Article pubs.acs.org/JPCA
Experimental and Modeling Study of the Temperature and Pressure Dependence of the Reaction C2H5 + O2 (+ M) → C2H5O2 (+ M) Ravi X. Fernandes,† Klaus Luther,‡ Gerd Marowsky,§ Matti P. Rissanen,∥ Raimo Timonen,⊥ and Jürgen Troe*,‡,§,# †
Physikalische-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany § Laser-Laboratorium Göttingen, Hans-Adolf-Krebs-Weg 1, D-37077 Göttingen, Germany ∥ Department of Physics, University of Helsinki, P.O. Box 64, FI 00014 Helsinki, Finland ⊥ Department of Chemistry, University of Helsinki, P.O. Box 55, FI 00014 Helsinki, Finland # Max-Planck-Institut für Biophysikalische Chemie, Am Fassberg 11, D-37077 Göttingen, Germany ‡
ABSTRACT: The reaction C2H5 + O2 (+ M) → C2H5O2 (+ M) was studied at 298 K at pressures of the bath gas M = Ar between 100 and 1000 bar. The transition from the falloff curve of an energy transfer mechanism to a high pressure range with contributions from the radical complex mechanism was observed. Further experiments were done between 188 and 298 K in the bath gas M = He at pressures in the range 0.7−2.0 Torr. The available data are analyzed in terms of unimolecular rate theory. An improved analytical representation of the temperature and pressure dependence of the rate constant is given for conditions where the chemical activation process C2H5 + O2 (+ M) → C2H4 + HO2 (+ M) is only of minor importance.
1. INTRODUCTION Because of its role in the oxidation of hydrocarbons in combustion and in atmospheric chemistry, the reaction between ethyl radicals and molecular oxygen has attracted considerable attention (for summaries of experimental results, see, e.g., refs 1 and 2). Being a complex-forming bimolecular reaction with several possible reaction channels, its rate and branching properties depend on the temperature, pressure, and the nature of the bath gas3 and even a multiexponential time dependence may be encountered.4 A full theoretical description of the kinetics is complicated and requires detailed knowledge of the potential energy surface as well as of collisional energy transfer. This knowledge is still fragmentary, and some empirical fitting is necessary. Nevertheless, the general features of the kinetics can be analyzed reasonably well (see, e.g., refs 4−9). On the experimental side, large ranges of pressure and temperature have been covered.1,2 At room temperature, the experimental pressures varied from very low values, such as realized in very low pressure reactor (VLPR) studies,10 up to 60 bar as applied in ref 11, with the range 1−1000 Torr being covered most extensively (see, e.g., refs 1, 2, 5, and 12−19). At pressures above about 1 bar, the reaction was found to be close to the high pressure limit of the “normal” falloff curve of the recombination reaction C2H5 + O2 ( +M) → C2H5O2 ( +M) © XXXX American Chemical Society
with a rate constant k1,∞. At moderately low pressures, the pressure-proportional low pressure rate constant k1,0 of the recombination reaction 1 was approached. At even lower pressures, a transition to the also pressure-dependent rate constant k2 of the chemical activation process C2H5 + O2 ( +M) → C2H4 + HO2 ( +M)
(2)
was observed. At high temperatures also the reactions C2H5O2 ( +M) → C2H4 + HO2 ( +M)
(3)
→C2H5 + O2 ( +M)
(4)
have to be considered. The present work first focuses on a particular aspect of the recombination reaction 1, i.e., in the approach of the high pressure rate constant k1,∞ at room temperature. At 1 bar, k1 in several bath gases was found15 to be within about 10% of an extrapolated k1,∞. No further pressure dependence was observed at pressures up to 60 bar.11 In the present work, we Special Issue: 100 Years of Combustion Kinetics at Argonne: A Festschrift for Lawrence B. Harding, Joe V. Michael, and Albert F. Wagner Received: November 21, 2014 Revised: January 22, 2015
(1) A
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profiles averaged over about 50 runs. We used the observation wavelengths 240 and 265 nm. Contrary to our HO2 and CH3O2 experiments,20,25 where absorptions of the reactants (H and CH3) at the observation wavelength 240 nm could be neglected, in the present experiments, absorptions of C2H5O2 and C2H5 always overlapped (more strongly at the absorption wavelength 240 nm than at 265 nm). This results in an absorption step from C2H5 directly after the photolysis flash, followed by the much slower exponential approach of the final absorption, which then was governed by overlapping C2H5O2 formation and C2H5 disappearance signals. As it was difficult to separate the initial absorption step from laser straylight, we reconstructed the initial step by means of literature values for the ratio of absorption coefficients ε(C2H5O2)/ε(C2H5). The absorption coefficient ε(C2H5O2) is well-known (see, e.g., ref 26), while there are more uncertainties in ε(C2H5) (see refs 14 and 27−29). We employed ε(C2H5O2)/ε(C2H5) ratios of 1.8 (±0.1) at 240 nm and of 7 (±1) at 265 nm. The signals S(t) then could be evaluated using the relationship S(t) = S(t=0) + [S(t=∞) − S(t=0)] exp(-k1t). Under our conditions, no selfreaction 2 C2H5O2 → C2H5O2C2H5 + O2 was observed.30 We estimate the accuracy of our measured values of k1 to be about ±10%, mostly determined by the uncertainty of the initial absorption step and, hence, the ratio ε(C2H5O2)/ε(C2H5). Most experiments, because of the larger absorption signals, were done at 240 nm. Table 1 summarizes the experimental high pressure values of k1 for 298 K from the present work. Our results between 100
extended the pressure even more (up to 1000 bar) in order to document the full transition to k1,∞. In doing this we were aware of the possibility that the reaction mechanism may change, becoming a superposition of the radical-complex (RC) and the “normal” energy transfer (ET) mechanism. High pressure experiments on other reactions have provided evidence for the onset of a RC mechanism, for the reaction CH3 + O2 → CH3O2 in Ar near 300 bar,20 for the reactions 2C7H7 → C14H14 and 2C7H6F → C14H12F2 in Ar near 10 bar.21−23 Reaction 1 might fall in between, such that studies of the high pressure end of the falloff curve of reaction 1 may provide information on the RC contribution as well. In order to test the expressions for the temperature and pressure dependence of k1 suggested, e.g., in ref 4 and to extend the database, we furthermore measured k1 down to temperatures of 188 K at pressures between 0.7 and 2 Torr. One expects that the falloff curves markedly shift with temperature such that details of recommended rate constants k1 can be well tested. In addition, we employed recent measurements24 of the rate constant k5 for the reaction C2H5 + Cl 2 → C2H5Cl + Cl
(5)
to recalibrate earlier measurements of the ratio k1/k5 from refs 15 and 17. On the basis of the previous and the present data, we intend to provide a refined representation of the pressure and temperature dependence of k1 in terms of unimolecular rate theory. As the theoretical modeling still requires empirical fitting of some parameters (such as average energies ⟨ΔE⟩ transferred in C2H5O2* + M collisions and transitional mode frequencies along the minimum-energy path (MEP) potential of reaction 1) and the experimental database is still limited, we employ simplistic but transparent models. In particular, we try to improve the falloff representation of k1. Properly analyzing k1,∞, which corresponds to the formation of the reaction complex C2H5O2*, is a necessary prerequisite also for the understanding of the full complex-forming bimolecular reaction, with competing back dissociation of C2H5O2* to C2H5 + O2 and forward rearrangement over an intrinsic barrier to C2H4 + HO2 (see refs 4−9). By considering only low temperature and medium to high pressure conditions, in the present work we limit ourselves to the addition reaction 1. Minor contributions from reaction 2 are accounted for by accepting its representation as given in ref 4.
Table 1. Experimental First-Order Rate Constants k1 at High Pressures from This Work (M = Ar, T ≈ 298 K) p/bar
k1/10−11 cm3 molecule−1 s−1
100 200 400 700 870 1000
0.79 0.83 1.02 1.03 1.08 1.23
and 200 bar of k1 ≈ 8(±1) 10−12 cm3 molecule−1 s−1 agree well with the results between 1 and 60 bar from refs 11, 14, and15 (an optimized value of k1 = 8.4 (±0.8) × 10−12 cm3 molecule−1 s−1 was obtained from fitting of the full falloff curve, see below; this value then was taken as the high pressure limit k1,∞ of the ET mechanism). We note that our measured k1 for higher pressures increase beyond this value of k1,∞. Analogous to our work20 on the reaction CH3 + O2 → CH3O2 we attribute this increase to a contribution from the RC mechanism. Figure 1 suggests that the data in M = N2 up to 60 bar from ref 11 do not show contributions from the RC mechanism, while the present results above about 400 bar do so. In order to support this conclusion, by plotting k/k∞ET, Figure 1 compares the present results with data for the CH3 + O2 → CH3O2 reaction20 in M = Ar and N2. It appears that the RC contribution sets in at slightly lower pressures for C2H5 + O2 → C2H5O2, but this conclusion will require further measurements. In particular, experiments with other bath gases appear desirable. Near to 1000 bar, the effects are of similar magnitude. We note that an onset of a decline of k1 due to diffusion control at the highest pressures, such as seen for benzyl dimerization in ref 22, was not yet observed.
2. EXPERIMENTAL TECHNIQUE AND RESULTS Our first series of experiments has been performed at Göttingen in the high pressure flow cell employed also in previous experiments over the pressure range 1−1000 bar (at temperatures 300−700 K for the reaction CH3 + O2 → CH3O2 in ref 20, and 300−900 K for the reaction H + O2 → HO2 in ref 25; for experimental details see these references). In the present work, ethyl radicals were generated by laser flash photolysis of azoethane. The latter was prepared by the method of Diels and Knoll. Argon (99.9999% from Air Liquide) was used as the bath gas M. The concentration of the reactant O2 was always kept high enough that nearly pseudo-first-order reaction conditions were realized; on the other hand, it was kept sufficiently low such that the reaction did not become too fast ([O2]/[Ar] = 10−5 − 10−4 and [C2H5]/[O2] ≈ 0.003 were employed). Azoethane was photolyzed with flashes from an ArF-laser at 193 nm. The formation of C2H5O2 was then recorded by UV absorption spectroscopy with absorption-time B
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k5 = 1.45(± 0.04) × 10−11(T /300 K)−1.73(±0.09) cm3 molecule−1 s−1
(6)
with the temperature dependence from ref 24.
3. MODELING OF RATE CONSTANTS IN THE FALLOFF RANGE In the present work we do not further analyze the experiments at very high pressures where the transition to the radical complex mechanism was observed and where binary collision models break down. Instead we only consider the pressure range where the usual energy transfer mechanism of thermal unimolecular reactions applies. In previous work like ref 5, the falloff curve of reaction 1 has been fitted in a simplified manner, equivalent to using the standard expression from ref 32 k /k∞ ≈ [x /(1 + x)]F(x)
with
Figure 1. High pressure rate constants k∞ at 298 K for the addition of O2 to CH3 and C2H5 with evidence of a contribution from the radicalcomplex mechanism (k∞ET = high pressure rate constant for the energy-transfer mechanism; data for O2 + C2H5 with k∞ET = 8.4 × 10−12 cm3 molecule−1 s−1, M = Ar (●) from present work and M = N2 (▲) from ref 11; data for O2 + CH3 with k∞ET = 2.2 × 10−12 cm3 molecule−1 s−1, M = Ar (○), and M = N2 (△) from ref 20; the [M] scale in ideal gas concentrations is for illustration only).
2
1/[1 + (log x / N ) ] F(x) ≈ Fcent
Table 2. Experimental First-Order Rate Constants k1 at Low Temperatures from This Work (M = He) P/Torr
[He]/1016 molecules cm−3
188
0.73 1.25 0.78 1.33 0.83 1.47 0.92 1.59 0.99 1.76 1.12 1.97
3.75 6.42 3.75 6.39 3.63 6.42 3.69 6.27 3.59 6.39 3.63 6.38
201 221 241 266 298
k1/10−12 cm3 molecule−1 s−1 4.12 4.59 4.30 4.51 3.04 3.96 2.76 2.76 1.87 2.09 1.36 1.98
(8)
where x = k0/k∞ and N ≈ 0.75−1.27 log Fcent. In previous studies, values of the center broadening factor Fcent in the range 0.3−0.7 were employed for 298 K, either obtained as fit parameters or derived from simple model calculations. In the present work, we refine the approach by modeling Fcent with the method of ref 33 for the strong collision broadening factor FSC cent and of refs 34 and 35 for the weak collision broadening factor SC WC WC FWC cent (with Fcent = Fcent Fcent, Fcent being derived from the analysis of master equation solutions for a series of model reactions). Suggested by the obtained value of Fcent, more advanced falloff expressions34,35 than eq 8 were also tested. Fitting the experimental falloff curves to eqs 7 and 8, highpressure rate constants k1,∞ and low-pressure rate constants k1,0 were obtained. In the following, we first analyze the experimental values of k1,∞, then proceed to falloff expressions for k1 ([M]), and finally analyze k1,0. We express k1,∞ as the product of kPST 1,∞ from phase space theory (PST) and a thermal rigidity factor f rigid(T),
Our second series of experiments was performed at Helsinki in a flow system using laser photolysis and photoionization mass spectrometry such as described in detail in ref 24. Experiments were done over the temperature range 188−298 K in the bath gas He. For each temperature, two pressures were studied, varying by about a factor of 1.7 within the range 0.7−2 Torr. Table 2 summarizes the results. Some increase of k1 with
T/K
(7)
(±0.25) (±0.33) (±0.33) (±0.29) (±0.34) (±0.25) (±0.12) (±0.06) (±0.15) (±0.06) (±0.04) (±0.17)
k1, ∞ = frigid (T )k1,PST ∞ kPST 1,∞
(9)
here is expressed by
3/2 ⎛ kBT ⎞⎛ h2 ⎞ ⎜ ⎟ = k1,PST ⎜ ⎟ Q cent ∞ ⎝ h ⎠⎝ 2πμkBT ⎠
(10)
with the centrifugal partition function Qcent given by ∞
Q cent =
∑ (2l + 1) exp[−E0(l)/kBT ] l=0
(11)
(see, e.g., ref 36). The centrifugal barriers E0(l) are derived from the MEP potential and hence depend on quantumchemical information. This is available for the O2−C2H5 potential from the B3LYP/6-31G* calculations of ref 37. Expressing the corresponding results in center-of-mass Morse form (with a bond energy of D/hc =14080 cm−1 and an equilibrium bond length re = 2.833 Å) leads to an effective Morse parameter β(r) expressed by
bath gas pressure is observed, but the increase is much weaker than proportional, corresponding to falloff curves nearer to the high pressure than to their low pressure limit. The results will be analyzed below in terms of unimolecular rate theory. We note that the measurements from ref 24, giving k5 = 1.45 (±0.04) × 10−11 cm3 molecule−1 s−1 at 298 K, for unknown reasons, led to markedly lower values than the older ones of k5 = 1.9 (±0.4) × 10−11 cm3 molecule−1 s−1 from ref 31. The latter were used in ref 15 to calibrate k1/k5 ratios. We recalibrate these ratios with our new k5, employing the expression
β(r ) ≈ 3.938 Å−1 exp[− 0.295 Å−1(r − re)]
(11)
The corresponding E0(l) then are expressed as C
DOI: 10.1021/jp511672v J. Phys. Chem. A XXXX, XXX, XXX−XXX
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n ≈ [ln 2/ln(2/Fcent)](0.8 + 0.2x q)
(12)
(for l > 80 and E0(l)/hc > 560 cm ; smaller values are obtained for l < 80). In addition, Qcent(T) is derived as Q cent(T ) ≈ 8890(T /300 K)n
(17)
where q = (Fcent − 1)/ln(Fcent/10). While the differences between the symmetric broadening factors of eq 8 and the asymmetric broadening factors of eq 16 are within the experimental uncertainty near room temperature (and thus are unimportant for the conditions considered in the present work), they matter when Fcent decreases with increasing temperature. Figure 2 compares experimental data obtained near 298 K in a variety of measurements, and the falloff expression of eqs 7
−1
(13)
with n ≈ 0.31 between 100 and 300 K (and n ≈ 0.55 between 300 and 2000 K). Combining eqs 10 and 13 gives kPST 1,∞ as −10 k1,PST (T /300 K)−0.19 cm 3 molecule−1 s−1 ∞ ≈ 9.6 × 10
(14)
(for T = 100−300 K and ∝ T for T = 300−2000 K). Comparing eq 14 with the experimental k1,∞ leads to the small experimental value of the thermal rigidity factor at 300 K of kPST 1,∞
0.05
exp f rigid (300 K) ≈ 10−2
(15)
f exp rigid
The magnitude of suggests that reaction 1 is governed by a rigid transition state with frequencies close to those of C2H5O2. The latter are chosen from ref 38, omitting the C−O stretching as the reaction coordinate. Calculating the effective number SK of transition state oscillators SK = 1 + Uvib‡/kBT and the effective parameter BK = [E0 + a(Eo)Ez‡]/kBT (with the zeropoint energy Ez‡ and the Whitten-Rabinovitch factor a(E0)), by the simple method of ref 33 leads to strong collision broadening factors FSC cent of the falloff curves k1([M]). One obtains the values given in Table 3. In order to determine weak Table 3. Modeled Center Broadening Factors Fcenta T/K
FSC cent
FWC cent,1 (M=N2)
FWC cent,1(M=He)
Fcent,1(M=N2)
100 200 300 400 500 600 800 1000
0.98 0.85 0.74 0.65 0.58 0.52 0.45 0.42
0.86 0.80 0.77 0.74 0.72 0.70 0.67 0.65
0.81 0.75 0.71 0.68 0.66 0.64 0.64 0.64
0.84 0.68 0.57 0.48 0.42 0.36 0.30 0.27
Figure 2. Dependence of k1 + k2 on the bath gas concentration [M] near 298 K (M = He: ⊗ (present work), ● (ref 15, absolute values), ○ (ref 15, from k1/k5 with k5 from eq 6); M = N2: □ (ref 15, from k1/ k5 with eq 6); M = Ar: □ with × inside (ref 18); M = C2H4: ■ (ref 11); solid line: modeled falloff curve for M = He with eqs 7, 8, 18, 19, and 24 for k1 and eq 20 for k2).
and 8 employing the calculated Fcent ≈ 0.53 (for M = He). The relative rate data k1/k5 from ref 15 were reevaluated with k5 from eq 6. Our new data together with measurements from refs 5, 12, and 15 fit well to the modeled falloff curve. We also note that measurements with M = Ar from ref 18 and with M = N2 from refs 11 and 15 within the scatter fall onto the same curve. The shown falloff curve corresponds to a fitted value of
See text, with −⟨ΔE⟩/hc ≈ 35 cm−1 for M = He and 60 cm−1 for M = N2.
a
collision broadening factors FWC cent, one needs to estimate the weak collision efficiencies βc, which follow from the analysis of the low pressure rate constants k1,0 given below. The collision efficiency βc depends on the temperature and the bath gas, see ref 32. The corresponding FWC cent, as calculated according to ref 34, are included in Table 3. We note that our modeled values of Fcent for M = N2, in spite of a quite different approach, for T = 100−600 K are not far from the results of ref 4. At larger temperatures, however, Fcent from ref 4 decreases markedly below the present results. We note that the present approach allows us to account for the bath gas dependence of Fcent. At 300 K, Fcent is large enough that eqs 7 and 8 still can be used; for larger temperatures, when Fcent falls below values of about 0.5, however, “broader” falloff curves are encountered with asymmetric broadening factors F(x) instead of eq 3.2 (i.e., F(x) then differs from F(1/x)). Here we tested the expression suggested in ref 36, i.e., F(x) ≈ (1 + x)/(1 + x n)1/ n
k1, ∞ = 8.4(± 0.5) × 10−12 cm 3 molecule−1 s−1
(18)
in agreement with the high pressure measurements above 1 bar given above. The fitted k1,0 = [He] 6.8(± 2) × 10−29 cm 6 molecule−2 s−1
(19)
will be analyzed below with respect to the collision efficiency βc. In obtaining k1,0, we note that Figure 2 shows k1 + k2 as a function of [M] such that the value and the pressure dependence of k2 needs also to be specified. As k2 under the present conditions is much smaller than k1, we employ the representation from ref 4, which gave k 2 = k 2,0(1 + x 2)−1F2(x 2)
(20)
with x2 = k2,0[M]/k2,∞, M = N2, F2(x2) given by F(x) from eq 8 with Fcent,2 = 0.45, k2,0 = 3.3 × 10−13 cm3 molecule−1 s−1, and k2,∞ = 4.0 × 105 s−1. With these values one estimates k2 (1 Torr) ≈ 2.4 × 10−13 cm3 molecule−1 s−1 such that k2/k1 ≈ 0.1 at 1 Torr in agreement with measurements from ref 16. As k2
(16)
with D
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Further considering the temperature dependence of k1,0, we compare experimental data from the present work and of ref 5 with theoretical predictions based on the formalism of ref 32 (after having fitted k1,0 for 298 K in Figure 2 by eq 19). First we calculate the low pressure strong collision rate constant kSC 1,0 following ref 33. The rotational factors Frot contained in kSC 1,0 employ the centrifugal energies E0(l) from eq 12. We assume the anharmonicity factor Fanh to have a typical value of Fanh ≈ 1.7. The calculation then is straightforward and leads to kSC 1,0. The comparison with the experimental k1,0 at 298 K gives the collision efficiencies of βc = 0.07−0.15 and 0.1−0.2 for M = He and N2, respectively. Using βc/(1 − βc1/2) ≈ −⟨ΔE⟩/kBT one obtains −⟨ΔE⟩/hc = (30−60) cm−1 for M = He and (60−120) cm−1 for M = N2 (we note that these values are not independent of the value chosen for Fanh and of the falloff fittings, but they appear to be of reasonable magnitude). We further assume that ⟨ΔE⟩ is temperature independent such as often observed, which allows us to predict the temperature dependence of βc and hence of k1,0 (a temperature independent ⟨ΔE⟩ corresponds to ⟨ΔEdown⟩ ∝ T−1). Table 4 gives the
decreases roughly proportional with the pressure, its influence on Figure 2 is only minor. Next we consider the temperature dependence of k1,∞. While k1,∞ could well be obtained for 298 K from the falloff curve of Figure 2, most studies of the pressure dependence of k1 for other temperatures were too limited to allow for an independent fit of k1,0 and k1,∞. Only the experiments from refs 11 and 17 were done sufficiently close to the high pressure limit such that falloff corrections k1/k1,∞ based on only semiquantitative k1,0 (see below) could be applied. For example, the measurements of ref 11 in 1.5 bar of C2H4 (with modeled k1,0 employing −⟨ΔE⟩/hc ≈200 cm−1, see below) required falloff corrections of k1/k1,∞ ≈ 0.95 and 0.90 for T = 387 and 475 K, respectively, while the measurements of ref 17 in 580 Torr of He (with modeled k1,0 employing −⟨ΔE⟩/hc ≈ 35 cm−1; see below) required k1/k1,∞ ≈ 0.91, 0.72, and 0.49 for T = 262, 362, and 463 K. The latter values illustrate the marked shift of the falloff curves along the pressure scale when the temperature changes. This explains why the present measurements at 201 K in 1.33 Torr of He with k1/k1,∞ ≈ 0.49 are similarly close to the high pressure limit as the measurements at 463 K in 580 Torr of He. Combining the falloff-corrected data from refs 11 and 17 and the present work, one obtains a clear indication of a negative temperature coefficient of k1,∞ such as shown in Figure 3 (one should note that recalibrations of the
Table 4. Modeled Limiting Low Pressure Rate Constants k1,0/[M]a T/K 100 200 300 400 500 600 800 1000
k1,0/[He] cm6 molecule−2 s−1 4.9 3.3 5.8 1.5 4.6 1.6 2.2 3.8
× × × × × × × ×
−27
10 10−28 10−29 10−29 10−30 10−30 10−31 10−32
k1,0/[N2] cm6 molecule−2 s−1 6.4 4.3 7.8 2.0 6.2 2.1 3.0 5.2
× × × × × × × ×
10−27 10−28 10−29 10−29 10−30 10−30 10−31 10−32
See text, with Fanh ≈ 1.7 and −⟨ΔE⟩/hc ≈ 35 cm−1 for M = He and 60 cm−1 for M = N2.
a
results. Having in hand k1,0, k1,∞, and Fcent,1, we construct falloff curves and compare these with experimental data for k1 + k2 in Figure 4 over the range 201−473 K. As in Figure 2, we take k2 from ref 4. We find that the present experiments between 188 and 298 K agree well with the falloff curves employing the modeled k1,0 and k1,∞ from eq 21. The temperature dependence of the corresponding k1,0 over the range 200−300 K then can be represented in the form
Figure 3. High-pressure rate constants k1,∞ for C2H5 + O2 → C2H5O2 (○: falloff extrapolation from Figure 2; ■: values in 1.5 bar of C2H4 from ref 11 after falloff correction; ⊗: values in 1−2 Torr of He from present work after falloff correction; --- and ___: representation of eq 21).
k1,0 = [He] 6.8 × 10−29(T /298 K)−4.2 cm6 molecule−2 s−1 (22)
Experimental results at T > 298 K agree less well with the modeled falloff curves after relative rate data for k1/k5 have been recalibrated with the present k5; see the dashed curves in Figure 4. For low pressures, with increasing temperatures k1 and k2 are increasingly difficult to disentangle. Calculating k1 with the present values of eqs 21 and 22, we cured the problem by employing k2,∞ ≈ 6.85 × 10−12 T6.53 exp(420 K/T) s−1 from ref 4 and by multiplying k2,0 ≈ 2.34 × 10−17 T1.09 exp(994 K/T) cm3 molecule−1 s−1 from ref 4 with a factor of 3. We note that the experimental C2H4 yields in He at 298 K from ref 16 are well reproduced with these rate constants. (The factor of 3 should be omitted for M = N2 to meet the C2H4 yields at 298 K from ref 16; we emphasize that this procedure has purely empirical fitting reasons as k2,0 is bath gas independent). Extending eq 22 to the range 300−500 K by our modeling gives
ratios k1/k5 from ref 17 were made using the present k5 from eq 6). Over the range 300−500 K, k1,∞ then is represented by k1, ∞ ≈ 8.4(± 0.5) × 10−12(T /298K)−1 cm3 molecule−1 s−1 (21)
while the temperature coefficient may change from −1 to −0.2 between 200 and 300 K. The negative temperature coefficient of k1,∞ is contrary to what would have been estimated with the statistical adiabatic channel model (combined with classical trajectories, SACM/CT) from ref 39 for a “standard potential” and a rigidity factor as low as the experimental value of 10−2 from eq 15. A positive temperature coefficient was also recommended in refs 4 and 7, contrary to ref 40. E
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k1,0 ≈ [He] 6.8 × 10−29(T /298 K)−4.2 cm6 molecule−2 s−1 (22)
between 200 and 300 K (with the exponent −4.2 changing to −4.9 for 300−500 K), k1, ∞ ≈ 8.4 × 10−12(T /298 K)−0.2 cm 3 molecule−1 s−1 (21)
between 200 and 300 K (with the exponent −0.2 changing to −1 for 300−500 K), and Fcent,1(M = He) ≈ exp( −T /470 K)
(24)
Fcent,1(M = N2) ≈ exp( −T /540 K)
(25)
or
A modification of eq 22 for M = N2 could not be made on the basis of the existing data. Employing eqs 21−24 for T = 300−500 K, we found that k2,0 from ref 4 had to be increased by a factor of 3 while k2,∞ could be retained.
Figure 4. Dependence of k1 + k2 on the bath gas concentration [He] for T = 201, 241, 298, 373(±12), and 468(±5) K (from top to bottom, ⊗: experiments from present work, at 201, 241, and 298 K; ○ experiments from ref 15 at 298 K; experiments from ref 17 at 362 (⊕) and 463 K (●); experiments from ref 5 at 385 (△) and 473 K (▲); experiments from ref 13 at 384 (▽) and 467 K (▼). Dashed curves: modeling with k1 from present work and k2 from ref 4; full curves with k1 from the present work and k2 of ref 4 empirically increased by a factor of 3; see text). −29
k1,0 = [He] 6.8 × 10
−4.9
(T /298K)
6
cm molecule
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: shoff@gwdg.de. Notes
The authors declare no competing financial interest.
■
−2 −1
s
ACKNOWLEDGMENTS Helpful discussions of this work with John Crowley, David Golden, Anatoly Maergoiz, James Miller, Michel Rossi, and Stephen Klippenstein are gratefully acknowledged.
(23)
A further extension of k1 to higher temperatures becomes less important because the overall reaction then is governed by reaction 2. The database appears too limited to specify the bath gas dependence of k1,0 and k2,∞ in further detail.
■
REFERENCES
(1) 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. Evaluated Kinetic Data for Combustion Modeling: Supplement II. J. Phys. Chem. Ref. Data 2005, 34, 757−1397. (2) Atkinson, R.; Baulch, D. L.; Cox, R. A.; Crowley, J. N.; Hampson, R. F.; Hynes, R. G.; Jenkin, M. E.; Rossi, M. J.; Troe, J. Evaluated Kinetic and Photochemical Data for Atmospheric Chemistry: Volume II − Gas Phase Reactions of Organic Species. Atmos. Chem. Phys. 2006, 6, 3625−4055. (3) Troe, J. The Colourful World of Complex-Forming Bimolecular Reactions. J. Chem. Soc. Faraday Trans. 1994, 90, 2303−2317. (4) Miller, J. A.; Klippenstein, S. J. The Reaction between Ethyl and Molecular Oxygen II. Further Analysis. Int. J. Chem. Kinet. 2001, 33, 654−668. (5) Wagner, A. F.; Slagle, I. R.; Sarzynski, D.; Gutman, D. Experimental and Theoretical Studies of the C2H5 + O2 Reaction Kinetics. J. Phys. Chem. 1990, 94, 1853−1858. (6) Ignatyev, I. S.; Xie, Y.; Allen, W. D.; Schaefer, H. F. Mechanism of the C2H5 + O2 Reaction. J. Chem. Phys. 1997, 107, 141−155. (7) Miller, J. A.; Klippenstein, S. J.; Robertson, S. H. A Theoretical Analysis of the Reaction Between Ethyl and Molecular Oxygen. Proc. Combust. Inst. 2000, 28, 1479−1486. (8) DeSain, J. D.; Klippenstein, S. J.; Miller, J. A.; Taatjes, C. A. Measurements, Theory, and Modeling of OH Formation in Ethyl + O2 and Propyl + O2 Reactions. J. Phys. Chem. A 2003, 107, 4415−4427. (9) Carstensen, H.-H.; Naik, C. V.; Dean, A. M. Detailed Modeling of the Reaction C2H5 + O2. J. Phys. Chem. A 2005, 109, 2264−2281. (10) Dobis, O.; Benson, S. W. Reaction of the Ethyl Radical with Oxygen at Millitorr Pressures at 243 − 368 K and a Study of the Cl + HO2, Ethyl + HO2, and HO2 + HO2 Reactions. J. Am. Chem. Soc. 1993, 115, 8798−8809. (11) Dilger, H.; Schwager, M.; Tregenna-Piggot, P. L. W.; Roduner, E.; Reid, I. D.; Arsenau, D. J.; Pan, J. J.; Senba, M.; Shelley, M.;
4. CONCLUSIONS The present work has documented a contribution from the radical complex mechanism to the recombination of C2H5 with O2 at room temperature and at bath gas pressures of several hundred bar. This observation is analogous to what was observed earlier20 in the recombination of CH3 with O2. Taking this into account, the falloff curve of the rate constant k1 for the reaction C2H5 + O2 → C2H5O2 could well be evaluated to fit the high pressure limiting value k1,∞ of the energy transfer mechanism. Having in hand the complete experimental falloff curve near 300 K, by analyzing k1,∞, strong collision broadening factors FSC cent,1 and, by analyzing k1,0, weak collision broadening factors FWC cent,1 were determined. Further analyzing k1,0 and assuming temperature independent ⟨ΔE⟩ for collisional energy transfer, WC the temperature dependences of Fcent,1 = FSC cent,1 Fcent,1 and of k1,0 were modeled. On one hand, this allowed us to estimate falloff corrections for high-pressure experiments at temperatures different from 298 K, which finally led to experimental k1,∞ between about 200 and 500 K. On the other hand, falloff curves could be modeled over this temperature range and compared with low pressure measurements such as shown in Figure 4. The good agreement between experiments and modeling in Figure 4 (after minor modification of k2,0) demonstrates the internal consistency of the present work. The results are then summarized in the following representation of the characteristic rate parameters F
DOI: 10.1021/jp511672v J. Phys. Chem. A XXXX, XXX, XXX−XXX
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C2H5O2 Self-Reactions and Cross Reactions. J. Phys. Chem. A 2010, 114, 6983−6995. (31) Timonen, R. S.; Gutman, D. Kinetics of the Reactions of Methyl, Ethyl, Isopropyl, and tert-Butyl Radicals with Molecular Chlorine. J. Phys. Chem. 1986, 90, 2987−2991. (32) Troe, J. Predictive Possibilities of Unimolecular Rate Theory. J. Phys. Chem. 1979, 83, 114−126. (33) Troe, J. Theory of Thermal Unimolecular Reactions in the Falloff Range, I. Strong Collision Rate Constants. Ber. Bunsenges. Phys. Chem. 1983, 87, 161−169. (34) Troe, J.; Ushakov, V. G. Revisiting Falloff Curves of Thermal Unimolecular Reactions. J. Chem. Phys. 2011, 135, 054304. (35) Troe, J.; Ushakov, R. G. Representation of “Broad” Falloff Curves for Dissociation and Recombination Reactions. Z. Phys. Chem. 2014, 228, 1−10. (36) Troe, J. Statistical Adiabatic Channel Model of Ion-Neutral Dipole Capture Rate Constants. Chem. Phys. Lett. 1985, 122, 425−430. (37) Hahn, D. K.; Klippenstein, S. J.; Miller, J. A. A Theoretical Analysis of the Reaction between Propargyl and Molecular Oxygen. J. Chem. Soc. Faraday Disc. 2001, 119, 79−100. (38) Rupper, P.; Sharp, E. N.; Tarczay, G.; Miller, T. A. Investigation of Ethyl Peroxy Radical Conformers via Cavity Ringdown Spectroscopy of the à -X Electronic Transition. J. Phys. Chem. A 2007, 111, 832−840. (39) Maergoiz, A. I.; Nikitin, E. E.; Troe, J.; Ushakov, V. G. Classical Trajectory and Statistical Adiabatic Channel Study of the Dynamics of Capture and Unimolecular Bond Fission. V. Valence Interactions between Two Linear Rotors. J. Chem. Phys. 1998, 108, 9987−9998. (40) Knepp, A. M.; Meloni, G.; Jusinski, L. E.; Taatjes, C. A.; Cavalotti, C.; Klippenstein, S. J. Theory, Measurements, and Modeling of OH and HO2 Formation in the Reaction of Cyclohexyl Radicals with O2. Phys. Chem. Chem. Phys. 2007, 9, 4315−4331.
Fleming, D. J. Addition Kinetics and Spin Exchange in the Gas Phase Reaction of the Ethyl Radical with Oxygen. J. Phys. Chem. 1996, 100, 6561−6571 and 16445. (12) Plumb, I. C.; Ryan, K. R. Kinetic Studies of the Reaction of C2H5 with O2 at 295 K. Int. J. Chem. Kinet. 1981, 13, 1011−1028. (13) Slagle, I. R.; Feng, Q.; Gutman, D. Kinetics of the Reaction of Ethyl Radicals with Molecular Oxygen from 294 to 1002 K. J. Phys. Chem. 1984, 88, 3648−3653. (14) Munk, J.; Pagsberg, P.; Ratajczak, E.; Sillesen, A. Spectrokinetic Studies of Ethyl and Ethylperoxy Radicals. J. Phys. Chem. 1986, 90, 2752−2757. (15) Kaiser, E. W.; Wallington, T. J.; Andino, J. M. Pressure Dependence of the Reaction C2H5 + O2. Chem. Phys. Lett. 1990, 168, 309−313. (16) Kaiser, E. W.; Lorkovic, I. M.; Wallington, T. J. Pressure Dependence of the C2H4 Yield from the Reaction C2H5 + O2. J. Phys. Chem. 1990, 94, 3352−3354. (17) Kaiser, E. W. Temperature and Pressure Dependence of the C2H4 Yield from the Reaction C2H5 + O2. J. Phys. Chem. 1995, 99, 707−711. (18) Atkinson, D. B.; Hudgens, J. W. Chemical Kinetic Studies Using Ultraviolet Cavity Ring-Down Spectroscopy Detection: Self-Reaction of Ethyl and Ethlyperoxy Radicals and the Reaction O2 + C2H5 → C2H5O2. J. Phys. Chem. A 1997, 101, 3901−3909. (19) Clifford, E. P.; Farrell, J. T.; DeSain, J. D.; Taatjes, C. A. Infrared Frequency-Modulation Probing of Product Formation in Alkyl + O2 Reactions: I. The Reaction of C2H5 with O2 between 295 and 698 K. J. Phys. Chem. A 2000, 104, 11549−11560. (20) Fernandes, R. X.; Luther, K.; Troe, J. Falloff Curves for the Reaction CH3 + O2 (+ M) → CH3O2 (+ M) in the Pressure Range 2− 1000 bar and the Temperature Range 300−700 K. J. Phys. Chem. A 2006, 110, 4442−4449. (21) Oum, K.; Sekiguchi, K.; Luther, K.; Troe, J. Observation of Unique Pressure Effects in the Combination Reaction of Benzyl Radicals in the Gas to Liquid Transition Region. Phys. Chem. Chem. Phys. 2003, 5, 2931−2933. (22) Luther, K.; Oum, K.; Sekiguchi, K.; Troe, J. Recombination of Benzyl Radicals: Dependence on the Bath Gas, Temperature, and Pressure. Phys. Chem. Chem. Phys. 2004, 6, 4133−4141. (23) Lee, C.; Luther, K.; Oum, K.; Troe, J. Pressure and Temperature Dependence of the Recombination of p-Fluorbenzyl Radicals. J. Phys. Chem. A 2006, 110, 2613−2612. (24) Eskola, A. J.; Lozovsky, V. A.; Timonen, R. S. Kinetics of the Reactions of C2H5, n-C3H7, and n-C4H9 Radicals with Cl2 at the Temperature Range 190−360 K. Int. J. Chem. Kinet. 2007, 39, 614− 619. (25) Fernandes, R. X.; Luther, K.; Troe, J. Experimental and Modelling Study of the Recombination Reaction H + O2 (+ M) → HO2 (+ M) between 300 and 900 K, 1.5 and 950 bar, and in the Bath Gases M = He, Ar, and N2. Phys. Chem. Chem. Phys. 208, 10, 4313− 4321. (26) Tyndall, G. S.; Cox, R. A.; Granier, C.; Lesclaux, R.; Moortgat, G.; Pilling, M. J.; Ravishankara, A. R.; Wallington, T. J. Atmospheric Chemistry of Small Organic Peroxy Radicals. J. Geophys. Res. 2001, 106, 12157−12182. (27) Wendt, H. R.; Hunziker, H. E. The UV Spectra of Primary, Secondary, and Tertiary Alkyl Radicals. J. Chem. Phys. 1984, 81, 717− 723. (28) Fagerström, K.; Lund, A.; Mahmoud, G.; Jodkowski, J. T.; Ratajczak, E. Kinetics of the Gas-Phase Reaction between Ethyl and Hydroxyl Radicals. Chem. Phys. Lett. 1993, 208, 321−327. (29) Khamaganov, V.; Karunanandan, R.; Rodriguez, A.; Crowley, J. N. Photolysis of CH3C(O)CH3 (248 nm, 266 nm), CH3C(O)C2H5 (248 nm) and CH3C(O)Br (248 nm): Pressure Dependent Quantum Yields of CH3 Formation. Phys. Chem. Chem. Phys. 2007, 9, 4098− 4113. (30) Noell, A. C.; Alconel, K. S.; Robichaud, D. J.; Okumura, M.; Sander, S. P. Near-Infrared Kinetic Spectroscopy of the HO2 and
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NOTE ADDED IN PROOF After submission of this article we extended our measurements, e.g. obtaining (k1 + k2)/10−13 cm3 molecule−1 s−1 = 2.24, 2.36, 3.28, 3.92, at [He]/1016 molecule cm−3 = 0.91, 1.55, 3.28, 6.41, resp., for T = 473 K. These points fit well to the dashed curve of Figure 4, modeled with k1 from the present work and k2 from ref 4. This suggests that the data from refs 5 and 13 are too high and no empirical modification of k2 from ref 4, which would have led to the solid curves in Figure 4, is required.
G
DOI: 10.1021/jp511672v J. Phys. Chem. A XXXX, XXX, XXX−XXX
Experimental and Modeling Study of the Temperature and Pressure Dependence of the Reaction C2H5 + O2 (+ M) → C2H5O2 (+ M) Ravi X. Fernandes,† Klaus Luther,‡ Gerd Marowsky,§ Matti P. Rissanen,∥ Raimo Timonen,⊥ and Jürgen Troe*,‡,§,# †
Physikalische-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany § Laser-Laboratorium Göttingen, Hans-Adolf-Krebs-Weg 1, D-37077 Göttingen, Germany ∥ Department of Physics, University of Helsinki, P.O. Box 64, FI 00014 Helsinki, Finland ⊥ Department of Chemistry, University of Helsinki, P.O. Box 55, FI 00014 Helsinki, Finland # Max-Planck-Institut für Biophysikalische Chemie, Am Fassberg 11, D-37077 Göttingen, Germany ‡
ABSTRACT: The reaction C2H5 + O2 (+ M) → C2H5O2 (+ M) was studied at 298 K at pressures of the bath gas M = Ar between 100 and 1000 bar. The transition from the falloff curve of an energy transfer mechanism to a high pressure range with contributions from the radical complex mechanism was observed. Further experiments were done between 188 and 298 K in the bath gas M = He at pressures in the range 0.7−2.0 Torr. The available data are analyzed in terms of unimolecular rate theory. An improved analytical representation of the temperature and pressure dependence of the rate constant is given for conditions where the chemical activation process C2H5 + O2 (+ M) → C2H4 + HO2 (+ M) is only of minor importance.
1. INTRODUCTION Because of its role in the oxidation of hydrocarbons in combustion and in atmospheric chemistry, the reaction between ethyl radicals and molecular oxygen has attracted considerable attention (for summaries of experimental results, see, e.g., refs 1 and 2). Being a complex-forming bimolecular reaction with several possible reaction channels, its rate and branching properties depend on the temperature, pressure, and the nature of the bath gas3 and even a multiexponential time dependence may be encountered.4 A full theoretical description of the kinetics is complicated and requires detailed knowledge of the potential energy surface as well as of collisional energy transfer. This knowledge is still fragmentary, and some empirical fitting is necessary. Nevertheless, the general features of the kinetics can be analyzed reasonably well (see, e.g., refs 4−9). On the experimental side, large ranges of pressure and temperature have been covered.1,2 At room temperature, the experimental pressures varied from very low values, such as realized in very low pressure reactor (VLPR) studies,10 up to 60 bar as applied in ref 11, with the range 1−1000 Torr being covered most extensively (see, e.g., refs 1, 2, 5, and 12−19). At pressures above about 1 bar, the reaction was found to be close to the high pressure limit of the “normal” falloff curve of the recombination reaction C2H5 + O2 ( +M) → C2H5O2 ( +M) © XXXX American Chemical Society
with a rate constant k1,∞. At moderately low pressures, the pressure-proportional low pressure rate constant k1,0 of the recombination reaction 1 was approached. At even lower pressures, a transition to the also pressure-dependent rate constant k2 of the chemical activation process C2H5 + O2 ( +M) → C2H4 + HO2 ( +M)
(2)
was observed. At high temperatures also the reactions C2H5O2 ( +M) → C2H4 + HO2 ( +M)
(3)
→C2H5 + O2 ( +M)
(4)
have to be considered. The present work first focuses on a particular aspect of the recombination reaction 1, i.e., in the approach of the high pressure rate constant k1,∞ at room temperature. At 1 bar, k1 in several bath gases was found15 to be within about 10% of an extrapolated k1,∞. No further pressure dependence was observed at pressures up to 60 bar.11 In the present work, we Special Issue: 100 Years of Combustion Kinetics at Argonne: A Festschrift for Lawrence B. Harding, Joe V. Michael, and Albert F. Wagner Received: November 21, 2014 Revised: January 22, 2015
(1) A
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profiles averaged over about 50 runs. We used the observation wavelengths 240 and 265 nm. Contrary to our HO2 and CH3O2 experiments,20,25 where absorptions of the reactants (H and CH3) at the observation wavelength 240 nm could be neglected, in the present experiments, absorptions of C2H5O2 and C2H5 always overlapped (more strongly at the absorption wavelength 240 nm than at 265 nm). This results in an absorption step from C2H5 directly after the photolysis flash, followed by the much slower exponential approach of the final absorption, which then was governed by overlapping C2H5O2 formation and C2H5 disappearance signals. As it was difficult to separate the initial absorption step from laser straylight, we reconstructed the initial step by means of literature values for the ratio of absorption coefficients ε(C2H5O2)/ε(C2H5). The absorption coefficient ε(C2H5O2) is well-known (see, e.g., ref 26), while there are more uncertainties in ε(C2H5) (see refs 14 and 27−29). We employed ε(C2H5O2)/ε(C2H5) ratios of 1.8 (±0.1) at 240 nm and of 7 (±1) at 265 nm. The signals S(t) then could be evaluated using the relationship S(t) = S(t=0) + [S(t=∞) − S(t=0)] exp(-k1t). Under our conditions, no selfreaction 2 C2H5O2 → C2H5O2C2H5 + O2 was observed.30 We estimate the accuracy of our measured values of k1 to be about ±10%, mostly determined by the uncertainty of the initial absorption step and, hence, the ratio ε(C2H5O2)/ε(C2H5). Most experiments, because of the larger absorption signals, were done at 240 nm. Table 1 summarizes the experimental high pressure values of k1 for 298 K from the present work. Our results between 100
extended the pressure even more (up to 1000 bar) in order to document the full transition to k1,∞. In doing this we were aware of the possibility that the reaction mechanism may change, becoming a superposition of the radical-complex (RC) and the “normal” energy transfer (ET) mechanism. High pressure experiments on other reactions have provided evidence for the onset of a RC mechanism, for the reaction CH3 + O2 → CH3O2 in Ar near 300 bar,20 for the reactions 2C7H7 → C14H14 and 2C7H6F → C14H12F2 in Ar near 10 bar.21−23 Reaction 1 might fall in between, such that studies of the high pressure end of the falloff curve of reaction 1 may provide information on the RC contribution as well. In order to test the expressions for the temperature and pressure dependence of k1 suggested, e.g., in ref 4 and to extend the database, we furthermore measured k1 down to temperatures of 188 K at pressures between 0.7 and 2 Torr. One expects that the falloff curves markedly shift with temperature such that details of recommended rate constants k1 can be well tested. In addition, we employed recent measurements24 of the rate constant k5 for the reaction C2H5 + Cl 2 → C2H5Cl + Cl
(5)
to recalibrate earlier measurements of the ratio k1/k5 from refs 15 and 17. On the basis of the previous and the present data, we intend to provide a refined representation of the pressure and temperature dependence of k1 in terms of unimolecular rate theory. As the theoretical modeling still requires empirical fitting of some parameters (such as average energies ⟨ΔE⟩ transferred in C2H5O2* + M collisions and transitional mode frequencies along the minimum-energy path (MEP) potential of reaction 1) and the experimental database is still limited, we employ simplistic but transparent models. In particular, we try to improve the falloff representation of k1. Properly analyzing k1,∞, which corresponds to the formation of the reaction complex C2H5O2*, is a necessary prerequisite also for the understanding of the full complex-forming bimolecular reaction, with competing back dissociation of C2H5O2* to C2H5 + O2 and forward rearrangement over an intrinsic barrier to C2H4 + HO2 (see refs 4−9). By considering only low temperature and medium to high pressure conditions, in the present work we limit ourselves to the addition reaction 1. Minor contributions from reaction 2 are accounted for by accepting its representation as given in ref 4.
Table 1. Experimental First-Order Rate Constants k1 at High Pressures from This Work (M = Ar, T ≈ 298 K) p/bar
k1/10−11 cm3 molecule−1 s−1
100 200 400 700 870 1000
0.79 0.83 1.02 1.03 1.08 1.23
and 200 bar of k1 ≈ 8(±1) 10−12 cm3 molecule−1 s−1 agree well with the results between 1 and 60 bar from refs 11, 14, and15 (an optimized value of k1 = 8.4 (±0.8) × 10−12 cm3 molecule−1 s−1 was obtained from fitting of the full falloff curve, see below; this value then was taken as the high pressure limit k1,∞ of the ET mechanism). We note that our measured k1 for higher pressures increase beyond this value of k1,∞. Analogous to our work20 on the reaction CH3 + O2 → CH3O2 we attribute this increase to a contribution from the RC mechanism. Figure 1 suggests that the data in M = N2 up to 60 bar from ref 11 do not show contributions from the RC mechanism, while the present results above about 400 bar do so. In order to support this conclusion, by plotting k/k∞ET, Figure 1 compares the present results with data for the CH3 + O2 → CH3O2 reaction20 in M = Ar and N2. It appears that the RC contribution sets in at slightly lower pressures for C2H5 + O2 → C2H5O2, but this conclusion will require further measurements. In particular, experiments with other bath gases appear desirable. Near to 1000 bar, the effects are of similar magnitude. We note that an onset of a decline of k1 due to diffusion control at the highest pressures, such as seen for benzyl dimerization in ref 22, was not yet observed.
2. EXPERIMENTAL TECHNIQUE AND RESULTS Our first series of experiments has been performed at Göttingen in the high pressure flow cell employed also in previous experiments over the pressure range 1−1000 bar (at temperatures 300−700 K for the reaction CH3 + O2 → CH3O2 in ref 20, and 300−900 K for the reaction H + O2 → HO2 in ref 25; for experimental details see these references). In the present work, ethyl radicals were generated by laser flash photolysis of azoethane. The latter was prepared by the method of Diels and Knoll. Argon (99.9999% from Air Liquide) was used as the bath gas M. The concentration of the reactant O2 was always kept high enough that nearly pseudo-first-order reaction conditions were realized; on the other hand, it was kept sufficiently low such that the reaction did not become too fast ([O2]/[Ar] = 10−5 − 10−4 and [C2H5]/[O2] ≈ 0.003 were employed). Azoethane was photolyzed with flashes from an ArF-laser at 193 nm. The formation of C2H5O2 was then recorded by UV absorption spectroscopy with absorption-time B
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k5 = 1.45(± 0.04) × 10−11(T /300 K)−1.73(±0.09) cm3 molecule−1 s−1
(6)
with the temperature dependence from ref 24.
3. MODELING OF RATE CONSTANTS IN THE FALLOFF RANGE In the present work we do not further analyze the experiments at very high pressures where the transition to the radical complex mechanism was observed and where binary collision models break down. Instead we only consider the pressure range where the usual energy transfer mechanism of thermal unimolecular reactions applies. In previous work like ref 5, the falloff curve of reaction 1 has been fitted in a simplified manner, equivalent to using the standard expression from ref 32 k /k∞ ≈ [x /(1 + x)]F(x)
with
Figure 1. High pressure rate constants k∞ at 298 K for the addition of O2 to CH3 and C2H5 with evidence of a contribution from the radicalcomplex mechanism (k∞ET = high pressure rate constant for the energy-transfer mechanism; data for O2 + C2H5 with k∞ET = 8.4 × 10−12 cm3 molecule−1 s−1, M = Ar (●) from present work and M = N2 (▲) from ref 11; data for O2 + CH3 with k∞ET = 2.2 × 10−12 cm3 molecule−1 s−1, M = Ar (○), and M = N2 (△) from ref 20; the [M] scale in ideal gas concentrations is for illustration only).
2
1/[1 + (log x / N ) ] F(x) ≈ Fcent
Table 2. Experimental First-Order Rate Constants k1 at Low Temperatures from This Work (M = He) P/Torr
[He]/1016 molecules cm−3
188
0.73 1.25 0.78 1.33 0.83 1.47 0.92 1.59 0.99 1.76 1.12 1.97
3.75 6.42 3.75 6.39 3.63 6.42 3.69 6.27 3.59 6.39 3.63 6.38
201 221 241 266 298
k1/10−12 cm3 molecule−1 s−1 4.12 4.59 4.30 4.51 3.04 3.96 2.76 2.76 1.87 2.09 1.36 1.98
(8)
where x = k0/k∞ and N ≈ 0.75−1.27 log Fcent. In previous studies, values of the center broadening factor Fcent in the range 0.3−0.7 were employed for 298 K, either obtained as fit parameters or derived from simple model calculations. In the present work, we refine the approach by modeling Fcent with the method of ref 33 for the strong collision broadening factor FSC cent and of refs 34 and 35 for the weak collision broadening factor SC WC WC FWC cent (with Fcent = Fcent Fcent, Fcent being derived from the analysis of master equation solutions for a series of model reactions). Suggested by the obtained value of Fcent, more advanced falloff expressions34,35 than eq 8 were also tested. Fitting the experimental falloff curves to eqs 7 and 8, highpressure rate constants k1,∞ and low-pressure rate constants k1,0 were obtained. In the following, we first analyze the experimental values of k1,∞, then proceed to falloff expressions for k1 ([M]), and finally analyze k1,0. We express k1,∞ as the product of kPST 1,∞ from phase space theory (PST) and a thermal rigidity factor f rigid(T),
Our second series of experiments was performed at Helsinki in a flow system using laser photolysis and photoionization mass spectrometry such as described in detail in ref 24. Experiments were done over the temperature range 188−298 K in the bath gas He. For each temperature, two pressures were studied, varying by about a factor of 1.7 within the range 0.7−2 Torr. Table 2 summarizes the results. Some increase of k1 with
T/K
(7)
(±0.25) (±0.33) (±0.33) (±0.29) (±0.34) (±0.25) (±0.12) (±0.06) (±0.15) (±0.06) (±0.04) (±0.17)
k1, ∞ = frigid (T )k1,PST ∞ kPST 1,∞
(9)
here is expressed by
3/2 ⎛ kBT ⎞⎛ h2 ⎞ ⎜ ⎟ = k1,PST ⎜ ⎟ Q cent ∞ ⎝ h ⎠⎝ 2πμkBT ⎠
(10)
with the centrifugal partition function Qcent given by ∞
Q cent =
∑ (2l + 1) exp[−E0(l)/kBT ] l=0
(11)
(see, e.g., ref 36). The centrifugal barriers E0(l) are derived from the MEP potential and hence depend on quantumchemical information. This is available for the O2−C2H5 potential from the B3LYP/6-31G* calculations of ref 37. Expressing the corresponding results in center-of-mass Morse form (with a bond energy of D/hc =14080 cm−1 and an equilibrium bond length re = 2.833 Å) leads to an effective Morse parameter β(r) expressed by
bath gas pressure is observed, but the increase is much weaker than proportional, corresponding to falloff curves nearer to the high pressure than to their low pressure limit. The results will be analyzed below in terms of unimolecular rate theory. We note that the measurements from ref 24, giving k5 = 1.45 (±0.04) × 10−11 cm3 molecule−1 s−1 at 298 K, for unknown reasons, led to markedly lower values than the older ones of k5 = 1.9 (±0.4) × 10−11 cm3 molecule−1 s−1 from ref 31. The latter were used in ref 15 to calibrate k1/k5 ratios. We recalibrate these ratios with our new k5, employing the expression
β(r ) ≈ 3.938 Å−1 exp[− 0.295 Å−1(r − re)]
(11)
The corresponding E0(l) then are expressed as C
DOI: 10.1021/jp511672v J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A E0(l)/hc ≈ 8600 cm−1[1(1 + 1)/105]1.4
n ≈ [ln 2/ln(2/Fcent)](0.8 + 0.2x q)
(12)
(for l > 80 and E0(l)/hc > 560 cm ; smaller values are obtained for l < 80). In addition, Qcent(T) is derived as Q cent(T ) ≈ 8890(T /300 K)n
(17)
where q = (Fcent − 1)/ln(Fcent/10). While the differences between the symmetric broadening factors of eq 8 and the asymmetric broadening factors of eq 16 are within the experimental uncertainty near room temperature (and thus are unimportant for the conditions considered in the present work), they matter when Fcent decreases with increasing temperature. Figure 2 compares experimental data obtained near 298 K in a variety of measurements, and the falloff expression of eqs 7
−1
(13)
with n ≈ 0.31 between 100 and 300 K (and n ≈ 0.55 between 300 and 2000 K). Combining eqs 10 and 13 gives kPST 1,∞ as −10 k1,PST (T /300 K)−0.19 cm 3 molecule−1 s−1 ∞ ≈ 9.6 × 10
(14)
(for T = 100−300 K and ∝ T for T = 300−2000 K). Comparing eq 14 with the experimental k1,∞ leads to the small experimental value of the thermal rigidity factor at 300 K of kPST 1,∞
0.05
exp f rigid (300 K) ≈ 10−2
(15)
f exp rigid
The magnitude of suggests that reaction 1 is governed by a rigid transition state with frequencies close to those of C2H5O2. The latter are chosen from ref 38, omitting the C−O stretching as the reaction coordinate. Calculating the effective number SK of transition state oscillators SK = 1 + Uvib‡/kBT and the effective parameter BK = [E0 + a(Eo)Ez‡]/kBT (with the zeropoint energy Ez‡ and the Whitten-Rabinovitch factor a(E0)), by the simple method of ref 33 leads to strong collision broadening factors FSC cent of the falloff curves k1([M]). One obtains the values given in Table 3. In order to determine weak Table 3. Modeled Center Broadening Factors Fcenta T/K
FSC cent
FWC cent,1 (M=N2)
FWC cent,1(M=He)
Fcent,1(M=N2)
100 200 300 400 500 600 800 1000
0.98 0.85 0.74 0.65 0.58 0.52 0.45 0.42
0.86 0.80 0.77 0.74 0.72 0.70 0.67 0.65
0.81 0.75 0.71 0.68 0.66 0.64 0.64 0.64
0.84 0.68 0.57 0.48 0.42 0.36 0.30 0.27
Figure 2. Dependence of k1 + k2 on the bath gas concentration [M] near 298 K (M = He: ⊗ (present work), ● (ref 15, absolute values), ○ (ref 15, from k1/k5 with k5 from eq 6); M = N2: □ (ref 15, from k1/ k5 with eq 6); M = Ar: □ with × inside (ref 18); M = C2H4: ■ (ref 11); solid line: modeled falloff curve for M = He with eqs 7, 8, 18, 19, and 24 for k1 and eq 20 for k2).
and 8 employing the calculated Fcent ≈ 0.53 (for M = He). The relative rate data k1/k5 from ref 15 were reevaluated with k5 from eq 6. Our new data together with measurements from refs 5, 12, and 15 fit well to the modeled falloff curve. We also note that measurements with M = Ar from ref 18 and with M = N2 from refs 11 and 15 within the scatter fall onto the same curve. The shown falloff curve corresponds to a fitted value of
See text, with −⟨ΔE⟩/hc ≈ 35 cm−1 for M = He and 60 cm−1 for M = N2.
a
collision broadening factors FWC cent, one needs to estimate the weak collision efficiencies βc, which follow from the analysis of the low pressure rate constants k1,0 given below. The collision efficiency βc depends on the temperature and the bath gas, see ref 32. The corresponding FWC cent, as calculated according to ref 34, are included in Table 3. We note that our modeled values of Fcent for M = N2, in spite of a quite different approach, for T = 100−600 K are not far from the results of ref 4. At larger temperatures, however, Fcent from ref 4 decreases markedly below the present results. We note that the present approach allows us to account for the bath gas dependence of Fcent. At 300 K, Fcent is large enough that eqs 7 and 8 still can be used; for larger temperatures, when Fcent falls below values of about 0.5, however, “broader” falloff curves are encountered with asymmetric broadening factors F(x) instead of eq 3.2 (i.e., F(x) then differs from F(1/x)). Here we tested the expression suggested in ref 36, i.e., F(x) ≈ (1 + x)/(1 + x n)1/ n
k1, ∞ = 8.4(± 0.5) × 10−12 cm 3 molecule−1 s−1
(18)
in agreement with the high pressure measurements above 1 bar given above. The fitted k1,0 = [He] 6.8(± 2) × 10−29 cm 6 molecule−2 s−1
(19)
will be analyzed below with respect to the collision efficiency βc. In obtaining k1,0, we note that Figure 2 shows k1 + k2 as a function of [M] such that the value and the pressure dependence of k2 needs also to be specified. As k2 under the present conditions is much smaller than k1, we employ the representation from ref 4, which gave k 2 = k 2,0(1 + x 2)−1F2(x 2)
(20)
with x2 = k2,0[M]/k2,∞, M = N2, F2(x2) given by F(x) from eq 8 with Fcent,2 = 0.45, k2,0 = 3.3 × 10−13 cm3 molecule−1 s−1, and k2,∞ = 4.0 × 105 s−1. With these values one estimates k2 (1 Torr) ≈ 2.4 × 10−13 cm3 molecule−1 s−1 such that k2/k1 ≈ 0.1 at 1 Torr in agreement with measurements from ref 16. As k2
(16)
with D
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Further considering the temperature dependence of k1,0, we compare experimental data from the present work and of ref 5 with theoretical predictions based on the formalism of ref 32 (after having fitted k1,0 for 298 K in Figure 2 by eq 19). First we calculate the low pressure strong collision rate constant kSC 1,0 following ref 33. The rotational factors Frot contained in kSC 1,0 employ the centrifugal energies E0(l) from eq 12. We assume the anharmonicity factor Fanh to have a typical value of Fanh ≈ 1.7. The calculation then is straightforward and leads to kSC 1,0. The comparison with the experimental k1,0 at 298 K gives the collision efficiencies of βc = 0.07−0.15 and 0.1−0.2 for M = He and N2, respectively. Using βc/(1 − βc1/2) ≈ −⟨ΔE⟩/kBT one obtains −⟨ΔE⟩/hc = (30−60) cm−1 for M = He and (60−120) cm−1 for M = N2 (we note that these values are not independent of the value chosen for Fanh and of the falloff fittings, but they appear to be of reasonable magnitude). We further assume that ⟨ΔE⟩ is temperature independent such as often observed, which allows us to predict the temperature dependence of βc and hence of k1,0 (a temperature independent ⟨ΔE⟩ corresponds to ⟨ΔEdown⟩ ∝ T−1). Table 4 gives the
decreases roughly proportional with the pressure, its influence on Figure 2 is only minor. Next we consider the temperature dependence of k1,∞. While k1,∞ could well be obtained for 298 K from the falloff curve of Figure 2, most studies of the pressure dependence of k1 for other temperatures were too limited to allow for an independent fit of k1,0 and k1,∞. Only the experiments from refs 11 and 17 were done sufficiently close to the high pressure limit such that falloff corrections k1/k1,∞ based on only semiquantitative k1,0 (see below) could be applied. For example, the measurements of ref 11 in 1.5 bar of C2H4 (with modeled k1,0 employing −⟨ΔE⟩/hc ≈200 cm−1, see below) required falloff corrections of k1/k1,∞ ≈ 0.95 and 0.90 for T = 387 and 475 K, respectively, while the measurements of ref 17 in 580 Torr of He (with modeled k1,0 employing −⟨ΔE⟩/hc ≈ 35 cm−1; see below) required k1/k1,∞ ≈ 0.91, 0.72, and 0.49 for T = 262, 362, and 463 K. The latter values illustrate the marked shift of the falloff curves along the pressure scale when the temperature changes. This explains why the present measurements at 201 K in 1.33 Torr of He with k1/k1,∞ ≈ 0.49 are similarly close to the high pressure limit as the measurements at 463 K in 580 Torr of He. Combining the falloff-corrected data from refs 11 and 17 and the present work, one obtains a clear indication of a negative temperature coefficient of k1,∞ such as shown in Figure 3 (one should note that recalibrations of the
Table 4. Modeled Limiting Low Pressure Rate Constants k1,0/[M]a T/K 100 200 300 400 500 600 800 1000
k1,0/[He] cm6 molecule−2 s−1 4.9 3.3 5.8 1.5 4.6 1.6 2.2 3.8
× × × × × × × ×
−27
10 10−28 10−29 10−29 10−30 10−30 10−31 10−32
k1,0/[N2] cm6 molecule−2 s−1 6.4 4.3 7.8 2.0 6.2 2.1 3.0 5.2
× × × × × × × ×
10−27 10−28 10−29 10−29 10−30 10−30 10−31 10−32
See text, with Fanh ≈ 1.7 and −⟨ΔE⟩/hc ≈ 35 cm−1 for M = He and 60 cm−1 for M = N2.
a
results. Having in hand k1,0, k1,∞, and Fcent,1, we construct falloff curves and compare these with experimental data for k1 + k2 in Figure 4 over the range 201−473 K. As in Figure 2, we take k2 from ref 4. We find that the present experiments between 188 and 298 K agree well with the falloff curves employing the modeled k1,0 and k1,∞ from eq 21. The temperature dependence of the corresponding k1,0 over the range 200−300 K then can be represented in the form
Figure 3. High-pressure rate constants k1,∞ for C2H5 + O2 → C2H5O2 (○: falloff extrapolation from Figure 2; ■: values in 1.5 bar of C2H4 from ref 11 after falloff correction; ⊗: values in 1−2 Torr of He from present work after falloff correction; --- and ___: representation of eq 21).
k1,0 = [He] 6.8 × 10−29(T /298 K)−4.2 cm6 molecule−2 s−1 (22)
Experimental results at T > 298 K agree less well with the modeled falloff curves after relative rate data for k1/k5 have been recalibrated with the present k5; see the dashed curves in Figure 4. For low pressures, with increasing temperatures k1 and k2 are increasingly difficult to disentangle. Calculating k1 with the present values of eqs 21 and 22, we cured the problem by employing k2,∞ ≈ 6.85 × 10−12 T6.53 exp(420 K/T) s−1 from ref 4 and by multiplying k2,0 ≈ 2.34 × 10−17 T1.09 exp(994 K/T) cm3 molecule−1 s−1 from ref 4 with a factor of 3. We note that the experimental C2H4 yields in He at 298 K from ref 16 are well reproduced with these rate constants. (The factor of 3 should be omitted for M = N2 to meet the C2H4 yields at 298 K from ref 16; we emphasize that this procedure has purely empirical fitting reasons as k2,0 is bath gas independent). Extending eq 22 to the range 300−500 K by our modeling gives
ratios k1/k5 from ref 17 were made using the present k5 from eq 6). Over the range 300−500 K, k1,∞ then is represented by k1, ∞ ≈ 8.4(± 0.5) × 10−12(T /298K)−1 cm3 molecule−1 s−1 (21)
while the temperature coefficient may change from −1 to −0.2 between 200 and 300 K. The negative temperature coefficient of k1,∞ is contrary to what would have been estimated with the statistical adiabatic channel model (combined with classical trajectories, SACM/CT) from ref 39 for a “standard potential” and a rigidity factor as low as the experimental value of 10−2 from eq 15. A positive temperature coefficient was also recommended in refs 4 and 7, contrary to ref 40. E
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k1,0 ≈ [He] 6.8 × 10−29(T /298 K)−4.2 cm6 molecule−2 s−1 (22)
between 200 and 300 K (with the exponent −4.2 changing to −4.9 for 300−500 K), k1, ∞ ≈ 8.4 × 10−12(T /298 K)−0.2 cm 3 molecule−1 s−1 (21)
between 200 and 300 K (with the exponent −0.2 changing to −1 for 300−500 K), and Fcent,1(M = He) ≈ exp( −T /470 K)
(24)
Fcent,1(M = N2) ≈ exp( −T /540 K)
(25)
or
A modification of eq 22 for M = N2 could not be made on the basis of the existing data. Employing eqs 21−24 for T = 300−500 K, we found that k2,0 from ref 4 had to be increased by a factor of 3 while k2,∞ could be retained.
Figure 4. Dependence of k1 + k2 on the bath gas concentration [He] for T = 201, 241, 298, 373(±12), and 468(±5) K (from top to bottom, ⊗: experiments from present work, at 201, 241, and 298 K; ○ experiments from ref 15 at 298 K; experiments from ref 17 at 362 (⊕) and 463 K (●); experiments from ref 5 at 385 (△) and 473 K (▲); experiments from ref 13 at 384 (▽) and 467 K (▼). Dashed curves: modeling with k1 from present work and k2 from ref 4; full curves with k1 from the present work and k2 of ref 4 empirically increased by a factor of 3; see text). −29
k1,0 = [He] 6.8 × 10
−4.9
(T /298K)
6
cm molecule
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: shoff@gwdg.de. Notes
The authors declare no competing financial interest.
■
−2 −1
s
ACKNOWLEDGMENTS Helpful discussions of this work with John Crowley, David Golden, Anatoly Maergoiz, James Miller, Michel Rossi, and Stephen Klippenstein are gratefully acknowledged.
(23)
A further extension of k1 to higher temperatures becomes less important because the overall reaction then is governed by reaction 2. The database appears too limited to specify the bath gas dependence of k1,0 and k2,∞ in further detail.
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REFERENCES
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4. CONCLUSIONS The present work has documented a contribution from the radical complex mechanism to the recombination of C2H5 with O2 at room temperature and at bath gas pressures of several hundred bar. This observation is analogous to what was observed earlier20 in the recombination of CH3 with O2. Taking this into account, the falloff curve of the rate constant k1 for the reaction C2H5 + O2 → C2H5O2 could well be evaluated to fit the high pressure limiting value k1,∞ of the energy transfer mechanism. Having in hand the complete experimental falloff curve near 300 K, by analyzing k1,∞, strong collision broadening factors FSC cent,1 and, by analyzing k1,0, weak collision broadening factors FWC cent,1 were determined. Further analyzing k1,0 and assuming temperature independent ⟨ΔE⟩ for collisional energy transfer, WC the temperature dependences of Fcent,1 = FSC cent,1 Fcent,1 and of k1,0 were modeled. On one hand, this allowed us to estimate falloff corrections for high-pressure experiments at temperatures different from 298 K, which finally led to experimental k1,∞ between about 200 and 500 K. On the other hand, falloff curves could be modeled over this temperature range and compared with low pressure measurements such as shown in Figure 4. The good agreement between experiments and modeling in Figure 4 (after minor modification of k2,0) demonstrates the internal consistency of the present work. The results are then summarized in the following representation of the characteristic rate parameters F
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NOTE ADDED IN PROOF After submission of this article we extended our measurements, e.g. obtaining (k1 + k2)/10−13 cm3 molecule−1 s−1 = 2.24, 2.36, 3.28, 3.92, at [He]/1016 molecule cm−3 = 0.91, 1.55, 3.28, 6.41, resp., for T = 473 K. These points fit well to the dashed curve of Figure 4, modeled with k1 from the present work and k2 from ref 4. This suggests that the data from refs 5 and 13 are too high and no empirical modification of k2 from ref 4, which would have led to the solid curves in Figure 4, is required.
G
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