Quantum mechanical and multichannel RRKM studies of the reaction N₂O + O (³P)

1 Introduction

The reaction of oxygen atom O (³P) with nitrous oxide (N₂O) has been studied considerably because of the importance of the reaction in the formation of NO_x pollutants during combustion (Brasseur et al., 1999). The O (³P) + N₂O reaction has been investigated by many laboratories (Fontijn et al., 2000; Meagher and Anderson, 2000; Nishida et al., 2004]. Reasons for interest in this reaction and relevant references are presented in more detail in Refs. (Fontijn et al., 2000; Meagher and Anderson, 2000). This reaction can take place via two possible exothermic channels (Fontijn et al., 2000):

In this paper, first geometrical parameters, frequencies and energies of the reactants, intermediates, transition states and products for O (³P) + N₂O reaction are calculated. In the second step, the kinetic calculations have been carried out using the multichannel RRKM (Rice–Ramsperger–Kassel–Marcus) (Holbrook et al., 1996) for total and individual rate constants. Several features of this work are following: (1) For N₂O (reactant) two structures (a) and (b) (N⚌N⚌O (a) and N⚌N⚌O (b)) are investigated. (2) The potential energy profile surface was calculated at the CCSD (T) (Pople et al., 1987) and CCSD (full) levels and Aug-cc-pVTZ (Kendall et al., 1992), 6-311++G (2df, 2pd) and 311++G (3df, 2p) basis sets. (3) The rate constants were obtained over a wide temperature range of 1000–5000 K, and the dominant products were studied. (4) The dependence of the reaction rate constants and branching ratios on the temperature is discussed.

2 Computation method

Ab initio calculation was carried out using Gauusian 03 programs (Frisch et al., 2004) and the geometries of the reactants, transition states, intermediates and products were optimized at the MP2 (Frisch et al., 1990)/full 6-311++G (2df, 2p) level. The vibrational frequencies were calculated at the same level of theory to determine the nature of different stationary points and the Zero-Point Energy (ZPE). All the stationary points have been positively identified for the minimum (number of imaginary frequencies NIMAG = 0) or transition states (NIMAG = 1). The intrinsic reaction coordinate (IRC) (Gonazales and Schlegel, 1990) calculation confirms that the transition state connects the designated reactions and products. To obtain more reliable energies higher levels, CCSD (T) and CCSD (full) and a more flexible basis set, Aug-cc-pVTZ, 6-311++G (2df, 2pd) and 311++G (3df, 2p), were employed to calculate the energies of various species.

3 Results and discussion

In this research, the mechanisms and kinetics of O (³P) to N₂O were investigated by two structures (a) and (b) for N₂O. Fig. 1 shows the optimized geometries of all the stationary points at the MP2 (full)/6-311++G (2df, 2p) level of theory. The calculated relative energies at the different levels of theory and Zero-Point Energies (ZPEs) are listed in Table 1.

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Figure 1

Optimized geometries of all the reactants, intermediates, transition states and products for the reaction of O (³P) + N₂O at the MP2 (full)/6-311++G (2df, 2p). Bond lengths are in angstroms and angles are in degrees.

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The reaction paths included in the calculation are shown in Fig. 2. Schematics of potential energy diagram at the CCSD (full)/6–311++G (2df, 2p) level of theory are presented in Fig. 3(R (a): O (³P) + N₂O (a)). Vibrational term values and moments of inertia for all species are listed in Table 2.

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Figure 2

The reaction paths were included in the calculation for the reaction of O (³P) + N₂O.

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Figure 3

Schematic energy profile of the potential energy surface at the CCSD (full)/6-311++G (2df, 2p) level of theory (R (a): O (³P) + N₂O (a)).

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4 Mechanism

As shown in Fig. 2, the reaction of N₂O with O (³P) starts with the formation of the N₂O₂ when the oxygen atom attaches to the nitrogen atom in N₂O via transition state TS1 to form energized intermediate IM1. Starting from IM1, three different reaction paths are explored.

Path 1:

In this path, IM1 can generate IM2 (NO⋯ON) via TS2, in TS2 the two 1N–3O and 4N–2O bonds are shortened by 0.101 and 0.109 Å, respectively, the 1N–4N bond is lengthened by 0.016 Å. The IM3 (NO⋯NO) is formed via TS3. The distances between atoms 2O–3O and 1N–4N increase from 2.100 and 2.140 Å in IM2 to 2.563 and 2.264 Å in TS3, respectively. As shown in Fig. 3, P1 was formed via IM3 by barrier less process.

Path 2:

This pathway proceeds by IM1 to form IM4 via TS4, in TS4 the lengths of bond 1N–3O increase to 1.674 Å and of 4N–2O to 1.561 Å, both bigger than in IM1, indicating almost complete breaking of these bonds. The IM4 may dissociate into N₂(S) + O₂(S) via channel IM4 → P3 (by TS5). Another possible path is formation of IM5 via TS6, in this channel the 1N–4N bond is shortened by 0.060 Å and the 2O–3O is lengthened by 0.132 Å. Then IM6 can be formed by TS7, in which the formation of 2O–3O bond is 1.354 Å, and the 1N–4N bond is lengthened by 0.064 Å. The P2, P4 and P5 were formed by barrier less process.

Path 3:

5 Rate constant calculations

As it is shown in Fig. 3 the reaction of N₂O + O (³P) can proceed via the saddle point TS1 leading to a common chemically activated intermediate (IM1). The various products originate from the unimolecular decomposition of this intermediate. In this research, the RRKM theory is used to calculate the rate constants for these products.

The individual rate constants for various product channels (R3–R6) are: $$k_{\text{IM}1}(\text{T,P})=\frac{{\alpha }_a}{h}\frac{{\text{Q}}_{\text{t}}^{+}{\text{Q}}_{\text{r}}^{+}}{{\text{Q}}_{{\text{N}}_2\text{O}}{\text{Q}}_{\text{O}}}{e}^{-E_{\text{a}}/\text{RT}\mathrm{\Delta }{E}^{+}}{\displaystyle \sum _{i=1}^{i_{\text{max}}}}\frac{\text{ω}}{Z}{\text{N}}_a(E^{+})e^{-E^{+}/\text{RT}}$$ $$k_{\text{IM}2}(\text{T,P})=\frac{{\alpha }_a}{h}\frac{{\text{Q}}_{\text{t}}^{+}{\text{Q}}_{\text{r}}^{+}}{{\text{Q}}_{{\text{N}}_2\text{O}}{\text{Q}}_{\text{O}}}{e}^{-E_{\text{a}}/\text{RT}\mathrm{\Delta }{E}^{+}}{\displaystyle \sum _{i=1}^{i_{\text{max}}}}\frac{k_2\text{ω}}{{\text{Y}}_{1}Z}{\text{N}}_a(E^{+})e^{-E^{+}/\text{RT}}$$ $$k_{\text{IM3}}(\text{T,P})=\frac{{\alpha }_a}{h}\frac{{\text{Q}}_{\text{t}}^{+}{\text{Q}}_{\text{r}}^{+}}{{\text{Q}}_{{\text{N}}_2\text{O}}{\text{Q}}_{\text{O}}}{e}^{-E_{\text{a}}/\text{RT}\mathrm{\Delta }{E}^{+}}{\displaystyle \sum _{i=1}^{i_{\text{max}}}}\frac{k_2{k}_4\omega }{X_3{Y}_{1}Z}{\text{N}}_a(E^{+})e^{-E^{+}/\text{RT}}$$ $$k_{\text{IM4}}(\text{T,P}k)=\frac{{\alpha }_a}{h}\frac{{\text{Q}}_{\text{t}}^{+}{\text{Q}}_{\text{r}}^{+}}{{\text{Q}}_{{\text{N}}_2\text{O}}{\text{Q}}_{\text{O}}}{e}^{-E_{\text{a}}/\text{RT}\mathrm{\Delta }{E}^{+}}{\displaystyle \sum _{i=1}^{i_{\text{max}}}}\frac{k_6\omega }{Y_{3}Z}{\text{N}}_a(E^{+})e^{-E^{+}/\text{RT}}$$ $$k_{\text{IM5}}(\text{T,P})=\frac{{\alpha }_a}{h}\frac{{\text{Q}}_{\text{t}}^{+}{\text{Q}}_{\text{r}}^{+}}{{\text{Q}}_{{\text{N}}_2\text{O}}{\text{Q}}_{\text{O}}}{e}^{-E_{\text{a}}/\text{RT}\mathrm{\Delta }{E}^{+}}{\displaystyle \sum _{i=1}^{i_{\text{max}}}}\frac{k_6{k}_9\omega }{Y_2{Y}_{3}Z}{N}_a(E^{+})e^{-E^{+}/\mathit{RT}}$$

With the following definition:

• X₁ = k₁ + k₂ + k₆ + k₁₃ + ω, X₂ = k₃ + k₄ + ω, X₃ = k₅ + ω,

• X₄ = k₇ + k₈ + k₉ + ω, X₅ = k₁₀ + k₁₁ + ω, X₆ = k₁₂ + ω,

In the above equations, α_a is the statistical factor (degeneracy) for the association step (Holbrook et al., 1996). ${\text{Q}}_{\text{t}}^{+}{\text{Q}}_{\text{r}}^{+}$ are the translational and rotational partition function of the variational “transition state” for the association. ${\text{Q}}_{{\text{N}}_2\text{O}}{\text{Q}}_{\text{O}}$ are the total partition function of N₂O and O, respectively. E_a is the barrier for the association. N_a(E⁺) is the number of state for the association “transition state” and s is the number of active degrees of freedom. In addition, in the RRKM calculations, a step size of ΔE⁺ = 0.8 kJ mol⁻¹is used. The energy-specific rate constants are calculated using the RRKM theory as follows:

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Figure 4

The MP2 (full)/6-311++G (2df, 2p) calculated intermolecular interaction energy between the IM1 and the helium bath gas. r_cm represents the separation of the center of mass of the IM1 and He.

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The RRKM calculations have been performed for both (a) and (b) conformations. Table 3 shows the individual rate constants over the temperature range of 1000–5000 K, at a pressure of 760 Torr and at the CCSD (T) and CCSD (full) levels and Aug-cc-pVTZ, 6-311++G (2df, 2pd) and 311++G (3df, 2p) basis sets, in comparison with the experimental results (Meagher and Anderson, 2000) and Ref. (Gonzalwz et al., 2001). In addition, Fig. 5 shows that the main products are 2NO, N₂ + O₂ and IM1, in both structure of (a) and (b). Also, From Figs. 2 and 3, we can see the decomposition of IM1 to yield 2NO and N₂ + O₂, thus the main channels are R (1) and R (2).

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Figure 5

The logarithm of calculated rate constants as a function of 10³/T for the O (³P) + N₂O reaction for two structures of N₂O N⚌N⚌O (a) and N≡N—O (b)).

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Table 3 indicates that at 1000 and 2000 K reaction rate of R(1) (k_2NO) and reaction rate of R(2) $(k_{{\text{N}}_2+{\text{O}}_2})$ for structure of (b) are in better agreement with available experimental value especially at the CCSD (T) level and Aug-cc-pVTZ, 6-311++G (2df, 2pd) and 311++G (3df, 2p) basis sets. Above 2000 K, the values of k_2NO for structure of (a) and the values of $(k_{{\text{N}}_2+{\text{O}}_2})$ for structure of (b) are in excellent agreement with available experimental values at different levels. In addition, the enthalpies of reaction paths for the formation of (a) and (b) at 298 K were calculated at the G1 (Pople et al., 1989; Curtiss et al., 1990) G2 (Curtiss et al., 1991), G3 (Curtiss et al., 1998), G2MP2 (Curtiss et al., 1993) and CBS-QB3 (Montgomery et al., 1999) levels of theory. These values are presented in Table 4 and compared with the experimental data. The values predicted by these levels show enthalpies of 2NO reaction for (a) and N₂ + O₂ reaction for (b) are in better agreement with available experimental data.

Therefore this reaction is dominated by two different reaction channels: channel O (³P) + N₂O → 2NO (R1), while N₂O has structure N⚌N⚌O and channel O (³P) + N₂O → N₂ + O₂ (R2) while N₂O has structure N≡N–O, these results match excellently with the experimental values and are shown in Fig. 6. Also, because of rate constants great discrepancies with experimental values for channel R (2) in Ref. (Curtiss et al., 1998) have been shown by these results. The branching ratios shown in Fig. 7 indicate that the major product is 2NO at temperature higher than 2000 K, while N₂ + O₂ is domain channel at temperature lower than 2000 K.

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Figure 6

The logarithm of calculated rate constants as a function of 10³/T for the O (³P) + N₂O reaction. ^aCalculations with structure of N⚌N⚌O (a), ^bCalculations with structure of N≡N—O (b), ^cRef. (Meagher and Anderson, 2000). ^dRef. (Nishida et al., 2004).

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Figure 7

The calculated branching ratio for the reaction of O (³P) + N₂O as a function of T (K), for two structures of N⚌N⚌O (a) and N≡N—O (b).

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6 Conclusions

In this research, the multichannel-RRKM theory is employed to calculate the rate coefficients for the N₂O + O (³P) reaction over the temperature range of 1000–5000 K. Various electronic structure theories were used to compute the energies of the stationary points on the potential energy surface of the reaction that lead to two major products. The results show the major channels are the addition of O to N₂O leading to an intermediate N₂O₂, which then decomposes to 2NO and N₂ + O₂, thus this reaction has two important product channels: 2NO and N₂ + O₂. The results show that at the temperatures higher than 2000 K, the formation of 2NO is the dominant channel, while N₂O has structure N⚌N⚌O, and at the temperatures lower than 2000 K, the formation of N₂ + O₂ is more favorable while N₂O has structure N≡N–O. These results match with the experimental values.