Proceedings

Proceedings of the Combustion Institute 30 (2005) 1049–1056

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of the

CombustionInstitute

A theoretical and experimental kinetic studyof phenyl radical addition to butadiene

Huzeifa Ismaila, J. Parkb, Bryan M. Wonga,William H. Green Jr.a,*, M.C. Linb

a Department of Chemical Engineering, Massachusetts Institute of Technology, 25 Ames Street,

Cambridge, MA 02139, USAb Department of Chemistry, Emory University, 1515 Dickey Drive, Atlanta, GA 30320, USA

Abstract

The reactions of phenyl radical (C6H5) are of growing technological and scientific interest. A betterunderstanding of phenyl radical addition to unsaturated hydrocarbons is of great practical interest becauseit is believed to be an essential component in both soot and fullerene formation. In this study, the rate ofphenyl radical addition to butadiene was measured, and the potential surface of the reaction C6H5 + C4H6

was explored using quantum chemistry with the B3LYP density functional. Vibrational analysis allowedthe determination of thermodynamic data and deduction of high-pressure-limit rate constants via transi-tion-state theory. The pressure and temperature dependences of this chemically activated reaction werecomputed using a weak collision master equation analysis. The comparison of the predictions for theC6H5 + C4H6 system with experimental data showed good agreement. The rate constant for disappear-ance of phenyl radical was found to be (3.16 ± 0.29) · 1012 cm3/mol-s exp [�(870 ± 30)/T] over thetemperature range 298–450 K. The dominant product at low temperature is the initial adduct, 4-phenyl-buten-3-yl. Around 1000 K, the dominant product is phenyl butadiene formed from the chemicallyactivated adduct, even at 10 atm. Above about 1400 K, bimolecular H-abstraction to form benzene isthe most important process. Other products such as 1,4-dihydronaphthalene are much less important.� 2004 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Master equation; 1,4-Dihydronaphthalene; Phenyl; 1,3-Butadiene

1. Introduction

A fundamental understanding of soot andpolycyclic aromatic hydrocarbon (PAH) forma-tion is of great practical importance. For example,PAHs such as benzo[a]pyrene have been impli-cated in the development of lung cancer [1]. Fur-thermore, the formation of soot is an importantcomponent in the chemistry of combustion [2].

1540-7489/$ - see front matter � 2004 The Combustion Institdoi:10.1016/j.proci.2004.08.127

* Corresponding author. Fax: +1 617 324 0066.E-mail address: whgreen@mit.edu (W.H. Green).

In particular, phenyl radical (C6H5) is believedto be a most important reactive precursor to sootformation in hydrocarbon combustion reactions.A quantitative understanding of the formationof larger PAH molecules, leading ultimately tosoot particles, is essential for a better design ofefficient and clean practical combustion devicessuch as engines or incinerators [3]. Extensiveresearch over the last decades has addressed path-ways leading to PAH of increasing size and ulti-mately soot has not included reaction of phenylwith 1,3-butadiene [4]. In particular, the simple

ute. Published by Elsevier Inc. All rights reserved.

1050 H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056

addition of phenyl radical to butadiene has notbeen reported to date in the literature.

Herein, we investigate the addition of phenylradical to butadiene using an experimental andcomputational approach. The reaction of C6H5

with C4H6 can initially form several different prod-ucts including C6H6 + CH2CHCHCH, C6H6 +CH2CHCCH2, C6H5CH2CHCHCH2, and C6H5

CH(CH2)CHCH2. The latter two adducts will behighly energized and might undergo several rapidunimolecular reactions, forming a wide variety ofpossible products before being collisionallyquenched. Such unimolecular reactions may sig-nificantly contribute to the formation of naphtha-lene and in a similar manner to larger PAH, likelyprecursors of carbonaceous material such as soot[4]. However, neither the overall reaction rate be-tween phenyl radical and butadiene nor the prod-uct branching ratios are known. In kineticmodeling of such large molecules, it is often as-sumed the reaction is in the high-pressure limit;that is, only the four channels listed above havesignificant yields. However, recent work has dem-onstrated that rather large molecules are indeed in-volved in chemically activated, pressure-dependentreactions, particularly at high temperatures [5]. Inthis work, the rate constants for important reac-tions in the C6H5 + C4H6 system are calculatedas an explicit function of the temperatures andpressures of interest.

The predictions of a theoretical model shouldbe consistent with that of the experimental obser-vations to which the model is applied. Therefore,a comparison of computed kinetic models withexperimental data is absolutely needed. Here, thecavity-ringdown spectroscopic (CRDS) methodis used to directly measure the total loss rate ofphenyl radical due to reaction with butadiene.This work concludes with a comparison of theo-retical rate constants against experimental data.

2. Experimental methodology

The experiment was carried out using the cav-ity-ringdown spectrometric technique at the totalpressure of 40 Torr. Detailed descriptions of theCRDS technique for kinetic applications havebeen reported in our earlier publications [6,7],and only a brief description will be given here.

All experiments were performed under slow-flow conditions using Ar as the carrier gas in atemperature-controlled reaction cell of about570 cm3 wrapped with resistive heaters. The reac-tor temperature, maintained within ±0.7 K, wasmeasured with a calibrated K-type thermocouple(Alumel–Chromel) located below the center ofthe probe region. A Pyrex glass flow reactor withtwo pairs of laser windows attached opposite eachother, permitting two-split photolysis laser beamsto cross at the center of the reactor at a 30� angle,

was vacuum-sealed at the ends with a pair ofhighly reflective mirrors (R = 0.9999 at 500 nm,radius of curvature 6 m). This high-quality opticalcavity approximately 50 cm in length can lengthena pulse of probing dye laser operating at 500 nmwith FWHM �10 ns to about 30 ls.

Two pulsed lasers were employed, one for thegeneration of the C6H5 radical and the other forits detection. For radical generation from nitroso-benzene (C6H5NO) as the precursor, we employeda Lambda Physik KrF excimer laser (EMG 102)at 248 nm with an unfocused beam and 30–50 mJin each�50 ns pulse. For probing theC6H5 radical,aLambdaPhysik excimer laser (EMG201)pumpedtunable dye laser (FL 3002) was used. As reportedby Porter and Ward [8], two strong absorptionpeaks were observed at 504.8 and 505.1 nm, andwe utilized the stronger absorption at 504.8 nm(2B1 ‹ 2A1) for all of our kinetic studies. The504.8 nmprobe laser light was injected directly intothe reactor cavity through one of the mirrors alongthe axis of the reactor tube.A fractionof the photonpulse transmitted through the secondmirrorwasfil-tered and detected with a Hamamatsu photomulti-plier tube (PMT). The photoelectric signal fromthe PMT was amplified with a fast preamplifier(SR445), and acquired and averaged with a multi-channel digital oscilloscope (LeCroy 140). Theaveraged signal was stored in a computer for futuredata analysis.Apulse-delay generator (SRDG535)interfaced with the computer using LabView soft-ware was employed to control the firing of the twolasers aswell as the triggeringof thedataacquisitionsystem.Tomeasure kinetics, the timedelaybetweenthe two lasers was varied.

The concentration of each individual moleculewas obtained by the following formula: [R] =9.66 · 1016 (%) PFR/TFT molecules/cm3, where% is the percentage of each molecule in its gasmixture, P is the total reaction pressure in Torr,T is the reaction temperature, FR is the flow rateof each gas mixture, and FT is the total flowrateof all gases. The typical resident time in the prob-ing zone was 10–40 ms, and the repetition rate of2 Hz allowed enough time for a fresh sample inthe reactor between pulses. The flowrates weremeasured by using mass flowmeters (MKS,0258C), and the gas pressure was measured withan MKS Baratron manometer. The initial concen-tration of C6H5NO was determined by UV/VISspectrometry (Shimadzu, UV-2401 PC) down-stream of the reaction cell using standard calibra-tion mixtures. Its concentrations ranged within0.5–5.0 · 1013 molecules/cm3.

Butadiene (Aldrich, 99%) was purified by trap-to-trap distillation using appropriate slush bathsbefore use. Nitrosobenzene (Aldrich, 97%) waspurified by re-crystallization with ethanol at icetemperature, filtered, and dried in vacuum. Ar(Specialty Gases, 99.995% UHP grade) was usedas the carrier gas without further purification.

H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056 1051

3. Theoretical methodology

3.1. Quantum chemistry

The potential energy surface was explored, andcomputations on the reactants, reaction interme-diates, products, and the corresponding transitionstates were performed using GAUSSIAN98 [9].Calculations on reactant radicals and open shelltransition states were done within the unrestrictedformalism. Harmonic vibrational frequencies andoptimized geometries were determined using theBecke [10] three-parameter hybrid functionalcombined with the Lee, Yang, and Parr [11] corre-lation (B3LYP) density functional theory methodwith the 6-31G(d,p) basis set. Although it isknown in the literature that B3LYP underesti-mates barrier heights in general, the underestima-tion is not usually very large [12]. B3LYP has beenrepeatedly demonstrated to give rather accurateresults for hydrocarbon structures, vibrationalfrequencies, and thermochemistry [13].

The standard rigid-rotor, harmonic-oscillatorapproximation was used to estimate the partitionfunctions needed to compute the thermochemistryand rate constants, except many of the torsionalmodes were treated as large-amplitude rotorsrather than small-amplitude harmonic vibrations.The low-frequency torsions about single bondswere treated as free internal rotors. Torsions withvibrational frequencies greater than 300 cm�1

were treated as harmonic oscillators (with statisti-cal corrections for the number of low-energy con-formations). A few of the higher-frequencytorsional vibrations were treated as hinderedinternal rotors. The computed B3LYP harmonicvibrational frequencies were scaled by 0.9613 asrecommended in the literature [14].

3.2. Rate constant calculations

The high-pressure-limit rate constants werecalculated from conventional (fixed geometry)canonical transition-state theory:

k ¼ ajðT Þ kBTh

� �ðQy=V ÞQN reactants

i¼1 ðQi=V Þe�E0=RT ; ð1Þ

where a is the reaction path degeneracy, j (T) isthe Wigner tunneling correction factor corre-sponding to the magnitude of the imaginary fre-quency at the activated complex, T is thetemperature, and E0 is the zero-point-correctedbarrier height. Qi/V denotes the total partitionfunction per unit volume of the ith reactant atthe zero-point energy of the motionless reactant.Q�/V is the corresponding partition function ofthe activated complex. The a was computed usingsymmetry numbers of the reactants and transitionstates such that it satisfied detailed balance as sug-gested by Pollak and Pechukas [15]. The momentsof inertia for all internal rotors were calculated

using the m = 2, n = 3, Pitzer–Gwinn formulation[16,17]. The resulting high-pressure-limit rate con-stants k1 (T) were fitted to the expressionk1 (T) = AT nexp(�Ea/RT) over a temperaturerange from 300 to 1800 K. The DFT energiesadded to the zero-point vibrational energy andthe temperature-dependent enthalpy correctionwere used to determine reaction enthalpies ofisodesmic reactions [18]; i.e., reactions thatmaintain the overall number and types of bonds.With a careful choice of isodesmic reactions, heatsof formation can be evaluated to accuraciesapproaching uncertainties in experimental values.The thermodynamic properties of all speciesconsidered in the present work are summarizedin Table 1 with available experimental values forphenyl and butadiene. Heats of formation werecalculated through isodesmic reactions usingmethane, methyl, ethane, ethyl, ethylene, and vi-nyl as the reference compounds.

Microcanonical RRKM rate constants fromstatistical theory were calculated with the Chem-Rate v.1.9 program available from NIST [20].Chemical activation, isomerization, and decompo-sition were accounted for in the program throughthe solution of the time-dependent master equation

dqai

dt¼x

Xj

P aijq

aj �x

Xj

P ajiq

ai �kai q

ai þ

Xb

kbl qbl þRi;

ð2Þwhere qa

i is the molecular population of isomer aat energy level Ei, x is the collision frequency,P aij is the collisional energy transfer probability

from energy level j to energy level i of isomer a,kai is the sum of rates for all the decompositionand isomerization channels from energy level Ei,kbl is the rate of isomerization from isomer b at en-ergy El to isomer a at energy level Ei, and Ri is thechemical activation flux, which is assumed to beconstant. Index l for isomer b corresponds tothe same energy as i for a, i.e. El=Ei.

The first step in the main reaction sequencewas the formation of the chemically activatedmolecule 4-phenyl-buten-3-yl from phenyl andbutadiene. Both of the reactants were assumedto have a canonical distribution, i.e., a Boltzmanndistribution. Energy transfer was modeled byweak molecular collisions with energy transferprobabilities given by the exponential-down mod-el [21] with ÆDEædown = 400 cm�1. The Lennard-Jones parameters used were [r (Ar) = 3.54 A,e/kB(Ar) = 93.3 K] [22] for Ar and r = 5.64 A,e/kB = 450.0 K for all the isomers [23]. The masterequation can be written in the matrix form

dqdt

¼ Bqþ R ð3Þ

and its solution is a function of the eigenvalues andeigenvectors of the relaxation matrix B [24].The energy bin size used was dE = 100 cm�1. The

Table 1Ideal gas thermochemical parameters of selected molecules and radicals relevant to this study computed at the B3LYP6-31G(d,p) level of theory

Species Formula DfH00

(kJ/mol)S0298(J mol�1 K �1)

DfH0298

(kJ/mol)DfH 0

298 expt(kJ/mol)

Hydrogen H 216.23 114.72 217.99a 217.99 [19]

1,3-Butadien-2-yl 308.90 292.14 299.07 —i-C4H4

1,3-Butadien-1-yl 372.48 289.44 362.16 —n-C4H4

1,3-Butadiene C4H6 121.48 284.03 107.00a 108.78 [19]

Phenyl C6H5 336.89 289.98 323.76a 337.23 [19]

1-Phenyl-1,3-butadiene 211.76 413.34 185.39 —

C10H10

1,4-Dihydro-naphthalene 171.86 364.43 142.55 —

C10H10

4-Phenyl-buten-3-yl 270.42 438.25 239.45 —

C10H11

3-Phenyl-buten-4-yl 356.61 438.67 326.69 —

C10H11

1,4,9-Trihydro-naphthalene 316.03 376.52 283.30 —

C10H11

a Other values have been reported in the literature, but these values are used on the potential energy surface for consistency.

1 With the ‘‘subtract incomes’’ option, the branchingratios need to be manually calculated in ChemRateversion 1.9; the numbers reported in the user interfaceare confusing. This bug will be corrected in futureversions of ChemRate.

1052 H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056

default for Emin was 64.63 kcal, and Emax was setat 370.00 kcal. The resulting matrix size of2098 · 2098 was tested to ensure convergence.

AsdiscussedbyKnyazevandTsang [25], the rateconstants inferred from the solution of the masterequationvarywith time.However, time ranges existwhere the computed rate constants are indeed con-stant, and if these timescales are used, the rate con-stants computed by solving the time-dependentequations are (at least in this case) very similar tothe values obtained from steady-state solutions. Inthe present case, solutions of the time-dependentequations were examined in the microsecond-to-millisecond time range, where the rate constantswere truly constant. At shorter timescales, the re-sults are convolved with vibrational relaxation; atlonger timescales, thermal dissociation of the stabi-lized adducts confuses the interpretation.

The decomposition reactions were treated asirreversible (‘‘infinite sink’’ approximation). Theisomerization reactions were defined as reversible.To correct for reversibility, the ‘‘subtract in-

comes’’ option in ChemRate v.1.9 was used to ob-tain effective (net) isomerization rate constantsfrom the total isomerization fluxes in the forwardand reverse directions. The use of the net isomer-ization rate constants simplifies the analysis ofproduct branching, because any reversible isomer-ization can be treated as an ‘‘irreversible’’ processwith an effective rate constant, which is alreadycorrected for the reversibility. Then for a kineticmodel of arbitrary complexity consisting of chem-ical activation followed by a sequence of effec-tively irreversible reactions, the apparent rateconstants are calculated as the products of the to-tal (chemical activation) rate constant and appro-priate branching fractions at each well.1

Fig. 2. Typical k 0 vs [C4H6] plot.

H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056 1053

4. Results and discussion

4.1. Experimental results

For kinetic measurements, the CRDS methodprovides the decay times of the injected probingphotons in the absence (t0c) and the presence (tc)of absorbing species. The typical photon decaycurves are shown in the inset of Fig. 1. These pho-ton decay times can be related to the concentra-tion of the species at time t after its generationby the equation

1=tc ¼ 1=t0c þ B½A�0e�kt ð4Þor

ln 1=tc � 1=t0c� �

¼ C � kt; ð5Þ

where [A]0 is the initial concentration of the radi-cal species of interest, C6H5, B is a constant thatcontains experimental parameters such as the cav-ity length (50 cm), the refractive index of theabsorbing medium, etc., and C = ln(B[A]0). Theinitial concentration of phenyl was estimated tobe in the range of 1–5 · 1012 radicals/cm3. Eq.(5) is valid, provided the decay time of the speciesof interest is much longer than that of photonswithin the cavity. This condition can be readilymet because the chemical decay time, typically inthe range of several hundreds of microsecondsto tens of milliseconds, can be controlled by theconcentration of the molecular reagent.

The slopes of the lnð1=tc � 1=t0cÞ vs t plots, asillustrated in Fig. 1 for the reactions of C6H5 withbutadiene, yield the pseudo-first-order rate coeffi-cients, k 0, for the decay of C6H5 in the presence ofknown excess 1,3-C4H6 concentrations as speci-fied. A standard plot of k 0 vs reagent concentra-tion is shown in Fig. 2, which gives the averaged

Fig. 1. Typical pseudo-first-order decay plots for thereaction of C6H5 with C4H6 at 301 K. The inset is atypical photon decay curves with (top) and without(below) absorbing species. The bold curves are expo-nentially fitted curves.

second-order rate constant k00 from its slopeaccording to the relationship

k0 ¼ k0 þ k00x ½X�; ð6Þwhere k0 is the radical decay constant in the ab-sence of the molecular reactant X; this is thoughtto be due to the loss of the radical by diffusionaway from the probing beam and from recombi-nation reactions (e.g., C6H5 + NO andC6H5 + C6H5).

The bimolecular rate constants determinedfrom the slopes of k 0 vs concentration plots are gi-ven in Fig. 3 and Table 2. Weighted least-squaresanalysis of the individual set of reaction rate con-stants gave rise to the following expression, inunits of cm3/(mol s):

k ¼ ð3:16� 0:29Þ � 1012 exp½�ð870� 30Þ=T �:ð7Þ

Fig. 3. Arrhenius plots of the overall rate constant forthe C6H5 + C4H6 reaction, from theory (n) and exper-iment (s).

1054 H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056

4.2. Theoretical results

The reaction pathways studied are shown inFig. 4. This network is of course not complete;the large number of isomers and an even largernumber of reactions interconnecting these iso-mers make a complete investigation of the poten-tial energy surface impractical. Therefore, onlythe most probable reaction pathways wereconsidered.

The addition of phenyl to the terminal carbonsof butadiene, via TS1, is the most important reac-tion path. According to our quantum calculations,this path is barrierless, but there is a significant T-dependence of the rate due to the difference betweenthe T-dependence of the partition function of theTS and the partition function of the reactants (pri-marily due to the T-dependence of the internalrotor partition functions). The theoreticalT-depen-dence is slightly weaker than that observed experi-mentally, suggesting that either there is a smallbarrier (�2 ± �5 kJ/mol). The error is due toaccumulation of uncertainties in our calculations.

The internal rotation of butadiene was treatedas a double minimum hindered rotor, where thebarrier between the trans to cis isomerization wasfound to be 27.41 kJ/mol. Species 3� is internallyexcited, and can potentially undergo several isom-erization and decomposition reactions before being

Table 2Measured bimolecular rate constant of C6H5 reactionwith butadiene (C4H6)

T (K) [C4H6] (Torr) k/1011 (cm3/mol s)a

298 0–0.073 1.74 ± 0.14301 0–0.700 1.75 ± 0.12313 0–0.077 1.96 ± 0.22331 0–0.070 2.16 ± 0.18351 0–0.064 2.53 ± 0.54363 0–0.054 2.96 ± 0.36381 0–0.074 3.37 ± 0.66400 0–0.072 3.58 ± 0.18

a All experiments were performed at 40 Torr using Arcarrier gas. The uncertainties represent 1r, evaluatedwith weighted least-squares analyses by convoluting theerrors in k 0 for k00.

Fig. 4. Network of reaction pathways.

stabilized. Species 3� was treated as a double mini-mumhindered rotorwith a high barrier of 64.31 kJ/mol because of the partial double bond. This is theinput channel for the main chemically activatedreaction network. The H-abstraction pathway pro-ducing benzene and 8, via TS5, is not competitiveuntil quite high temperatures since its barrier is�8 kJ/mol higher than the near-zero barrier forthe addition to the terminal carbons. Addition tothe interior carbons on butadiene forming 10�, viaTS7, was considered but it is even less significantsince its barrier is higher (17 kJ/mol). Similarly,H-abstraction from the terminal carbons to pro-duce benzene and 9 has a much higher barrier(9 kJ/mol) than H-abstraction from the interiorcarbons; therefore, it is not significant except at ex-tremely high temperatures.

The most important portions of the potentialenergy surface are shown schematically in Fig. 5.The reaction channel through TS1 is very exo-thermic due to the thermodynamically favorableresonance-stabilized product. Therefore, in accor-dance with Hammond�s postulate, the transition-state structure is similar to that of the separatedreactants. The energized adduct 3� can lose ahydrogen atom, via TS3, to form 1-phenyl-1,3-bu-tadiene (5). Alternatively, the energized adductcan cyclize to form the bicyclic radical 1,4,9-trihy-dronaphthalene (4) via TS2. The bicyclic radicalthen rapidly loses hydrogen to form 1,4-dihydro-naphthalene (6).

The computed thermodynamic properties of allthe species considered in the present work are gi-ven in Table 1. The molecular geometries andvibrational frequencies are available from theauthors upon request. The high-pressure-limit rateconstants corresponding to the transition stateslisted in Fig. 4 are shown in Table 3. At pressuresof 0.01 and 10 atm, the computed apparent rate

Fig. 5. Potential energy diagram for the chemicallyactivated C6H5 + C4H6 reaction. ZPE-corrected energies(kJ/mol) at 0 K, relative to reaction 1 + 2, are calculatedby B3LYP 6-31G(d,p) basis.

Table 3Transition-state theory rate constantsa for elementary reactions included in Fig. 4

Reaction Log10 (A) n Ea (kJ/mol)

1 + 2fi 3 5.91 2.56 0b

1 + 2fi 10 4.68 2.65 16.701 + 2fi 7 + 8 3.94 3.12 8.101 + 2fi 7 + 9 4.50 3.11 16.703fi 4 5.51 1.64 110.613fi 5 + H 7.82 2.11 161.624fi 6 + H 10.14 1.25 92.60

a Fitted to the modified Arrhenius form k = ATnexp(�Ea/RT). A is in mol, cm3, s units.b The slope in Fig. 3 is primarily due to the Tn dependence of the partition functions. Combining the experimental

T-dependence with the theoretical partition functions suggests Ea � 2 ± 5 kJ/mol.

Fig. 6. Computed rates for the reaction C6H5 + C4H6 toform various products (A) P = 7.6 Torr, (BP = 7600 Torr. The thermalized initial adduct 1 is themain product at low temperature, phenyl butadiene ithe main product around 1000 K, and H-abstraction toform benzene becomes dominant at high temperaturesRing closure is much slower than the main reactionchannels. The keff is the effective rate constant for theformation of all the products from the initial adductThe line labeled with 3-phenyl-buten-4-yl represents anupper bound on the sum of all the minor productswhich can be formed through that adduct, see text.

H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056 1055

constants for the formation of the stabilized ad-duct (3) and of several isomers are shown in Figs.6A and B. The experimental and computed A-fac-tors are in excellent agreement where the formerequals 3.31 · 1012 cm3/mol s and the latter equals2.72 · 1012 cm3/mol s. The computed overall ratefor loss of phenyl at 0.052 atm is compared withexperiment in Fig. 3. The computed overall reac-tion rate is higher than the experimental data byabout 50% at room temperature, with a slightlysmaller deviation at higher temperatures. This isan excellent agreement considering computationaluncertainties for this barrierless reaction, whichwas modeled rather simply using conventional(rather than variational) transition-state theory.The deviation may stem primarily from the under-estimation of the addition barrier; an increase of0.5 kcal/mol, for example, would result in a quan-titative agreement between experiment andtheory.

The master equation analysis shows the initialadduct (3) is the dominant product at low temper-atures, and this reaction is in its high-pressurelimit even at the low-pressure experimental condi-tions of 0.052 atm. According to the calculations,the rate of this reaction begins to fall off signifi-cantly above T = 580 K at 0.052 atm, and evenat 10 atm the reaction is far from the high-pres-sure limit above 1000 K. This is consistent withpublished analyses of other chemically activatedreactions of molecules in this size range [3].

The calculations show that 1-phenyl-1,3-buta-diene (5) becomes the dominant product when 3begins to fall off, but that at higher temperatures(T > 1400 K) the channels with significant en-trance barriers, particularly H-abstraction fromthe interior carbons, become even more impor-tant. The bicyclic species 1,4-dihydronaphthalene(6), an isomer of 1-phenyl-1,3-butadiene, is pre-dicted to be most important in the temperaturerange around 1000 K, where 1-phenyl-1,3-butadi-ene is the dominant channel; however, the chem-ically activated reactions forming the bicyclicspecies are always more than an order of magni-tude slower than the main channels. It seemslikely there are other more important reaction

)

s

.

.

,

paths to bicyclics even at 1000 K, possiblyincluding subsequent bimolecular reactions ofthe major product thermalized 1-phenyl-1,3-butadiene.

1056 H. Ismail et al. / Proceedings of the Combustion Institute 30 (2005) 1049–1056

5. Conclusions

The overall rate of phenyl addition to butadi-ene was measured directly with laser flash photol-ysis, using cavity-ringdown spectroscopy to probethe phenyl radical. In 40 Torr of Ar, over the tem-perature range 298–450 K, the rate constant is

k ¼ ð3:16� 0:29Þ� 1012 cm3=mol s exp½�ð870� 30Þ=T �: ð8Þ

According to our calculations, this reaction is inthe high-pressure limit under the experimentalconditions, and the dominant product under theseconditions is the adduct 4-phenyl-buten-3-yl.

The rate constant and product branching ra-tios for C6H5 + C4H6 under other conditions werepredicted based on ab initio calculations. The po-tential energy surface was explored using B3LYPcalculations. Transition states were determined,and rate constants were obtained by means oftransition-state theory. Apparent rate constantsfor the various product channels were obtainedfrom a master equation analysis at different pres-sures. The computed apparent rate constant forthe disappearance of phenyl was within 50% ofthe experimental values. The master equationanalysis showed the formation of the stabilizedadduct is in its high-pressure limit for tempera-tures lower than 600 K even at P = 0.01 atm.Chemically activated product channels dominatearound 1000 K, even at 10 atm, but simple H-ab-straction to form benzene dominates above1400 K. The bicyclic species 1,4,9-trihydronaph-thalene and 1,4-dihydronaphthalene are minorproducts under all reaction conditions.

Acknowledgments

J.P. and M.C.L. acknowledge support by theBasic Energy Sciences, Department of Energy, un-der Grant No. DE-FG-97ER14784 to Emory Uni-versity. H. Ismail, B.M. Wong, and W.H. Greenacknowledge the support Office of Basic EnergySciences, Office of Energy Research, U.S., DOEGrant No. DE-FG02-98ER14914 and NSF underGrant CTS-0123345. The authors are grateful toJoanna Yu for providing the isodesmic reactionsemployed in this study.We also gratefully acknowl-edge Dr. Igor Tokmakov for several helpfulconversations.

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