Temperature and Pressure-Dependent Rate Constants for the Reaction of the Propargyl Radical with Molecular Oxygen.

Ab initio CCSD(T)/CBS(T,Q,5)//B3LYP/6-311++G(3df,2p) calculations have been conducted to map the C3H3O2 potential energy surface. The temperature- and pressure-dependent reaction rate constants have been calculated using the Rice-Ramsperger-Kassel-Marcus Master Equation model. The calculated results indicate that the prevailing reaction channels lead to CH3CO + CO and CH2CO + HCO products. The branching ratios of CH3CO + CO and CH2CO + HCO increase both from 18 to 29% with reducing temperatures in the range of 300-2000 K, whereas CCCHO + H2O (0-10%) and CHCCO + H2O (0-17%) are significant minor products. The desirable products OH and H2O have been found for the first time. The individual rate constant of the C3H3 + O2 → CH2CO + HCO channel, 4.8 × 10-14 exp[(-2.92 kcal·mol-1)/(RT)], is pressure independent; however, the total rate constant, 2.05 × 10-14 T0.33 exp[(-2.8 ± 0.03 kcal·mol-1)/(RT)], of the C3H3 + O2 reaction leading to the bimolecular products strongly depends on pressure. At P = 0.7-5.56 Torr, the calculated rate constants of the reaction agree closely with the laboratory values measured by Slagle and Gutman [Symp. (Int.) Combust.1988, 21, 875-883] with the uncertainty being less than 7.8%. At T < 500 K, the C3H3 + O2 reaction proceeds by simple addition, making an equilibrium of C3H3 + O2 ⇌ C3H3O2. The calculated equilibrium constants, 2.60 × 10-16-8.52 × 10-16 cm3·molecule-1, were found to be in good agreement with the experimental data, being 2.48 × 10-16-8.36 × 10-16 cm3·molecule-1. The title reaction is concluded to play a substantial role in the oxidation of the five-member radicals and the present results corroborate the assertion that molecular oxygen is an efficient oxidizer of the propargyl radical.

Introduction

Small hydrogen-deficient free radicals (SHDFRs) such as C2H, C2H3, C4H5, and C3H3 are generally thought to play a vital role in the formation of aromatic compounds, polycyclic aromatic hydrocarbons (PAHs), and soot in combustion hydrocarbon fuels.[1−6] It is not difficult to distinguish the SHDFRs from ordinary free radicals due to the delocalization of the unpaired electrons in SHDFRs, i.e., they are localized at two or more sites in the radicals, leading to at least two resonant electronic structures that have equivalent properties. Because of the delocalization of the unpaired electrons, SHDFRs are more stable than other free radicals, e.g., having lower standard heat of formation. As a consequence, SHDFRs can exist in flames longer than ordinary free radicals, making their concentrations increase quickly in combustion environments. In such high concentrations, the SHDFRs can easily combine together to form larger hydrocarbon compounds in fuel-rich flames.[6−8] The reactions between SHDFRs and molecular oxygen have been paid attention to both experimentally and theoretically[9−18] in which the measured rate coefficients for the C2H + O2 reaction are found to be in the ∼4 × 10–11–5 × 10–10 cm3·molecule–1·s–1 range[9−11] with the branching ratio of CO:CO2 to be 9:1,[11] while the calculated rate coefficients for C2H3 + O2[14] and C4H5 + O2[19] reactions are (2–8) × 10–12 cm3·molecule–1·s–1 (500–2500 K) and around 2 × 10–13 cm3·molecule–1·s–1 (1000–2000 K), respectively. In addition, the rate constants for the reaction of C3H3 with O2, the most crucial competing reaction with the ring-forming process, have also been made by several experimental and theoretical studies, e.g., the high-pressure limit value k∞ = (2.3 ± 0.5) × 10–13 cm–3·molecule–1·s–1 at 295 K was measured by Atkinson and Hudgens[15] utilizing cavity ring-down spectroscopy, whereas the measured overall rate coefficient ktotal = 5 × 10–14 exp(−2.87 kcal·mol–1/RT) cm–3·molecule–1·s–1 was implemented by Slagle and Gutman[16] using a tubular reactor coupled with a photoionization mass spectrometer. The RRKM/ME high-temperature rate constant k(T) = 2.83 × 10–19 T1.7 exp(−1500 kcal·mol–1/RT) cm–3·molecule–1·s–1 (500 < T < 2000 K) was calculated by Hahn et al.[17] using the QCIST(T,full)/6-311++G(3df, 2pd) energies. At a high-temperature region, however, their data are found to be larger than the experimental values of Slagle and Gutman.[16] One possibility for the inconsistency between the experiment and theory may result from the activation-barrier deviation of some main transition states located on the low-lying energy channels leading to products of the C3H3 + O2 system. Hence, their relative energy values should be reinvestigated at higher level of theory to examine whether the calculated rate coefficients are indeed larger than the experimentally reported values at high temperatures. Moreover, in Hahn’s study, two reaction paths leading to CH2CO + HCO and HCCO + H2CO products were mentioned to be dominant at high temperatures. Nevertheless, several energetically accessible product channels, namely, CHCCO + H2O (−70.6 kcal·mol–1), CCCHO + H2O (−29.9 kcal·mol–1), and CHCCHO + OH (−44.2 kcal·mol–1), have not been investigated yet, even though similar reaction channels have been detected in the C2H3 + O2 system.[14] Another investigation on the C3H3 + O2 reaction performed by Dong et al.[18] in 2003 indicated that three nascent vibrationally excited products HCO, CO2, and CO were explored using time-resolved Fourier transform infrared spectroscopy. These new products were also confirmed by theoretical calculations at the B3LYP/6-31+G(d,p) level in their study.[18] Similar to Hahn’s study, Dong and co-workers did not discover important products such as OH and H2O of the reaction between the propargyl radical and molecular oxygen as mentioned above.

In the present work, the most comprehensive analysis to date of the C3H3O2 potential energy surface has been carried out. The desirable products OH and H2O have been found for the first time. The detailed reaction mechanism was clarified by quantum-chemical calculations at a high-level approach, CCSD(T)/CBS(T,Q,5)//B3LYP/6-311++G(3df,2p). The calculated high-accuracy relative energy values in conjunction with state-of-the-art RRKM/ME methods were employed to extract information about the temperature- and pressure-dependent rate constants and product branching ratios for all elementary reaction channels on the C3H3 + O2 PES. A comparison between the calculated results and the previous experimental and theoretical data[15−17] has also been carried out.

Computational Methods

The geometric structures and vibrational frequencies as well as zero-point vibrational energy (ZPE) corrections for all stationary points considered were obtained via the density functional theory (DFT) using the Becke-3 Lee–Yang–Parr (B3LYP) functional[20] in conjunction with the 6-311++G(3df,2p) basis set.[21] Intrinsic reaction coordinate (IRC)[22,23] calculations were carried out by the same optimization method above for each well-defined transition state to confirm the connection with the local minima (i.e., the reactants, intermediates, and products). The local minima were identified by all positive frequencies, whereas each saddle point must contain one imaginary frequency. The computed B3LYP/6-311++G(3df,2p) harmonic frequencies were scaled with a 0.971 factor[24] before being utilized for thermodynamic property computations. This factor was also used for various reaction systems,[25−37] and the energy values obtained were proved to agree closely with the experimental data. Single-point energies for all species were calculated by the CCSD(T) method[38] together with the aug-cc-pVnZ (n = T, Q and 5) basis sets[39−41] using the B3LYP/6-311++G(3df,2p) geometric structures as the input data. The CCSD(T)/aug-cc-pVnZ (n = T, Q, and 5) single-point energies were then extrapolated to the desired values at the complete basis set (CBS) limit and corrected with the ZPEs. The extrapolation model can be found in our previous study.[42] The CCSD(T) method is often considered to be the gold standard method of computational chemistry that gives accurate reproduction of experimental results with an error margin of about ±1 kcal·mol–1, especially when a complete basis set (CBS) extrapolation is utilized.[43,44] To check the accuracy of the CCSD(T)/CBS(T,Q,5) level, the thermodynamic properties, i.e., ΔfH298K, for all species involved in the C3H3 + O2 (X3∑g–) system were calculated and compared with the available literature data as shown in Table . The multireference character of the wavefunctions of each substance on the PES was checked via T1 diagnostic tests[45,46] at the RCCSD(T) and UCCSD(T) levels for the closed-shell and open-shell species, respectively, relied on the B3LYP/6-311++G(3df,2p) geometric structures. The Gaussian 16 software package[47] was used in all of the present quantum-chemical computations.

Table 1

Comparison of Formation Heats (at 298 K, in kcal·mol–1) of All Structures Related to the Title Reactiona with the Literature Datab

species	ΔH298K	species	ΔH298K
C3H3 (propargyl radical)	83.56 84.02 ± 0.39b	T4/15	36.47
IS1	–19.24	T5/6	1.87
IS2	–18.68	T6/6	–16.44
IS3	–1.77	T6/7	–41.66
IS4	–10.1	T6/10	–64.79
IS5	–11.94	T6/14	–57.65
IS6	–107.14	T6P5	–45.42
IS7	–82.7	T6P9	–79.67
IS8	–29.35	T7/14	–53.10
IS9	13.96	T7P3	–43.65
IS10	–112.36	T7P7	–11.56
IS11	–97.43	T8/6	–25.46
IS12	–9.89	T9/13	23.39
T0/1	9.43	T9P2	45.59
T0/2	6.64	T10P6	–108.16
T1/2	80.02	T12/11	13.55
T1/4	4.66	T14/10	–51.88
T1/6	29.01	T14P8	–23.18
T1/8	5.27	T15P11	–95.69
T1/9	46.28	HCCCH	131.23
			130.60 ± 0.16b
T1/12	3.36	CH3CHO	–39.32
			–39.57 ± 0.06b
T1/13	18.51	C2H3	70.43
			70.99 ± 0.07b
T1P4	29.59	H2O	–58.95
			–57.80 ± 0.01b
T2/3	15.13	CH2O	–25.63
			–26.10 ± 0.01b
T2/9	51.27	HCO	9.17
			9.98 ± 0.02b
T2P3	23.65	OH	8.15
			8.96 ± 0.01b
T3/5	22.66	CO	–51.78
			–52.10 ± 0.70b
T3/8	66.14	CO2	–95.02
T4/6	–5.38		–94.04 ± 0.01b

This work calculated at CCSD(T)/CBS(T,Q,5)//B3LYP/6-311++G(3df,2p) level of theory.

Values collected from active thermochemical tables (ATcT).68,69

The second-order P-dependent rate coefficients and product branching ratios for the C3H3 + O2 (X3∑g–) reaction were calculated with the computer code MESMER[48] using the state-of-the-art statistic Rice–Ramsperger–Kassel–Marcus (RRKM),[49−51] which solves the master equation (ME)[52−54] involving multistep vibrational energy transfers for the excited intermediate (C3H3O2)*. In the kinetic predictions for the reaction paths controlled by H-shift processes, the tunneling effect[55] has been considered, utilizing a one-dimensional asymmetrical Eckart potential. Density and sum of states (DOS/SOS) were computed by the Beyer–Swinehart algorithm[56,57] using the needed parameters including activation barriers, moments of inertia, and vibrational frequencies of the species involved. Several low-frequency vibrational modes of single bonds (C–C and C–O) were treated by hindered internal rotors (HIR) in which the V(θ) hindrance potentials as a function of torsional angle, θ, in agreement with the single bonds (i.e., C–C, C–O bonds) were definitely obtained at the B3LYP/6-311++G(d,p) level via the relaxed scans with an interval size of 10° for dihedral angles related to the rotations. The energy-transfer scheme was computed with the temperature-dependent exponential-down model with ⟨ΔEdown⟩ = 250 × (T/298)0.8 cm–1 for N2 as the bath gas.[58] In this work, the L–J parameters (ε/kB = 82 K and σ = 3.74 Å)[59] were set for N2, whereas the ε/kB = 389.4 K and σ = 5.14 Å values were estimated for the [C3H3–O2] system relied on CH3CO2CH3.[60] The T-, P-dependent rate coefficients have been calculated under the conditions of 300 ≤ T ≤ 2000 K and 0.1 ≤ P ≤ 760,000 Torr using the calculated CCSD(T)/CBS(T,Q,5)//B3LYP/6-311++G(3df,2p) energy values.

Results and Discussion

Equilibrium Constants

According to the study of Slagle and Gutman,[16] the C3H3 + O2 reaction proceeds by simple addition at a low-temperature region (T < 500 K); hence, an equilibrium between C3H3 and C3H3O2 was observed (C3H3 + O2 ⇌ C3H3O2). In the temperature range of 380–430 K, the equilibrium constants for this reaction were measured, being 2.60 × 10–16–8.52 × 10–16 cm3·molecule–1, which are found to be in good agreement with our calculated values, 2.48 × 10–16–8.36 × 10–16 cm3·molecule–1, as shown in Figure . The predicted equilibrium constants in the study of Hahn et al.[17] were also in accord with the experimental data of Slagle and Gutman;[16] however, the energy of C3H3O2 was adjusted to be 18.2 instead of 19.2 kcal·mol–1 calculated by the HL method (the high-level method described in detail in the study of Hahn et al.[17]). In this study, there is no need to adjust the C3H3–O2 bond energy because its value is 18.1 kcal·mol–1, calculated at the CCSD(T)/CBS level of theory. Moreover, association enthalpy at 298 K for the C3H3O2 complex was also in excellent agreement with the experimental value, 19.24 versus 18.88 ± 1.43 kcal·mol–1.

Figure 1

Equilibrium constant of the C3H3 + O2 ⇌ C3H3O2 reaction in the 450–368 K temperature range.

Potential Energy Surface and Reaction Mechanism

The CCSD(T)/CBS(T,Q,5)//B3LYP/6-311++G(3df,2p) potential energy surface (PES) for the C3H3 + O2 system is illustrated in Figure . The corresponding stationary-point geometric structures for the reactants, intermediates, and products optimized at the B3LYP/6-311++G(3df,2p) level are shown in Figure S1, while those of the saddle points are geometrically displayed in Figure S2. To be convenient for readers, geometric structures of some main species on the PES are shown in Figure . Single-point energies (atomic units) for all species computed at different quantum-chemical levels are summarized in Table S1. The calculated heats of formation (ΔH298K) of all stationary points in comparison with the available experimental values are documented in Table . Also in the SI file, the vibrational frequencies of all species involved are presented in Table S2, whereas their Cartesian reaction coordinates are shown in Table S3.

Figure 2

Detailed potential energy surface of the C3H3 + O2 system calculated at the CCSD(T,Q,5)/CBS//B3LYP/6-311++G(3df,2p) + ZPEs level of theory (energies are in kcal·mol–1).

Figure 3

Geometries of some main species optimized at the B3LYP/6-311++G(3df,2p) level (bond lengths are in Å, bond angles are in degrees).

In this study, the T1 diagnostics values of the open-shell and closed-shell species were considered at the restricted coupled-cluster RCCSD(T)/aug-cc-pVTZ level and the unrestricted coupled-cluster UCCSD(T)/aug-cc-pVTZ level, respectively, to probe the multireference character of the used wavefunctions. The computed T1 diagnostics values shown in Table S4 reveal that most of the closed-shell species hold the numbers less than 0.02 except for the 1HCCHCO (P2) species whose value is up to 0.029, while the T1 diagnostics values of almost open-shell species are in the range of 0.017–0.043 except for several values of 0.078, 0.057, 0.058, 0.082, 0.057, and 0.049 owned by T1/2, T1/13, T2/9, T3/5, T3/8, and T4/15, respectively. However, these transition states can be ignored when exploring the C3H3O2 system because their relative energies are so high, as discussed below. Hence, it can be said that the single-reference methods are absolutely suitable for the title reaction. In addition, the spin contamination numbers for all species are also presented in the table, which are in the range of 0.75–0.77 for the stationary points, while those for transition states appear to be more variable, ranging from 0.75 to 1.41 in which T0/1, T0/2, T1/2, and T5/6 have extreme spin contaminations, suggesting some need for caution in interpreting the energies for these species.

The C3H3 + O2 mechanism illustrated in Figure reveals that the oxygen molecule can attack either of the radical sites from the two resonance structures of propargyl (•CH2-C≡CH and CH2=C=C•H). On one hand, if the attack takes place on the •CH2 side, the reaction will proceed via a well-defined saddle point T0/2 with a barrier height of ∼3.5 kcal·mol–1. On the other hand, it must overcome a barrier height T0/1 of about 7 kcal·mol–1 as one of the oxygen atom of O2 adds on the C•H side. The values of T0/1 and T0/2 have been found to agree closely with the data of ts2 and ts1, being 7.1 and 3.7 kcal·mol–1, respectively, calculated by Hahn and co-workers[17] at the HL level. The presence of T0/1 and T0/2 for the C3H3 + O2 additions can be seen as a result of the loss of the resonance stabilization of both C3H3 and O2 upon the formation of the two addition complexes CH2=C=CH–OO (I1, −18.1 kcal·mol–1) and CH≡C–CH2-OO (I2, −17.6 kcal·mol–1). It can be realized from the PES that the addition barrier on the •CH2 side of C3H3 is a half lower than on the C•H side, which is suitable with the dominance of the corresponding resonance structure in the propargyl radical (60% of •CH2-C≡CH). The well depths of the two adducts I1 and I2 in this study are found to be in good agreement with those in the previous studies of Hahn et al.[17] (∼19 kcal·mol–1 each) and Slagle and Gutman[16] (18.88 ± 1.43 kcal·mol–1).

The T0/1 structure displayed in Figure S2 indicates that an O atom approaches the CH group at a distance of 2.13 Å, and the bond angle of ∠HCC is bent down to 154.6° from 180° to facilitate the bond formation of CH–O. The IRC scan result illustrated in Figure S3 confirmed that T0/1 was located at the maximum point on the curve connecting the reactants and the I1 adduct. In the structure of T0/2, an oxygen atom comes close to the CH2 carbon atom of the propargyl radical at a distance of 2.22 Å, while the bond angle of ∠CCC slightly decreases by about 8° to facilitate the bond formation of CH2–O. The calculated vibrational frequencies also show that T0/2 has only one imaginary frequency with the value of 230i cm–1 (see Table S2). It should be noted that I1 can isomerize to I2 and vice versa via a well-defined transition state T1/2 characterized by an H-shift between the CH2 group and the CH group with the CH–H and C–H distances of 1.313 and 1.593 Å (see Figure S2), respectively. The relative energy of T1/2 was calculated to be 8.15 kcal·mol–1, which is in good agreement with the previous value (8.9 kcal·mol–1, ts7) shown in Hahn’s study.[17]

From the I1 intermediate state, there are various channels leading to different isomers and bimolecular products as can be seen in the PES in which the most notable is the reaction pathway giving a much stable isomer I6 (CH2COCHO) with the relative energy of nearly −106 kcal·mol–1. Compared to similar species (denoted as VIII) reported in Hahn’s study, the I6 intermediate in this study is about 2 kcal·mol–1 higher. There are two possibilities to make connection between I1 and I6. The first path goes directly via T1/6, which was not shown in Hahn’s study, while the second one must proceed via two TSs, namely, T1/4 and T4/6. In terms of energy, the latter is more favorable than the former because the relative energies of T1/4 and T4/6 are relatively lower than the T1/6 relative energy (6.0 and −4.2 versus 30.7 kcal·mol–1). According to the study of Hahn and co-workers, the energies of ts9 (II → IV) and IV (COO ring; CH side) calculated at the HL level are about 2.4 kcal·mol–1 lower than those of T1/4 and I4 in this study, while ts5 (IV → VIII) is 5.7 kcal·mol–1 higher than T4/6. It is worth noting that the I6 intermediate can also be formed by the multistep isomerization I2 → I3 → I5 → I6 in which the first and second steps must go over the high TSs T2/3 and T3/5 of 16.6 and 24.5 kcal·mol–1, respectively, while the third step needs to overcome the T5/6 saddle point of 3.2 kcal·mol–1. Obviously, compared to the prior two-step channel I1 → I4 → I6, the multistep channel is less dominant due to the high barriers. It should be noted that the two TSs T2/3 and T3/5 appear to be 1.5 and 6.5 kcal·mol–1 higher than the corresponding TSs ts8 and ts4 of the prior investigation carried out by Hahn and co-workers at the HL level, and the step I5 → I6 going via T5/6 was not shown in Hahn’s study. In addition to the two isomers I2 and I6 as mentioned, several intermediates I8, I9, I12, and I13 were also created from I1 by one-step isomerization processes for which I8 (CH2COOCH, 4-membered −COOC– ring) was formed via a transition state T1/8 by a ring closure between the outermost O atom and the single C atom in the structure of I1. The structure of T1/8 (Figure S2) shows that the new bond of C–O was formed at a distance of 1.986 Å with the imaginary frequency of 597i cm–1, while the ∠CCC bond angle decreases ∼18° from 178.9° in the I1 geometry. The I1 → I8 isomerization can proceed easily because the relative energy of T1/8 is only 6.9 kcal·mol–1, which is 1 kcal·mol–1 larger than the ts3 (II → VII) of Hahn et al.,[17] and this process was calculated to be exothermic by ∼8.5 kcal·mol–1. After forming the I8 isomer, the reaction channel can go to the I6 via the T8/6 saddle point by cutting the bond between two oxygen atoms of I8. The optimized structure of T8/6 shows that the O–O bond stretches at a distance of 1.623 Å with a vibrational mode of 1908i cm–1. In terms of activation energy, the forward isomerization I8 → I6 is roughly five times smaller than the backward one, I8 → I1; therefore, it can be said that the former takes place more quickly than the latter (∼80% faster). The relative energies of T8/6 and I8, −23.7 and −27.7 kcal·mol–1, have been found to be rougly 3 kcal·mol–1 higher than those of ts12 (−26.9 kcal·mol–1) and VII (−30.5 kcal·mol–1) indicated in the study of Hahn et al. The highest intermediate I9 (CHCHCHOO) located at 15 kcal·mol–1 above the reactants was produced when I1 went through the saddle point T1/9 with the 57.6 kcal·mol–1 relative energy. Apparently, this intermediate is difficult to be formed at room temperature due to the high activation barrier (∼57 kcal·mol–1). This is reasonable because the transfer of an H atom from the CH2 group to the single C-atom position will make the structure of I9 much less stable than that of I1. The I9 intermediate is also created by I2 if it isomerizes via the T2/9 saddle point at 52.5 kcal·mol–1. Following the I9 stationary point is an unstable isomer, I13 (CH2COCHO), holding an over 10 kcal·mol–1 relative energy, which was created via a relatively high transition state T9/13 at a position of 25 kcal·mol–1. Moreover, the I13 isomer was also made by the I1 → I13 isomerization through the T1/13 transition state located at 20 kcal·mol–1. Compared to the I1 → I9 → I13 channel, the reaction path I1 → I13 is more favorable in energy. The I13 isomer can then give out the first product P1 (HCCCH + HO2) if it goes across the 80 kcal·mol–1 saddle point T13/P1. The reaction path giving the P1 product is the endothermic process by about 62 kcal·mol–1; therefore, this product cannot be formed at normal conditions. Unlike the P1 product, the exothermic product P2 (CHCHCHO + OH) lying at nearly 14 kcal·mol–1 below the reactants can be produced by the decomposition process I2 → P2 via the T9/P2 saddle point with about 10 kcal·mol–1 activation barrier. Another exothermic product, P3 (CHCCHO + OH), was produced either by the channel I2 → T2/P3 (25 kcal·mol–1) → P3 (−44 kcal·mol–1) or by the channel I6 → T6/7 (−40 kcal·mol–1) → I7 (−81 kcal·mol–1) → T7/P3 (−43 kcal·mol–1) → P3, with the second channel providing the dominant pathway. The PES shows that this product was made easier than the P1 and P2 products. However, the bimolecular product P4 (CH2CCO + OH), an isomer product of P2 and P3, was formed by an H-shift from the CH group to the outermost O atom of I1; this process was confirmed at the geometric structure of T1/P4 whose relative energy is 31 kcal·mol–1. The bond distances between the hydrogen atom and the C atom as well as the O atom were recorded to be 1.445 and 1.241 Å, while the imaginary frequency of T1/P4 was 2010i cm–1 calculated at the B3LYP/6-311++G(3df,2p) level. An isomer of I1, namely, I12 (5-membered −CH2CCHOO– ring), although less stable than I6 (−8.2 versus −105.8 kcal·mol–1), is formed quite easily because the transition state, T1/12, that it must overcome has only 5.3 kcal·mol–1 relative energy. The I12 was then converted to either a much stable intermediate I11 (5-membered −CHCHCHOO– ring) via an H-shift TS, T12/11, at 15.5 kcal·mol–1 (this process was not indicated in Hahn’s study) or a 3-membered −COC– ring intermediate I16 via a splitting TS, T12/16, at 8 kcal·mol–1. The T12/11 saddle point shows that the hydrogen atom moves from the CH2 group to the center C atom with the CH–H and C–H distances of 1.307 and 1.356 Å. This process makes I11 become more stable with −95.7 kcal·mol–1 relative energy. However, the T12/16 shows the splitting of the O–O bond at a distance of 1.825 Å. The route forming the I16 intermediate from the reactants also appears to be highly exothermic, being 80.5 kcal·mol–1. It is worth noting that the I16 stationary point can also be created by another dominant process I6 → I16 via the T6/16 transition state whose relative energy is −70.2 kcal·mol–1. This reation path is realized to be in good agreement with the corresponding path (VIII → X) in the study of Hahn and co-workers.

The two isomer bimolecular products, P5 (CH2CHO + CO) and P6 (CH3CO + CO), can be established from the I6 intermediate via different channels in which the P5 was created by an H-shift between the CHO group and the CO group of I6 with the shift distances of 1.570 and 1.396 Å as shown in the structure of the saddle point T6/P5 in Figure S2. Although this TS stands at a position that is about 62 kcal·mol–1 higher than the I6 stationary point, it is still lower than the entrance point ∼44 kcal·mol–1; hence, the P5 can be formed easily at ambient conditions. The reaction path giving rise to this product was considered to be the most exothermic by 116.5 kcal·mol–1, indicating that P5 is the most stable product. Unlike P5, the isomer product P6 can be formed either by multichannel I6 → I10 → P6 or I6 → I7 → I14 → I10 → P6 or I6 → I14 → I10 → P6. The first channel must go through the two transition states T6/10 (−63 kcal·mol–1) and T10/P6 (−107.5 kcal·mol–1), while the second and third channels proceed via four TSs (T6/7, T7/14, T14/10, T10/P6) and three TSs (T6/14, T14/10, T10/P6), respectively. All three channels pass through the same step I10 → P6, but the second and third channels must overcome the saddle points T6/7 (−40 kcal·mol–1) and T6/14 (−56 kcal·mol–1), which are higher than T6/10. Therefore, it can be confirmed that the P6 product was vigorously produced via the first channel. Compared to the I6 → P5 channel, the I6 → I10 → P6 channel prevails in energy; thus, the P6 product is more favorable than the P5 product. The two other isomer products P7 (CCCHO + H2O) and P8 (CHCCO + H2O) were also produced by the channels originating from the I6 intermediate for which the branch I6 → I7 → P7 made the P7 product, while the branch I6 → I14 → P8 made the P8 product. In terms of thermodynamics, P7 (−29.9 kcal·mol–1) is less stable than P8 (−70.6 kcal·mol–1), and the former is also unfavorable in comparison with the latter because the transition state T7/P7, which the former must overcome, is higher than the T14/P8 (−22 kcal·mol–1) on the I14 → P8 path about 12 kcal·mol–1. It should be noted that the T6/P9 saddle point whose relative energy is −78.8 kcal·mol–1 was identified to be very convenient for the conversion process I6 → P9 (CH2CO + HCO). Therefore, P9 is considered to be the most dominant product, even though this product is not as stable as P5, being about 29 kcal·mol–1 higher. Compared to the ts11 (VIII → P1) and P1 species in the study of Hahn et al., the corresponding T6/P9 and P9 species in this study are nearly 2 kcal·mol–1 larger. Another bimolecular product P10 (CH2O + CHCO) might be present if the I5 (CH2(O)C(O)CH) intermediate passes through the T5/P10 TS with the activation barrier of 4 kcal·mol–1. This barrier is seen to be 3 kcal·mol–1 lower than a similar barrier (IV → ts14) in the study of Hahn et al. Although the step I5 → P10 is advantageous in energy, the prior steps forming I5 from the reactants were disadvantageous; hence, this product can be ignored in the kinetic treatment for the C3H3 + O2 reaction. Similarly, the last bimolecular product P11 (C2H3 + CO2) was easily created in the I15 → P11 step with only 7 kcal·mol–1 barrier (−101.9 kcal·mol–1 of I15 versus −94.5 kcal·mol–1 of T15/P11), but it was difficult to form in the I1 → I15 step due to the high activation barrier of 57 kcal·mol–1 at the T1/15 TS. This product is also a very stable product with the relative energy being −108.5 kcal·mol–1; however, it is expected to be kinetically insignificant.

From the above-analyzed results, it can be concluded that among 11 bimolecular products of the title system, only several products (P3, P5, P6, P7, P8, and P9) can be established in normal conditions, the remaining products being negligible due to overcoming of the much high activation barriers. Of those products, P6 and P9 are considered to be the key products of the system. This conclusion will be clarified in the Rate Constants and Product Branching Ratios section.

Thermochemical Properties

To check the accuracy of the calculations, the calculated thermodynamic property (ΔH298K) for all species involved in the C3H3 + O2 system is presented in Table and compared to the literature data for several limited species (e.g., C3H3, CH3CO, HCCCH, CH2O, HCO, C2H3, H2O, HO2, CO, CO2, OH, and O2). It can be seen from Table that the computed values agree well with the available literature data within their deviations, e.g., the difference between our values and the ATcT data does not exceed 1.0 kcal·mol–1. Such good agreements on the calculated thermodynamic parameter show that the methods used in this study are a reasonably suitable choice for the title reaction.

Rate Constants and Product Branching Ratios

As discussed above, the reaction paths going through high energy barriers (such as T1/6, T1/9, T1/13, T1P4, T2/3, T2/9, T2P3, T3/5, T3/8, T4/15, T9/13, T9P2, T12/11, T12/16, T13P1) should not be accounted for in the rate-constant calculations; therefore, the second-order rate constants of the C3H3 + O2 reaction have been computed based on the following reaction pathsThe bimolecular rate constants of the addition reactions C3H3 + O2 → I1/I2 (k01 and k02) have been computed using the TST approach, while the second-order rate constants of the reactions C3H3 + O2 → products (k1–k6) have been calculated by utilizing the RRKM theory. All of the rate-constant calculations were implemented by the MESMER program.

The calculated results for k1–k6 in the temperature range of 300–2000 K at different pressures 7.6–760,000 Torr (N2) are shown in Tables S5–S9, whereas the plots of the temperature- and pressure-dependent rate constants and the branching ratios for the channels indicated are graphically shown in Figures –9.

Figure 4

Plots of the predicted individual rate constants of the C3H3 + O2 (RA) reaction forming P3, P5–P9 products in the 300–2000 K range and 7.6 Torr (N2). The k1, k2, and k3 lines are hidden by the k6 line.

Figure 9

Branching ratios of C3H3 + O2 (RA) → P3, P5–P9 reactions in the 300–2000 K temperature range and P = 760 Torr (N2). The k3 curve is hidden by the k6 curve.

Plots of the predicted individual rate constants of the C3H3 + O2 (RA) reaction forming the P3, P5–P9 products in the 300–2000 K range and 7600 Torr (N2). The k3 line is hidden by the k6 line.

Overall, as can be seen from Figures –8, the bimolecular rate coefficients (k1–k6) of the C3H3 + O2 reaction depend on both temperature and pressure. However, the dependence tends to be opposite, increasing with the increase in temperatures but reducing with increasing pressures except for the C3H3 + O2 → I1 → I6 → P9 (CH2CO + HCO) (k6) channel. The pressure-independent value of k6 increases from 3.61 × 10–16 to 2.30 × 10–14 cm3·molecule–1·s–1 when temperature goes up, covering the considered T-range. At P = 760 Torr, k6 holds the same value and shares the highest position with k3, with branching ratio increasing from 18 to 29% in the descending temperature range of 2000–300 K. Hence, it can be firmly said that P6 (CH3CO + CO) and P9 (CH2CO + HCO) are the two key products of the title reaction at ambient conditions. This result is found to be in good agreement with the analysis of the C3H3O2 PES (cf. Figure ) as discussed in the above section. Also, at 760 Torr, when the temperature increases up to 600 K, the rate constant of the channel C3H3 + O2 → I1 → I6 → P5 (CH2CHO + CO) (k2) becomes competitive with the k3 and k6 channels. Moreover, the competition for the first place will add the k1 channel if the temperature reaches 1100 K, leading to equal branching ratios of 18–21% in the reducing temperature regime of 2000–1100 K owned by four channels (k1, k2, k3, and k6). The 760 Torr rate constants of the k4 and k5 channels are less competitive compared to the above channels in the considered temperature region (cf. Figure ) in which the former occupies the lowest values starting from 6.37 × 10–21 to 1.22 × 10–14 cm3·molecule–1·s–1, followed by higher values of the latter holding the range of 1.51 × 10–18–2.14 × 10–14 cm3·molecule–1·s–1. Apparently, at the low-temperature region, the k4 and k5 rate constants were found to be much smaller than the first group, e.g., the 300 K values of k4 and k5 underestimate the k6 by about two and five orders of magnitude, indicating that the two products P7 (CCCHO + H2O) and P8 (CHCCO + H2O) have an inconsequential contribution to the general product formation of the C3H3 + O2 → product system. At the high-temperature region (T > 1800 K), however, the indirect formation of the P7 and P8 final products from the reactants becomes sufficient to compete with the formation of the remaining bimolecular products (see Figure ). This appears reasonable because high temperatures can help the reactants easily pass the T7P7 and T14P8 transition states. In the 300–200K range, branching ratio of CCCHO + H2O was computed to be 0–10%, whereas the value of CHCCO + H2O was recorded to be 0–17%.

Figure 8

Plots of the predicted individual rate constants of the C3H3 + O2 (RA) reaction forming the P3, P5–P9 products in the 300–2000 K range and 76,000 Torr (N2).

Figure 6

Plots of the predicted individual rate constants of the C3H3 + O2 (RA) reaction forming the P3, P5–P9 products in the 300–2000 K range and 760 Torr (N2). The k2 and k3 lines are hidden by the k6 line.

At higher pressures (P > 760 Torr), the dominance of the leading position is only the k6 as the competitors dwindled in value. Specifically, at 7600 Torr, the 300 K rate constants of k1–k5 were calculated to be 2.73 × 10–16, 1.04 × 10–17, 3.59 × 10–14, 6.37 × 10–22, and 1.52 × 10–19 cm3·molecule–1·s–1, respectively, while the value of k6 was still unchanged (3.61 × 10–16 cm3·molecule–1·s–1). At 400 K, k3 begins reaching the k6 value, being 1.22 × 10–15 cm3·molecule–1·s–1, whereas k1 and k2 only acquire the k6 value at 1700 and 1100 K, being 2.02 × 10–14 and 1.26 × 10–14 cm3·molecule–1·s–1, respectively. It is easy to realize that the rate constants k4 and k5 always underestimate the remaining values in the considered temperature and pressure ranges; the deviation between them increases with increasing pressure but reduces when the temperature increases (cf. Figures –8). At 76,000 Torr, the k3 rate coefficient cannot compete with the k6 value until the temperature reaches 1100 K, while the competition between the k2 and k6 values only happens at T ≥ 1700 K. For its part, the value of k1 lags behind in its inability to compete with k6 across the given temperature domain, e.g., the 300 and 2000 K rate constants of k1 were computed to be 2.86 × 10–18 and 2.28 × 10–14 cm3·molecule–1·s–1, respectively, which are 126 and 1.01 times less than those of k6, indicating that the k1–k5 rate constants show a negative pressure dependence. At lower pressures (P < 760 Torr), the k1–k3 values significantly increase, e.g., the 7.6 and 76 Torr rate coefficients of these channels have the same values as k6 in the whole temperature region (see Tables S5 and S6 and Figures and 5), reflecting that the products P3 (CHCCHO + OH), P5 (CH2CHO + CO), and P6 (CH3CO + CO) have a significant contribution to the total products of the bimolecular C3H3 + O2 → products reaction.

Figure 5

Plots of the predicted individual rate constants of the C3H3 + O2 (RA) reaction forming the P3, P5–P9 products in the 300–2000 K range and 76 Torr (N2). The k1, k2, and k3 lines are hidden by the k6 line.

The individual and total rate constants (in units of cm3 molecule–1 s–1) for the bimolecular C3H3 + O2 reaction in the temperature range of 300–2000 K at 760 Torr (N2) can be presented by the modified Arrhenius equation as followsRate constants at very low pressures (P = 0.7–5.56 Torr) in this study were also calculated and compared with the experimental data measured by Slagle and Gutman[16] in which the values at T < 380 K and P = 0.7 and 2 Torr are plotted in Figure , while those at T > 400 K and P = 1.04–5.56 Torr are graphically shown in Figure . It should be noted that at near room temperature, the C3H3 + O2 reaction proceeds by simple addition, forming the intermediate states CH2=C=CH–OO (I1) and CH≡C–CH2-OO (I2) as indicated in the PES section. These processes were also confirmed in the study of Slagle and Gutman. Therefore, the rate constants at T < 380 K of the title reaction were calculated for the channels C3H3 + O2 → I1 via T0/1 (k01) and C3H3 + O2 → I2 via T0/2 (k02). In the 250–375 K temperature range, the total rate constant (ktotal = k01 + k02) has been recorded in the ranges of 5.81 × 10–15–3.90 × 10–14 cm3·molecule–1·s–1 at P = 0.7 Torr and 2.33 × 10–14–5.41 × 10–14 cm3·molecule–1·s–1 at P = 2 Torr, i.e., ktotal (0.7 Torr) = 2.68 × 1060T–25.98±1.61 exp[(−13.22 ± 1.16) kcal·mol–1/RT] and ktotal (2 Torr) = 7.04 × 109T–8.25±0.22 exp[(−3.8 ± 0.13) kcal·mol–1/RT] cm3·molecule–1·s–1, T = 250–375 K. As can be seen from Figure , the computed values are in excellent agreement with the laboratory data, e.g., at 333 K, the calculated total rate constants at 0.7 and 2 Torr have been predicted to be 1.84 × 10–14 and 3.42 × 10–14 cm3·molecule–1·s–1, respectively, whereas the experimental values are 1.78 × 10–14 and 3.31 × 10–14 cm3·molecule–1·s–1, respectively. The maximum deviation between theory and experiment in this situation is only 4.8% occurring at T = 333 K and P = 0.7 Torr. Apparently, at near room temperature, the rate constants of the C3H3 + O2 reaction strongly depend on the pressure. This dependence was further evident when considering the rate constants of the reaction at T = 295 K and P = 0.1–100 Torr as shown in Figure . According to the figure, the calculated rate constants were found to be in good agreement with the experimental values measured by Slagle and Gutman[16] and Atkinson and Hudgens.[15] At T > 400 K, the calculated rate constants for the channel C3H3 + O2 → I1 → I6 → P9 (CH2CO + HCO) (k6) are found to be in good agreement with the measured data of Slagle and Gutman, cf. Figure , with the deviation being less than 7.8%, e.g., at 800 K, the predicted and experimental values are 7.67 × 10–15 and 7.55 × 10–15 cm3·molecule–1·s–1, respectively.

Figure 10

Calculated rate constants of the C3H3 + O2 reaction at very low pressures (P = 0.7 and 2 Torr) and T = 250–375 K in comparison with the experimental data measured by Slagle and Gutman.

Figure 11

Calculated rate constants of the C3H3 + O2 reaction at very low pressures (P = 1.04–5.56 Torr) and T > 400 K in comparison with the experimental data measured by Slagle and Gutman.

Figure 12

Calculated P-dependent rate constants of the C3H3 + O2 reaction at T = 295 K in comparison with the experimental data measured by Slagle and Gutman and Atkinson and Hudgens.

Conclusions

The detailed mechanism and kinetics of the C3H3 + O2 reaction have been intensively studied employing the accurate electronic structure calculations at the CCSD(T)/CBS(T,Q,5)//B3LYP/6-311++G(3df,2p) level of theory and the rigorous ME/RRKM kinetic model. The obtained PES indicated that the C3H3 + O2 reaction can first proceed via the addition mechanisms and then isomerize/dissociate to possibly create the bimolecular products, namely, P1–P11, in which P6 (CH3CO + CO) and P9 (CH2CO + HCO) were found to be the major products of the reaction. Several bimolecular products P2 (HCCHCO + OH), P3 (HCCCHO + OH), P4 (H2CCCO + OH), P7 (CCCHO + H2O), and P8 (HCCCO + H2O) have been found for the first time in this study. At a low temperature (T < 500 K), the reaction takes place by simple addition forming the C3H3O2 intermediate state. The equilibrium constants of the C3H3 + O2 ⇌ C3H3O2 reaction were identified to be in good accordance with the available literature data. The calculated results in this study were proved to be better than those of Hahn et al. at two points: (i) there is no need to adjust the energy of C3H3O2 as Hahn et al. did, and (ii) many new reaction paths and bimolecular products have been found in this study but not shown in the study of Hahn et al.

According to the predicted results, it can be concluded that the rate constants of the reaction generally depend on both temperature and pressure, increasing with increasing temperatures but reducing with increasing pressures except for the k6 channel whose rate constant, 4.8 × 10–14 exp[(−2.92 kcal·mol–1)/(RT)], only depends on temperature. At P = 760 Torr, k3 together with k6 has the largest values among the individual rate constants with the branching ratio increasing from 18 to 29% in the descending temperature range of 2000–300 K, confirming again that P6 and P9 are the two key products of the title reaction at ambient conditions. At higher pressures (P > 760 Torr), k6 monopolizes the number one spot. At lower pressures (P < 760 Torr), k1–k3 and k6 values strongly compete with each other, showing that the products P3 (CHCCHO + OH), P5 (CH2CHO + CO), and P6 (CH3CO + CO) have a significant contribution to the total products of the bimolecular C3H3 + O2 → products reaction. At very low pressures (P = 0.7–5.56 Torr), the calculated rate constants of the reaction have been found to be in good agreement with the experimental data measured by Slagle and Gutman with the maximum deviation being only 7.8%. It is recommended that the given detailed kinetic mechanism along with the computed rate constants and thermodynamic properties of the title reaction should be used for the modeling/simulation of both atmospheric and combustion applications.