 Torr)

5.5 ( 0.5

32 (relative rate measurement; 50
1500 Torr)

4.16

34 (flash photolysis/GC)

4.6+1.1
0.9

38 (muon spin relaxation; 1.5
60 bar)

4.82+2.79
1.77

55 (review)

3.3 ( 0.5

31 (tubular reactor/PIMS; 1
4 Torr)

3.40 ( 0.27 4.52 ( 0.84

40 (tubular reactor/PIMS; 0.3
4 Torr) 30 (tubular reactor/PIMS; 1
4 Torr) 9.3c

secondary C2C•

CC•CC

6.62+6.62
3.31

55 (review)

8.5 ( 1.5

31 (tubular reactor/PIMS 1
4 Torr)

5.00 ( 0.20

56 (pulse radiolysis/UV absorption; 1.01 bar)

10.0 ( 0.13

30 (tubular reactor/PIMS 1
4 Torr) 14d

tertiary C3C•

10.7+0.73
0.48 14.1 ( 0.39

38 (muon spin relaxation; 1.5 bar) 30 (tubular reactor/PIMS; 1
4 Torr)

a In this notation, the hydrogen atoms are assumed, and a “•” symbolizes a radical site. b Average of CC• + O2, average of refs 32
34, 38; CCC• + O2, ref 31; and CCCC• + O2, ref 30. c Average of C2C• + O2, ref 31; and CC•CC + O2, ref 30. d C3C• + O2, ref 30.

agreement (see Table 1). Both sets of results agree well with experimentally determined BDEs for methyl peroxy radical (32.7 ( 0.9 kcal mol
1), ethyl peroxy radical (35.5 ( 2.0 kcal mol
1), and i-propyl peroxy radical (37.1 ( 2.3 kcal mol
1), which are derived from the temperature dependence measurements of the equilibrium constant.24 Using this same method for the t-butyl peroxy radical, the BDE was found to be 36.5 ( 0.9 kcal mol
1,24 which seems anomalous because this value is lower than that for the i-propyl peroxy radical. More recently though, two studies have measured the heat of formation of the t-butyl peroxy radical using a negative ion cycle27 and threshold photoelectron-photo ion coincidence spectroscopy.28 Combined with the well-known heat of formation of the t-butyl radical,29 the BDE of t-butyl peroxy radical is determined to be higher at 37.5 ( 2.5 and 38.5 ( 1.0 kcal mol
1, respectively. These new values are

consistent with the current CBS-QB3 results and the trend seen experimentally for methyl, ethyl, and i-propyl peroxy radicals. The opposing trends in the R
O2 and R
H BDEs suggest that the ignition properties of branched alkanes might differ from those of linear alkanes. Because tertiary C
H bonds are weaker than primary and secondary C
H bonds, H atom abstraction from a branched alkane will lead to a higher concentration of tertiary radicals than suggested by the fraction of tertiary C
H bonds in the parent alkane. Subsequent reactions of these tertiary radicals with O2 lead to the formation of a relatively stable tertiary alkyl peroxy adduct, which has a greater well depth than a primary or secondary alkyl peroxy radical. As a result, the barriers for concerted elimination and isomerization reactions are lowered relative to the energy of the R + O2 entrance channel, and we might expect this to lead to different reactivities. However, as 13428

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Table 3. RO2 Dissociation Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set Used To Derive Rate Rulesa modified Arrhenius parameters E

ΔHrxn(298)

ΔSrxn(298)

(kcal mol
1)

(kcal mol
1)

(cal mol
1 K
1)

A reaction

(s
1)

n

kTST/krate rule

thermochemistry

500 K

750 K

1000 K

COO w C• + O2

1.09  1014

0.25

33.3

33.1

31.1

primary rate rule CCOO• w CC• + O2

8.80  1021 9.49  1021


2.40
2.41

35.9 35.8

35.4

34.1

1.06

1.04

1.03

CCCOO• w CCC• + O2

1.52  1023


2.71

36.4

35.7

35.6

1.47

1.54

1.53

CCCCOO• w CCCC• + O2

9.13  1022


2.67

36.0

35.5

34.8

1.78

1.63

1.52

C2CCOO• w C2CC• + O2

5.93  1022


2.60

36.5

35.9

35.4

1.10

1.24

1.29

C2CCCOO• w C2CCC• + O2

9.37  1021


2.38

35.8

35.4

34.6

1.33

1.29

1.27

C3CCOO• w C3CC• + O2

6.66  1022


2.56

37.3

36.8

36.9

0.67

1.00

1.20

CCC(C)COO• w C2•CCC + O2

2.23  1022


2.46

36.2

35.8

35.3

1.29

1.39

1.44

CCCCCOO• w CCCCC• + O2 C2CC(C)COO• w C2CCC2• + O2

2.17  1022 1.49  1022


2.50
2.43

35.8 36.0

35.5 35.5

34.8 34.9

1.50 1.32

1.39 1.34

1.33 1.35

C2CCCCOO• w C2CCCC• + O2

5.01  1022


2.65

36.2

35.6

34.3

0.88

0.88

0.86

C3CCCOO• w C3CCC• + O2

6.30  1021


2.32

35.3

34.9

34.5

2.09

1.78

1.66

CCC(C)CCOO• w CCC(C)CC• + O2

2.03  1022


2.53

35.8

35.3

34.2

1.15

1.06

1.00

CCC(CC)COO• w CCC(CC)C• + O2

3.46  1021


2.23

35.7

35.4

34.6

1.36

1.38

1.41

CCC(C2)COO• w CCC(C2)C• + O2

4.29  1022


2.43

36.9

36.4

36.6

1.45

2.03

2.41

CCCC(C)COO• w C2•CCCC + O2

3.14  1022


2.53

36.4

35.6

35.0

0.93

1.05

1.11

CCCCCCOO• w CCCCCC• + O2 secondary rate rule

2.04  1022 6.85  1025


2.58
3.29

36.0 38.2

35.5

33.6

0.69

0.67

0.65

C2COO• w CC•C + O2

7.65  1024


3.10

38.0

37.3

37.0

0.45

0.46

0.47

CCC(C)OO• w CCC•C + O2

1.19  1026


3.40

38.5

37.5

38.0

0.66

0.70

0.71

CCC(CC)OO• w CCC•CC + O2

4.30  1026


3.55

38.6

37.4

40.7

0.89

0.91

0.89

C2CC(C)OO• w C2CC•C + O2

4.96  1025


3.24

38.3

37.4

38.3

0.92

0.98

1.02

C2CCC(C)OO• w C2CCC•C + O2

3.77  1025


3.30

38.3

37.5

37.8

0.50

0.51

0.51

C2CC(CC)Q w C2CC•CC + O2

9.74  1026


3.62

38.6

37.6

39.9

1.26

1.26

1.23

C3CC(C)OO• w C3CC•C + O2 CCC(C)C(C)OO• w CCC(C)C•C + O2

3.16  1026 5.15  1024


3.42
2.86

38.3 37.7

37.4 37.0

40.0 39.1

1.85 1.88

1.84 1.87

1.81 1.94

CCCCC(C)OO• w CCCCC•C + O2

3.38  1025


3.28

38.1

37.3

37.8

0.59

0.57

0.56

CCCC(CC)OO• w CCCC•CC + O2

5.08  1026


3.59

38.4

37.5

39.3

1.01

0.94

0.88

tertiary rate rule

8.04  1028


4.15

39.8

C3COO• w C3C• + O2

1.54  1028


3.98

40.2

38.9

38.7

0.38

0.46

0.52

CCC(C2)OO• w C2C•CC + O2

5.70  1027


3.83

39.9

38.8

40.2

0.52

0.60

0.66

C2CC(C2)OO• w C2C•CC2 + O2

3.99  1028


4.08

39.5

38.4

40.9

1.09

1.00

0.97

CCCC(C2)OO• w C2C•CCC + O2 (CC)2C(C)OO• w (CC)2(C)C• + O2

3.90  1028 1.71  1030


4.10
4.47

39.8 40.1

38.7 38.8

40.7 44.8

0.71 2.31

0.72 2.23

0.73 2.13

a

The rate estimation rules are also provided along with ratios of the rate rule and the TST rate constants. In this notation, the hydrogen atoms are assumed, and a “•” symbolizes a radical site.

shown in Table 1, only a small difference is observed in the equilibrium constants for primary, secondary, and tertiary alkyl peroxy radicals at 750 and 1000 K, with no obvious trend with structure. This is due to a corresponding increase in ΔSrxn in going from the primary, to secondary, to tertiary alkyl substrate. Thus, despite the deeper well associated with tertiary alkyl peroxy radicals, we expect that for a given radical concentration there is little difference in the RO2 to R ratio for the various radicals at typical NTC (near equilibrium) conditions. (The ratio of RO2 to R, however, does depend on the concentration of O2; at higher concentrations, the equilibrium is shifted toward RO2. Thus, at higher pressures, the NTC region is shifted toward higher temperature.) In contrast, at 300 K there are notable structure

effects on the equilibrium constant. There the equilibrium constants differ by almost an order of magnitude in going from primary to tertiary radicals. These equilibrium constants agree to within a factor of 2 with those reported by Miyoshi.1 II. Rate Rule Assignments. In this section, we present rate rules for the dissociation, isomerization, and concerted elimination reaction of RO2. These rate rules have been developed from sets of reactions that consist of all C1
C5 alkyl peroxy radical reactants and, in addition, several selected reactions of C6 and C7 peroxy radicals. Next, we compare the results for isomerization and concerted elimination to other previously reported values. Dissociation of RO2. While the alkyl peroxy radicals are stable at low temperatures with respect to dissociation to alkyl + O2, 13429

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The Journal of Physical Chemistry A redissociation starts to become important at elevated temperatures. Because this reaction type proceeds with no barrier above the reaction endothermicity, calculation of the rate constant requires the use of a variational transition state method, which is beyond the scope of this work. Alternatively, rate constants for RO2 dissociation can be obtained from the above calculated equilibrium constants provided that reliable experimental values for the corresponding association rate constants are available. Room-temperature measurements at or near the high-pressure limit for the reactions of ethyl, propyl, and butyl radicals with O2 indicate that the association rate constants depend on the nature of the alkyl radical (see Table 2).30,31 The reactions of primary alkyl radicals with O2 have distinctly smaller rate constants than those of the secondary alkyl radicals. Likewise, the rate constants for the reactions of the secondary alkyl radicals with O2 are smaller than those of tertiary alkyl radicals with O2. The reaction of ethyl radical plus O2 has been investigated over a pressure range of 1.5
60 bar using relative rate measurements in a highpressure flow tube,32,33 flash photolysis,34 and muon spin relaxation.35
37 The results of these studies are in good agreement with each other, and the average rate constant is 4.8  1012 cm3 mol
1 s
1. Reactions of propyl radical and butyl radical with O2 have been investigated in a cylindrical reaction cell using photoionization mass spectrometry.30,31 Pressure variations in the range of 1
4 Torr suggest that these values are near the high-pressure limit and that formation of the RO2 adduct is the major reaction channel. The rate constants for the reactions of n-propyl radical and n-butyl radical with O2 were found to be 3.3  1012 and 4.52  1012 cm3 mol
1 s
1, respectively, similar to that of ethyl radical plus O2. On the basis of these measurements, we estimate the association rate constant for a primary radical with O2 to be 4.2  1012 cm3 mol
1 s
1. The reactions of i-propyl radical and s-butyl radical with O2 are reported to be 8.5  1012 and 1.0  1013 cm3 mol
1 s
1, respectively. On the basis of these measurements, we estimate the association rate constant for a secondary radical with O2 to be 9.2  1012 cm3 mol
1 s
1. The rate constant for the reaction of tertiary radicals with O2 is estimated on the basis of the rate constant of 1.41  1013 cm3 mol
1 s
1 for t-butyl radical reacting with O2. This same reaction was also studied using muon spin relaxation at a pressure of 1.5 bar.38 The results of these two studied are in good agreement with one another, providing evidence that the high-pressure limit has been reached. Significantly less data are available regarding the temperature dependence of these association reactions. The variational TST calculations of Miller and Klippenstein suggest a positive temperature dependence (k µ AT 0.5 ) for the reaction of ethyl plus O2.39 In contrast, experimental data for larger alkyl radical plus O2 reactions indicate that those rate constants decline with increasing temperature. Falloff curves for the reaction of n-propyl radical with O2 have been measured using a heated tubular flow tube over a temperature range of 297
635 K.40 Subsequent QRRK/MSC calculations by Huynh et al. showed that these falloff data can be reproduced using a negative temperature dependence.10 The reaction of t-butyl radical with O2 was studied over a temperature range of 241
462 K at 1.5 bar.38 A temperature dependence of T
0.96(0.21 was observed. Miyoshi calculated variational TST rate constants for the association of representative primary, secondary, and tertiary alkyl radicals with O2.1 He reports a negative temperature dependence for rate constants of the reactions involving primary and secondary alkyl radicals and a positive dependence for tertiary alkyl radical reactions. Given the conflicting information present in the

ARTICLE

literature, it is difficult to ascertain the correct temperature dependence of these R + O2 recombination reactions with confidence. Fortunately, within the temperature range of interest, the effect of temperature on the rate expressions appears to be small and, therefore, will only have a minor impact on model predictions for many applications. For the purposes of this work, we use the averaged roomtemperature rate constants for the association reactions discussed above and assume a T
1.0 temperature dependence for all three reactions to obtain the following rules: kprim ¼ 1:3  1015 cm3 K mol
1 s
1 T
1 ksec ¼ 2:8  1015 cm3 K mol
1 s
1 T
1 ktert ¼ 4:2  1015 cm3 K mol
1 s
1 T
1 Combined with the equilibrium constants presented earlier, we obtained the dissociation rate constants for C1
C6 alkyl peroxy radicals that are given in Table 3. The rate constant for the dissociation of a secondary alkyl peroxy radical is almost 5 times larger than that of a primary radical at 750 K. The dissociation rate constant for tertiary radicals is slightly larger than the secondary at this temperature. These rules are also provided in Table 3 along with the ratios of individual rate constants and the rate rule derived rate constant at 500, 750, and 1000 K. The rate rules reproduce roughly 80% of the individual rate constants within a factor of 2. Concerted HO2 Elimination from RO2. The concerted elimination reaction has the potential to inhibit ignition through the formation of the relatively unreactive HO2 radical. This reaction type proceeds via a planar five-membered ring transition state, with simultaneous breaking of the Cα
OO and H
CβCαOO bonds. Because both the C
OO and C
H bond dissociation energies are sensitive to the structure of the RO2 radical, one may expect that this reaction type requires a classification with respect to the nature of the peroxy moiety and the type of C
H bond that is broken. However, our calculations reveal that all rate constants group together regardless of the nature of the reacting C
H and C
OO bonds. The detailed results are presented in Table 4 on a per H-atom basis. While there is no obvious systematic differentiation of the rate constants with respect to the molecular structure of the reactant, a subtle distinction can be made regarding the nature of the olefin that is formed. Concerted HO2 reactions that result in the formation of a highly substituted olefin proceed approximately 2
3 times as fast as those that produce less substituted olefins. The faster reactions involve tertiary alkyl peroxy radicals abstracting from a secondary or tertiary C
H site, or secondary alkyl peroxy radicals abstracting from a tertiary C
H site. Thus, the concerted HO2 elimination reaction class can be described by two rate rules (provided in Table 4) depending upon the degree of substitution found in the product olefin. The quality of these rate rules is demonstrated by the ratio of the individually calculated TST rate constants and the rate estimation rule (Table 4). The rate rules predict the rate constants of more than 90% of these reactions within a factor of 2. All of the outliers involve either ethyl peroxy or i-propyl peroxy. The rate constant for the reaction of the ethyl peroxy radical deviates from the rate rule by almost a factor of 5 at 500 K, and, therefore, this reaction rate constant should not be estimated by the rate rule. One possible explanation for the lower 13430

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The Journal of Physical Chemistry A

ARTICLE

Table 4. TST Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set Used To Derive Rate Rules for the Concerted Elimination Reactiona modified Arrhenius parameters no. reaction

type of

AH

type of

of H C
OO site C
H site

kTST/krule

thermochemistry E

(s
1)

n 1.20 1.67

29.4 29.7

ΔHrxn

ΔSrxn

(kcal mol
1) (kcal mol
1) (cal mol
1 K
1)

500

750

1000

K

K

K

rate rule CCOO• w CdC + HO2

1 3

p

p

3.16  108 4.84  106

21.7

32.9

0.22 0.29

0.34

C2COO• w CCdC + HO2

6

s

p

1.36  108

1.28

30.0

23.7

37.3

0.40 0.50

0.56

CCCOO• w CCdC + HO2

2

p

s

2.08  108

1.25

29.6

19.0

35.5

0.74 0.80

0.84

C3COO• w C2C dC + HO2

9

t

p

3.66  1010 0.62

30.1

25.2

40.5

1.57 1.57

1.49

C2CCOO• w C2C dC + HO2

1

p

t

1.94  108

1.27

29.6

17.1

34.7

0.83 0.90

0.94

CCCCOO• w CCCdC + HO2

2

p

s

6.29  107

1.38

28.9

19.0

35.4

1.03 0.94

0.91

CCC(C)OO• w CCCdC + HO2

3

s

p

1.71  109

1.00

30.4

24.1

37.5

0.65 0.82

0.90

CCC(C)OO• w t-CCdCC + HO2 CCC(C)OO• w c-CCdCC + HO2

2 2

s s

s s

7.25  109 0.80 1.70  1010 0.67

29.9 30.7

21.4 22.5

35.1 35.6

1.21 1.20 0.52 0.66

1.15 0.70

C2CC(C)OO• w C2CCdC + HO2

3

s

p

4.92  108

1.21

29.8

23.7

37.7

1.10 1.27

1.36

C2CCCOO• w C2CCdC + HO2

2

p

s

7.59  106

1.66

28.4

19.2

34.7

1.20 1.03

0.99

CCC(CC)OO• w t-CCCdCC + HO2

4

s

s

1.49  1010 0.68

29.8

21.3

38.3

1.25 1.16

1.07

CCC(CC)OO• w c-CCCdCC + HO2

4

s

s

6.25  109

0.77

30.1

22.5

38.0

0.69 0.73

0.72

CCCC(C)OO• w CCCCdC + HO2

3

s

p

8.01  108

1.13

30.1

23.9

37.5

0.85 1.04

1.14

CCCC(C)OO• w t-CCCdCC + HO2

2

s

s

2.24  109

0.95

29.2

21.4

36.9

1.90 1.60

1.43

CCCC(C)OO• w c-CCCdCC + HO2 CCCCCOO• w CCCCdC + HO2

2 2

s p

s s

1.84  109 1.29  107

0.96 1.58

30.0 28.5

22.6 18.8

36.5 34.8

0.75 0.82 1.12 0.95

0.85 0.91

CCC(C)COO• w CCC(C)dC + HO2

1

p

t

1.26  108

1.32

28.9

17.1

35.4

1.50 1.30

1.23

CCC(C2)OO• w CCC(C)dC + HO2

6

t

p

4.08  109

0.89

29.5

25.3

40.1

1.85 1.66

1.54

rate rule (highly substituted olefins)

1

1.3  1011

0.50

30.0

C2CC(C)OO• w C2CdCC + HO2

1

s

t

3.50  1010 0.71

30.1

20.2

37.6

0.87 1.00

1.09

CCC(C2)OO• w C2CdCC + HO2

2

t

s

5.62  1010 0.58

29.6

23.4

39.0

0.97 0.89

0.86

C2CC(C2)OO• w C2CdCC2 + HO2

1

t

t

6.25  1012 0.02

30.7

23.5

39.0

1.21 1.25

1.22

C2CC(CC)OO• w CCCdCC2 + HO2

1

s

t

7.26  1010 0.54

29.6

20.2

37.9

1.01 0.91

0.86

a

The rate estimation rules are also provided along with ratios of the rate rule and the individually calculated TST rate constants. In this notation, the hydrogen atoms are assumed, and a “•” symbolizes a radical site. The “p” refers to primary, “s” to secondary, and “t” to tertiary.

rate constant found for ethyl peroxy may be that the lack of substituents on the transition state ring structure may lead to a lower pre-exponential factor. The reaction of i-propyl peroxy radical is a factor of 2.5 slower at 500 K than the rate rule. At present, the reason for this deviation is not clear. For reactions that can form both a cis- and a trans-olefin, we find that the transproduct is produced approximately 2 times faster than the cisproduct. Because this difference falls within the spread of the data set, we find it sufficient to describe these two channels with the same rate rule. Furthermore, most mechanisms do not differentiate between cis- and trans-isomers. RO2 Isomerization. The isomerization reactions of alkyl peroxy radicals play a key role in hydrocarbon ignition chemistry because the subsequent reactions of the QOOH isomers with O2 lead to chain branching. We have evaluated rate constants for the 1
3, 1
4, 1
5, and 1
6 isomerization reactions of RO2. In this notation, the first number refers to the peroxy radical site, while the second number refers to the location of the radical site in the product relative to that in the reactant. Thus, a 1
3 isomerization refers to an intramolecular H shift from the α-carbon to the peroxy moiety, while 1
4 H migration involves an H atom at the β-carbon position, and so forth. The 1
3 isomerization reaction leads to the formation of an unstable isomer that immediately

dissociates to form a carbonyl plus OH. Consistent with the results of previous studies,3,41 the isomerization rate constants group according to the size of the ring formed in the transition state structure (see Figure 2). They further depend on the nature of the abstracted hydrogen. For a given ring size, isomerization reactions that involve a hydrogen atom located on a primary carbon atom have distinctly lower rate constants than those originating from a secondary position. Likewise, the rate coefficients for reactions originating from a secondary position are distinctly lower than those from a tertiary position. The derived rate rules along with the individual TST rate constants are presented in Table 5 on a per H-atom basis. The rate estimation rules predict a large majority of the calculated individual TST rate constants within a factor of 2 (see Table 5 for the ratios of the rate constants). As seen above, the outliers included the small ethyl and i-propyl radicals; these have smaller rate constants than those of larger alkyl peroxy radicals. This again illustrates the importance of varying the size of the alkyl peroxy radical in the formulation of these rate rules. The largest deviations are found for the 1
4 isomerization reactions. Fortunately, these reactions are slow as compared to the 1
5 and 1
6 isomerization reactions, and, therefore, this uncertainty should not affect ignition time predictions for most practical fuels. 13431

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The Journal of Physical Chemistry A

Figure 2. Arrhenius plots of the derived rate rules for the 1
4 (black), 1
5 (red), and 1
6 (blue) RO2 isomerization reactions.

Figure 3. Evans
Polanyi relationships for the 1
3 through 1
7 isomerization reactions. The Ea is calculated from Ea = E + nRT at T = 750 K, with “n” and “E” being the parameters in the corresponding the modified Arrhenius equations. Note that the heats of reaction for the 1
3 isomerization reactions are provided on the upper x-axis. (These are exothermic because the initially formed isomer is unstable and dissociates to a carbonyl and OH.)

As shown in Figure 2, for a given type of abstracted H-atom, the 1
5 isomerization reactions (red) are faster than the 1
6 isomerization reactions (blue). However, these rate constants are sufficiently close to one another that they can potentially compete depending upon the structure of the alkyl peroxy radical. For instance, consider the isomerization reactions of the 5-methylpentyl peroxy radical (C2CCCCOO•). At temperatures below 800 K, the 1
6 isomerization reaction is favored over the 1
5 isomerization, because the former occurs from a tertiary carbon atom rather than from a secondary. However, as the temperature is increased, entropic effects become more important and the 1
5 isomerization reaction is favored. The 1
3 (not shown) and 1
4 (black lines) isomerization reactions are sufficiently slow at typical NTC temperatures such that these channels can be neglected. The exceptions to this are isomerization reactions of small alkyl peroxy radicals that cannot isomerize via a six-membered ring transition state. Isomerization reactions

ARTICLE

Figure 4. Pre-exponential factors as a function of temperature for the various RO2 isomerization reactions. These values are calculated from the modified Arrhenius fits using: A = k(T)/exp(
(E + nRT)/RT)), with “n” and “E” being the parameters in the corresponding modified Arrhenius equations. The black lines correspond to the rate rules, while the gray lines are for the 1
7 isomerization reactions.

that proceed via an eight-membered ring are also expected to be slow. Although we have not performed a systematic investigation of these reactions, the corresponding rate constants can be estimated on the basis of the observed trends in the preexponential factors and barrier heights. This will be discussed in more detail in the following paragraphs. The activation energies for the isomerization reactions follow Evans
Polanyi relationships,3,41 as shown in Figure 3. The Arrhenius activation energy Ea was calculated from n and E parameters in the modified Arrhenius fits to the TST-derived rate constants. The activation energies associated with the 1
3 isomerization reactions (upper x-axis) are ∼10 kcal mol
1 higher than those for the 1
4 reactions (for transfers of a secondary H in each case). The activation energies for the 1
4 isomerization reactions are, on average, 11 kcal mol
1 higher than those associated with the 1
5 reactions. These differences can mostly be attributed to the differences in ring strain in the corresponding transition states. However, we note that the Cα
H bond is weaker than a typical C
H bond (95 vs 98.4 kcal mol
1 for a secondary C
H bond).4 In contrast, the Cβ
H bond is slightly stronger than a typical C
H bond (101 vs 99.6 kcal mol
1 for a secondary C
H bond).4 Thus, some of the differences in activation energies can be traced to the differences in the Cα
H and Cβ
H bond dissociation energies, but most of it is due to ring strain differences between four- and five-member transition states. The activation energies are very similar for the 1
5 and 1
6 transfers, as well as the limited number of calculations involving a 1
7 transfer. Thus, it seems safe to expect that the barrier height for isomerization via eight-membered or larger sized cyclic transition states can be approximated to be that of the 1
6 H isomerization reaction. In Figure 4, we present plots of the pre-exponential factors of the isomerization reactions as a function of temperature. This plot reveals that the temperature dependences for the 1
4 through 1
6 H-atom migration reactions are very similar, while the pre-exponential factors for the 1
3 isomerization reactions exhibit a stronger temperature dependence. This difference is attributed to the unique nature of the 1
3 H migration reaction, which leads to the formation of an unstable α-QOOH molecule that spontaneously dissociates to an aldehyde or a ketone plus a hydroxyl 13432

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The Journal of Physical Chemistry A

ARTICLE

Table 5. TST Rate Constants, Heats of Reaction, and Entropies of Reaction for the Training Set Used To Derive Rate Rules for the RO2 Isomerization Reactionsa modified Arrhenius parameters E

ΔHrxn

ΔSrxn

(kcal mol
1)

(kcal mol
1)

(cal mol
1 K
1)

2.98

39.0


20.1

30.4

2.51 2.63

36.5 37.0


24.8

2.31

36.9


24.7

9.22  104

2.37

36.5

2

5.61  104

2.42

CCCCCOO• w CCCCCdO + OH

2

3.28  104

CCC(C)COO• w CCC(C)CdO + OH

2

C2CCCOO• w C2CCCdO + OH

AH no. of H

(s
1)

3

8.25  102

1
3s rate rule CCOO• w CCdO + OH

1 2

3.25  104 1.13  104

CCCOO• w CCCdO + OH

2

1.68  105

CCCCOO• w CCCCdO + OH

2

C2CCOO• w C2CCdO + OH

reactionb

n

kTST/krate rule

thermochemistry

500 K

750 K

1000 K

31.5

0.43

0.53

0.60

32.8

0.89

0.96

0.98


24.8

32.5

1.10

1.06

1.03

36.5


24.5

32.7

0.97

0.94

0.92

2.57

36.4


24.9

32.9

1.57

1.55

1.54

3.42  104

2.47

36.2


24.7

32.2

1.08

0.97

0.92

2

1.84  104

2.59

36.6


25.0

31.2

0.85

0.91

0.95

C3CCOO• w C3CCddO + OH 1
3t rate rule

2 1

4.69  104 4.87  105

2.47 2.18

36.3 35.1


24.4

32.8

1.26

1.18

1.13

C2COO• w C2CdO + OH

1

5.22  104

2.45

35.7


27.4

32.4

0.31

0.42

0.50

CCC(C)OO• w CCC(C)dO + OH

1

1.25  106

2.08

35.6


27.6

33.5

0.86

0.98

1.04

CCCC(C)OO• w CCCC(C)dO + OH

1

5.09  105

2.20

35.3


27.5

32.9

0.98

1.06

1.10

C2CC(C)OO• w C2CC(C)dO + OH

1

6.81  105

2.13

35.0


27.3

34.0

1.17

1.11

1.08

CCC(CC)OO w CCC(CC)dO +OH

1

1.03  107

1.77

35.1


27.9

33.5

1.76

1.48

1.31

1
4p rate rule

1

2.17  106

1.73

32.0

CCOO• w C•COOH C2COO• w C2•COOH

3 6

4.60  105 1.84  106

1.88 1.75

32.8 32.3

17.7 18.2

4.9 5.9

0.27 0.75

0.37 0.82

0.43 0.86

CCC(C)OO• w CCC(C•)OOH

3

1.84  107

1.49

32.4

18.3

4.9

1.40

1.43

1.42

C3COO• w C3•COOH

9

6.94  106

1.58

32.6

19.0

7.1

0.70

0.80

0.84

CCCC(C)OO• w CCCC(C•)OOH

3

4.12  106

1.61

32.2

18.4

4.4

0.81

0.81

0.80

CCC(C2)OO• w CCC(C2•)OOH

6

6.55  105

1.8

31.9

18.3

3.4

0.70

0.71

0.72

C2CC(C)OO• w C2CC(C•)OOH

3

6.19  106

1.65

31.7

18.3

4.7

2.47

2.15

2.00

1
4s rate rule

1

4.78  107

1.36

28.6

CCCOO• w CC•COOH CCCCOO• w CCC•COOH

2 2

1.43  107 9.61  106

1.48 1.53

29.1 28.6

14.0 14.0

6.4 5.4

0.36 0.52

0.45 0.57

0.51 0.61

CCC(C)OO• w CC•C(C)OOH

2

3.54  108

1.16

28.8

14.6

5.6

1.58

1.61

1.59

CCCCCOO• w CCCC•COOH

2

3.03  106

1.65

28.3

13.8

5.1

0.46

0.48

0.50

CCCC(C)OO• w CCC•C(C)OOH

2

2.16  108

1.26

28.6

14.6

6.1

2.31

2.23

2.17

CCC(CC)OO• w CC•C(CC)OOH

4

6.15  108

1.05

28.6

14.4

8.5

1.78

1.59

1.47

CCC(C2)OO• w CC•C(C2)OOH

1

8.14  106

1.52

28.1

14.5

5.2

0.71

0.65

0.63

C2CCCOO• w C2CC•COOH

2

2.62  106

1.72

28.8

14.2

4.4

0.40

0.50

0.58

1
4t rate rule C2CCOO• w C2C•COOH

1 1

2.52  107 3.07  107

1.39 1.38

25.3 25.9

11.0

6.4

0.64

0.78

0.85

CCC(C)COO• w CCC•(C)COOH

1

2.86  107

1.38

25.6

11.0

5.2

0.82

0.89

0.93

C2CC(C)OO• w C2C•C(C)OOH

1

1.57  108

1.14

25.2

11.6

6.3

1.57

1.36

1.24

1
5p rate rule

1

1.62  107

1.23

21.5

CCCOO• w C•CCOOH

3

6.83  106

1.31

21.7

16.4

3.7

0.50

0.56

0.60

CCC(C)OO• w C•CC(C)OOH

3

4.98  107

1.12

21.5

16.7

5.1

1.44

1.41

1.38

C2CCOO• w C2•CCOOH

6

2.42  106

1.42

21.3

16.6

4.3

1.84

1.80

1.75

CCC(CC)OO• w C•CC(CC)OOH CCC(C)COO• w CCC(C•)COOH

6 3

6.41  107 1.78  106

1.06 1.46

21.3 21.0

16.6 15.9

5.4 1.4

1.53 0.69

1.37 0.65

1.27 0.64

CCC(C2)OO• w C•CC(C2)OOH

3

6.54  106

1.31

21.4

16.5

3.3

0.65

0.67

0.68

C2CC(C)OO• w C2•CC(C)OOH

6

9.58  107

1.11

21.6

17.1

6.7

2.24

2.26

2.25

C3CCOO• w C3•CCOOH

9

7.76  106

1.30

21.9

17.6

5.1

0.50

0.59

0.65

1
5s rate rule

1

4.65  107

1.11

18.2

CCCCOO• w CC•CCOOH

2

4.67  107

1.08

18.6

13.3

5.2

0.54

0.612

0.66

CCCCCOO• w CCC•CCOOH

2

4.26  106

1.37

17.9

13.1

4.0

0.64

0.64

0.65

CCCC(C)OO• w CC•CC(C)OOH

2

3.09  108

0.93

18.4

13.4

5.7

1.76

1.78

1.76

COO• w CdO + OHb b

13433

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Table 5. Continued modified Arrhenius parameters

kTST/krate rule

thermochemistry ΔSrxn (cal mol
1 K
1)

500 K

750 K

1000 K

13.0

4.3

1.09

0.99

0.95

16.1

11.1

5.1

0.46

0.56

0.62

15.4

10.7

4.0

0.83

0.81

0.80

1.14

15.2

10.8

3.1

1.11

1.02

0.99

6.11  109

0.51

16.3

11.8

6.3

1.62

1.65

1.61

1

3.42  105

1.48

20.2

CCCCOO• w C•CCCOOH C2CCCOO• w C2•CCCOOH

3 6

4.88  105 1.78  105

1.40 1.54

20.5 20.1

15.9 16.6

3.5 4.6

0.64 0.82

0.68 0.82

0.70 0.82

CCC(C)COO• w C•CC(C)COOH

3

3.75  105

1.45

19.8

15.6

3.0

1.41

1.22

1.14

CCCC(C)OO• w C•CCC(C)OOH

3

2.61  106

1.27

20.8

16.4

4.7

1.16

1.31

1.37

1
6s rate rule

1

5.16  105

1.41

16.4

CCCCCOO• w CC•CCCOOH

2

1.43  106

1.27

17.0

12.7

4.7

0.69

0.79

0.84

CCCCCCOO• w CCC•CCCOOH

2

1.76  106

1.20

16.9

13.4

5.1

0.55

0.61

0.63

CCC(C)CCOO• w CC•C(C)CCOOH

2

6.07  105

1.43

16.1

13.0

4.3

1.77

1.62

1.56

1
6t rate rule C2CCCCOO• w C2C•CCCOOH

1 1

2.02  107 1.63  107

0.90 0.91

14.6 14.7

10.8

2.9

0.75

0.78

0.80

C2CCC(C)COO• w C2C•CCC(C)COOH

1

3.36  106

1.05

14.0

10.6

2.9

0.79

0.67

0.62

C2CCCC(C)OO• w C2C•CCC(C)OOH

1

1.49  108

0.71

15.0

11.5

3.6

1.48

1.57

1.60

1
7p rate rulec

1

3.42  104

1.48

19.7

CCCCCOO• w C•CCCCCOOH

3

1.65  104

1.51

18.8

15.8

3.2

1.62

1.18

1.01

1
7s rate rule

1

5.16  104

1.41

16.2

CCCCCCOO• w CC•CCCCCOOH

2

2.67  105

1.12

15.7

12.8

2.4

1.47

1.11

0.94

1 1

2.02  106 1.13  106

0.90 0.82

14.8 13.8

11.2

4.8

0.96

0.66

0.54

no. of H

AH (s
1)

n

E (kcal mol
1)

CCC(C)COO• w CC•C(C)COOH

2

3.11  107

1.14

17.9

1
5t rate rule

1

1.45  108

0.94

15.8

C2CCCOO• w C2C•CCOOH

1

2.69  107

1.14

CCC(C)CCOO• w CCC•(C)CCOOH

1

2.79  107

1.13

C2CC(C)COO• w C2CC(C)COOH

1

2.83  107

C2CCC(C)OO• w C2C•CC(C)OOH

1

1
6p rate rule

reactionb

c

c

1
7t rate rule C2CCCCCOO• w C2C•CCCCCOOH

ΔHrxn (kcal mol
1)

a

The rate estimation rules are also provided along with ratios of the rate rule and the individually calculated TST rate constants. In this notation, the hydrogen atoms are assumed, and a “•” symbolizes a radical site. b For the reaction class notation, the first number refers to the peroxy radical site, while the second number refers to the location of the radical site in the product relative to that in the reactant. The letters “p”, “s”, and “t” refer to reacting primary, secondary, and tertiary sites. c Estimated based on trends in the pre-exponential factors and activation energies (see text).

radical. The transition state structure reflects the subsequent O
O fission as its O
O bond is already substantially elongated. This leads to different temperature dependent values for the reaction entropy and, thus, a different temperature dependence as compared to the larger ring sized transition states. Figure 4 also illustrates the dramatic impact of the ring size in the transition state on the pre-exponential factor. For every additional rotor that is tied up in the cyclic transition state, the pre-exponential factor decreases by about an order of magnitude. This provides the opportunity to estimate the A-factor for 1
7 H migration reactions, which proceeds via eight-membered ring transition states. Following the trend, we expect that the preexponential factor for 1
7 H transfer reactions is approximately 1 order of magnitude smaller than that of the 1
6 isomerization reaction. To test how well these trend-based rate constants compare to the actual calculated TST rate constants, we include in Table 5 rate constants for three selected 1
7 isomerization reactions. The estimated rate constants agree with the TST results within a factor of 2. The corresponding barrier heights (gray solid symbols) and pre-exponential factors (gray lines) are included in Figures 3 and 4, respectively, to demonstrate how well these estimates match the expectations. For the evaluated reactions, the barrier heights are slightly lower than expected on the basis of the estimation from the 1
6 Evans
Polanyi relationship. Similarly, the pre-exponential factors are also slightly lower than expected on

the basis of the observed trends for the 1
4, 1
5, and 1
6 isomerizations. The deviations in the activation energy and preexponential factor offset each other, and, as a result, the estimated rate rule can satisfactorily predict the actual rate constant. Given that these isomerization reactions are slow as compared to the 1
5 and 1
6 isomerization reactions, a more thorough investigation of these reactions does not seem warranted at this time. In the past, when electronic structure calculations were not available, the rate rules for these RO2 isomerizations were derived on the basis of established trends for the analogous isomerization reactions in alkyl radicals. However, as pointed out by Davis and Francisco,41 incorporation of the peroxy oxygen atoms into the transition state ring significantly changes the energetics. For a pure hydrocarbon reaction, the difference of the barrier heights for isomerization via a five- versus a six-membered ring transition state is approximately 7 kcal mol
1.20 In contrast, for the alkyl peroxy reactions, we find roughly a 10
11 kcal mol
1 difference between the two types of isomerizations. Figure 5 compares the 1
4, 1
5, and 1
6 isomerization rate rules derived in this work (solid lines) to those incorporated in the Lawrence Livermore (LLNL) iso-octane mechanism42 (dashed lines), which are derived by analogy to the corresponding alkyl radical isomerization reactions. A secondary hydrogen atom is abstracted in each case. The 1
4 isomerization rate constants (red) are in reasonable agreement. However, for the 13434

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1
5 and 1
6 isomerization reactions (blue and black, respectively), our rate rules-based rate constants are significantly higher (∼30 times) than those obtained with the corresponding LLNL rules. Comparison to Other Work. Higher level electronic structure calculations have been performed for the reactions of small alkyl peroxy radicals. They provide a benchmark to test the accuracy of lower level calculations such as the CBS-QB3 method. Table 6 summarizes some selected results for the reactions of ethyl peroxy and propyl peroxy radicals. For ease of comparison, we have referenced the energy of the RO2 adduct and transition states relative to that of the alkyl radical + O2 entrance channel, and in parentheses we provided the barrier heights (E0K + ZPE) relative to that of the stabilized RO2 adduct. Schaefer and coworkers43 have investigated the reactions of ethyl peroxy radical using CCSD and CCSD(T) methods with a range of basis sets. Their results show that the barrier for both concerted elimination and isomerization decreases (relative to ethyl + O2) with increased levels of correlation and basis set size. At their highest level of theory (CCSD(T)/TZ2Pf//CCSD(T)/TZ2P), these barriers were found to be
0.9 and 5.3 kcal mol
1, respectively.

However, the authors note that these barriers might further decrease with more sophisticated levels of theory. This observation prompted a subsequent study44 of the ethyl peroxy concerted elimination channel using rigorous focal point methods to converge to the ab initio limit. Consistent with their expectations, these higher level calculations lowered the barrier for the concerted elimination to
3.0 kcal mol
1. This result is in good agreement with the G2 results of Miller and co-workers45 and with the CBS-QB3 results presented here. Subsequent master equation kinetic modeling studies based on the unadjusted calculations of Miller and co-workers show good agreement with experimental data.46 So far the barrier for isomerization has not been recalculated at the focal point level, and it would be interesting to know if such a calculation brought this barrier closer to that obtained with the G2 method. The CBS-QB3 calculations predict a slightly lower isomerization barrier than the G2 calculations; however, the barriers for isomerization from the stabilized ethyl peroxy radical are within 1 kcal mol
1 of each other. Recently, focal point methods have been extended to the reactions of the n-propyl peroxy radical.47 These results predict a barrier of
3.9 kcal mol
1 for the concerted elimination pathway and
0.8 kcal mol
1 for the 1
4 isomerization channel. Reactions of the n-propyl peroxy and i-propyl peroxy radicals have also been studied using QCISD(T)/6-311G(d,p)//B3LYP/6-31G* calculations with MP2 corrections.48 These potential energy surfaces provided the basis for master equation kinetic modeling studies, although we note that these barriers have been adjusted to achieve good fits with experimental data. In Table 6, we provide the direct ab initio values and the most recent set of adjusted values.49 The QCISD(T) and CBS-QB3 isomerization barriers are in good agreement with each other. The QCISD(T) concerted elimination barriers are roughly 2 kcal mol
1 lower than the CBS-QB3 and focal point predictions, although the adjusted barriers are in reasonable agreement. On the basis of these comparisons, it seems that the CBS-QB3 values are reasonable and that this method is sufficient to describe these two reaction classes adequately. The CBS-QB3 method has previously been used to investigate both the concerted elimination1,2,4 and the isomerization channels.1
4 A comparison of the rate rules derived in this work (red lines) to previously reported rate rules and other individually calculated values is provided in Figure 6a and b for the “low substituted” concerted elimination and the secondary 1
5-isomerization reactions, respectively. Comparisons for the

Figure 5. Comparison of the 1
4 (red), 1
5 (blue), and 1
6 (black) isomerization reaction rate rules for H-atom abstraction from a secondary carbon atom derived in this work (solid lines) to those incorporated in the LLNL iso-octane mechanism (dashed lines),42 which are formulated on the basis of the analogous alkyl radical isomerization reactions.

Table 6. Comparison of the Current CBS-QB3 Barrier Heights (E0K + ZPE) for Concerted Elimination (TSCE) and Isomerization (TSIsom) to Other Selected Values in the Literaturea reaction CC• + O2

CCC• + O2

C2C• + O2

RO2

TSCE


34.2


2.9 (31.2)


33.0


3.0 (30.0)

TSIsom(1
4)

TSIsom(1
5)

reference; method

1.8 (36.0)

this work; CBS-QB3


33.9


0.9
3.0 (30.9)

5.3 3.1 (37.0)

43; CCSD(T)/TZ2Pf//CCSD(T)/TZ2P 45; modified G2


34.8


3.9 (30.9)


2.7 (32.1)


3.9


0.8


34.9


5.2 (29.7)


2.6 (32.3)


11.2 (23.7)


33.9


3.8 (30.1)


2.1 (31.8)


11.2 (22.7)


36.3


5.1 (31.2)


0.9 (35.3)

this work; CBS-QB3


36.8


7.0 (29.8)


1.4 (35.4)

46; QCISD(T)/6-311G(d,p)//B3LYP/6-31G* with MP2 corrections


34.8


5.0 (29.8)


1.7 (33.1)

49; adjusted in ME kinetic modeling studies to fit experimental data

44; focal point analysis


11.0 (23.8)

this work; CBS-QB3 47; focal point analysis 46; QCISD(T)/6-311G(d,p)//B3LYP/6-31G* with MP2 corrections 49; adjusted in ME kinetic modeling studies to fit experimental data

The values are given in kcal mol
1. In this notation, the hydrogen atoms are assumed, and a “•” symbolizes a radical site. The energies are given relative to the R + O2 entrance channel. The values in parentheses are given relative to the RO2 stabilized adduct. a

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Figure 6. Comparisons of the rate rules derived in this work to previously reported values for (a) the concerted elimination reactions leading to low-substituted olefins and (b) the secondary 1
5 isomerization reactions. Red lines: rate rules derived in this work. Blue lines: ref 1. Green lines: ref 4. Gray lines: ref 3. Black lines: ref 2.

other isomerization reactions are provided in the Supporting Information. For the concerted elimination reaction, differences in the calculated rate constants span 1 order of magnitude. The agreement among calculations for the isomerization reactions is only slightly better. Although there is a considerable spread in the available values, in both cases our rate rules fall in the middle of these values, providing some confidence as to the reliability of these results. There could be many reasons for the observed spread in the data. In some cases, the differences in the calculated rate constants may be attributed to the unintended use of higher energy reactant and/or transition state conformers. Explicit calculation of the hindrance potentials for each internal rotation around a single bond as it has been done in this work increases the likelihood that the lowest energy conformer has been identified. However, because these motions are treated as uncoupled modes, not all of the phase space is sampled and, therefore, a small possibility exists that in a few cases the lowest energy conformer might have been missed. The spread in the available rate constants may also arise due to different ways to correct for tunneling, variations in accounting for optical isomers, and differences in the hindered rotor treatments. Because the observed spread in Figure 6 appears to be largely due to calculation of the pre-exponential factor, the differences are more likely due to optical isomer assignments and hindered rotors treatments. Several different methods of dealing with internal rotation have been presented in the literature, and at present no standard or recommended procedure exists. For an overview of

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the popular 1D-hindered rotor approximation, which is employed in this work, as well as a discussion of how the different methods of obtaining the hindrance potential and reduced moments of inertial impact the resulting partition function, we point the reader to a useful review by Pfaendtner et al.50 Rate rules for both the concerted elimination and the isomerization channels have recently been reported by Miyoshi.1 Miyoshi also used the CBS-QB3 level of theory but a somewhat different way of postprocessing the ab initio data. The most significant difference between the two studies is in the hindered rotor treatments. In Miyoshi’s work, hindrance potentials of C4 and smaller species were calculated at the CBS-QB3 level of theory, and the rotational contributions to the partition functions were determined using the Pitzer
Gwinn approximation.51 The hindrance potentials of C5 and larger alkyl peroxy radicals were then estimated from the results of smaller alkyl peroxy radicals. In this work, although we calculate the hindrance potentials using a lower level of theory (B3LYP/6-31G(d)), the averaged rotational moments of inertia are determined on the basis of the work of Kilpatrick and Pitzer,52 which extends the original work of Pitzer and Gwinn toward asymmetric rotors. A second important difference is in the number of evaluated reactions from which the rate rules are derived. Miyoshi’s rules are based on the “minimum sized representative molecule”, and, therefore, there is no indication as to the uncertainty in the evaluated data. Despite these differences, there is generally good agreement between the two sets of rules. A comparison is provided in Figure 6a and b; the blue lines correspond to Miyoshi’s rules and the red lines to this work. For both the concerted elimination and the isomerization reactions, Miyoshi provides nine rate rules because he subdivides the reaction classes on the basis of the nature of the peroxy group and on the C
H bond type of abstracted hydrogen. However, in several instances, his nine distinct rate rules group closely together and effectively could be collapsed into a single rule, consistent with our assignments. For instance, similar rate constants are obtained for the concerted elimination reactions of primary and secondary alkyl radicals. The concerted elimination reactions of tertiary alkyl peroxy radicals were found to be faster. This in part agrees with our finding that HO2 elimination reactions leading to higher substituted olefins are faster. The number of suggested rate rules for isomerization also may be reduced. As is evident in the 1
5 isomerization reactions (Figure 6b), the rate constants do not seem to depend on the nature of the peroxy group. The exceptions to this are Miyoshi’s rate constants for 1
4 and 1
6 isomerization reactions (provided in the Supporting Information). It is not clear why these rate constants do not collapse to three groups. In summary, comparison of the calculated barrier heights for the reactions of ethyl peroxy and propyl peroxy radicals shows reasonable agreement with the results of higher level calculations, providing support to the reliability of the method used in this work. Comparison of our calculated rate rules to other rate constants calculated at the CBS-QB3 level of theory shows that there is considerable spread among the available values, especially in the pre-exponential factors. The rate rules recommended here fall in the middle of these values. The observed spread in the various literature values illustrates the importance of using an internally consistent method to derive a rate rule for a given class of reactions. III. Analysis of the Effect of Pressure. Thus far, we have developed high-pressure rate rules for the unimolecular reactions 13436

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of RO2. In this section, we investigate the pressure dependence of these rate constants in an effort to provide some guidance as to the conditions for which these high-pressure rules can be directly applied. The pressure dependent submechanism is composed of six reactions: R þ O2 T RO2

ð1=
1Þ

RO2 f olefin þ HO2

ð2Þ

RO2 f QOOH

ð3Þ

R þ O 2 f R þ O2

ð4Þ

R þ O2 f olefin þ HO2

ð5Þ

R þ O2 f QOOH

ð6Þ

The products of the isomerizations and concerted elimination reactions can be produced by two distinct routes: chemically activated pathways (reactions 5 and 6) and thermal dissociation pathways (reactions 2 and 3). The reactants can be reformed via a thermally activated route (reaction 1/
1) or via dissociation of the activated RO2* adduct (reaction 4). The chemically activated reactions proceed directly from the initially formed energized RO2* adduct prior to collisional stabilization, and hence they occur on a rapid time scale and are referred to as “prompt”. Collisional stabilization results in the formation of a thermally equilibrated RO2, and the subsequent dissociation reactions of this thermalized adduct proceed generally on a slower time scale. We refer to those products as “delayed”. The pressure dependence arises because of the competition between the unimolecular reactions of the energized adduct (including reaction back to reactants) and its bimolecular collisional stabilization. At the high-pressure limit, all of the energized RO2* adduct is collisionally stabilized, and only the thermal dissociation reactions occur. To simplify this analysis, we ignore the subsequent reactions of QOOH in the above mechanism. In an effort to thoroughly explore the effect of pressure, we investigated a series of different sized alkyl radical plus O2 reactions over a wide range of temperatures and pressures. Specifically, we obtain pressure and temperature dependent apparent rate constants for representative small (n-C4H9), medium (n-C8H17), and large (n-C12H25) alkyl radicals reacting with O2 using QRRK/MSC calculations. (We refer to these rate constants as “apparent rate constants” because they result from a steady-state analysis of the reaction network and do not describe specific elementary reaction steps.) Use of the QRRK/MSC approach is especially useful for calculating pressure dependent rate constants in this context because we can utilize the highpressure rate rules as input parameters. Because we ignore subsequent reactions of QOOH, the calculated pressure dependent rate constants are not suitable for use in hydrocarbon oxidation mechanisms, but do allow a comparison of the trends for the various sized species. (Values appropriate for inclusion in an oxidation mechanism would require expansion of the network to account for the unimolecular reactions of QOOH, including the reverse reaction to RO2, as well as its bimolecular reaction with O2.) In our analysis, the reactions of n-C8H17 and n-C12H25 plus O2 proceed via the same pathways. The reaction of n-C4H9 + O2 is slightly different because the corresponding alkyl peroxy radical cannot undergo a 1
7 isomerization reaction and the

1
6 isomerization results in the abstraction of a primary hydrogen atom versus a secondary hydrogen as in the larger sized alkyl peroxy radicals. Qualitatively, we expect two opposing effects to occur as the size of the alkyl radical increases: (1) The number of atoms and bonds increases significantly, thereby increasing the number of vibrational modes that can store energy. This effect should decrease the probability of accumulating sufficient energy in the one mode that leads to reaction, thereby decreasing the rate constant. (2) The amount of internal energy will increase, which should increase the rate constant. Figure 7 shows the calculated apparent pressure dependent rate constants at 750 K for the representative small, medium, and large alkyl radical reactions with O2. The top three panels show the impact of size on the pressure dependence of the chemically activated reactions. There is a clear trend toward increasing stabilization (formation of RO2) with increasing size. This result indicates that the unimolecular rate constants decrease with increasing size, consistent with the first expectation above. Another impact of increasing size is a shift of the predicted branching ratios. For the C4 case, the isomerization channels are only slightly faster than redissociation, while for the C12 reaction isomerization dominates. Examination of the three lower panels in Figure 7 shows similar patterns in terms of the impact of molecular size on the thermal dissociation reactions of RO2. For C12, the rate constants are almost independent of pressure over the entire range, while for C4 falloff effects are evident at the lower pressures. A difference in the branching ratios of thermal versus chemically activated pathways is also evident. For the thermalized C4H9O2 adduct, note that the isomerization channel dominates, while for the energized C4H9O2* adduct the branching ratios of dissociation and isomerization are similar. This is attributed to the difference in energy distributions between the thermalized and chemical activated RO2 adduct. Because all of the chemically activated adduct population has enough energy to dissociate, the product distribution is more directly influenced by the pre-exponential factors. In contrast, only a small fraction of the thermalized RO2 population has enough energy to dissociate, and, therefore, the lowest energy pathway (i.e., isomerization) is favored. Similar size dependent trends are observed for the rate constants at 500 and 1000 K; these results are provided in the Supporting Information. As expected, at a higher temperature of 1000 K the chemically activated pathways become more significant, and pressure falloff effects are more likely to be important. However, the thermal dissociation rate constants also increase in magnitude, and, even though they are sensitive to pressure effects, these channels now compete with the chemically activated channels. This combined effect will be further investigated in the following paragraphs. In summary, at lower pressures the chemically activated reactions are more significant than at higher pressures; the rate constants for the chemically activated reactions decrease and the rate constants for stabilization and thermal dissociation channels increase at higher pressure. As the species increase in size, the corresponding rate constants approach their highpressure values at lower pressures. To investigate the sensitivity of the results to the value of ΔEall, we have also calculated the apparent rate constants for the reaction of n-C4H9 + O2 using a lower value of
80 cm
1, rather than the above value of
154 cm
1. These energy transfer parameters correspond to ΔEdown values of
70 and
125 cm
1, respectively, at 750 K in master equation calculations; such values are consistent with typical literature values.53 Use of a lower ΔEall 13437

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Figure 7. Calculated pressure dependent rate constants at 750 K for the chemically activated reactions of (a) C4H9 + O2, (b) C8H17 + O2, and (c) C12H25 + O2, and the corresponding thermal dissociation reactions of RO2 ((d), (e), and (f), respectively). The solid line corresponds to k∞ for R + O2 f RO2, the dotted line to the stabilization channel, the small dashed lines to the sum of the isomerization channels, the dot
dash lines to the dissociation channel, and the large dashed lines to the concerted elimination channel. The black lines correspond to calculations performed using a ΔEall value of
154 cm
1, while the red lines are for
80 cm
1.

value may lead to more falloff, especially at higher temperatures. However, comparison of the red and black lines in Figures 7a and d shows that the two sets of results at 750 K are virtually identical to one another. At 500 and 1000 K, the two sets of data are also indistinguishable from each other (see the Supporting Information). On the basis of this, we expect that the C8 and C12 results are also not sensitive to the value of ΔEall. We can use these rate constants to quantitatively assess the need to explicitly account for pressure effects by comparing predicted concentration
time profiles with two mechanisms: (1) a mechanism consisting of reactions 1/
1 to 3 using the high-pressure rate rules [high-P mechanism], and (2) a mechanism containing all six reactions using the pressure dependent rate constants computed as described above [variable-P mechanism]. Isothermal calculations were performed at 500, 750, and 1000 K and at two pressures: at 0.1 atm, which is relevant to many laboratory experiments, and at 10 atm, which is close to combustion/ignition conditions. The initial mole fraction of the alkyl radical was arbitrarily set to 0.2% with the remainder being air. Note that as the pressure is increased, the initial absolute concentrations of the reactants also increase, and, therefore, the R + O2 reaction proceeds on a faster time scale. For the selected initial gas composition, the reaction is pseudofirst-order in the alkyl radical concentration, and an increase in pressure from 0.1 to 10 atm will decrease the half-life by a factor of 100. Thus, the impact of pressure is 2-fold; one effect is changes in the rate due to the different concentrations, and the other is changes in the rate constants due to the changing relative contributions from the chemically activated versus thermal dissociation channels.

The normalized reactant and product mole fractions predicted by the two mechanisms are shown in Figure 8 for the 500 K case. The results using the high-P mechanism are indicated by solid lines, while those using the variable-P mechanism are shown by dashed lines. The results shown in the top panel are for simulations performed at 0.1 atm. The low-pressure results for n-butyl radical (top left panel) clearly illustrate the impact of the chemically activated channels. At very short reaction times (seen in the inset of Figure 8a), the variable-P mechanism predicts the prompt formation of γ-QOOH (from the 1
5 isomerization reaction) in addition to the formation of the stabilized adduct. At longer times, the stabilized adduct also isomerizes to γ-QOOH. Because the significance of this chemically activated isomerization channel, the RO2 adduct is converted to γ-QOOH more rapidly with the variable-P mechanism than with the high-P mechanism. However, the final steady-state concentrations are comparable in both cases. Interestingly, even though the discrepancy between the two simulations is largest at short reaction times, both simulations show similar rates for alkyl radical decay (seen in the inset). This indicates that the dissociation rate of the activated RO2 adduct back to the reactants is small, relative to isomerization, under these conditions. As the size of the alkyl substrate increases, the agreement between the variable-P and high-P simulations improves. This is consistent with the fact that larger species approach the high-pressure limit at lower pressures (see Figure 7). At 10 atm, the two simulations, shown in the bottom panels, are virtually identical to each other. In particular, we see no impact of molecular size for this case. The results of the simulations at 750 K are shown in Figure 9. There are some significant differences relative to the results 13438

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Figure 8. Normalized concentration
time profiles at 500 K for the reactions of C4H9 + O2, C8H17 + O2, and C12H25 + O2. The dotted lines display the results of the variable-pressure simulation, while the solid lines display the results of the high-pressure simulation. The top panel is calculated at 0.1 atm and the bottom at 10 atm.

Figure 9. Normalized concentration
time profiles at 750 K for the reactions of C4H9 + O2, C8H17 + O2, and C12H25 + O2. The dotted lines display the results of the variable-pressure simulation, while the solid lines display the results of the high-pressure simulation. The top panel is calculated at 0.1 atm and the bottom at 10 atm.

at 500 K. Note that the time scale is now measured in microseconds rather than milliseconds. This is due to the much larger rates for the reactions of the thermalized RO2 adduct. The results at low pressure for C4 (top left panel) show that the decay of the

n-butyl radical predicted with the variable-P mechanism is now slower than that predicted with the high-pressure mechanism. This is due to the increased dissociation of the energized adduct back to the reactants (reaction 4), which reduces the 13439

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Figure 10. Normalized concentration
time profiles at 1000 K for the reactions of C4H9 + O2, C8H17 + O2, and C12H25 + O2. The dotted lines display the results of the variable-pressure simulation, while the solid lines display the results of the high-pressure simulation. The top panel is calculated at 0.1 atm and the bottom at 10 atm.

consumption of alkyl radicals. This effect diminishes with increased pressure. The predicted increase in rate of isomer production at early times with the variable-P mechanism relative to the high-P mechanism is also a consequence of the chemically activated channel. Similarly, the significant differences in the RO2 profiles between the two simulations (which is more pronounced for smaller alkyl radicals) are a direct result of the importance of chemical activation. However, it is important to note that, even though the occurrence of these chemically activated processes results in a different time evolution of the reaction products, the final concentrations predicted with both mechanisms are very similar. The results shown in the lower panels of Figure 9 indicate that molecular size has only a minimal impact at 10 atm and that the high-P mechanism is adequate for all cases considered. Note that the much lower predicted concentration of the δ-QOOH isomer (from the 1
6 isomerization reaction) for the n-butyl case is due to the fact that this reaction involves transfer of a primary H-atom, and the rate rule for this step is significantly slower than the secondary H-transfers for the larger radicals. Figure 10 displays the results obtained with the variable-P and high-P mechanisms at 1000 K. Once again, we see that the largest deviations occur for the C4 reactant. At 0.1 atm, dissociation of the energized adduct back to the reactants leads to a slower decay of the n-butyl radical than at the high-pressure limit. Additionally, small differences are observed in the final product distribution. The variable-P mechanism shows slightly less formation of the γQOOH isomer and increased formation of the concerted elimination products (and β-QOOH from 1 to 4 isomerization; not shown) relative to the high-P mechanism. This is the result of the different branching ratios for the chemically activated versus thermal dissociation channels. At 10 atm, the agreement between

the simulations improves in terms of the steady-state product distributions; however, the time evolution of the products differs. This is attributed to the chemically activated isomerization reactions, which increase the rate of formation of the isomers at early times and decreases intermediate concentration of the stabilized adduct. As the size of the reactant increases, the agreement between the two simulations improves. For the C12 reaction, the results of the two simulations are almost indistinguishable. In summary, we have investigated the effect of pressure for representative small, medium, and larger alkyl radical plus O 2 reactions over a wide range of temperatures and pressures by explicit comparisons of product concentration
time profiles using two mechanisms, one that used pressure dependent rate constants and one that used the corresponding high-pressure values. For most of the investigated conditions, we found that the predictions with both mechanisms were similar in terms of the final product distributions, even though in some instances there were differences in the early time profiles. Small differences in the steady-state product distributions occurred at high temperature and low pressure. The largest differences occurred for the reaction of n-C4H9 radical with O2. The predictions of the two mechanisms for the three radicals are in good agreement at 10 atm, suggesting that under these conditions (typical for many ignition problems) the high-pressure rate rules can be used directly to describe the reactions of RO2. We demonstrated that this conclusion is not contingent upon the assignment of ΔEall. For those applications where pressure effects might be important, the same rate rules can easily be utilized in QRRK/MSC calculations (along with the subsequent reactions of QOOH) to predict the appropriate pressure dependent rate constants. 13440

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’ SUMMARY We have developed an internally consistent set of rate estimation rules for the dissociation, concerted elimination, and isomerization reactions of alkyl peroxy radicals by systematically calculating rate constants for a series of small to intermediate sized alkyl peroxy species. Some surprising results have emerged from this analysis such as: (1) the opposite ordering of the alkyl peroxy R
O2 and corresponding alkane R
H bond dissociation energies, (2) the insensitivity of the concerted elimination rate constants to the structure of the RO2 radical, and (3) the large barrier height differences between the 1
4 and 1
5 isomerization reactions of RO2 when compared to the corresponding alkyl radical reactions. In general, the rate rules derived in this work are in reasonable agreement with other published results at the CBS-QB3 level of theory. However, in some cases, they differ significantly from rate rules that were originally derived on the basis of analogous isomerization reactions of hydrocarbon radicals. As a result, substitution of the rate rules derived in this work into hydrocarbon oxidation mechanisms that currently utilize isomerization rate rules based on analogous alkyl radical reactions may degrade the predictions, thereby indicating that other changes in the mechanism might also be needed. We also examined the effect of pressure by comparing concentration
time profiles obtained using a pressure dependent mechanism to those obtained using a highpressure mechanism. These results suggest that under typical combustion/ignition conditions the high-pressure rules can be used directly without any further consideration of pressure effects. ’ ASSOCIATED CONTENT

bS

Supporting Information. Further comparison of the isomerization rate rules derived in this work to previously reported values is provided in Figure S1. Calculated pressure dependent rate constants at 500 and 1000 K for the chemically activated reactions of C4H9, C8H17, and C12H25 + O2 and the corresponding thermal dissociation reactions of RO2 are shown in Figures S2 and S3, respectively. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We thank Dr. William J. Pitz, Dr. Charles K. Westbrook, Dr. Marco Mehl, and Dr. S. Mani Surathy at Lawrence Livermore National Laboratories for helpful discussions. We thank Prof. Theodore S. Dibble for providing a preprint of his paper on tunneling corrections.16 We also thank Mr. Eric Kosovich for his assistance with the BDE calculations. This work is supported by the Office of Naval Research (N00014-08-1-0539; program manager: Dr. Sharon Beermann-Curtin). ’ REFERENCES (1) Miyoshi, A. J. Phys. Chem. A 2011, 115, 3301–3325. (2) Zhang, F.; Dibble, T. S. J. Phys. Chem. A 2011, 115, 655–663. (3) Sharma, S.; Raman, S.; Green, W. H. J. Phys. Chem. A 2010, 114, 5689–5701.

ARTICLE

(4) Zhu, L.; Bozzelli, J. W.; Kardos, L. M. J. Phys. Chem. A 2007, 111, 6361–6377. (5) Montgomery, J. A., Jr.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 1999, 110, 2822–2827. (6) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; T., V.; Kudin, K. N.; Burant, J. C.; et al. Gaussian 03, revision A.1; Gaussian, Inc.: Pittsburgh, PA, 2003. (7) Saeys, M.; Reyniers, M. F.; Marin, G. B.; Van Speybroeck, V.; Waroquier, M. J. Phys. Chem. A 2003, 107, 9147–9159. (8) Vandeputte, A. G.; Sabbe, M. K.; Reyniers, M. F.; Van Speybroeck, V.; Waroquier, M.; Marin, G. B. J. Phys. Chem. A 2007, 111, 11771–11786. (9) Carstensen, H. H.; Dean, A. M. J. Phys. Chem. A 2009, 113, 367–380. (10) Huynh, L. K.; Carstensen, H. H.; Dean, A. M. J. Phys. Chem. A 2010, 114, 6594–6607. (11) Carstensen, H. H.; Naik, C. V.; Dean, A. M. J. Phys. Chem. A 2005, 109, 2264–2281. (12) Chen, C.-J.; Bozzelli, J. W. J. Phys. Chem. A 1999, 103, 9731–9769. (13) Asatryan, R.; Bozzelli, J. W. J. Phys. Chem. A 2010, 114, 7693–7708. (14) East, A. L. L.; Radom, L. J. Chem. Phys. 1997, 106, 6655–6674. (15) Eckart, C. Phys. Rev. 1930, 35, 1303–1309. (16) Zhang, F.; Dibble, T. S. Phys. Chem. Chem. Phys. 2011, 113, 17969–17977. (17) Chang, A. Y.; Bozzelli, J. W.; Dean, A. M. Z. Phys. Chem. 2000, 214, 1533–1568. (18) Ritter, E. R.; Bozzelli, J. W. Int. J. Chem. Kinet. 1991, 23, 767–778. (19) Mourits, F. M.; Rummens, F. H. A. Can. J. Chem. 1977, 55, 3007–3020. (20) Carstensen, H. H.; Dean, A. M. In Comprehensive Chemical Kinetics; Carr, R., Ed.; Elsevier: New York, 2007; Vol. 42, pp 105
187. (21) Carstensen, H. H.; Dean, A. M. Int. J. Chem. Kinet. 2011in press. (22) ChemKin-Pro, Reaction Design; San Diego, 2008. (23) Simmie, J. M.; Black, G.; Curran, H. J.; Hinde, J. P. J. Phys. Chem. A 2008, 112, 5010–5016. (24) Knyazev, V. D.; Slagle, I. R. J. Phys. Chem. A 1998, 102, 1770–1778. (25) Porter, N. A.; Pratt, D. A. Org. Lett. 2003, 5, 387–390. (26) Lay, T. H.; Bozzelli, J. W. J. Phys. Chem. A 1997, 101, 9505–9510. (27) Blanksby, S. J.; Ramond, T. M.; Davico, G. E.; Nimlos, M. R.; Kato, S.; Bierbaum, V. M.; Lineberger, W. C.; Ellison, G. B.; Okumura, M. J. Am. Chem. Soc. 2001, 123, 9585–9596. (28) Shuman, N. S.; Bodi, A.; Baer, T. J. Phys. Chem. A 2010, 114, 232–240. (29) Blanksby, S. J.; Ellison, G. B. Acc. Chem. Res. 2003, 36, 255–263. (30) Lenhardt, T. M.; McDade, C. E.; Bayes, K. D. J. Chem. Phys. 1980, 72, 304–310. (31) Ruiz, R. P.; Bayes, K. D. J. Phys. Chem. 1984, 88, 2592–2595. (32) Kaiser, E. W.; Wallington, T. J.; Andino, J. M. Chem. Phys. Lett. 1990, 168, 309–313. (33) Kaiser, E. W. J. Phys. Chem. 1995, 99, 707–711. (34) Dingledy, D. P.; Calvert, J. G. J. Am. Chem. Soc. 1963, 85, 856–861. (35) Dilger, H.; Schwager, M.; Roduner, E.; Reid, I. D.; Fleming, D. G. Hyperfine Interact. 1994, 87, 899–904. (36) Dilger, H.; Schwager, M.; Tregenna-Piggott, P. L. W.; Roduner, E.; Reid, I. D.; Arseneau, D. J.; Pan, J. J.; Senba, M.; Shelley, M.; Fleming, D. G. J. Phys. Chem. 1996, 100, 6561–6571. (37) Dilger, H.; Schwager, M.; Tregenna-Piggott, P. L. W.; Roduner, E.; Reid, I. D.; Arseneau, D. J.; Pan, J. J.; Senba, M.; Shelley, M.; Fleming, D. G. J. Phys. Chem. 1996, 100, 16445–16445. (38) Dilger, H.; Stolmar, M.; Tregenna-Piggott, P. L. W.; Roduner, E.; Reid, I. D. Ber. Bunsen-Ges. Phys. Chem. 1997, 101, 956–960. 13441

dx.doi.org/10.1021/jp2079204 |J. Phys. Chem. A 2011, 115, 13425–13442

The Journal of Physical Chemistry A

ARTICLE

(39) Miller, J. A.; Klippenstein, S. J. Int. J. Chem. Kinet. 2001, 33, 654–668. (40) Slagle, I. R.; Park, J.-Y.; Gutman, D. Proc. Combust. Inst. 1984, 20, 733–741. (41) Davis, A. C.; Francisco, J. S. J. Phys. Chem. A 2011, 115, 2966–2977. (42) Curran, H. J.; Gaffuri, P.; Pitz, W. J.; Westbrook, C. K. Combust. Flame 2002, 129, 253–280. (43) Rienstra-Kiracofe, J. C.; Allen, W. D.; Schafer, H. F., III. J. Phys. Chem. A 2000, 104, 9823–9840. (44) Wilke, J. J.; Allen, W. D.; Schafer, H. F., III. J. Chem. Phys. 2008, 128, 074308. (45) Miller, J. A.; Klippenstein, S. J.; Robertson, S. H. Proc. Combust. Inst. 2000, 28, 1479–1486. (46) DeSain, J. D.; Klippenstein, S. J.; Miller, J. A.; Taatjes, C. A. J. Phys. Chem. A 2003, 107, 4415–4427. (47) Liang, T.; Allen, W. D. The 7th International Conference on Chemical Kinetics; Cambridge, MA, 2011. (48) DeSain, J. D.; Taatjes, C. A.; Miller, J. A.; Klippenstein, S. J.; Hahn, D. K. Faraday Discuss. 2001, 119, 101–120. (49) Huang, H.; Merthe, D. J.; Zador, J.; Jusinski, L. E.; Taatjes, C. A. Proc. Combust. Inst. 2011, 33, 293–299. (50) Pfaendtner, J.; Yu, X.; Broadbelt, L. J. Theor. Chem. Acc. 2007, 118, 881–898. (51) Pitzer, K. S.; Gwinn, W. D. J. Chem. Phys. 1942, 10, 428–440. (52) Kilpatrick, J. E.; Pitzer, K. S. J. Chem. Phys. 1949, 17, 1064– 1075. (53) Golden, D. M.; Barker, J. R. Combust. Flame 2011, 158, 602–617. (54) DeMore, W. B.; Sander, S. P.; Golden, D. M.; Hampson, R. F.; Kurylo, M. J.; Howard, C. J.; Ravishankara, A. R.; Kolb, C. E.; Molina, M. J. JPL Publication 97-4, 1997; pp 1
266. (55) Atkinson, R.; Baulch, D. L.; Cox, R. A.; Hampson, R. F.; Kerr, J. A.; Rossi, M. J.; Troe, J. J. Phys. Chem. Ref. Data 1997, 26, 521–1011. (56) Munk, J.; Pagsberg, P.; Ratajczak, E.; Sillesen, A. Chem. Phys. Lett. 1986, 132, 417–421.

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