Theoretical Researches of Kinetics and Anharmonic Effect for the Reactions Related to NO in the Ozone Denitration Process

For the reactions of NO with NO, NO₂, NO₃ and N₂O, the geometries of the reactants and TSs were optimized at the B3LYP/6-311++G(d,p) level. The B3LYP method is currently most cost-effective approach for anharmonic corrections.³² However, B3LYP method is not suit to calculate the reactions of NO with O, O₂ and O₃. M062X is the one of best functionals for a combination of main-group thermochemistry and kinetics. At the same time, the M062X functionals is recommended most highly for the study of main-group thermochemistry and kinetics.³³ Therefore, for the reactions of NO with O, O₂ and O₃, the geometries of the reactants and TSs were optimized at the M062X/6-311++G(d,p) level. Here, Vibrational harmonic and anharmonic frequencies were used to identify all of the stationary points. In order to obtain the valid geometry of TS, intrinsic reaction coordinate (IRC) was traced at the same level. Then, the single-point energies (SPEs) were worked out by the coupled cluster QCISD(T) method with 6-311++G(d,p) basis set. At least, in order to obtain the more exact and trustworthy analog data, the calculated barriers were revised by zero-point energy (ZPE). All of the ab initio calculations of frequency and energy were carried out by Gaussian 09 program.^29,34,35

Besides, the reactions calculated in this paper are mainly ground state reactions. States of relative species are considered by spin multiplicity (spin multiplicity=2S+1, S=u/2, S is spin angular momentum, u is single electron number) in the Table 1.³⁶ It is worth noting that spin multiplicity of ground states for O and O₂ are 3 (³O and ³O₂), and spin multiplicity of excited states for O and O₂ are 1 (¹O and ¹O₂). Therefore, for calculating relative energies and kinetics data, these two reactions (NO with O and O₂) are worked out considering ground state (²NO+³O→²NO₂, ²NO+³O₂→²NO₂+³O) and excited state (²NO+¹O→⁴NO₂, ²NO+¹O₂→²NO₂+¹O). Other reactions of this paper calculated are ground state reactions, furthermore, atoms/molecules without providing the information about the spin multiplicity are all ground state species.

According to the TS theory, it is well known that the rate constant k(T) expression for unimolecular reaction is expressed as:³⁷⁻⁴²

where k_b is Boltzmann’s constant, T is the temperature of the system, h is the Planck’s constant, and E^≠ is the activation energy. Q(T) and Q^≠(T) are partition functions of reactant and TS, respectively.

Similarly, the rate constant k(T) expression for bimolecular reaction can be expressed as below:

where Q_A(T) and Q_B(T) represents the partition function for the two reactants of A and B, respectively. The expression of the partition function is:

where q_i is the ith partition function of vibration mode, N is the number of the vibrational modes of reactant, b=1/k_bT, E_ni is ith vibrational mode energy. And the logarithmic function of Q can be expressed as follows:

Here, applies YL method to calculate the partition function, which takes advantage of Laplace transformation, inverse Laplace transformation and Morse oscillators (MO).^29,34,43 MO potential is applied to calculate the partition function. For MO,

where ω_i is the frequency of the ith vibrational mode, n_i is the vibrational quantum number of ith vibrational mode, and n_i(m)=1/2x_i-1/2. x_i is the anharmonic constant for various molecules, and ℏ=h/2π. Hence, the rate constant can be worked out by the corresponding expressions, and the anharmonic effect in the canonical case can be discussed in this paper.

According to the Arrhenius equation, the rate constant expression in the canonical ensemble is expressed as below:^44,45

where A represents pre-exponential factor, n represents the temperature exponent. E represents the activation energy of reaction, and R=8.314 J·mol^-1·K^-1. Then, the equation can be expressed as follows:

where, lnk=y, lnT=x, T=e^x, a₀=lnA, a₁=n, a₂=-E/R. Based on the least square method:

where m is the least numerical value of the right side of the equation, z is the total number of the rate constants corresponding to different temperatures. Subsequently, it is obvious that the differentiating values of b₀, b₁ and b₂ are zero, respectively.

So far, the kinetics parameters of Arrhenius formula can be worked out.

Concerning this reaction, the reaction equation was NO+O→NO₂, and the graph of the potential energy surface was shown in the Fig. 1. It can be seen that the barrier was 22.98 kcal·mol^-1 when the reaction was in ground state. Besides, this reaction was predicted to be exothermic and exergonic with ΔH=−65.32 kcal·mol^-1 and ΔG=−49.32 kcal·mol^-1 calculated at M062X method as shown in the Table 2. According to the obtained data, the harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were worked and listed in the Fig. 7. It can be seen that harmonic and anharmonic rate constants had the same order of magnitudes at the same temperature, and these two curves of rate constants were very close, which means the anharmonic effect was not obvious. Besides, in order to verify this calculation, the calculated rate constants by this work were compared with the researches by others. The obtained rate constants by Tsang were 1.81×10¹³ cm³·mol^-1·s^-1 at 300 K and 3.69×10¹² cm³·mol^-1·s^-1 at 2500 K,⁴⁶ and the result by Cobos was 13 order of magnitudes from 300 K to 1900 K.⁴⁷ Although the rate constants were up with the temperature rising, the rate constant of this reaction was only 10 order of magnitudes when the temperature reached 2800 K. Therefore, the excited state reaction of NO+¹O→⁴NO₂ were worked out. It was clear that the calculated values of excited state reaction were closer to references than that ground state. And, the rate constants were 13 order of magnitudes when the temperature reached 1900 K. Therefore, the calculated results by this work were reliable.

For the reaction channel of NO+O₂→NO₂+O, from the Table 3, this reaction was not easy to occur because of ΔH=48.43 kcal·mol^-1 and ΔG=50.16 kcal·mol^-1 calculated at M062X method, and the barrier was 83.49 kcal·mol^-1 at QCISD(T) method. The potential energy surface was shown in the Fig. 2, and the harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K were given in the Fig. 8. When the temperature was above 1300 K, the reaction can occur slowly. When the temperature was 1600 K, the anharmonic rate constant was 9.80×10³ cm³·mol^-1·s^-1. Besides, the rate constants calculated by this work were compared with Dumas’s for this reaction channel at 300 K.⁴⁸ Similarly, it can be seen that the calculated values of excited state reaction were closer to references than that ground state. Therefore, it was reliable that the ground state reaction of NO+O₂→NO₂+O was not easy to occur.

This reaction was not a one-step reaction. With regard to the detailed reaction progress, O₃ combined with NO to generate intermediate 1 (IM1) via the channel of TS3a firstly, and then IM1 cleaved into NO₂ and O₂ by the channel of TS3b. For the whole reaction process, the potential energy surface was shown in the Fig. 3. The related geometries of stationary point of this reaction channel were similar to that of trans PES channels by Julio.⁴⁹ Among them, the barrier for reaction process of channel TS3a was 6.12 kcal·mol^-1, and the barrier of channel TS3b was −4.11 kcal·mol^-1. Owing to the barrier-less reaction of the second step, the whole rate constant can be expressed by the first step. Besides, from the Table 4, this reaction was predicted to be exothermic and exergonic with ΔH=−64.54 kcal·mol^-1 and ΔG=−64.83 kcal·mol^-1 calculated at M062X method and ΔE=−50.01 kcal·mol^-1 at QCISD(T) method. Therefore, this reaction was crucial for the conversion process of NO.

The harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were given in the Fig. 9. Moreover, the rate constants obtained by this study were compared with DeMore’s and Borders’s for this reaction channel.^50,51 The rate constants of DeMore’s paper were 1.10×10⁹ cm³·mol^-1·s^-1 at 200 K and 1.13×10¹⁰ cm³·mol^-1·s^-1 at 300 K. The rate constants of Borders’s paper were 1.35×10⁹ cm³·mol^-1·s^-1 at 200 K, 1.18×10¹⁰ cm³·mol^-1·s^-1 at 300 K and 2.83×10¹⁰ cm³·mol^-1·s^-1 at 350 K. In this paper, the anharmonic and harmonic rate constant were 6 order of magnitudes at 200 K, and 9 order of magnitudes at 300 K, respectively. When the temperature was 350 K, the harmonic and anharmonic rate constant was 3.32×10¹⁰ cm³·mol^-1·s^-1 and 1.10×10¹⁰ cm³·mol^-1·s^-1. The rate constant calculated by this work were very close to that by Borders at 350 K. Therefore, this calculation was reliable. Moreover, when the temperature was 4000 K, the harmonic rate constant (1.10×10¹⁹ cm³·mol^-1·s^-1) differed from anahrmonic rate constant (1.31×10¹⁶ cm³·mol^-1·s^-1) 829.69 times. It can be seen that effect of anharmonic factor on rate constant were up with the increasing temperature, and the anharmonic effect at the high temperature range was more obvious than the low temperature range. Therefore, the anharmonic effect was very obvious and cannot be ignored especially in high temperature environment.

The researches on the reaction between NO and NO have not been studied deeply,^37,52,53 this work conducted research on this reaction generating the final products. As shown in the Table 5 and Fig. 4, firstly, NO combined with NO to form IM2 via the reaction channel of TS4a, then IM2 broke into two products via TS4b and TS4c, respectively. The barrier for reaction processes of channel TS4a, 4b, 4c were 74.58 kcal·mol^-1, 43.15 kcal·mol^-1, 42.46 kcal·mol^-1 at QCISD(T) method, respectively. Moreover, the reaction of NO+NO→N₂+O₂ was predicted to be exothermic and exergonic with ΔH=−41.58 kcal·mol^-1 and ΔG=−39.61 kcal·mol^-1, and the reaction of NO+NO→N₂O+O was predicted to be endothermic and endergonic with ΔH=36.39 kcal·mol^-1 and ΔG=41.78 kcal·mol^-1 calculated at M062X method. However, owing to the high barrier, the reaction NO+NO→N₂+O₂ was not easy to occur, even though this reaction was endothermic and endergonic.

According to the obtained data, the anharmonic and harmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were calculated. For TS4a, the rate constants were given in the Fig. 10(a). When the temperature was above 1300 K, the reaction can slowly occur. The rate constants were 4 order of magnitudes at 1600 K. And, when the temperature was 4000 K, the anharmonic rate constant was 1.32×10¹² cm³·mol^-1·s^-1. The harmonic rate constant was 8.79×10¹¹ cm³·mol^-1·s^-1. It can be seen that the curves of harmonic and anharmonic rate constants at the same temperature were relatively close, which indicates the anharmonic effect was not obvious. However, the rate constant changed rapidly at the low temperature range (300 K - 1000 K), and the rate constant changed gradually at the high temperature range (1000 K - 4000 K). In general, the rate constant at the high temperature range was still higher than the low temperature range. Besides, in order to verify this calculation, the rate constants obtained by Yuan, Freedman and Trung were compared with this calculation.⁵⁴⁻⁵⁶ It can be seen that the ate constant calculated by this work were similar to these references. Therefore, this work was reliable.

For TS4b and TS4c, the rate constants were given in the Fig. 10(b). It was clear that the rate constants were similar at the same temperature for these two reaction channels. As shown in the Fig. 10(b), the rate constant changed rapidly at the low temperature range (300 K - 1000 K), and the rate constant changed gradually at the high temperature range (1000 K - 4000 K). In general, the anharmonic effect was not obvious. However, the anharmonic effect at the high temperature range was still more obvious than the low temperature range. On the whole, the rate constant at the high temperature range was still higher than the low temperature range.

This reaction was a two-step reaction. Firstly, NO₂ combined with NO to form IM3 via the passage of TS5a. Then, IM3 was cleaved into N₂O and O₂ by TS5b. For the whole reaction process, the potential energy surface was shown in the Fig. 5. Among them, the barrier for reaction process of channel TS5a was 78.64 kcal·mol^-1, and the barrier for reaction process of channel TS5b was 82.93 kcal·mol^-1. It was clear that the reaction was difficult to occur because of the high barrier, even though this whole reaction process was exothermic and exergonic as shown in the Table 6 and Fig. 5.

For TS5a, the harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were given in the Fig. 11(a). the harmonic rate constant was 12 order of magnitudes at above 3100 K. the anharmonic rate constant was 10 order of magnitudes at above 3100 K. Therefore, the anharmonic effect of this reaction was obvious and cannot be ignored especially at high temperature range. And, the anharmonic effect at the high temperature range was still more obvious than the low temperature range. On the whole, the rate constant at the high temperature range was still higher than the low temperature range.

For TS5b, the harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were given in the Fig. 11(b). As shown in the figure, the rate constant changed rapidly at the low temperature range (300 K - 1000 K), and the rate constant changed gradually at the high temperature range (1000 K - 4000 K). In general, the anharmonic effect was not obvious. However, the anharmonic effect at the high temperature range was still more obvious than the low temperature range. On the whole, the rate constant at the high temperature range was still higher than the low temperature range.

For this reaction, the reaction equation was NO+NO₃→2NO₂, and the barrier of reaction channel was 1.94 kcal·mol^-1 from the Table 7. Therefore, this reaction was very easy to occur. The potential energy surface was shown in the Fig. 6, and the harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were given in the Fig. 12. Furthermore, rate constants calculated by this work were compared with the Atkinsons’ and Ashmores’ for this reaction channel.^57,58 When the temperature range was from 300 K to 4000 K, the reference value of rate constants obtained by Atkinson reached 13 orders of magnitude. And the data obtained by Ashmore was 11-12 orders of magnitude at 500-700 K. In this paper, the rate constant was 13 orders of magnitude at about 600 K. Therefore, this result was similar to that reference at low temperature range. Moreover, the rate constant was 17 orders of magnitude at about 4000 K. The anharmonic and harmonic rate constant was close at the same temperature. Therefore, the anharmonic effect was not obvious. In addition, the rate constants were up with the temperature rising, and the curve of rate constants changing with temperature was steep from 300 K to 1000 K (low temperature range), whereas the curve was gentle at 1000 K - 4000 K (high temperature range). In general, the rate constant at the high temperature range was still higher than the low temperature range.

For the reaction channel of NO+N₂O→NO₂+N₂, as shown in the Table 8, the reaction of was predicted to be exothermic and exergonic with ΔH=-32.87 kcal·mol^-1 and ΔG=-16.66 kcal·mol^-1 calculated at M062X method, and the barrier and ΔE of this reaction was 50.73 kcal·mol^-1 and -34.53 kcal·mol^-1 at QCISD(T) method. According to the obtained data, the potential energy surface was shown in the Fig. 6, and the harmonic and anharmonic rate constants at the temperature range from 300 K to 4000 K in the canonical ensemble were given in the Fig. 13. The harmonic and anharmonic rate constants were -24 orders of magnitude at about 300 K. When the temperature was 1600 K, the harmonic rate constant (7.63×10⁹ cm³·mol^-1·s^-1) was great 713.08 times than that anharmonic (1.07×10⁷ cm³·mol^-1·s^-1). When the temperature was 4000 K, the harmonic rate constant (1.14×10¹⁶ cm³·mol^-1·s^-1) was great 57000 times than that anharmonic (2.00×10¹¹ cm³·mol^-1·s^-1). Therefore, the anharmonic effect was obvious and cannot be ignored especially in high temperature range. And, the anharmonic effect at the high temperature range was still more obvious than the low temperature range. Besides, in order to verify this calculation, the rate constants were compared with other data.⁵⁹⁻⁶² As shown in the Fig. 13, the reference of others mainly was corresponding with the harmonic rate constant of this work. Therefore, this calculation was credible and anharmonic effect of this reaction cannot be ignored.

In addition, according to above calculation, A, n, E of Arrhenius equation related to NO in the ozone denitration reactions respectively were fitted and listed in the Table 9 through obtained anharmonic and harmonic rate constants at the temperature range from 300 K to 4000 K. Therefore, kinetic parameters of these reactions were worked out. Next, the numerical simulation by CHEMKIN can be accomplished through the fitted kinetic results. Besides, it was worth noting that the fitted anharmonic and harmonic parameters were different. Therefore, the research of anharmonic effect cannot be ignored.