Introduction

Volatile organic compounds (VOCs) have been a major concern due to their chemical compositions within the atmosphere in the lower atmosphere and troposphere where aromatic hydrocarbons always contribute a significant fraction (about 20 %) of the total VOCs in the polluted atmosphere in the urban areas [1]. The main emission sources of the aromatics are motor vehicle exhaust, evaporation of solvents, and the emission in the gasoline stations [2–4]. In addition to the important role in the photochemical production of ozone, oxidation of aromatic compounds leads to the formation of secondary organic aerosol (SOA) in the urban air [5], which is well known to be harmful to human health [6].

Toluene is one of the most abundant aromatic hydrocarbons in the urban atmosphere [7]. Its most important loss process is initiated by OH radical during daylight hours and by nitrogen trioxide NO3 radical during evening and nighttime hours [8–10]. The reactions of toluene with OH have been studied widely both experimentally [11–17] and theoretically [18, 19]. The NO3-initiated toluene oxidation reactions have also been investigated in the experiments of Carter and co-workers [20] and Atkinson and co-workers [21–24]. The upper limit of the rate constant of NO3 + C6H5CH3 was reported as ≦3 × 10−15-8 × 10−15 cm3⋅molecule−1⋅s−1 determined at 300 ± 1 K in 1981 [20]. A smaller value of (1.8 ± 1.0) × 10−17 cm3⋅molecule−1⋅s−1 at 296 ± 2 K determined by using a relative rate technique was also reported, and suggested for the methyl H-abstraction constant in 1984 [21]. Shortly in the same year, it was refined to be (2.0 ± 1.1) × 10−17 cm3⋅molecule−1⋅s−1 at 298 ± 1 K with two different techniques by monitoring the enhanced decay rates of N2O5 and by employing a relative rate in the reaction system [22]. For the temperature at 296 ± 2 K, a slightly larger rate constant (7.8 ± 1.5) × 10−17 cm3⋅molecule−1⋅s−1 was reported in 1988 [23]. Recently in 2007, the reaction of NO3 radical with deuterium substituted toluene-d3 (C6H5CD3) and toluene-d8 (C6D5CD3) implied [24] that both hydrogen abstraction from the substituent group and the ortho addition may proceed at room temperature. It was also found that the ortho addition may decompose back to the reactants [24] but the rate constants have not been developed.

In the point of view of chemical reactions, the hydrogen atom can be abstracted either from the -CH3 group or from the benzene ring, and four different NO3-toluene adduct isomeric radicals can be formed. The H-abstraction abstractions will lead to the formation of benzyl and methylphenyl radicals which can associate with O2 and produce benzylperoxy and methylphenylperoxy radicals in the presence of molecular oxygen. These association processes are exothermic and the reaction enthalpies are 91.4 ± 4 kJ mol-1 [25] and 196.1 kJ mol-1 [26], respectively. Ref. [26] also reported that the methylphenyl radical can isomerize into the more stable benzyl radical via intramolecular hydrogen shift reaction at higher temperatures. Although the reaction paths and rate constants of benzyl and methylphenyl radicals with O2 have been explored [26, 27] the detailed information of the formation of benzyl and methylphenyl radicals in the reaction system of NO3-toluene has not been obtained so far.

This work is, thus, aimed at carrying out a theoretical investigation on all of the possible initial steps of NO3-toluene hydrogen abstraction reactions, in which some of the results can be compared with the available experiments, develop the temperature dependent rate constants, and explore the over-all reaction rate. In order to emphasize the details of the H-abstractions, the addition reactions of toluene and NO3 radical will be reported in a separate work.It should be noted that just after this paper had been accepted for publication we noted Huang and coworkers [28] published a theoretical study of a similar reaction system with CCSD(T)/6-311++G(d,p)//BHandHLYP/6-311++G(d,p) method. The main results will also be compared.

Computational methods

All the geometries of the stable species and the transition states (TSs) were fully optimized with the B3LYP [29, 30] density functional theory combined with the 6-311G(d,p) basis sets. The vibrational frequency analyses were performed at the same level to identify the minimum or the transition state with no or solely an imaginary normal mode vibration. The intrinsic reaction coordinates (IRCs) [31] were calculated to confirm the connection of the transition state and the respective minima with a step-size of 0.02 amu1/2bohr. In order to exam the reactions more reliably, the energies of all the species as well as the minimum energy paths (MEPs) were further refined with the accurate model chemistry method G3(MP2) [32, 33] at the B3LYP/6-311G(d,p) geometries, namely G3(MP2)//B3LYP/6-311G(d,p). Non-dynamic electronic correlations for all the species were examined with the complete active space self-consistent field (CASSCF) calculations [34–39] at CASSCF/6-311G(d,p) level with the active space (7,7) (seven active electrons distribute in seven active orbitals) for radicals and (6,6) for closed shell species. The dominant configuration coefficients of all the species are larger than 0.93 which indicates that the multi-reference-state feature can approximately be eliminated. Thermochemistry data were developed with the standard statistical thermodynamics in which the electron excitation energies (vertical) were calculated with the time dependent TD-B3LYP/6-311G(d,p) method and were truncated at 1.860 eV [40]. All electronic structure calculations were performed using the Gaussian-09 program [41]. The rate constants were evaluated in the temperature range of 200–2000 K by using the canonical variational transition-state theory (CVT) [42–45] and that incorporated with the small-curvature-tunneling (SCT) [46, 47] corrections proposed by Truhlar and co-workers combined in the POLYRATE 8.2 program [48], in which the necessary information should be supplied including the geometries and energies of the reactants, products, transition states, and those of a number of the selected points around the TS in the MEP as well as the energy gradients and the Hessian matrix for each of the points. For the reactions that do not have an energy barrier, the rate constants were only calculated with the CVT method by the VKLab program [49], where the SCT effect was neglected [50]. The over-all reaction rate (within 300–2000 K) was explored with the COMSOL software [51] in which a batch reactor was chosen.

Results and discussion

The structure as well as the symmetry and the electronic state for each of the reactants (Rs), intermediates (IMs), and the transition states (TSs) optimized with B3LYP/6-311G(d,p) are shown in Fig. 1, in which the numbering of the 19 species will be used in the further discussions. The G3(MP2) energies (internal energy at 0 K), enthalpy, and the Gibbs free energy corrections (H corr. and G corr. at 298.15 K without zero point energy, ZPE) and their relative (with respect to the sum of the reactants) values are listed in Table 1. The energy profiles in the potential energy surface are illustrated in Fig. 2 that involves the internal energies at 0 K and the Gibbs free energies at 298.15 K.

Fig. 1
figure 1

Structure, symmetry and the electronic state of the reactants, products, intermediates and transition states for the reaction of toluene with NO3 obtained with B3LYP/6-311G(d,p) calculations

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Table 1 G3(MP2) energy (sum of electronic energy and ZPE, in atomic unit: a.u.), the enthalpy correction (H corr., without ZPE) (a.u.), the Gibbs free energy correction (G corr., without ZPE) (a.u.) at 298.15 K and their relative (with respect to the sum of the reactants) values (in kJ mol-1) calculated with the standard statistical thermodynamics
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Fig. 2
figure 2

Energy profile for the hydrogen abstractions obtained with the G3(MP2)//B3LYP/6-311G(d,p) theoretical method. The data shown in the parentheses is the relative energy with respect to that of the sum of the reactants

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Hydrogen abstraction

For the reaction of toluene with NO3, a hydrogen atom can be abstracted from the side chain (methyl group) or from the benzene ring. It is shown in Fig. 2a that the lowest energy path is for the side chain H-abstraction reaction. At the entrance, a pre-reactive complex (CR, numbered 8, in 2A’ state) is formed with a weak van der Waals interaction. The internal energy is 2.4 kJ mol-1 lower than that of the sum of the reactants (R1 + R2). It should be helpful for a detailed analysis of the electronic structure of 8.CR. For this purpose, we performed a complete active space self-consistent field CASSCF calculation and obtained a dominant state 2A” that corresponds to an orbital alteration of the HOMO-3 and the LUMO of a beta electron. However, this state was finally confirmed less stable either at B3LYP/6-311G(d,p) or at MP2/6-311G(d,p) level of theory.

Subsequently, 8.CR may decompose into the products (3.HNO3 + 4.C6H5CH2) via 14.TS-s with a small energy barrier of 18.4 kJ mol-1. In 14.TS-s, the new forming O-H bond is about 1.416 Å, which is about 0.445 Å longer than that in the nitric acid molecule HNO3. The breaking C-H bond is 1.220 Å, about 0.124 Å longer than the equilibrium distance in toluene. The change of the C-H distance is thus much smaller than that of the O-H. This may predict that 14.TS-s is a reactant-like transition state or the reaction proceeds via an early barrier. The HNO3 molecule is now a less active species but the benzyl radical, a C2v perfect planar structure, has the spin density 0.7887 mainly localized on the dehydrogenated carbon atom in the -CH2 group. Its further degradation reactions that have not been involved in the scope of the present work must be a complicated but very interesting topic for future investigations.

The data in Table 1 also show that the whole reaction is thermodynamically favorable with 43.3 kJ mol-1 lowering of the energy, or is exothermic with 45.8 kJ mol-1 lowering of the enthalpy and is spontaneous with 40.4 kJ mol-1 lowering of the Gibbs free energy at 298.15 K.

For the abstraction of an ortho, para, and meta hydrogen atom from the benzene ring, the pre-reactive complex was not found but five transition states 15.TS-oa, 16.TS-ob, 17.TS-p, 18.TS-ma, and 19.TS-mb were located as shown in Fig. 1. Among those, both 15.TS-oa and 16.TS-ob are for the H-abstraction on the ortho position but distinguished with the positions of the H-bonds. This is due to that, in the rotation of the O18-N bond, different hydrogen bonds O19-H9 and O19-H14 can be formed. The relative energies of 15.TS-oa and 16.TS-ob are 50.1 and 47.3 kJ mol-1 with respect to the reactants, which are so close to each other that both of the paths should be taken into account for the ortho hydrogen abstraction reaction rate evaluations. In the 15.TS-oa and 16.TS-ob structures, the breaking C4-H12 bonds are 0.226 and 0.237 Å longer than the equilibrium distances, and the forming H12a-O18 and H12b-O18 bonds are 0.247 and 0.242 Å longer than the O-H bond in HNO3, respectively. These similar changes may not support an obvious early or later barrier. By performing the IRC calculations starting from the corresponding transition states, the van der Waals complexes 9.IM-oa and 10.IM-ob at the exit of the channels are found which have energies of 23.3 and 19.8 kJ mol-1 lower than their corresponding transition states. The C4-H12 bond distances are 2.012 and 2.011 Å in 9.IM-oa and 10.IM-ob showing a considerable interaction of the H-bond. By decomposing 9.IM-oa and 10.IM-ob, the same products, a less active 3.HNO3 and the 5.o-C6H4CH3 radical, can be formed with increasing energies of 22.8 and 22.1 kJ mol-1 but with decreasing Gibbs free energies of 10.3 and 12.0 kJ mol-1.

For the para position, only one transition state 17.TS-p, and the associated product complex 11.IM-p, was found due to the symmetry of toluene. The energy barrier is 51.4 kJ mol-1 and is also the highest barrier found in this work and, is also a value that is well known for a reaction to proceed at an adequate rate around room temperatures. The complex 11.IM-p is 29.0 kJ mol-1 higher in energy with respect to the reactants, and it may further decompose into 3.HNO3 and 6.p-C6H4CH3 with an energy absorption of 22.6 kJ mol-1 but with a Gibbs energy releasing of 12.8 kJ mol-1.

For the meta position, this work also found two transition states 18.TS-ma and 19.TS-mb as well as their produced complexes 12.IM-ma and 13.IM-mb are distinguished by the position of the H-bonds. The relative energies of these transition states are 48.6 and 50.4 kJ mol-1 higher than the reactants. Again, the small difference of the barriers represents that both of the paths should be taken into account in the further investigations.

It should be noted that, although the five complexes (numbered 9.IM-oa to 13.IM-mb) are stable in energy, their Gibbs free energies at 298.15 K are higher than those of the sum of the dissociated HNO3 and a radical (5.o-C6H4CH3 or 6.p-C6H4CH3 or 7.m-C6H4CH3) due to the increase of the entropy as shown in Fig. 2b. The relative (with respect to the reactants) Gibbs free energy for the dissociations are 48.0, 49.5, and 47.4 kJ mol-1, corresponding to the equilibrium constant in the magnitude of ppb, a value that is difficult to be observed experimentally. Compared with the process of side chain H-abstraction, the ring H-abstractions are less favorable since the energy barriers are at least 31.3 kJ mol-1 higher. Additionally, the products 3.HNO3 + 4.C6H5CH2 from the side chain abstraction are much more stable than either of the 9.IM-oa to 13.IM-mb complexes or their dissociated species. The negative relative Gibbs free energy (−40.4 kJ mol-1) corresponds to an equilibrium constant of 1.2 × 107 for the side chain H-abstraction reaction. This, therefore, confirmed thermodynamically that the side chain H-abstraction is dominant at least for 298.15 K.

Rate constants

The rate constants for all of the reaction steps were calculated with the CVT and CVT/SCT methods as mentioned in the previous section in which the G3(MP2)//B3LYP/6-311G(d,p) energies were employed both for the stationary points and for the minimum energy path (MEP). The temperature range is examined within 200–2000 K.

For the side chain H-abstraction, since a pre-reactive complex (CR) is produced with a very small energy decrease of 2.4 kJ mol-1, a two-step mechanism that involves a fast equilibrium between the reactants and the pre-reactive complex is assumed as:

$$ {\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{C}\mathrm{H}}_3+{\mathrm{NO}}_3\overset{K_{\mathrm{eq}.}}{\rightleftarrows}\mathrm{C}\mathrm{R}\underset{k_1}{\overset{k_{-1}}{\rightleftarrows }}{\mathrm{H}\mathrm{NO}}_3+{\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{C}\mathrm{H}}_2 $$
(1)

The forward rate constant for side chain hydrogen abstraction, k s, becomes:

$$ {k}_{\mathrm{s}}={K}_{\mathrm{eq}}{k}_1=\sigma \frac{Q_{\mathrm{CR}}}{Q_{\mathrm{toluene}}\cdot {Q}_{{\mathrm{NO}}_3}} \exp \left(\frac{E_{\mathrm{R}}-{E}_{\mathrm{CR}}}{RT}\right){k}_1 $$
(2)

where K eq. is the equilibrium constant between the reactants and the complex (CR), Q denotes the partition function of the related species, E R and E CR are the total energies (internal energy at 0 K) of reactants and CR, σ is the reaction symmetry number and R is the molar gaseous constant 8.314472 J · mol−1 · K−1.

For the ring H-abstractions, the processes should be described as:

$$ {\mathrm{C}}_6{\mathrm{H}}_5{\mathrm{C}\mathrm{H}}_3+{\mathrm{NO}}_3\left\{\begin{array}{l}\left.\begin{array}{l}\underset{k_{-2}}{\overset{k_2}{\rightleftarrows }}\mathrm{I}\mathrm{M}\hbox{-} \mathrm{o}\mathrm{a}\underset{k_{-2}\hbox{'}}{\overset{k_2\hbox{'}}{\rightleftarrows }}\\ {}\underset{k_{-3}}{\overset{k_3}{\rightleftarrows }}\mathrm{I}\mathrm{M}\hbox{-} \mathrm{o}\mathrm{b}\underset{k_{-3}\hbox{'}}{\overset{k_3\hbox{'}}{\rightleftarrows }}\end{array}\right\}\mathrm{o}\hbox{-} {\mathrm{C}}_6{\mathrm{H}}_4{\mathrm{C}\mathrm{H}}_3+{\mathrm{H}\mathrm{NO}}_3\\ {}\underset{k_{-4}}{\overset{k_4}{\rightleftarrows }}\mathrm{I}\mathrm{M}\hbox{-} \mathrm{p}\underset{k_{-4}\hbox{'}}{\overset{k_4\hbox{'}}{\rightleftarrows }}\mathrm{p}\hbox{-} {\mathrm{C}}_6{\mathrm{H}}_4{\mathrm{C}\mathrm{H}}_3+{\mathrm{H}\mathrm{NO}}_3\\ {}\left.\begin{array}{l}\underset{k_{-5}}{\overset{k_5}{\rightleftarrows }}\mathrm{I}\mathrm{M}\hbox{-} \mathrm{m}\mathrm{a}\underset{k_{-5}\hbox{'}}{\overset{k_5\hbox{'}}{\rightleftarrows }}\\ {}\underset{k_{-6}}{\overset{k_6}{\rightleftarrows }}\mathrm{I}\mathrm{M}\hbox{-} \mathrm{m}\mathrm{b}\underset{k_{-6}\hbox{'}}{\overset{k_6\hbox{'}}{\rightleftarrows }}\end{array}\right\}\mathrm{m}\hbox{-} {\mathrm{C}}_6{\mathrm{H}}_4{\mathrm{C}\mathrm{H}}_3+{\mathrm{H}\mathrm{NO}}_3\end{array}\right. $$
(3)

The numerical results of all the rate constants as a function of temperature within 200–2000 K are listed in Tables 2 and 3 and are shown in Fig. 3. In order to facilitate the reaction branching, a side chain rate branching ratio Γ that assumes all the reactions are irreversible can be defined as:

Table 2 Rate constants (CVT/SCT, in cm3 · molecule−1 · s−1) of the side-chain and the first step ring H-abstractions of toluene and NO3 radical as a function of temperature within 200–2000 K
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Table 3 Decomposition rate constants (CVT, in s−1) of the IMs formed in the first step ring H-abstractions
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Fig. 3
figure 3

Plots of the H-abstraction rate constants versus 1000/T (K−1) in the temperature range 200–2000 K. (a) is for the side-chain, (b) for ortho-a, (c) for ortho-b, (d) for para, (e) for meta-a, and (f) for the meta-b H-abstractions. Experimental values shown in panel (a) are from refs. [21, 22]

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$$ \varGamma =\frac{k_{\mathrm{s}}}{k}=\frac{k_{\mathrm{s}}}{{\displaystyle \sum {k}_i}} $$
(4)

with i = s, 2, 3, …, 6 as shown in Eqs. (2) and (3). The results are also listed in Table 2.

Figure 3 shows that all the CVT and CVT/SCT rate constants increase with increasing temperature or have positive temperature dependence. The CVT and CVT/SCT rate constants are basically close to each other within the temperature range 200–2000 K with the exception of the side chain H-abstraction that is quite different in its order of magnitude in the low temperature range. Interestingly, the CVT/SCT values of k s shown in Fig. 3a are very close to the available experimental points [21, 22].

In investigating the magnitude of the rate constants in Table 2, it is found that the reaction proceeds exclusively through side chain H-abstraction below room temperature or mainly through this path within 400–700 K. This result is consistent with the experimental findings that the NO3 radical only abstracts the side chain hydrogen at low temperatures [21, 24]. Our values, 1.40 × 10−17 cm3 · molecule−1 · s−1 at 296 K and 1.42 × 10−17 cm3 · molecule−1 · s−1 at 298 K are also in very good agreement with the observed side chain H-abstraction rate constants (1.8 ± 1.0) × 10−17 cm3⋅molecule−1⋅s−1 [21] and (2.0 ± 1.1) × 10−17 cm3⋅molecule−1⋅s−1 [22] at the correspondence temperatures. The theoretical value from ref. [28] is slightly larger in digit by 6.96 with the same order.

Although the experimental reaction rates for higher temperatures are unavailable, our results show that the ratio Γ for the side chain H-abstractions above 700 K decreases with increasing temperatures. For example, the ratio is 82.0 % at 700 K and is below 50 % (47.2 % at 1000 K, a smaller value from two more reaction channels (15.TS-oa and 19.TS-mb) on the ring compared with the larger theoretical value 68% of ref. [28]).

The relative values of the rate constants for the ring H-abstractions, as expected, are consistent with the magnitudes of the relative Gibbs free energies of the transition states that govern the rate constants in the CVT principles [42–48]. For example, k 2 for the rate constant via TS-oa is always smaller than the others due to TS-oa has the highest activation Gibbs free energy shown in Fig. 2b. Consequently, k 3 to k 6 have only slightly different values also due to the similar activation Gibbs free energies.

For the next decomposition of the ring H-abstractions intermediates, it is found in Table 3 that the rate constants k 2’ to k 6’ have very large values due to very small activation energies (Fig. 2a). This reason also leads to the very small change of the rate constants with increasing temperature according to the Arrhenius expression. It is interesting to find that the values for the decompositions of IM-ob (k 3’) and IM-mb (k 6’) are about 2–3 orders smaller than the others although the activation energies (22.1 to 23.6 kJ mol-1) are similar. This may mainly be from the differences of the entropy increases and of the positions of the variational transition state in the potential energy surfaces.

It should be helpful to have an analytical expression of the rate constants for the convenience of further reference. All the rate constants for both of the forward and the reverse reactions have been fitted into three-parameter Arrhenius expressions:

$$ k=A\times {T}^n\times \exp \left(-\frac{E}{T}\right) $$
(5)

The results within 200–2000 K are listed in Table 4.

Table 4 Fitted parameters A (in cm3 · molecule−1 · s−1 if not specified), n and E for Eq.(4), and the fitting correlation coefficient R obtained within 200–2000 K for all the reactions in Eqs.(1) and (3)
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It is found that the correlation coefficients of the fit are larger than 0.94 and most of them are larger than 0.99. The smaller values (about 0.95) are found for k 1, k −1 and k s in the side chain H-abstraction. This shows that the three-parameter expressions are in very good quality for further applications. However, the parameter E for either k s or k 1 is a negative value indicating that the rate constant for the decomposition of 8.CR as well as the over-all forward reaction of the side chain H-abstraction has non-Arrhenius characteristics. This might be from the larger tunneling effect at lower temperatures as shown in Fig. 3a. For the other elementary reactions having a transition state, the maximum error for the activation energies between the calculated in the fitted expressions and those shown in Table 1 (or Fig. 2a) is 10.2 kJ mol-1. For the elementary reactions without transition state, the activation energies are also comparable with the energy differences between the products and the intermediates. The results for the ring H-abstractions and dissociation reactions imply that the rate constant expressions are well Arrhenius characterized in the cases where the tunneling effect is small.

Reaction rate

In order to explore the over-all reaction rate, the concentration change of a reactant as a function of time needs to be developed. In the case of a complicated reaction, one needs to solve the coupled differential rate equations. Occasionally, an analytical expression can be obtained with, for example, the steady state approximation. However, the present work encountered that the concentrations of some intermediates were unable to be eliminated. Instead, numerical method can be employed and we simulated the reaction rate of NO3 and toluene in a batch reactor with the COMSOL [51] package. The temperature range of 300–1500 K is examined for detailed reference especially for the significance of the ring H-abstractions. All the reactions are considered reversible and the initial concentrations of toluene and NO3 radical are 4.9 × 10−5 and 8.3 × 10−5 mol/m3, respectively. These values are the equivalent concentrations of the experiments of ref. [22]. The results for the over-all, i.e., all the reactions in both Eqs. (1) and (3), consumption of toluene are plotted in Fig. 4, where 4(a) is for temperatures of 300–600 K and 4(b) is for 700–1500 K.

Fig. 4
figure 4

Concentration of the reactant toluene versus the reaction time for 300–1500 K calculated with COMSOL [51], (a) for 300–600 K, (b) for 700–1500 K

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Although a radical attacking reaction, possessing very low energy barrier, is generally a fast process, the reaction may also be slow if the concentration of a reactant is small. It is shown in Fig. 4a that the reactions are significant at room temperature with most of the reactant toluene to be consumed in about 30 minutes. The reaction rate increases, as expected, with increasing temperatures. It should be noted that the near-equilibrium concentration of toluene may have a larger value for a sufficient time at higher temperatures. For example, Fig. 4b shows that this occurs at about 15 s for 1500 K compared with that for 1300 K. This predicts that the over-all reaction is exothermic. For the reactions below 700 K, it is shown that the side chain H-abstraction is dominant which is consistent with the results obtained previously, e.g., the ratio Γ in Table 2, and the experiments [21, 24]. It should also be noted that the side chain H-abstraction is nearly irreversible since the concentration of toluene will approach a very low value (close to zero) for an extent of time below 700 K. For 700 K at 100 s, for example, the concentrations have a very small deviation of less than 0.17 % if the reaction is considered irreversible. This result is in good agreement of the much higher energy barrier in the reverse process shown in Fig. 2a.

The over-all reactions may have more complicated properties in the ring H-abstractions since three different products could be produced via five different parallel channels shown in Eq.(3). The concentrations of the products C6H5CH2, o-C6H4CH3, p-C6H4CH3 and m-C6H4CH3 as a function of reaction time for different temperatures of 300–1700 K are plotted in Fig. 5. It is shown that the concentration for the product of methyl H-abstraction C6H5CH2 is always larger than others. The concentrations for those of the ring H-abstractions increase with temperature but they are not significant for temperatures below 700 K. For temperatures above 800 K (which has not been shown in Fig. 5 but 700 and 900 K can be referred), it is expectable to have the ring H-abstraction products observed. However, their concentrations have a maximum at a very short reaction time. The existence of the maximum concentration implies that the ring H-abstraction reactions are obviously reversible. Within the ring products, the concentration of the meta H-abstraction m-C6H4CH3 is larger than the others for 900 K or 1100 K. However, that of the o-C6H4CH3 becomes larger above 1300 K.

Fig. 5
figure 5

Concentrations of the different products (C6H5CH2, o-C6H4CH3, p-C6H4CH3, and m-C6H4CH3) versus reaction time at 300–1700 K. (a) for 300 K, (b) for 500 K, (c) for 700 K, (d) for 900 K, (e) for 1100 K, (f) for 1300 K, (g) for 1500 K and (h) for 1700 K

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Therefore, the reactions within the system can be, qualitatively, regarded as the typical reaction below 700 K is the side chain H-abstraction and those above 800 K are the side chain and the ring H-abstraction challenging.

The logarithm overall reaction rates, donated by the consumption rates of toluene, versus the reciprocal temperatures are presented in Fig. 6. The nonlinearity relationship also shows that the over-all reaction follows a complex procedure. However, the apparent activation energy E app. of the reaction can be roughly estimated if the curve is divided into two parts, the side chain H-abstraction part below 700 K and the challenging reactions above 800 K. The fitted reaction rate r is lnr = −10.951–1862.9/T for 300–700 K and is lnr = −2.8811–7981.3/T for 800–2000 K with the correlation coefficients 0.9716 and 0.9934, respectively. This corresponds to the apparent activation energies E app. of 15.5 and 66.4 kJ mol-1, and the values are comparable with the activation energies shown previously in Fig. 2a.

Fig. 6
figure 6

Plots and fits of the overall reaction rate (denoted by the concentration decreasing rate of toluene for its initial concentration 4.9 × 10−5 mol/m3) versus 10,000/T (K−1) within 300–2000 K

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Conclusions

The hydrogen abstraction reactions of toluene by NO3 radical are studied theoretically by means of accurate ab initio and reaction kinetics methods. Both the side chain and the ring H-abstractions are examined and the properties of the reactions are explored in detail. It is found that the side chain H-abstraction has a larger tunneling effect and shows a non-Arrhenius characteristic. In contrast, the ring H-abstractions have smaller tunneling effect and follow the Arrhenius behavior well. The theoretical rate constant for the side chain H-abstraction at room temperature is in very good agreement with the available experiments. The over all reaction possesses a complicated reaction mechanism in which the side chain H-abstraction is dominant below 700 K and is nearly irreversible, while the ring H-abstractions are competitive above 800 K with observable products. The ring H-abstractions are also found typically reversible. The approximate apparent activation energies E app are 15.5 and 66.4 kJ mol-1 at 300–700 K and 800–2000 K, respectively.