Rate Constant for the Reaction C2H5 + HBr ... - ACS Publications

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Rate Constant for the Reaction C2H5 + HBr → C2H6 + Br David M. Golden,*,† Jingping Peng,‡ A. Goumri,‡ J. Yuan,‡ and Paul Marshall*,‡ †

Department of Mechanical Engineering, Stanford University, Stanford, California 94305, United States Department of Chemistry and Center for Advanced Scientific Computing and Modeling (CASCaM), University of North Texas, Denton, Texas 76203, United States

‡

S Supporting Information *

ABSTRACT: RRKM theory has been employed to analyze the kinetics of the title reaction, in particular, the once-controversial negative activation energy. Stationary points along the reaction coordinate were characterized with coupled cluster theory combined with basis set extrapolation to the complete basis set limit. A shallow minimum, bound by 9.7 kJ mol−1 relative to C2H5 + HBr, was located, with a very small energy barrier to dissociation to Br + C2H6. The transition state is tight compared to the adduct. The influence of vibrational anharmonicity on the kinetics and thermochemistry of the title reaction were explored quantitatively. With adjustment of the adduct binding energy by ∼4 kJ mol−1, the computed rate constants may be brought into agreement with most experimental data in the literature, including new room-temperature results described here. There are indications that at temperatures above those studied experimentally, the activation energy may switch from negative to positive.

■

INTRODUCTION In the past, heats of formation of several radicals R had been determined by measuring the temperature dependence of the reactions1 (1) RH + X → R + HX or RX + X → R + X2 (2)

was required to accommodate the measured negative activation energy. Notable prior theoretical work on the title reaction C2H5 + HBr → C2H6 + Br

was carried out by Chen and Tschuikow-Roux, who used MP4/ 6-31G(d) energies at MP2/6-31G(d) geometries for stationary points.14 They concluded that a loosely bound intermediate complex bound through the H atom of HBr caused the negative activation energy, and they calculated rate constants k3 using transition-state theory (TST) and RRKM theory. Their results were intermediate between the higher and lower measured values. Recently, Sheng et al.15 have presented a calculated potential surface for reaction 3. They found that there is a weakly bound (by about 13 kJ mol−1) complex between the reactants, followed by a transition state to products that is much lower in entropy (tighter) than, but at essentially the same energy as, the complex. They used ICVT (improved canonical variational transition-state theory) to explain the rate constants. Their calculated rate constants exhibit the negative activation energy found in many recent studies, but their suggested fitting parameters would seem to have a typographical error. We have also performed quantum calculations on this surface, with results not too different from those of Sheng et al. (It might be noted that one of the senior authors (P.M.) sent the other (D.M.G.) these essential results over 10 years ago, but we did not have the wherewithal to perform the chemical activation calculations that we discuss herein. A poster was presented at the 16th International Symposium on Gas Kinetics in 2000.)

where X is I or Br. It was thought that the rate constant for the reverse of reaction 1 (k−1) could be characterized by activation energies (E−1) of 4−8 kJ mol−1 for X = I and Br, and that for the reverse of reaction 2 could be characterized by an activation energy (E−2) of 0−4 kJ mol−1. In addition, in many of the studies of these first two reactions, it was often possible to characterize the ratio of the rate constants for the reverse reactions, and it was usually the case that the difference between E−1 and E−2 was in the range of 4−8 kJ mol−1. All of this notwithstanding, it has become apparent from several direct studies2−5 of the reverse of reaction 1 that for many species R, E−1 can be negative. (There is a study6 that indicates a positive activation energy for R = C2H5 and X = Br, as well as studies7,8 with R = t-butyl and X = Br and I and R = CH2Br and CHBrCl with HBr9 that report positive activation energies.) This negative activation energy, together with the values of E1, has led to values of the heats of formation of several simple radicals that are higher than first thought and that are in accord with modern quantum chemistry calculations.10 Understanding of the reasons for the negative activation energy of the reverse of reaction 1 has been somewhat elusive. Some discussions for the case where R = CH3 have been presented.11,12 McEwen and Golden attempted an explanation for the case of t-butyl + HI by positing a complex between t-butyl and HI at the I atom.13 Even then, a quite deep complex © 2012 American Chemical Society

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Special Issue: A. R. Ravishankara Festschrift Received: September 20, 2011 Revised: January 17, 2012 Published: January 23, 2012 5847

dx.doi.org/10.1021/jp209081v | J. Phys. Chem. A 2012, 116, 5847−5855

The Journal of Physical Chemistry A

Article

of pressure. A calcium fluoride window blocked any Lyman-α radiation from the lamp. Experiments were conducted in a stainless steel reactor, initially passivated with a flow of HBr. As described previously,21 vacuum UV photometry was used to confirm that HBr was not lost significantly by surface adsorption in the reactor. Following the pulsed generation of C2H5 radicals, [Br] is influenced by reaction 3 and

Newly recomputed ab initio data are employed in RRKM/ master equation calculations in order to interpret the negative activation energy quantitatively, as well as the magnitude of the rate constant k3. The value of k3 has in the past been somewhat controversial. The work of Gutman and co-workers based on laser flash photolysis/ time-resolved photoionization mass spectrometric detection (LFPPIMS) of C2H5 indicated a fast reaction, with k3 around 8 × 10−12 cm3 molecule−1 s−1, and a then-unexpected negative activation energy.2,16 Repeated experiments with the same apparatus by Seetula gave the same results.3 Laser flash photolysis experiments by Wine and co-workers4 and Ferrell,5 who employed time-resolved resonance fluorescence spectroscopic detection (LFP-RF) of atomic Br, also yielded similar results (see Table 1), but the very low

k/10−12 cm3 molecule−1 s−1

ref

LFP-RF LFP-RF LFP-PIMS VLPR LFP-PIMS FP-RF theory theory

8.69 ± 0.33, 7.81 ± 0.51, 8.24 ± 0.65 8.30 ± 0.20 9.39 ± 0.50, 8.20 ± 1.07 0.667 ± 0.014 8.47 ± 0.29, 8.59 ± 0.38 7.0 ± 0.8 2.28, 1.62 2.7

4 5 2 6 3 this work 14 15

Br → loss by diffusion

(5)

[Br] = A exp(−k5t ) − B exp( −k ps1t )

(6)

where kps1 is the pseudo-first-order coefficient k3[HBr] + k4. For each [HBr] employed, the four parameters of eq 6 were obtained in two steps. k5, which describes the slow loss of Br mainly through diffusion to the reactor walls, and the constant background arising from scattered resonance light were obtained by fitting to data obtained with a long time base. Then, the value of k5 was fixed and employed in a fit to data obtained with a short time base, focused on the rapid growth of [Br] from reaction 3. Linear plots of kps1 versus [HBr] yield the slope, k1. Ar (Big Three, 99.998%) was used directly from the cylinder. HBr (Matheson, 99.8% nominal) was separated from a substantial fraction of noncondensable gas by repeated freeze−pump−thaw cycles at 77 K and from less volatile impurities by distillation from 156 K. Iodoethane (Lancaster, 99%) was purified by repeated freeze−pump−thaw cycles at 77 K and stored in the dark over copper foil. Gas mixtures were prepared in darkened glass bulbs. Table 2 summarizes results of 11 experiments carried out at 297 K and about 30 mbar of total pressure. Variation of the Ar

pressure reactor (VLPR) technique of Benson and co-workers yielded much smaller rate constants (see Table 1),17 with a more regular positive activation energy. They argued that previous k3 measurements were too high.18 There are also discrepancies in rate constants for the reverse of reaction 3 C2H6 + Br → C2H5 + HBr

(4)

This mechanism implies growth and decay of atomic bromine according to

Table 1. Summary of Data for the Reaction C2H5 + HBr → C2H6 + Br at 296−301 K method

C2H5 → loss without generation of Br

(−3)

where LFP-RF results2,5 are about a factor of 1.4 larger than results from steady-state experiments19 at the lowest overlapping temperature of ∼500 K. Here, we present new measurements of k3 at room temperature and, in combination with the now precisely determined thermochemistry of the ethyl radical,10 reevaluate the kinetics of reaction −3. Finally, the reactions of C2H5 and other alkyl radicals with HBr form part of the mechanism of combustion inhibition by Brcontaining agents.20 Quantitative flame models require k3 values at temperatures higher than the experimental upper limit of 677 K, and therefore, a reliable and theoretically based extrapolation is needed. Complications in extrapolating to higher temperatures arise as the rotational energy of the adduct and TS-2 increase faster than that of the hindered Gorin transition state due to the differences in their respective moments of inertia. Vide infra. At some point, considering the shallow depth of the adduct well, the well disappears, and the barrier to reaction becomes the centrifugal barrier.

Table 2. Rate Constant Measurements for the C2H5 + HBr Reaction at 297 K

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τres/s

F/J

[C2H5I]/1014 molecule cm−3

0.5 0.5 0.6 0.5 0.5 1.0 0.5 0.5 0.5 0.5 0.9

4.05 4.05 4.05 5.00 2.45 4.05 4.05 4.05 4.05 4.05 4.05

1.1 2.1 1.4 1.1 1.1 2.1 1.2 2.3 2.3 1.2 2.1

[HBr]max/ 1013 molecule cm−3 14.4 14.3 9.1 13.7 13.7 13.4 14.0 13.9 12.4 13.1 16.5

k3 ± σk3/10−12 cm3 molecule−1 s−1 7.65 7.20 6.62 6.80 6.61 7.82 6.63 6.34 6.20 8.27 8.87

± ± ± ± ± ± ± ± ± ± ±

0.34 0.05 0.15 0.11 0.11 0.23 0.10 0.16 0.26 0.48 0.21

bath gas flow changed the average residence time for gases within the reactor before photolysis, τres, while variation of the energy discharged through the flashlamp, F, and [C2H5I] changed the initial concentration of ethyl radicals. Typically, six values of [HBr] were employed at each set of conditions, up to the maximum value given in Table 2. The 1σ uncertainties in k3 are statistical only and are derived from weighted linear plots of kps1 and its uncertainty versus [HBr]. The weighted mean value is k3 = 7.0 × 10−12 cm3 molecule−1 s−1. The 2σ statistical precision is ±0.3 × 10−12 cm3 molecule−1 s−1, and an allowance

EXPERIMENTAL MEASUREMENTS The experiments were carried out as described in work on trimethylsilyl radical reactions with HBr.21 C2H5 radicals were generated by flashlamp photolysis of C2H5I through magnesium fluoride optics (λ > 120 nm) in an argon bath gas. Atomic bromine, a product of reaction 3, was monitored by time-resolved resonance fluorescence at λ = 134−140 nm, with photon counting and signal averaging, excited by a microwave-powered discharge lamp operating with a flow of 0.5% CH2Br2 in Ar at about 1 mbar 5848



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vibrational perturbation theory (VPT2), based on the full cubic force field and the semidiagonal part of the quartic force field, to obtain the anharmonicity constants xij. These data were derived with the CFour program, 24 and those for TS-2 were derived with a version of CFour extended to apply VPT2 to transition states, 25 and they are given in the Supporting Information. The geometries of the stationary points were refined through geometry optimization at the spin-unrestricted coupled cluster CCSD(T)/cc-pVTZ-PP level of theory, based on restricted open-shell Hartree−Fock wave functions. The effective core potential of Peterson et al. included in the cc-pVTZ-PP basis set allows, in part, for scalar relativistic effects.26 The CCSD(T) calculations were made with the Molpro 2006 program suite.27 Then single-point energy calculations were made with CCSD(T) theory and the systematic series of all-electron basis sets aug-cc-pVTZ, aug-cc-pVQZ, and aug-cc-pV5Z, which were extrapolated to the infinite or complete basis set limit with an exponential function. CCSD(T)/cc-pwCVQZ calculations were employed to find core−valence correlation corrections as the difference between calculations employing all electrons and orbitals in the correlation treatment and calculations, where the core orbitals were frozen. Scalar relativistic effects were derived at the Douglas−Kroll−Hess level with the cc-pVTZ-DK basis set. For atomic Br, an empirical spin−orbit correction was also included. The anharmonic zero-point vibrational energy was derived from the VPT2 analysis noted above. The energies are summarized in Table 3. The Cartesian coordinates of the reactants, the intermediate complex, and the transition state (TS-2) leading to C2H6 + Br are provided in the Supporting Information, and the vibrational frequencies and moments of inertia are discussed later. Geometries are shown in Figure 2. The relative enthalpies are plotted in Figure 3. It is important to notice that the intermediate complex is bonded by 9.7 kJ mol−1 according to this calculation, and the transition state (TS-2) is at a similar energy, 0.2 kJ mol−1 above the adduct. Compared to the data shown in Figure 1, the energy profile in the region of the adduct and the TS is smoothed. That is, there is what seems to be a shelf in the potential energy surface along the reaction path. Also, notice must be taken of the fact that when the structure and frequencies of the complex and the transition state are compared, it is apparent that the transition state is tighter (less entropy) than the complex. Sheng et al.15 found a binding energy for the adduct closer to 13 kJ mol−1 and also that it had a similar energy to the TS. One test of the accuracy of the present calculations is the enthalpy of the overall reaction, computed here as −56.0 kJ mol−1. The NASA-JPL review of thermochemistry, 28 which includes Δ fH 298(C 2H 5) = 120.9 ± 1.7 kJ mol −1, yields Δ rH 298 = −49.6 kJ mol −1. The 0 K quantities quoted there, plus the temperature dependence of the enthalpy of ethane,29 yield ΔrH0 = −53.7 kJ mol−1, which is 2.3 kJ mol−1 more positive than the present CCSD(T) result. Breaking down this enthalpy change into individual bond strengths at 0 K, the computed (experimental) values are 360.2 (362.4) kJ mol−1 for H−Br and 416.2 (416.1) kJ mol−1 for C2H5−H. These errors are well within the ∼4 kJ mol−1 accuracy expected for this level of computation.26 We also note there is some “noise” in these calculations. For example, had we employed core−valence corrections at the MP2 rather than CCSD(T) level, the computed reaction enthalpy would have become −54.9 kJ mol−1,

for possible systematic errors increases the accuracy limits to ±0.8 × 10−12 cm3 molecule−1 s−1. This k3 supports the higher values listed in Table 1.

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QUANTUM AND TST CALCULATIONS Geometries and frequencies at stationary points on the potential energy surface for reaction 3 were initially characterized with QCISD/6-311G(d,p) theory. Energies were obtained via QCISD(T)/6-311+G(3df,2p) theory. These two basis sets included an effective core potential for bromine,22 which partially accounts for relativistic effects. These spin-unrestricted calculations were carried out with the Gaussian 94 program suite.23 Spin contamination was found to be negligible (⟨S2⟩ ≈ 0.76 rather than the ideal value of 0.75). The reaction coordinate was defined at the simple HF/631G(d) level of theory and was found to pass through the QCISD/6-311G(d,p) geometries. G2(MP2)ECP energies22 were evaluated along this reaction path and were verified to reproduce the QCISD(T)/6-311+G(3df,2p) data closely, to within 0.7 kJ mol−1. Figure 1 shows the classical energy profile

Figure 1. Classical energies at the G2(MP2)ECP level of theory along the HF/6-31G(d) reaction path for C2H5 + HBr → C2H6 + Br, shown as a function of the length of the forming C−H bond. Zero-point vibrational energy is not included.

(without zero-point vibrational energy, ZPE) obtained as a function of the length of the forming C−H bond, R. There is a local maximum in the classical energy at R = 1.70 Å (1 Å = 10−10 m). This initial analysis shows the bound complex identified in prior work and confirms that there is no energy barrier to the addition of C2H5 to HBr. Stationary points on this PES were then investigated at higher levels of theory and used as the basis for the RRKM/master equation calculations, described in the next section. The stationary points on the PES were next optimized at the all-electron MP2/6-311G(d,p) level of theory, for the purpose of determining vibrational frequencies and investigating the importance of anharmonicity. Torsion about the C−C bond was treated as a hindered rotor, as detailed in the next section. The lowest frequencies in the adduct and TS-2 correspond to bending of the C2H5−H−Br angle either in the plane of the C, C, and Br atoms or perpendicular to this plane. In the adduct, the harmonic values are 54 and 91 cm −1, and in TS-2, they are 95 and 149 cm −1, respectively. The vibrational modes in all species were analyzed in terms of second-order 5849



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Table 3. Coupled Cluster Results for the Reaction C2H5 + HBr → C2H6 + Br

species C2H5

CCSD(T)/ aug-cc-pVTZ au −79.007973

CCSD(T)/ aug-cc-pVQZ au −79.027616

CCSD(T)/ aug-cc-pV5Z au

CCSD(T)/ CBS au

−79.033330

−79.035674

relative anharmonic ZPE kJ mol−1 0

relative core− valence correlation kJ mol−1 0

relative DK scalar relativity kJ mol−1 0

relative 0 K enthalpy kJ mol−1 0

HBr

−2573.252493

−2573.271271

−2573.276984

−2573.279481

adduct

−2652.266479

−2652.304575

−2652.315948

−2652.320788

5.4

−0.2

−0.1

−9.7

TS

−2652.266270

−2652.304353

−2652.315724

−2652.320566

5.3

−0.1

−0.5

−9.5

−79.679947

−79.700537

−79.706596

−79.709123

23.2

2.2

−2.3

−56.0

−2572.606199

−2572.623161

−2572.628319

−2572.630574

C2H6 Br

Figure 3. Potential energy diagram based on CCSD(T)/CBS data (see text).

codes.30−32 An argument against such a treatment might be that the intermediate complex (adduct) is apt to be too short-lived to statistically distribute the internal energy. This presupposes that the complex needs to be completely statistical. In fact, in many systems where RRKM theory fits the data, it is not really possible to discern whether or not the energy is truly statistically distributed. This is because the higher frequencies have such minimal statistical weight that they may not be participating at all. Figure 4 shows the density and sum of states

Figure 2. Structures of HBr, C2H5, the adduct, the transition state, and ethane derived at the CCSD(T)/cc-pVTZ-PP level of theory.

somewhat closer to experiment. This change would make the adduct and TS-2 enthalpies lower by 0.5 kJ mol−1.

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Figure 4. The sum of states and density of states calculated for the complex (adduct) using all of its frequencies and the K-rotor are given by the orange and light blue curves. The same quantities computed without using the frequencies above 1088 cm−1 are shown in black and pink.

RRKM RATE CONSTANT CALCULATIONS We have treated this chemical system exactly as if it were a standard chemical activation system using the Multiwell suite of 5850



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Table 4. Properties of TS-1 for (a) ΔH0 = 9.70 and (b) 5.50 kJ mol−1 (a) T 200 300 400 500 600 800 1000 1200 1500 (b)

I‡1/AMU-A2 (Morse)

harmonic Keq = kca/kback; ΔH0 = 9.7 kJ mol−1

1017 924 859 811 771 710 662 623 575

T

I‡1/AMU-A2 (Morse)

200 300 400 500 600 800 1000 1200 1500

1017 924 859 811 771 710 662 623 575

1.14 2.02 1.01 7.40 6.51 6.38 7.12 8.32 1.08

−21

× 10 × 10−22 × 10−22 × 10−23 × 10−23 × 10−23 × 10−23 × 10−23 × 10−22

harmonic Keq = kca/kback; ΔH0 = 5.5 kJ mol−1 9.14 3.79 3.76 2.85 2.70 2.81 3.40 4.31 5.48

× 10−23 × 10−23 × 10−23 × 10−23 × 10−23 × 10−23 × 10−23 × 10−23 × 10−23

for the complex up to 3500 cm−1 with and without the participation of any of the frequencies above 1088 cm−1. Clearly, these higher frequencies have no weight at the energies shown; therefore, they may or may not be participating in the distribution of energy. (The same sort of considerations apply to the two transition states.) If the energy is distributed only among the lowest frequencies and the K-rotor, RRKM theory can still be sufficient. (Of course, this does not address the question as to whether internal vibrational relaxation (IVR) is fast enough such that RRKM theory applies.)

kback/s−1 =1 × 10−10/Keq 8.76 4.94 9.92 1.35 1.54 1.57 1.40 1.20 9.29

× 1010 × 1011 × 1011 × 1012 × 1012 × 1012 × 1012 × 1012 × 1011

kback/s−1 = 1.0 × 10−9T−0.7/Keq 2.68 4.89 4.91 5.29 4.79 4.05 2.73 1.84 1.27

× 1011 × 1011 × 1011 × 1011 × 1011 × 1011 × 1011 × 1011 × 1011

η 92.90% 93.60% 93.80% 94.10% 94.30% 94.60% 94.80% 95.00% 95.10% η 91.58% 92.76% 93.12% 92.29% 93.38%

In order to treat the system as a standard chemical activation problem, information is needed on the transition state (TS-1) between C2H5 + HBr and the complex. This was first treated as a hindered Gorin transition state such that the capture rate constant for the C2H5 + HBr collision forming the complex was equal to kca = 1 × 10−10 cm3 molecule−1 s−1. Because the Multiwell calculations refer to the rate constants for the complex dissociating either back to C2H5 and HBr or on to C2H6 and Br, the former was computed from the value of kca above and the equilibrium constant for the formation of the

Table 5. Parameters for Overall Equilibrium Constant Calculation C2H6 heat of formation at 0 K/kJ mol−1 vibrational frequencies/cm−1 (J-rotor) adiabatic moments of inertia/AMU Å2 (K-rotor) active external rotor/AMU Å2 Symmetry; optical isomers; electronic levels Br heat of formation at 0K/kJ mol−1 symmetry; optical isomers; electronic levels electronic degeneracy C2H5 heat of formation at 0 K/kJ mol−1 vibrational frequencies/cm−1 (J-rotor) adiabatic moments of inertia/AMU Å2 (K-rotor) active external rotor/AMU Å2 symmetry; optical isomers; electronic levels HBr heat of formation at 0 K/kJ mol−1 vibrational frequencies/cm−1 (J-rotor) adiabatic moments of inertia/AMU Å2 symmetry; optical isomers; electronic levels

−68.4 332.6046,a 833.5883, 833.5883, 1034.172, 1241.424, 1241.424, 1420.633, 1448.511, 1524.472, 1524.472, 1526.3, 1526.3, 3081.161, 3081.232, 3155.563, 3155.563, 3177.442, 3177.442 25.2984 6.2889 6; 1; 1 117.9 1; 1; 1 4 131.8 162.62,b 461.5385, 820.2583, 999.341, 1090.7297, 1218.3813, 1421.5334, 1496.4448, 1510.4911, 1511.0793, 3038.1168, 3122.2853, 3167.6673, 3207.3428, 3318.8351 23.4 4.92 2; 1; 1 −28.44 2740.9124 with anharmonicity −43.2007 2.08 1; 1; 1

Treated as a hindered internal rotor using the “hrd” method in Multiwell: hrd 4, 1, 3; Vhrd2 3, 3.141900783, 542.5460689, −558.5092507, 16.85238199, −0.926709024; Ihrd1 3, 3.141900783, 1.572. bTreated as a hindered internal rotor using the “hrc” method in Multiwell: hrc 1.126, 0.0, 3. a

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Table 6. Parameters for Multiwell Calculations C2H5−HBr (adduct) critical energy at 0 K/kJ mol−1 vibrational frequencies/cm−1 (J-rotor) adiabatic moments of inertia/AMU Å2 (K-rotor) active external rotor/AMU Å2 symmetry; electronic degeneracy; optical isomers collisions: (σ/Å; ε/K)

C2H5*HBr N2

⟨ΔEd⟩ all temperatures C2H5···HBr (TS-1 Gorin transition state) frequencies/cm−1 (J-rotor) adiabatic moments of inertia/AMU Å2 200, 300, 400, 500, 600, 800, 1000, 1200, 1500 K (K-rotor) active external rotor/AMU Å2 moments of inertia active C2H5 2-D rotor/AMU Å2 moments of inertia active HBr 2-D rotor/AMU Å2 % hindrance (to match kca = 1 × 10−10) 200, 300, 400, 500, 600, 800, 1000, 1200, 1500 K symmetry; electronic degeneracy; optical isomers C2H5−H···Br (TS-2 transition state) critical energy at 0 K/kJ mol−1 frequencies/cm−1 (J-rotor) adiabatic moments of inertia/AMU Å2 (K-rotor) active external rotor/AMU Å2 symmetry; electronic degeneracy; optical isomers

9.70 53.7339, 88.4678, 90.5998, 236.4037,a 371.2521, 374.3163, 619.9212, 821.4042, 1011.467, 1088.06, 1225.228, 1421.482, 1495.222, 1506.35, 1511.556, 2515.573, 3043.518, 3130.456, 3175.24, 3193.494, 3304.788 304.679 23.2732 1; 2; 1 4.5; 400 3.74; 82.0 400 162.62,b 461.5385, 820.2583, 999.3414, 1090.7297, 1218.3813, 1421.5334, 1496.4448, 1510.4911, 1511.0793, 3038.1168, 3122.2853, 3167.6673, 3207.3428, 3318.8351, 2740.9124 1017, 924, 859, 811, 771, 710, 662, 623, 575 5.0 23.4 2.05 92.90, 93.60, 93.80, 94.10, 94.30, 94.60, 94.80, 95.00, 95.10 1; 2; 1 0.02 96.56355, 149.0749, 368.1181c 655.4832, 730.9119, 792.9926, 827.4368, 1012.7, 1089.875, 1235.517, 1412.524, 1494.443, 1500.847, 1508.877, 1528.115, 3054.257, 3142.631, 3170.832, 3184.472, 3278.09 262 21.6 1; 2; 1

Treated as a hindered internal rotor using the “hrd” method in Multiwell: hrd 4, 1, 1; Vhrd2 3, 3.144286725, 142.7779246, −150.7537996, 9.510636945, −1.273030429; Ihrd1 3 0.0, 3.139. bTreated as a hindered internal rotor using the “hrc” method in Multiwell: hrc 1.126, 0.0, 3. cTreated as a hindered internal rotor using the “hrd” method in Multiwell: hrd 4, 1, 1; Vhrd2 3, 3.1445868427, 244.9700387, −243.6030743, 1.283447581; −2.68952829, −1.273030429; Ihrd1 3, 0.0, 3.147. a

complex (Keq = kca/kback). Keq is shown in Table 4a and was computed using the “Thermo” code in Multiwell and the values shown in Tables 5 and 6 with ΔH0 = 9.70 kJ mol−1. The moment of inertia of TS-1 (I‡1) was computed as a function of temperature from a pseudodiatomic attractive Morse potential. Values of I‡1 are shown in Table 4a. Values in bold italics highlight the change in energy of the equilibrium constants for the formation of the adduct at temperatures where E0 + kT(1 − I‡1/I) becomes negative. Essentially, the energy at the adduct and the transition state relative to the reactants, C2H5 and HBr, has become positive, and the temperature dependence of the rate constant should become positive. An illustration is shown in Figure 5. This corresponds to what Troe has called “rotational channel switching”.33 Others34 have discussed the same phenomenon by describing a two-transition-state model. The hindrance, η, at each temperature between 200 and 1000 K was computed from the “fit” function in the “Thermo” code such that kca = 1 × 10−10 cm3 molecule−1 s−1 at all temperatures. These hindrances, as a percentage of the 4π steradians available for free rotation, are also given in Table 4a along with the values of kback at each temperature. When computing the sum of states in TS-1, the twodimensional, or J-rotors, of C2H5 and HBr are each multiplied by (1 − η/100)1/2.

Figure 5. Vibrationally adiabatic potential energy curves plus rotational energy, based on the G2(MP2)ECP data shown in Figure 1 together with the HF/6-31G(d) zero-point vibrational energy scaled by 0.9074.

The calculations were performed at six pressures of N2 between 10−5 and 1 atm to check for any pressure dependence. 5852



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Figure 6. Arrhenius plot. Data from Nicovich et al.,4 black diamonds; Seetula,3 pink squares; Seakins et al.,2 green triangles, and Ferrell,5 blue circles. Linear fit to experimental data, black line. Calculation using ΔH0 = 9.7 kJ mol−1 and k1 = 1 × 10−10 cm3 molecule−1 s−1, orange filled circles. (Also shown is the effect of treating the anharmonicity, blue filled triangles.) Calculation using ΔH0 = 5.5 kJ mol−1 and k1 = 1 × 10−9T−0.7 cm3 molecule−1 s−1, brown filled squares; linear fit to the brown filled squares, red line. The red vertical line indicates temperatures above which E0 = kT(1 − I‡1/I) becomes negative for ΔH0 = 9.7 kJ mol−1, while the green vertical line corresponds to the same phenomenon for ΔH0 = 5.5 kJ mol−1.

Table 7. Equilibrium Constant as a Function of Temperature T

1000/T

200 298 300 400 500 600 800 1000 1200 1500

5.00 3.36 3.33 2.50 2.00 1.67 1.25 1.00 0.83 0.67

K anharmonic 3.64 5.16 4.43 1.40 4.20 4.02 2.18 4.00 1.34 4.77

× × × × × × ×

1012 107 107 105 103 102 101

× 10−1

K harmonic

anh/har

× × × × × × ×

1.04 1.02 1.02 0.99 0.96 0.94 0.95 0.99 1.05 1.16

3.52 5.04 4.33 1.41 4.36 4.25 2.30 4.03 1.28 4.13

1012 107 107 105 103 102 101

× 10−1

Figure 7. Equilibrium constant for C2H5−HBr (adduct) formation from HBr + C2H5, computed with just harmonic frequencies and with taking anharmonicities into account.

There was essentially none using estimated Lennard-Jones parameters and ΔEd = 400 cm−1. It is assumed that the J-rotors are adiabatic, and the Multiwell code makes a correction to the critical energy to account for conservation of angular momentum. The results are shown in Figure 6. Note that using a binding energy for the complex of 9.70 kJ mol−1, taken from this work, yields values higher than the experimental values. Lowering the binding energy to 5.5 kJ mol −1, a change within the uncertainty of this work, and arbitrarily introducing some temperature dependence in the value of kback = 1 × 10−9T−0.7 cm3 molecule−1 s−1, yields a reasonable fit. Values of quantities needed for this calculation are shown in Table 4b. Table 6 remains the same except for the critical energy and the hindrance parameters. If the experimental data sets are fit to an Arrhenius expression, the result is k−1 = 1.76 × 10 −12 exp(470.6/T). The computation with a binding energy of 5.50 kJ mol−1 yields k−1 = 2.86 × 10−12 exp(319.2/T). The change in slope for the upper curves

Figure 8. Equilibrium constant for HBr + C2H5 = C2H6 + Br, computed with and without anharmonicities.

in Figure 6 reflects the change to a negative value of E0 + kT(1 − I‡1/I), referred to above. It is not totally clear whether the results of the Multiwell code under these conditions are appropriate, but it does seem as though the temperature dependence of the overall process becomes positive. This was also observed by Sheng et al.15 5853



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Figure 9. The points are from Seakins et al.,2 King et al.,19 and Ferrell.5 The corresponding Arrhenius parameters are the fits to those points. The blue line is from the Arrhenius parameters of the combined data for the reverse reaction of Nicovich et al.,4 Seetula,3 Seakins et al.,2 and Ferrell5 combined with the equilibrium constants in Table 7.

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EFFECT OF INCLUDING ANHARMONICITY The anharmonicity matrix has been calculated along with the frequencies. The data are provided in the Supporting Information. The Multiwell suite, through the code “Adensum”,35 allows the calculation of densities and sums of states using the anharmonicities. With the output from this code, thermochemical quantities may be computed. The equilibrium constant for the process HBr + C2H5 → C2H5−HBr (adduct) has been computed using just harmonic frequencies and with the addition of anharmonicites. The results are shown in Figure 7. Interestingly, only the thermochemical values of the adduct are seriously affected by including anharmonicities. As can be seen from Figure 6, the transition states are almost equally affected, and the rate constant hardly changes. (The hindered Gorin transition state anharmonicity matrix was calculated using the anharmonicity matrix of ethyl radicals and adding the HBr vibration as separable in the calculation of sums and densities.) As a result, all other computations herein are performed using harmonic frequencies.

equilibrium constants in Table 7 are correct). There are differences of between factors of 1.3 and 2 between the directly observed reverse rate constant and the values derived via the equilibrium constant.

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CONCLUSIONS The experimental data and the results of a master equation analysis based on a computed potential energy surface can be brought into accord. The surface indicates a bound complex and a transition state to products at about the same energy, with the transition state being considerably tighter (having less entropy). The computed energy is within 5 kJ mol−1 of the value needed to fit the data. There is some indication that the slope of the Arrhenius plot changes sign at higher temperatures.

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ASSOCIATED CONTENT

* Supporting Information S

Cartesian coordinates and anharmonicity constants for stationary points on the C2H5 + HBr potential energy surface are listed in Tables S1 and S2, respectively. This material is available free of charge via the Internet at http://pubs.acs.org.

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REACTION Br + C2H6 → C2H5 + HBr The equilibrium constant of the overall reaction Br + C2H6 → C2H5 + HBr was computed as a function of temperature using the “Thermo” code in the Multiwell suite. Values for C2H5 and HBr are as given in Table 5. Measurements of the forward and reverse reactions can be compared with a direct computation of the equilibrium constant. The values of the thermochemical quantities in Table 6 agree with literature values. When anharmonic densities replace the harmonic ones in the calculation of the overall equilibrium constant, differences in values obtained are small, but as can be seen from Table 7 and Figure 8, differences grow at higher temperatures. Figure 9 shows the calculations of the rate constant for Br + C2H6 compared with some data for the reaction. This figure illustrates the difficulty of extracting an equilibrium constant from data for reverse reactions (assuming of course that the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.M.G.); marshall@unt. edu (P.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank John Stanton for the anharmonic analysis of the transition state using CFour. This work was supported through Grant # NNG06GF98G “Critical Evaluation of Kinetic Data/ Applied Theory” from The NASA Upper Atmosphere Research Program (DMG) and the R.A. Welch Foundation with Grant 5854



Article

for Use in Atmospheric Studies, Evaluation Number 17; Jet Propulsion Laboratory: Pasadena, CA, 2011. (29) Gurvich, L. V.; Veyts, I. V.; Alcock, C. B. Thermodynamic Properties of Individual Substances, 4th ed.; Hemisphere Publishing Corp.: New York, 1991; Vol. 2. (30) Barker, J. R. Int. J. Chem. Kinet. 2001, 33, 232−245. (31) Barker, J. R. MultiWell-2011.1 Software; designed and maintained by John R. Barker with contributors Ortiz, N. F., Preses, J. M.; Lohr, L. L.; Maranzana, A.; Stimac, P. J.; Nguyen, T. L., Dhilip Kumar, T. J.; University of Michigan: Ann Arbor, MI, 2011; http://aoss.engin. umich.edu/multiwell/. (32) Golden, D. M.; Barker, J. R. Combust. Flame 2011, 158, 602− 617. (33) Troe, J. J. Chem. Soc., Faraday Trans. 1994, 90, 2303−2317. (34) Greenwald, E. E.; North, S. W.; Georgievskii, Y.; Klippenstein, S. J. J. Phys. Chem. A 2005, 109, 6031−6044. (35) Nguyen, T. L.; Barker, J. R. J. Phys. Chem. A 2010, 114, 3718− 3730.

B-1174 (P.M.) and the UNT Faculty Research Fund (P.M.). Computer facilities in part were purchased with NSF Grant CHE-0741936.

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REFERENCES

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