Shock Tube Explorations of Roaming Radical Mechanisms: The

Article pubs.acs.org/JPCA

Shock Tube Explorations of Roaming Radical Mechanisms: The Decompositions of Isobutane and Neopentane R. Sivaramakrishnan, J. V. Michael,* L. B. Harding,* and S. J. Klippenstein* Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States S Supporting Information *

ABSTRACT: The thermal decompositions of isobutane and neopentane have been studied using both shock tube experiments and ab initio transition state theory based master equation calculations. Dissociation rate constants for these molecules have been measured at high temperatures (1260−1566 K) behind reflected shock waves using high-sensitivity H-ARAS detection. The two major dissociation channels at high temperature are iso-C4H10 → CH3 + i-C3H7 (1a) and neo-C5H12 → CH3 + t-C4H9 (2a). Ultrahigh-sensitivity ARAS detection of H-atoms produced from the rapid decomposition of the product radicals, i-C3H7 in (1a) and t-C4H9 in (2a), through iC3H7 + M → H + C3H6 + M (3a) and t-C4H9 + M → H + i-C4H8 + M (4a) allowed measurements of both the total decomposition rate constants, ktotal, and the branching to radical products, which were observed to be equivalent in both systems, k1a/ktotal and k2a/ktotal = 0.79 ± 0.05. Theoretical analyses indicate that in isobutane, the non-H-atom fraction has two contributions, the dominant fraction being due to the roaming radical mechanism leading to molecular products through iso-C4H10 → CH4 + C3H6 (1b) with k1b/ktotal = 0.16, and a minor fraction that involves the isomerization of i-C3H7 to n-C3H7 that then subsequently forms methyl radicals, i-C3H7 + M → n-C3H7 + M → CH3 + C2H4 + M (3b). In contrast to isobutane, in neopentane, the contribution to the non-H-atom fraction is exclusively through the roaming radical mechanism that leads to neo-C5H12 → CH4 + i-C4H8 (2b) with k2b/ktotal = 0.21. These quantitative measurements of larger contributions from the roaming mechanism for larger molecules are in agreement with the qualitative theoretical arguments that suggest long-range dispersion interactions (which become increasingly important for larger molecules) may enhance roaming.

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neo‐C5H12 + M → CH3 + t‐C4H 9 + M

INTRODUCTION Traditional transportation fuels are composed of a variety of chemical classes with the largest contributions from linear and branched alkanes.1−3 Consequently, the thermal decompositions of these linear and branched alkanes represent important initiation processes in the combustion chemistry of real fuels. Furthermore, these thermal dissociations represent a class of reactions whose rate parameters are pressure and temperature dependent. Hence, alkane thermal decompositions have been experimentally studied using a variety of methods over the last 50 years.4 In recent work from this laboratory,5 the thermal decomposition of a simple n-alkane, propane, was studied and compared to previous experimental and theoretical work. The large difference in octane numbers for iso-octane (100) and n-heptane (0) suggests that branching strongly affects the combustion properties of alkanes. Here, results are presented for the decompositions of the two smallest branched isoalkanes and neoalkanes: isobutane and neopentane. For these two simple branched alkanes, the only decomposition pathway that was thought to be significant was the C− C bond fission forming methyl radicals and an alkyl radical, i.e., for isobutane and neopentane, respectively, iso‐C4 H10 + M → CH3 + i‐C3H7 + M (1a) © 2012 American Chemical Society

(2a)

Furthermore, the product alkyl radicals, i-C3H7 (isopropyl) and t-C4H9 (tert-butyl), are generally expected to instantaneously decompose at high temperature to give H-atoms and the respective olefins, propene and isobutene. i‐C3H7 + M → H + i‐C3H6 + M

(3a)

t‐C4 H 9 + M → H + i‐C4 H8 + M

(4a)

With these assumptions the yield of H-atoms (in the absence of secondary loss of H due to abstractions with the unreacted alkane) would be equal to the total loss of alkane from decomposition and the overall H-atom branching ratio would be unity. In recent years, another outcome5 has been documented both experimentally6,7 and theoretically.8,9 In particular, the departing atom or radical (CH3 in these cases) can roam around the partner radical, subsequently abstracting an H-atom yielding CH4 and, for the present cases, the respective olefins, Special Issue: A. R. Ravishankara Festschrift Received: November 14, 2011 Revised: March 5, 2012 Published: March 6, 2012 5981

dx.doi.org/10.1021/jp210959j | J. Phys. Chem. A 2012, 116, 5981−5989

The Journal of Physical Chemistry A

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bimolecular reactions, and, therefore, conclusive arguments cannot be based on such experiments. Here, the possible role of these two isomerization reactions is explored with modern ab initio rate theory. First, the potential energy surfaces are explored with high-level electronic structure calculations. The branching between dissociation and isomerization in i-C3H7 and in t-C4H9 is then predicted with ab initio transition state theory. The possible role of C−H fissions in the initial reactants, iso-C4H10 and neo-C5H12, is also briefly considered using more limited ab initio transition state theory calculations.

propene and isobutene. With this roaming mechanism, total yields of H-atoms will drop below unity depending on the branching ratios of the radical products to those of the stable disproportionation products, that is, iso‐C4 H10 + M → CH 4 + C3H6 + M

(1b)

neo‐C5H12 + M → CH 4 + i‐C4H8 + M

(2b)

A detailed discussion of the roaming radical pathways for alkane decompositions was recently provided.10 The electronic structure calculations presented there show that the process is a “frustrated dissociation” to radicals. Then, for the “barrierless” internal abstraction to be important, the roaming saddle point must lie below the radical dissociation threshold. A statistical theory based description of the roaming kinetics has also been presented recently.11 These studies suggest that long-range dispersion interactions, which become increasingly important for larger molecules, may enhance the effect of the “roaming” pathway. The roaming saddle points are predicted to be ∼l kcal/mol below separated radicals for propane, isobutane, n-butane, and neopentane,10 thereby suggesting that roaming pathways should contribute to the dissociations of each of these molecules. In our recent propane study,5 the measured branching ratio for roaming was 0.10 ± 0.08, indicating a small roaming effect. For n-butane, the competition between two separate CC fissions (to form two ethyl radicals or methyl + n-propyl) poses an experimental challenge for identifying and quantifying “roaming” contributions. Thus, this dissociation was not studied here. For isobutane and neopentane there is only one CC fission channel, thereby allowing the quantification of the roaming contribution for these two reactions. This prior theoretical analysis10 and the earlier experimental result5 for propane, coupled with the potential for an enhanced “roaming” contribution in larger molecules, has supplied the esoteric motivation for the present experiments on isobutane and neopentane. In these experiments, the absolute H-atom rates of production and H-atom yields have been measured using ultrasensitive H-atom atomic resonance absorption spectrometry (ARAS).12−14 From the data, H-atom branching ratios and dissociation rate constants can be obtained. The measured dissociation rate constants are compared to earlier studies. A potential complication that may affect the interpretation of these dilute H-atom yield experiments is the possibility of isomerization of the radicals, i-C3H7 formed in (1a) and t-C4H9 in (2a).

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EXPERIMENT Techniques. The present experiments, in Kr diluent, were performed with the reflected shock tube technique using Hatom atomic resonance absorption spectrometric (ARAS) detection. The methods and the apparatus currently being used have been previously described12,13 and only a brief description of the experiment will be presented here. The shock tube was constructed entirely from a 7 m (10.2 cm o.d.) 304 stainless steel tube with the 10.2 cm o.d. cylindrical section being separated from the He driver chamber by a 4 mil unscored 1100-H18 aluminum diaphragm. The tube was routinely pumped between experiments to less than 1.3 × 10−11 bar by an Edwards Vacuum Products Model CR100P packaged pumping system. Shock wave velocities were measured with eight equally spaced pressure transducers (PCB Piezotronics, Inc., Model 113A21) mounted along the downstream part of the test section and recorded with a 4094C Nicolet digital oscilloscope. Temperature and density in the reflected shock wave regime were calculated from this velocity. This procedure has been given previously, and corrections for boundary layer perturbations have been applied.16−18 The oscilloscope was triggered by a pulse derived from the last velocity gauge signal on the end plate. The photometer system was radially located at 6 cm from the end plate. H-atom ARAS detection was used to follow [H t ] quantitatively. The optical components (windows and lenses) were crystalline MgF2, and the resonance lamp beam intensity (filtered through 4 cm of dry air (21% O2) at 1 atm to isolate the Lyman-α wavelength at 121.6 nm), was measured by an EMR G14 solar blind photomultiplier tube, as described previously.14,19,20 The atmospheric O2 filter serves as a monochromator because there is a narrow region of high transmittance in the O2 absorption spectrum at 121.6 nm. The signal was recorded with a LeCroy model LC334A oscilloscope. Due to hydrogeneous impurities in the lamp gas, Lyman-α radiation will mostly be emitted from the lamp along with a few percent of radiation that is extraneous (nonresonant). To measure the fraction of Lyman-α present in the lamp, an H2 discharge flow system, an atom filter, is used to create large [H] (∼1 × 1014 atoms cm−3) between the lamp and shock tube lens14,21−23 thereby removing all of the Lyman-α in the emission lamp. It can be shown using line absorption theory14,21,24 that [H] = 1 × 1014 atoms cm−3 at room temperature will remove 99.6% of Lyman-α. The use of ultrahigh-purity He as the lamp gas ensures that [H] will be very low ((1−4) × 1011 atoms cm−3) in the lamp, and this will then give a completely defined unreversed Ladenburg-Reiche Gaussian line shape.14,24 To determine absolute [H] and/or [D], the curve-of-growth for each experiment is calculated using a parameter-free threelayer model for the source that was experimentally validated.14,21 The oscillator strength of the Lyman-αH transition was measured21

i‐C3H7 + M → n‐C3H7 + M → C2H 4 + CH3 + M (3b)

t‐C4 H 9 + M → i‐C4 H 9 + M → C3H6 + CH3 + M (4b)

i-C3H7 can isomerize to n-C3H7 via a 1,2 H-shift, which then undergoes C−C fission to yield non-H atom products, CH3 + C2H4. Though such a 1,2 H-shift is energetically unfavorable, a minor contribution at high temperatures cannot be immediately ruled out. Furthermore, because the roaming pathway is only a secondary channel, such minor channels need to be considered. Similarly, a 1,2 H-shift in t-C4H9 yields i-C4H9, which predominantly undergoes C−C bond fission to yield CH3 + C3H6. Therefore, both of these isomerizations may have a small effect on observed H-atom yields. Such reactions have been postulated on the basis of observed product formation studies,15 but these studies are complicated by the presence of numerous 5982



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to within 10% of the theoretical value (Bethe and Salpeter25) of 0.1387 for the weakest transition (this is arguably the best known transition probability value in all of spectroscopy). H-atom cross calibrations have additionally been performed with I (C2H5I20), O (H+O226), and Cl (Cl + H227), where curves-of-growth for these atoms were determined from complete dissociation of known concentrations of precursors (CF3I,28 N2O,29 and CCl430). From this experience, absolute [H] can be specified to within ∼±5%. A more detailed description of H-atom atomic resonance absorption spectroscopy measurements can be found in a recent review.14 Gases. High-purity He (99.995%), used as the driver gas, was from AGA Gases. Research grade Kr (99.999%), the diluent gas in reactant mixtures, was from Praxair, Inc. The ∼10 ppm impurities (N2 < 5 ppm, O2 < 2 ppm, Ar < 1 ppm, CO2 < 0.5 ppm, H2 < 1 ppm, H2O < 3 ppm, Xe < 2 ppm, and THC < 0.2 ppm) either are all inert or are in sufficiently low concentration so as to not perturb H-atom profiles. The microwave driven resonance lamp operated at 35 W and 1.9 Torr of ultrahigh-purity He (99.999%) (effective Doppler temperature, 470 K,14,21). This grade of He contains a trace of hydrogenous impurities that are sufficient to give Lyman-α radiation, the fraction of which we measure with the atomic filter. Iso-C4H10 (purity of 99.99%) was obtained from Aldrich Chemical Co., Inc., and neo-C5H12 (purity of 99.997%) was obtained from the American Petroleum Institute (API Standard Sample). Both compounds were further purified by bulb-tobulb distillation, retaining only middle thirds for mixture preparation. Gas mixtures were accurately prepared from pressure measurements using a Baratron capacitance manometer in an all glass high-purity vacuum line.

and aug-cc-pVQZ calculations. All calculations were done with the MOLPRO program package.31 Kinetics Calculations. The rate coefficients were predicted with conventional transition state theory employing rigid rotor harmonic oscillator assumptions for all but the methyl rotors, which were treated as hindered internal rotors. The rovibrational properties of the reactants and transition state were obtained from the above-described electronic structure calculations. Asymmetric Eckart tunneling corrections were included. The pressure dependence of the branching between propyl dissociation and isomerization was studied with the 1dimensional master equation. An exponential down energy transfer form was used with a temperature dependent average downward energy transfer of ⟨ΔEd⟩ = 125 (T/300)0.85 cm−1, where T is the temperature in K.

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EXPERIMENTAL RESULTS Isobutane. In the present isobutane experiments, we measured H-atoms from the subsequent instantaneous isoC3H7-radical decomposition from (1a) to H + C3H6. In the absence of reactions that deplete [H], the results will then be direct measures of reactions 1a and 1b. Figure 1 shows a typical

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THEORY A potential complication to the interpretation of these experiments is the possible existence of competing pathways that either do not lead to the production of H atoms or lead to the production of more than one H atom. In our previous study on propane5 we considered five alternative pathways to molecular products. Most of these involved the formation of carbenes via reverse insertion reactions. None of these were found to compete significantly with CC bond cleavage. We assume that these carbene-type pathways will also not be important in the decomposition of either isobutane or neopentane. As noted above, for isobutane or neopentane there is another possible competing pathway that would complicate the interpretation of the experiments. This involves the isomerization of the initially formed hydrocarbon radical through a (1,2) hydrogen migration followed by loss of a methyl radical to make an alkene. Theoretical calculations were carried out to estimate the importance of this alternate pathway for both i-C3H7 and t-C4H9. CH bond fissions in isobutane and neopentane would also complicate the interpretation of the present experiments, and theoretical calculations are again performed to estimate their role. Electronic Structure Calculations. The approach used in the electronic structure calculations was the same as that used in our previous study of alternative molecular pathways for decomposition of propane.5 Geometry optimizations (and zero point evaluations) were done with CCSD(T)/cc-pVTZ followed by single point CCSD(T)/aug-cc-pVTZ and CCSD(T)/ aug-cc-pVQZ calculations. The final energies were then obtained from a complete basis set extrapolation of the aug-cc-pVTZ

Figure 1. [H] profile from iso-C4H10 at 1408 K. The solid line is a fit over the entire time range using eq E1 with k1a + k1b = 3750 s−1 and BR1a = 0.79. The dashed lines represent changes in k1a + k1b by ±20% with BR1a = 0.79. The dotted lines represent changes in BR1a by ±0.05 with k1a + k1b fixed at 3750 s−1. The conditions for the experiment at T5 = 1408 K are P1 = 30.93 Torr, Ms = 2.368, ρ5 = 5.449 × 1018 molecules cm−3, and [iso-C4H10]0 = 1.631 × 1012 molecules cm−3.

H-atom profile at T = 1408 K using [iso-C4H10]0 = 1.631 × 1012 molecules cm−3, yielding [H]∞ = 1.288 × 1012 atoms cm−3. The solid line shown in the figure is determined from the mechanism shown in Table 1 where rate constants for (1a) and (1b) are obtained from a first-order analysis. If the sensitivity for H-detection is high, secondary reaction perturbations become negligible, and then the temporal behavior is only dependent on (1a) and (1b); i.e., the profile will be given by the first-order buildup expression, [H]t = {k1a[iso‐C4 H10]0 /(k1a + k1b)} × {1 − exp(− (k1a + k1b)t )}

(E1)

In this experiment, changes in either the total rates k1 total = k1a + k1b = 3750 s−1 by ±20% (depicted by the dashed lines) or the branching ratios BR1a = k1a/(k1a + k1b) = 0.79 by ±0.05 (shown as the dotted lines) degrade the fit to the experimental profile. 5983



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Table 1. Mechanism for the Thermal Decompositions of iso-C4H10 and neo-C5H12a

a

1a

i‐C4 H10 → i‐C3H7 + CH3

k1a = to be fitted

present

1b

i‐C4 H10 → C3H6 + CH 4

k1b = to be fitted

present

2a

neo‐C5H12 → t‐C4 H 9 + CH3

k2a = to be fitted

present

2b

neo‐C5H12 → i‐C4 H8 + CH 4

k2b = to be fitted

present

3

CH3 + CH3 → C2H6

k3 = (ρ, T)

60

4

CH3 + CH3 → C2H 4 + 2H

k4 = 5.26 × 10−11 exp(−7392 K/T)

19

5

i‐C3H7(+ M) → C3H6 + H(+ M)

k5 = (ρ, T)

61

6

n‐C3H7 (+ M) → C2H 4 + CH3 (+ M)

k6 = (ρ, T)

62 −20 3.06

7

H + C2H6 → C2H 4 + H + H2

k7 = 1.42 × 10

exp(−2555 K/T)

63

8

i‐C4 H10 + H → C3H6 + CH3 + H2

k8 = 3.00 × 10−18T2.54 exp(−3401 K/T)

44

T

−18 2.40

9

i‐C4 H10 + H → i‐C4 H8 + H + H2

k9 = 1.00 × 10

10

neo‐C5H12 + H → i‐C4 H8 + CH3 + H2

k10 = 2.79 × 10−16T1.97 exp(−4227 K/T)

3

All bimolecular rate constants are in cm molecule

−1

T

exp(−1300 K/T)

44 64

−1

s .

Using the Table 1 mechanism, a simulation was performed with the SENKIN32 suite of programs in the CHEMKIN package. The corresponding sensitivity plot is shown in Figure 2 where it

varies substantially with density, but the branching ratio appears to increase with increasing temperature. However, for the full temperature and pressure range of the experiments, the average branching ratio BR1a is measured to be 0.79 ± 0.05. Neopentane. In these experiments, we measured H-atoms from the subsequent instantaneous t-C4H9 decomposition, 2a, to give H + iso-C4H8. In the absence of reactions that deplete [H], these results are then a direct measure of reactions 2a and 2b. A typical neopentane experiment at 1339 K with [neoC5H12]0 = 1.203 × 1012 molecules cm−3 and yielding [H]∞ = 9.500 × 1011 atoms cm−3, is shown in Figure 6. The solid line shown in the figure is determined from the mechanism shown in Table 1. In this case, changes in either the total rate constant k2 total = k2a + k2b = 2000 s−1 by ±20% (the dashed lines) or the branching ratio BR2a = 0.79 by ±0.05 (the dotted lines) also significantly degrade the fit to this profile. Figure 7 shows the H-atom sensitivity analyses for the experiment in Figure 6 using the mechanism of Table 1. Similar to the case of isobutane, the neopentane experiments are not affected by secondary reactions (Figure 7), thereby indicating the appropriateness of using first-order analyses, i.e., the same as given in eq E1 with the initial alkane concentration being that of neopentane. The experimental results for neopentane are summarized in Table 3. The bimolecular rate constants for neopentane decomposition at various densities are shown in Figures 8 and 9 as Arrhenius plots. The major decomposition pathway is reaction 2a, as assumed in all previous work. However, we find a significant contribution that is likely due to the roaming mechanism, reaction 2b. The temperature and density dependences of BR2a are shown in Figure 10. Within experimental error, there is really no clear indication that BR2a varies over a 3-fold change in density. There is, however, evidence that BR2a increases with increasing temperature. Again, for the full temperature and pressure range of the experiments, the average branching ratio BR2a is measured to be 0.79 ± 0.05.

Figure 2. H-atom sensitivity analysis for the 1408 K profile shown in Figure 3 using a full reaction mechanism scheme. The three most sensitive reactions are shown in the figure.

is seen that the profile depends only on k1a and k1b; i.e., secondary reactions involving H are completely unimportant because the detectability for [H] is so low. This confirms that first-order analysis, eq E1, is appropriate. The normalized sensitivity coefficients are defined as S = ∂ ln[H]/∂ ln ki, where [H] is the H-atom concentration and ki is the rate constant for reaction i. Hence, the present results are a direct measure of dissociation. In the illustration shown in Figure 1, inspection shows that BR1a = 1.288 × 1012/1.631 × 1012 = 0.79. We have determined the total decomposition rate constant from temporal profiles like that shown in Figure 1. Subsequently, we determine k1a and k1b from the measured BR1a for the same experiment. The results are given in Table 2 along with the measured BR1a. The decomposition rate constants at various densities are shown in Figures 3 and 4 as Arrhenius plots. The major decomposition pathway is reaction 1a, as assumed in all previous work. However, we find a contribution that is almost certainly due to the roaming mechanism, reaction 1b. The temperature and density dependences of BR1a are shown in Figure 5. Within experimental error, there is really no clear indication that BR1a

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THEORETICAL RESULTS The calculated zero-point corrected relative energies are summarized in Tables 4 and 5 for i-C3H7 and t-C4H9, respectively. Frequencies and rotational constants are given in Table S1 and S2 (Supporting Information). In each case only two pathways were considered, loss of a hydrogen atom and isomerization via (1,2) hydrogen migration. For i-C3H7 the H loss barrier is 5984



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Table 2. High-T Rate Data: iso-C4H10 → CH3 + i-C3H7 and iso-C4H10 → CH4 + C3H6 P1/ Torr

Msa

ρ5/ (1018 cm−3)b

T5/Kb

XC4H10 = 6.114 × 10

k1ac

k1bc

BR1ad

−7

10.95 10.91 10.94 10.93 10.95 10.95 10.95 10.96

2.332 2.432 2.330 2.272 2.312 2.390 2.418 2.496

1.939 1388 1404 2.019 1498 7200 1.936 1387 1560 1.880 1325 450 1.927 1362 962.5 1.990 1451 3705 2.015 1483 5880 2.089 1566 12880 XC4H10 = 5.371 × 10−7

396 800 440 150 287.5 1045 1120 1120

0.78 0.90 0.78 0.75 0.77 0.78 0.84 0.92

10.93 10.95 10.96 10.96 10.96 10.95 10.96

2.384 2.346 2.319 2.358 2.422 2.371 2.398

1.981 1445 1.952 1404 1.929 1374 1.964 1417 2.014 1493 1.967 1436 1.999 1461 XC4H10 = 6.114 × 10−7

2635 1400 1264 1482 5330 3078 3400

465 350 336 418 1170 722 600

0.85 0.80 0.79 0.78 0.82 0.81 0.85

15.93 15.91 15.92 15.94 15.92 15.93 15.94 15.91 15.92

2.230 2.301 2.346 2.336 2.353 2.398 2.264 2.347 2.325

2.699 1279 2.788 1351 2.836 1403 2.828 1393 2.845 1411 2.901 1459 2.728 1323 2.827 1409 2.810 1381 XC4H10 = 5.371 × 10−7

105 551 1620 1480 2145 3444 180 1443 777

45 174 630 520 605 756 60 507 273

0.7 0.76 0.72 0.74 0.78 0.82 0.75 0.74 0.74

15.92 15.91 15.92 15.93 15.94 15.94

2.336 2.382 2.410 2.333 2.296 2.303

2.825 1393 2.869 1447 2.914 1472 2.822 1389 2.777 1351 2.786 1358 XC4H10 = 2.992 × 10−7

1387.5 3527.5 4176 720 370.5 745.5

462.5 722.5 624 280 104.5 304.5

0.75 0.83 0.87 0.72 0.78 0.71

30.89 30.93 30.87 30.92 30.96 30.87 30.96 30.91 30.79

2.299 2.368 2.456 2.362 2.364 2.284 2.302 2.365 2.344

5.274 5.450 5.641 5.433 5.445 5.234 5.294 5.439 5.366

924 2962.5 10375 2997 2200 382.5 1440 2028 1518

276 787.5 2125 703 550 67.5 360 572 682

0.77 0.79 0.83 0.81 0.80 0.85 0.80 0.78 0.69

1334 1408 1503 1401 1403 1319 1338 1404 1381

Figure 3. Bimolecular rate constants for iso-C4H10 → CH3 + i-C3H7. The three panels represent data obtained at three different reflected shock densities from (2−6) × 1018 molecules cm−3.

Figure 4. Bimolecular rate constants for iso-C4H10 → CH4 + C3H6. The three panels represent data obtained at three different reflected shock densities from (2−6) × 1018 molecules cm−3.

a

The error in measuring the Mach number, Ms, is typically 0.5−1.0% at the one standard deviation level. bQuantities with the subscript 5 refer to the thermodynamic state of the gas in the reflected shock region. cRate constants: First order in s−1. dBR1a = k1a/(k1a + k1b).

Figure 5. Branching ratios in the thermal decomposition of iso-C4H10. The three panels represent data obtained at three different reflected shock densities from (2−6) × 1018 molecules cm−3.

predicted to be 3.9 kcal/mol below the migration barrier. For t-C4H9 the difference is 4.8 kcal/mol. In the high-pressure limit, the branching between isomerization and dissociation in i-C3H7, k3b/(k3a + k3b), is predicted to vary from 0.073 to 0.083 for temperatures ranging from 1250 to 1500 K. At pressures of 0.1−1 atm, this branching is reduced to values between 0.04 and 0.05 for this same temperature range. These predictions of the branching between isomerization and dissociation have uncertainties of roughly a factor of 2.33

The dissociation of n-C3H7 yields primarily CH3 + C2H4, with each of these products stable to further dissociation under the present experimental conditions. Thus, the isomerization branching fraction contributes directly to the measured non-Hatom branching fraction of 0.21. Correspondingly, the roaming fraction should be estimated as the difference between the measured non-H-atom fraction and the theoretical prediction for the isomerization fraction, 0.21 − 0.05 = 0.16. 5985



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Table 3. High-T Rate Data: neo-C5H12 → CH3 + t-C4H9 and neo-C5H12 → CH4 + iso-C4H8 P1/Torr

Ms a

ρ5/(1018 cm−3)b

T5/Kb

k2ac

k2bc

BR2ad

−7

XC5H12 = 6.033 × 10

Figure 6. [H] profile from neo-C5H12 at 1339 K. The solid line is a fit over the entire time range using an analogue of eq E1 with k2a + k2b = 2000 s−1 and BR2a = 0.79. The dashed lines represent changes in k2a + k2b by ±20% with BR2a = 0.79. The dotted lines represent changes in BR2a by ±0.05 with k2a + k2b fixed at 2000 s−1. The conditions for the experiment at T5 = 1339 K are P1 = 15.98 Torr, Ms = 2.397, ρ5 = 2.770 × 1018 molecules cm−3, and [neo-C5H12] 0 = 1.203 × 1012 molecules cm−3.

Figure 7. H-atom sensitivity analysis for the 1339 K profile shown in Figure 5 using a full reaction mechanism scheme. The three most sensitive reactions are shown in the figure.

For t-C4H9 the predicted high-pressure limit for the isomerization to dissociation branching, k4b/(k4a + k4b), is much smaller, ranging from only 0.005 to 0.008 over the 1250− 1500 K temperature range. Furthermore, this branching will again be reduced somewhat at lower pressures. Thus, for the neopentane dissociation case, the non-H-atom fraction should correlate closely with the roaming fraction. The possible role of CH fission was directly examined for the iso-C4H10 case. For this case, there are two types of CH bonds that can be broken. The CH fissions to yield tert-butyl or isobutyl are 8.9 and 13.9 kcal/mol endothermic relative to the CC fission to yield methyl + isopropyl. Our prior predictions of the related recombination rate coefficients for a variety of alkyl + H34 and alkyl + alkyl35 reactions allow us to predict the branching to these two CH fissions relative to that for the CC fission. These predictions imply that the CH fission can be safely ignored because its branching ratio is predicted to be less than 0.002 for temperatures up to 1500 K. One may similarly conclude that the CH fission process for neopentane is insignificant.

10.92 10.92 10.94 10.92 10.93 10.92

2.240 2.299 2.330 2.293 2.396 2.379

1.848 1291 1.903 1353 1.936 1387 1.898 1347 1.992 1459 1.975 1439 XC5H12 = 4.345 × 10−7

562.5 1800 3713 988 6160 6205

187.5 450 987 312 840 2295

0.75 0.80 0.79 0.76 0.88 0.73

10.94 10.94 10.96 10.95 10.95

2.315 2.238 2.355 2.377 2.245

1.921 1370 1.850 1289 1.961 1413 1.979 1437 1.858 1296 XC5H12 = 6.033 × 10−7

1870 440.8 5655 5312.5 624

330 139.2 845 937.5 176

0.85 0.76 0.87 0.85 0.78

15.94 15.90 15.92 15.93 15.98 15.96 15.97 15.95

2.386 2.304 2.210 2.266 2.319 2.368 2.332 2.394

2.889 1446 2.781 1359 2.661 1262 2.737 1320 2.814 1375 2.871 1427 2.828 1388 2.910 1450 XC5H12 = 2.871 × 10−7

8880 1000 456 781 1840 3465 2625 5904

3120 250 144 319 460 1035 875 1296

0.74 0.80 0.76 0.71 0.8 0.77 0.75 0.82

15.85

2.234

2.682 1287 255 XC5H12 = 4.345 × 10−7

120

0.68

15.98 15.97 15.98 15.93 15.93 15.96 15.96 15.94

2.397 2.207 2.308 2.336 2.401 2.278 2.211 2.247

2.770 1339 2.666 1260 2.800 1364 2.826 1392 2.905 1462 2.758 1332 2.670 1264 2.714 1300 XC5H12 = 2.871 × 10−7

1580 300 1764 2407 5915 908.5 255.5 440

420 100 336 493 585 241.5 94.5 110

0.79 0.75 0.84 0.83 0.91 0.79 0.73 0.80

30.82 30.96 30.95 30.87 30.96 30.95 30.95

2.343 2.399 2.281 2.289 2.239 2.268 2.347

5.369 5.526 5.242 5.247 5.135 5.207 5.401

3822 5785 740 819 450 720 3915

1380 1440 1317 1324 1273 1302 1384

1078 715 260 231 150 180 585

0.78 0.89 0.74 0.78 0.75 0.8 0.87

a

The error in measuring the Mach number, Ms, is typically 0.5−1.0% at the one standard deviation level. bQuantities with the subscript 5 refer to the thermodynamic state of the gas in the reflected shock region. cRate constants: First order in s−1. dBR2a = k2a/(k2a + k2b).

■

DISCUSSION A first-order plot of the experimental rate constants obtained in this study for isobutane, along with measurements from earlier studies,36−43 is shown in Figure 11. An Arrhenius fit to the first order rate constants (Table 2) for k1a over the temperature (1279−1566 K) and pressure (0.3−1.0 atm) range of the present experiments gives k1a = 1.53 × 1013 exp( − 32211 K/T ) s−1

(E2)

The present results are approximately a factor of 2−3 lower than the most recent laser-absorption measurements of 5986



Article

Table 4. Calculated, CCSD(T)/aug-CBS//CCSD(T)/ccpVTZ Energies (kcal/mol) Relative to i-C3H7 for the Stationary Points on the C3H7 Potential Surface species

energy

n-C3H7 H + C3H6 i-C3H7 → n-C3H7 i-C3H7 → H + C3H6

3.04 34.78 40.88 36.97

Table 5. Calculated, CCSD(T)/aug-CBS//CCSD(T)/ccpVTZ Energies (kcal/mol) Relative to t-C4H9 for the Stationary Points on the C4H9 Potential Surface Figure 8. Bimolecular rate constants for neo-C5H12 → CH3 + t-C4H9. The three panels represent data obtained at three different reflected shock densities from (2−6) × 1018 molecules cm−3.

Figure 9. Bimolecular rate constants for neo-C5H12 → CH4 + i-C4H8. The three panels represent data obtained at three different reflected shock densities from (2−6) × 1018 molecules cm−3.

species

energy

i-C4H9 H + C4H8 t-C4H9 → i-C4H9 t-C4H9 → H + C4H8

5.04 34.63 40.97 36.20

Figure 11. Arrhenius plot of the iso-C4H10 decomposition rate constants at high-T (1000−1650 K): filled symbols, H-atom ARAS experiments (Table 2); [●] ρ5 ∼ 2.0 × 1018 molecules cm−3; [▲] ρ5 ∼ 3.0 × 1018 molecules cm−3; [◆] ρ5 ∼ 6.0 × 1018 molecules cm−3; (---) Tsang44 k∞ (750−1650 K); (red triangles) Brooks36 (823−853 K); (red asterisks) Konar et al.37 (770−855 K); (blue circles) Bradley39 (1200−1500 K); (◇···◇) Pratt and Rogers40 (970−1030 K); (blue triangles) Hidaka et al.42 (1000−1500 K); (red dashed line) Golden et al.38 (1100−1280 K); (blue dashed line) Koike and Morinaga41 (1300−1600 K). Open symbols, Oehlschlaeger et al.:43 [□] 0.23 atm; [○] 1.7 atm; [Δ] 4.4 atm; [◇] 8.3 atm; [···] k∞. The figure in the inset depicts an expanded version of the Arrhenius plot summarizing available experiments over the T-range 750−1650 K.

the secondary chemistry. The available literature database suggests that the static reactor studies of Konar et al.37 may be the most reliable estimates of the high-pressure-limiting rate constants. Again, deviations by a factor of 3 are observed between the k∞ recommendations of Tsang44 and the modeled k∞ from Oehlschlaeger et al.43 With the observed temperature dependence in the radical branching ratio, k1a/ktotal, and the theoretical predictions for the nearly T-independent (over the T-range of the present experiments) ratio ∼0.05 for isomerization through (3b), the roaming fraction predicted for each experiment in Table 2 was calculated as BR1b = 1 − BR1a − 0.05. Reflecting the observed temperature dependence in the roaming branching ratios in isobutane, an Arrhenius-type fit to BR1b gives

Figure 10. Branching ratios in the thermal decomposition of neoC5H12. The three panels represent data obtained at three different reflected shock densities from (2−6) × 1018 molecules cm−3.

Oehlschlaeger et al.43 Over the T-range of overlap, larger deviations are observed at higher T (>1500 K). Interestingly, the present measurements are in good agreement with the earlier CH3-lamp UV-absorption studies of Koike and Morinaga41 and the VLPP measurements of Golden et al.38 The IR-laser absorption data of Hidaka et al.42 show better agreement with the Oehlschlaeger et al.43 measurements. There are larger discrepancies among the lower-T studies primarily due to uncertainties in

(1/BR1b) − 1 = 1503 × exp( −7930 K/T ) 5987

(E3)



Article

These observations of a roaming branching that decreases with increasing temperature are in good accord with theoretical expectations.11,59 For the present reactions, the threshold for the roaming channel is about 1 kcal/mol lower than that for the radical channel. Correspondingly, in the low-temperature limit the branching to roaming should be essentially unity. As the temperature increases, the distribution of reacting states includes an increasing population above the radical threshold. Furthermore, with increasing energy above the radical threshold there is an increasing probability of the dissociation from the long-range intermediate region out to radicals occurring before the reorientation that leads to roaming products. The combination of these effects implies that the roaming branching should monotonically decrease with increasing energy or temperature.

With eq E3, BR1b is predicted to be negligible at high temperatures and approaches 1 at low temperatures. A first-order plot of the experimental rate constants obtained in this study for neopentane decomposition along with measurements from earlier studies45−58 is shown in Figure 12. An

■

CONCLUSIONS The present high-temperature shock tube studies utilizing Hatom ARAS have allowed rate constant measurements of the primary C−C bond fission channels 1a in isobutane and 2a in neopentane over the T-range 1260−1566 K and at pressures ∼0.3−1.0 atm. The ultrasensitive H-atom ARAS data have been used to derive a branching ratio of 0.21 ± 0.05 for a non-Hatom producing channel in isobutane and neopentane. The present ab initio transition state theory based master equation theoretical analysis was used to analyze the decompositions and (1,2) H-atom shifts in i-C3H7 and t-C4H9 which are the primary radicals that form in the bond fission channels 1a and 2a. The theoretical analysis indicates that the (1,2) H-atom shift in iC3H7 yields k3b ∼ 0.05 k3a, over the temperature and pressure ranges of the present experiments, suggesting that the average roaming fraction in isobutane is ∼0.16 ± 0.05. In neopentane, the corresponding (1,2) H-atom shift in t-C4H9 contributes a negligible amount in comparison to CH fission, i.e., k4b < 0.01 k4a, and therefore the average non-H atom fraction, 0.21 ± 0.05, should correspond to the contribution from the roaming channel. The increasing contributions of the roaming mechanism with the larger size of the molecule is in agreement with the qualitative theoretical arguments that suggest long-range dispersion interactions, which become increasingly important for larger molecules, may enhance the effect of the “roaming” pathway.

Figure 12. Arrhenius plot of the neo-C5H12 decomposition rate constants at high-T (1000−2000 K): (red solid circles) experiments, present work (Table 3) (1260−1462 K); (---) Srinivasan et al.,58 k∞ (1300−2000 K); (gren open circles) Bernfeld and Skinner55 (1140− 1300 K); (blue open circles) Rao and Skinner56 (1230−1460 K); (■−■) Pacey49 (793−953 K); (●−●) Tsang45 (1000−1100 K); (□---□) Baldwin et al.53 (1000−1260 K); (blue triangles) Bradley and West51 (1030−1300 K); (red diamonds) Marshall et al.50 (1030− 1300 K); (blue x's) Marquaire and Come52 (703−743 K); (green boxes) Baronnet et al.48 (723−803 K); (○···○) Taylor et al.47 (923− 1070 K); (pink triangles) Halstead et al.46 (700−833 K); (◇···◇) Pratt and Rogers54 (1030−1300 K); (red asterisks) Mitchell and Benson57 (700−800 K). The inset in the figure is an expanded version of the Arrhenius plot summarizing available experiments over the T-range 700−2000 K.

Arrhenius fit to the first-order rate constants (Table 3) for k2a over the temperature (1260−1462 K) and pressure (0.3−1.0 atm) range of the present experiments gives k2a = 8.66 × 1012 × exp( −30423 K/T ) s−1

(E4)

■

The present results for (2a) agree quite well with the most recent high-T laser schlieren-shock tube measurements of Srinivasan et al.58 It is evident from this recent study that the high-pressure-limiting rate coefficient for neopentane decomposition is well established over a wide-T range. Although there is no disagreement between the laser-schlieren data and the present H-ARAS data on the rate constants and the degree of observed falloff, the present results confirm the existence of a roaming pathway with a significant contribution leading directly to CH4 + C4H8 in neopentane decomposition, i.e., (2b). The conclusion that neopentane decomposition is not a simple one-channel dissociation but rather a two-channel dissociation (2a and 2b) could have some implications for its use as a thermal source for H-atoms. As with isobutane, the roaming fraction for each experiment in neopentane was calculated as BR2b = 1 − BR2a (Table 3). An Arrhenius-type fit to the roaming branching ratios, BR2b, in neopentane gives (1/BR2b) − 1 = 696 × exp( −7146 K/T )

ASSOCIATED CONTENT

S Supporting Information *

Tables S1 and S2 contain frequencies and rotational constants. This information is available free of charge via the Internet at http://pubs.acs.org

■

AUTHOR INFORMATION

Corresponding Author

*Phone: (630) 252-3171. Fax: (630) 252-9570. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, under Contract No. DE-AC0206CH11357 as part of the Argonne-Sandia Consortium on High-Pressure Combustion Chemistry; FWP# 2009 ANL 59044.

(E5)

Again, this function leads to limiting values for BR2b between 0 and 1 at extremely high and low temperatures, respectively. 5988



■

Article

(30) Michael, J. V.; Lim, K. P.; Kumaran, S. S.; Kiefer, J. H. J. Phys. Chem. 1993, 97, 1914−1919. (31) MOLPRO version 2010.1, a package of ab initio programs written by Werner, H.-J.; Knowles, P. J.; Knizia, G; Manby, F. R.; Schütz, M.; Celani, P.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.;et al. (32) Lutz, A. E.; Kee, R. J.; Miller, J. A. SAND87-8248 Revised, Sandia National Laboratory, November 1997. (33) Klippenstein, S. J. Advances in Theory of Combustion. Plenary Lecture, 7th US National Combustion Meeting, Atlanta, Georgia, March 23, 2011. (34) Harding, L. B.; Georgievskii, Y.; Klippenstein, S. J. J. Phys. Chem. A 2005, 109, 4646−4656. (35) Klippenstein, S. J.; Georgievskii, Y.; Harding, L. B. Phys. Chem. Chem. Phys. 2006, 8, 1133−1147. (36) Brooks, C. T. Trans. Faraday Soc. 1966, 62, 935−944. (37) Konar, R. S.; Marshall, R. M.; Purnell, J. H. Int. J. Chem. Kinet. 1973, 5, 1007−1021. (38) Golden, D. M.; Alfassi, Z. B.; Beedle, P. C. Int. J. Chem. Kinet. 1974, 6, 359−370. (39) Bradley, J. N. Proc. R. Soc. London A 1974, 337, 199−216. (40) Pratt, G. L.; Rogers, D. J. Chem. Soc., Faraday Trans. 1 1980, 76, 1694−1703. (41) Koike, T.; Morinaga, K. Bull. Chem. Soc. Jpn. 1982, 55, 690−695. (42) Hidaka, Y.; Fujiwara, M.; Oki, T.; Kawano, H. Chem. Phys. Lett. 1988, 144, 570−572. (43) Oehlschlaeger, M. A.; Davidson, D. F.; Hanson, R. K. J. Phys. Chem. A 2004, 108, 4247−4253. (44) Tsang, W. J. Phys. Chem. Ref. Data 1990, 19, 1−68. (45) Tsang, W. J. Chem. Phys. 1966, 44, 4283−4295. (46) Halstead, M. P.; Konar, R. S.; Leathard, D. A.; Marshall, R. M.; Purnell, J. H. Proc. R. Soc. London A 1969, 310, 525−539. (47) Taylor, J. E.; Hutchings, D. A.; Frech, K. J. J. Am. Chem. Soc. 1969, 91, 2215−2219. (48) Baronnet, F.; Dzierzynski, M.; Come, G. M.; Martin, R.; Niclause, M. Int. J. Chem. Kinet. 1971, 3, 197−213. (49) Pacey, P. D. Can. J. Chem. 1973, 51, 2415−2422. (50) Marshall, R. M.; Purnell, H.; Storey, P. D. J. Chem. Soc., Faraday Trans. 1 1976, 72, 85−92. (51) Bradley, J. N.; West, K. O. J. Chem. Soc., Faraday Trans. 1 1976, 72, 8−19. (52) Marquaire, P. M.; Come, G. M. React. Kinet. Catal. Lett. 1978, 9, 171−175. (53) Baldwin, A. C.; Lewis, K. E.; Golden, D. M. Int. J. Chem. Kinet. 1979, 11, 529−542. (54) Pratt, G. L.; Rogers, D. J. Chem. Soc., Faraday Trans. 1 1981, 77, 2751−2761. (55) Bernfeld, D.; Skinner, G. B. J. Phys. Chem. 1983, 87, 3732−3735. (56) Rao, V. S.; Skinner, G. B. Int. J. Chem. Kinet. 1988, 20, 165−175. (57) Mitchell, T. J.; Benson, S. W. Int. J. Chem. Kinet. 1993, 25, 931− 955. (58) Srinivasan, N. K.; Kiefer, J. H.; Tranter, R. S. J. Phys. Chem. A 2003, 107, 1532−1539. (59) Harding, L. B.; Georgievskii, Y.; Klippenstein, S. J. J. Phys. Chem. A 2010, 114, 765−777. (60) Kiefer, J. H.; Santhanam, S.; Srinivasan, N. K.; Tranter, R. S.; Klippenstein, S. J.; Oehlschlaeger, M. A. Proc. Combust. Inst. 2005, 30, 1129−1135. (61) Baulch, D. L.; Bowman, C. T.; Cobos, C. J.; Cox, R. A.; Just, Th.; Kerr, J. A.; Pilling, M. J.; Stocker, D.; Troe, J.; Tsang, W.; Walker, R. W.; Warnatz, J. J. Phys. Chem. Ref. Data 2005, 34, 757−1397. (62) Bencsura, A.; Knyazev, V. D.; Xing, X.-B.; Slagle, I. R.; Gutman, D. Proc. Combust. Inst. 1992, 24, 629−635. (63) Sivaramakrishnan, R.; Michael, J. V.; Ruscic, B. Int. J. Chem. Kinet. 2012, 44, 194−205. (64) Carstensen, H.-H.; Dean, A. M. J. Phys. Chem. A 2009, 113, 367−380.

REFERENCES

(1) Pitz, W. J.; Cernansky, N. P.; Dryer, F. L.; Egolfopoulos, F. N.; Farrell, J. T.; Friend, D. G.; Pitsch, H. SAE paper, SAE-2007-01-0175 2007. (2) Farrell, J. T.; Cernansky, N. P.; Dryer, F. L.; Friend, D. G.; Hergart, C.-A.; Law, C. K.; McDavid, R. M.; Mueller, C. J.; Patel, A. K.; Pitsch, H. SAE paper, SAE-2007-01-0201 2007. (3) Colket, M.; Edwards, T.; Williams, S.; Cernansky, N. P.; Miller, D. L.; Egolfopoulos, F.; Lindstedt, P.; Seshadri, K.; Dryer, F. L.; Law, C. K. ;et al. AIAA J. AIAA-2007-0770, 2007. (4) Manion, J. A.; Huie, R. E.; Levin, R. D.; Burgess, D. R., Jr.; Orkin, V. L.; Tsang, W.; McGivern, W. S.; Hudgens, J. W.; Knyazev, V. D.; Atkinson, D. B.; et al. NIST Chemical Kinetics Database; NIST Standard Reference Database 17; version 7.0 (web version), release 1.4; National Institute of Standards and Technology (NIST): Gaithersburg, MD, 2000; available at http://kinetics.nist.gov. (5) Sivaramakrishnan, R.; Su, M.-C.; Michael, J. V.; Klippenstein, S. J.; Harding, L. B.; Ruscic, B. J. Phys. Chem. A 2011, 115, 3366−3379. (6) Townsend, D.; Lahankar, S. A.; Lee, S. K.; Chambreau, S. D.; Suits, A. G.; Zhang, X.; Rheinecker, J.; Harding, L. B.; Bowman, J. M. Science 2004, 306, 1158−1161. (7) Houston, P. L.; Kable, S. H. Proc. Natl. Acad. Sci. 2006, 103, 16079−16082. (8) Marcy, T. P.; Diaz, R. R.; Heard, D.; Leone, S. R.; Harding, L. B.; Klippenstein, S. J. J. Phys. Chem. A 2001, 105, 8361−8369. (9) Klippenstein, S. J.; Miller, J. A.; Harding, L. B. Proc. Combust. Inst. 2002, 29, 1209−1217. (10) Harding, L. B.; Klippenstein, S. J. J. Phys. Chem. Lett. 2010, 1, 3016−3020. (11) Klippenstein, S. J.; Georgievskii, Y.; Harding, L. B. J. Phys. Chem. A 2011, 115, 14370−14381. (12) Michael, J. V. Prog. Energy Combust. Sci. 1992, 18, 327−347. (13) Michael, J. V. In Advances in Chemical Kinetics and Dynamics; Barker, J. R., Ed.; JAI: Greenwich, CT, Vol. I, pp 47−112, for original references, 1992. (14) Michael, J. V.; Lifshitz, A. In Handbook of Shock Waves; Ben-Dor, G., Igra, O., Elperin, T., Lifshitz, A., Eds.; Academic Press: New York, 2001; pp 77−105; Vol. 3. (15) Birrell, R. N.; Trotman-Dickenson, A. F. J. Chem. Soc. 1960, 4218−4224. (16) Michael, J. V.; Sutherland, J. W. Int. J. Chem. Kinet. 1986, 18, 409−436. (17) Michael, J. V. J. Chem. Phys. 1989, 90, 189−198. (18) Michael, J. V.; Fisher, J. R. In Seventeenth International Symposium on Shock Waves and Shock Tubes; Kim, Y. W., Ed.; AIP Conference Proceedings 208; American Institute of Physics: New York, 1990; pp 210−215. (19) Lim, K. P.; Michael, J. V. Proc. Combust. Inst. 1994, 25, 713−719 and references therein. (20) Kumaran, S. S.; Su, M.-C.; Lim, K. P.; Michael, J. V. Proc. Combust. Inst. 1996, 26, 605−611. (21) Maki, R. G.; Michael, J. V.; Sutherland, J. W. J. Phys. Chem. 1985, 89, 4815−4821. (22) Mielke, S. L.; Peterson, K. A.; Schwenke, D. W.; Garrett, B. C.; Truhlar, D. G.; Michael, J. V.; Su, M.-C.; Sutherland, J. W. Phys. Rev. Lett. 2003, 91, 063201−4. (23) Michael, J. V.; Su, M.-C.; Sutherland, J. W.; Harding, L. B.; Wagner, A. F. Proc. Combust. Inst. 2005, 30, 965−973. (24) Mitchell, A. C. G.; Zemansky, M. W. Resonance Radiation and Excited States; Cambridge University Press: Cambridge, England, 1934. (25) Bethe, H.; Salpeter, E. E. Quantum Mechanics of One- and TwoElectron Atoms; Plenum: New York, 1977. (26) Michael, J. V.; Kumaran, S. S.; Su, M.-C. J. Phys. Chem. A 1999, 103, 5942−5948. (27) Kumaran, S. S.; Lim, K. P.; Michael, J. V. J. Chem. Phys. 1994, 101, 9487−9498. (28) Kumaran, S. S.; Su, M.-C.; Lim, K. P.; Michael, J. V. Chem. Phys. Lett. 1995, 243, 59−63. (29) Michael, J. V.; Lim, K. P. J. Chem. Phys. 1992, 97, 3228−3234. 5989


Shock Tube Explorations of Roaming Radical Mechanisms: The

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