The Theoretical Calculations of Kinetic and Thermodynamics Parameters and Anharmonic Correction for the Related Reactions of NO₃

The ab initio calculations were accomplished by using Gaussian 09 and Gauss View 5.0 software. In this paper, the geometries, frequencies and related parameters respectively were calculated by 6-311++G(d,p) set with B3LYP (R1, R5-R11) or M062X (R2-R4) method, firstly. For anharmonic corrections, the B3LYP is most ideal function at pressent.²⁷

However, this function is not suit to solve the reactions of NO₃ with NO₃, O₃, and O₂. M062X is the one of best functions for thermodynamic and kinetic calculation of main-group combination.^28,29

Next, intrinsic reaction coordinate (IRC) was achieved for verifying TS. At last, the single-point energy (SPE) was obtained by QCISD(T) method, and the related energy was corrected by zero-point energy (ZPE) for obtaining the more exact data. Therefore, related potential energy surfaces for calculated reactions in this paper can be obtained by SPE and ZPE.¹⁸⁻²⁰

Based on the TS theory, for unimolecular reaction, the rate constant k(T) is:³⁰⁻³⁵

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

Similarly, k(T) of bimolecular reaction (such as C+D→E+F) is,

where, Q_C(T) and Q_D(T) respectively represents the partition function for reactants C and D. For partition function,

Here, Q was calculated by using Morse oscillator (MO) potential. For MO, Q of i-th vibration mode is,

where, n represents the number of vibrational modes for reactant, β=1/(k_bT), energy of i-th vibrational mode (E_ni) is,

where w_i is the frequency of i-th vibrational mode, ħ=h/2π and x_i is the i-th anharmonic constant. Hence, according to above equations, rate constant and anharmonic correction can be achieved by this research. The details of derivation can be found in Refs. 18-20.

The Fitting of A, n and E. According to the Arrhenius equation, the expression with three parameters of rate constant is:^24,36−39

The three-parameter Arrhenius equation is also the most widely used for reaction mechanism at present.^38,39 Where, the R is molar gas constant, A represents pre-exponential/frequency factor, E represents activation energy. n is the exponent of temperature, and it is commonly called as temperature exponent.

Based on the ordinary least squares:

A, n, E of Arrhenius equation can be fitted. And, the kinetics parameters considering anharmonic and harmonic factor can be obtained, respectively.

The Calculating and Fitting of the Thermodynamic Parameters. The thermodynamic functions (standard state molar constant-pressure specific heat capacity (C^o_p), standard state molar enthalpies of formation (Δ_fH^o) and standard state molar entropy (S^o)) in CHEMKIN are shown as follows,³⁹

where a₁-a₇ respectively represent thermodynamic parameters. The thermodynamic functions in the form of least-squares coefficients for each reactant can be described as,⁴⁰

where q_i and a_i represent the factors of least-squares, respectively. In this work, the fourth-order polynomial are used for coefficients of least-squares. Therefore, the values of a₁-a₇ are fitted by the least square idea. The details of derivation can be found in Ref. 24.²⁴

Moreover, the value of C^o_p, S^o, H (enthalpy) of reactants can be worked out at different temperatures by Gaussian 09. Δ_fH^o can be got by H and as following conversions,⁴¹

According to above equations, the kinetic and thermodynamic parameters can be worked out and fitted.

The Dissociation and Formation Reaction of NO₃. For the dissociation reaction of NO₃ (NO₃→TS1→NO₂+O, R1-1), the detailed dissociation process can be described as that the NO₃ can directly cleave into NO₂ and O via the transition state of TS1. As shown in the Fig. 1, the relative energies of the TS, reactant and products for this reaction was worked out at QCISD(T)/6-311++G** level. According to the obtained data, PES for NO₃→TS1→NO₂+O was drawn in the Fig. 1. Meanwhile, the configurations of reactant, TS and products were pasted into the figure. The decompose reaction was an endothermal process (37.77 kcal·mol^-1), and its barrier was 42.67 kcal·mol^-1. In other words, this reaction was not easy to occur because of the relatively high barrier. In addition, the related rate constants were calculated for this reaction from 300 K to 4000 K in the Fig. 2(a), and the rate constants were relatively slow. Therefore, the reaction of NO₃→NO₂+O was not easy to occur.

On the contrary, as shown in the Fig. 1, the formation reaction of NO₃ (NO₂+O→TS1→NO₃, R1-2) was low barrier (4.90 kcal·mol^-1) and exothermic reaction (37.77 kcal·mol^-1). Therefore, the NO₃ was easy to form where existed NO₂ and O. Next, the rate constants of R1-2 were calculated and shown in the figure 2(b) (calculated results (point), the fitting Arrhenius equations (line), the detailed illustration of the fitting results were in 3.2.). It can be seen that the harmonic (1.19×10⁷ cm³·mol^-1·s^-1) and anharmonic (1.43×10⁷ cm³·mol^-1·s^-1) rate constants all were 7 order of magnitudes at 300 K. With the increasing temperature, the rate constants were up. The harmonic rate constant (3.01×10¹³ cm³·mol^-1·s^-1) was greater than that anharmonic (7.71×10¹¹ cm³·mol^-1·s^-1) 390 times at 4000 K. Therefore, as the raised temperature, the anharmonic effect of this reaction was more and more clear, and the anharmonic correction on rate constant was necessary, especially in high temperature range. Besides, the reaction can easily occur by obtaining rate constants. (In addition, this reaction may not be the only one for formation reaction of NO₃, and the calculations of the related reactions will be accomplished in the future.)

Figure1.

The potential energy surfaces for R1 by QCISD(T) method.

Figure2.

The rate constants of anharmonic and harmonic for R1 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation. (a) R1-1, NO₃→TS1→NO₂+O, (b) R1-2, NO₂+O→TS1→NO₃.

Therefore, the formation NO₃ was relatively easy and the dissociation NO₃ was relatively difficult. However, relative results showed that the concentration of NO₃ was low in related environment.¹⁰ Next, some bimolecular reactions referring to NO₃ ((NO₃+NO₃, R2), (NO₃+O₃, R3), (NO₃+O₂, R4), (NO₃+O, R5), (NO₃+N, R6), (NO₃+H, R7), (NO₃+N₂, R8), (NO₃+HO₂, R9), (NO₃+H₂O₂, R10), (NO₃+OH, R11)) were researched in this paper, respectively. The details were displayed as follows.

The Reaction of NO₃+NO₃. Concerning this reaction, owing to the high activity and strong oxidizing properties of NO₃ in chemical reaction, NO₃ can react with itself at the atmosphere of only NO₃ existing. When two NO₃ had collision, there was 11.23 kcal·mol^-1 to be obtained through the three transition states (TS2a, TS2b and TS2c) and two intermediates (IM1 and IM2), 2NO₂ and O₂ will be generated as end products from the Fig. 3. By the way, some main configurations were stuck on the Fig. 3. Meanwhile, energy of 31.66 kcal·mol^-1 will be released on this reaction process and they can provide power for others. It was to say that NO³ can be converted to NO² and O² within a certain period of time, as long as the environment can provide the energy of 11.23 kcal·mol^-1. Moreover, the rate constant of R2 can be expressed by that TS2a. According to these calculated parameters, such as related energies and frequencies, rate constants were listed in the Fig. 4 (point). With the rising temperature, the anharmonic effect was more and more clear. To be specific, the harmonic (1.98×10⁵ cm³·mol^-1·s^-1) and anharmonic (6.84×10⁵ cm³·mol^-1·s^-1) rate constants were 5 order of magnitudes at 300 K. However, the anharmonic (3.52×10²¹ cm³·mol^-1·s^-1) and harmonic (1.15×10¹⁹ cm³·mol^-1·s^-1) rate constants differed from about 2 order of magnitudes at 4000 K. Besides, as shown in the Fig. 4, the rate constants obtained by this work were compared with others, and it can be seen that this calculation was reliable.⁴²⁻⁴⁵

Figure3.

The potential energy surfaces for R2 by QCISD(T) method.

Figure4.

The rate constants of anharmonic and harmonic for R2 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

Therefore, when two NO₃ had collision, the NO₃ will be easy to decompose. This may be one of reasons why low concentration and short life span of NO₃ in flue gas.

The reaction of NO₃+O₃. Concerning this reaction, according to the Fig. 5, the relative energies of reactants, products, IM3, TS3a and TS3b had been calculated, respectively. By the way, some main configurations were stuck on the Fig. 5. The barrier of TS3a was 11.01 kcal·mol^-1. Therefore, the reaction was relatively easy to occur. According to the method of rate-determining step, the whole rate constant can be expressed by the first step. In other words, the rate constant of R3 can be expressed by that of TS3a. From the obtained data by Gaussian, rate constants of TS3a were worked out in the Fig. 6 (point) at 298 K-4000 K. Besides, the result of Hjorth (6.02×10² cm³·mol^-1·s^-1) was shown in the Fig. 6 at 298 K.⁴⁶ Similarly, the harmonic rate constant was 3.41×10⁶ cm³·mol^-1·s^-1. Therefore, this calculation was reliable. Besides, the anharmonic rate constant (5.57×10⁴ cm³·mol^-1·s^-1) differed from that harmonic about 2 order of magnitudes at 298 K, but the difference (anharmonic one was 2.11×10¹⁵ cm³·mol^-1·s^-1, harmonic one was 1.74×10²⁰ cm³·mol^-1·s^-1) was about 5 order of magnitudes at 4000 K. Therefore, the anharmonic correction was necessary.

The Reaction of NO₃+O₂. Concerning this reaction, it was two-step reaction. As shown in the Fig. 5, the reaction process and PES which included the two reactants (NO₃ and O₂), two transition states (TS4a and TS4b), two products (NO₂ and O₃), and one intermediate (IM4) were displayed. From the obtained data, it can be seen that this reaction was an endothermic reaction and the whole reaction process needed the energy of 27.49 kcal·mol^-1, and, the reaction of NO₃ with O₂ was not easy to occur rather than that NO₂ with O₃. According to the calculated parameters, rate constants were respectively worked out at 300-4000 K in the Fig. 7 (point). For the Fig. 7, with the increasing temperature, the rate constants and anharmonic effect all respectively were up, and the growth trends of rate constants were faster at the low temperature range than that at high temperatures. Therefore, the anharmonic correction was necessary.

The Reaction of NO₃+O. From the Fig. 5, it was clear that NO₃ combined with O to generate NO₂ and O₂ via the unique transition state of TS5 (NO₃+O→TS5→NO₂+O₂), and the barrier of R5 was 3.58 kcal·mol^-1. Moreover, it was an exothermal reaction and the energy of 35.34 kcal·mol^-1 can be released. Therefore, the reaction of NO₃ with O had a great influence on the related reaction of NO₃. Moreover, the anharmonic and harmonic rate constants were obtained at 300-4000 K in the Fig. 7 (point). From the graph, it was clear that the harmonic rate constants were greater than that anharmonic one at identical temperature. And the harmonic rate constant (3.72×10⁸ cm³·mol^-1·s^-1) was 8.44 times than that anharmonic one (4.41×10⁷ cm³·mol^-1·s^-1) at 300 K, and the harmonic rate constant (1.11×10¹⁴ cm³·mol^-1· s^-1) was 7.94 times than that anharmonic one (1.40×10¹³ cm³·mol^-1·s^-1) at 4000 K. When the temperature was 700 K, the difference of anharmonic (8.50×10⁹ cm³·mol^-1·s^-1) and harmonic rate constant (8.90×10¹⁰ cm³·mol^-1·s^-1) was 10.47 times what the maximum value was. It was to say that the anharmonic effect of this reaction was not very clear at the whole calculated temperatures and it was the most obvious at 700 K. Besides, the results can illustrate that the reaction of NO₃ with O was easy to occur, and its rate constant was relatively high even in room temperature.

Figure5.

The potential energy surfaces for R3-R5 by QCISD(T) method.

Figure6.

The rate constants of anharmonic and harmonic for R3 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

Figure7.

The rate constants of anharmonic and harmonic for R4 and R5 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

The Reaction of NO₃+N. In terms of the reaction, NO₃ with N reacted to form NO and O₂ via TS6. Related details had been published previous article.⁴⁷ The reaction process and potential energy surface were showed in the Fig. 8, and the rate constants of NO₃ combined with N were listed in the Fig. 9. From the calculated results, the reaction can occur easily and provide a lot heat for others. In another aspect, the anharmonic correction on rate constant of this reaction was necessary and cannot be neglected, especially at high temperatures.

The Reaction of NO₃+H. For this reaction process, the Fig. 8 had shown the detailed PES and related energies. NO₃ with H reacted to form IM5 by the reaction channel of TS7a firstly, and then IM5 was cracked into NO₂ and OH by the reaction channel of TS7b. These two reaction channels all were low barriers and exothermal reactions. The barriers respectively were 11.92 kcal·mol^-1 and 6.26 kcal·mol^-1 for the reaction channel of TS7a and TS7b, and the whole reaction process will release the energy of 56.44 kcal·mol^-1. Therefore, this reaction can occur easily. According to the method of rate-determining step, the rate constant of R7 can be represented by that TS7a. As shown in the Fig. 9 (point), the rate constants were worked out. As the temperature raised, the rate constants and anharmonic effect were up. The harmonic rate constant (1.13×10¹² cm³·mol^-1·s^-1) was about 4 times than that anharmonic one (4.12×10¹² cm³·mol^-1·s^-1) at 4000 K. Therefore, the anharmonic correction cannot be neglected especially at high temperatures. Besides, from the relative energy and rate constant, the reaction was easy to occur.

Figure8.

The potential energy surfaces for R6-R7 by QCISD(T) method.

Figure9.

The rate constants of anharmonic and harmonic for R6 and R7 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

The Reaction of NO₃+N₂. For the reaction process of NO₃+N₂, as shown in the Fig. 10, NO₃ with N₂ reacted to form NO₂ and N₂O via TS8, and the barriers was 59.86 kcal·mol^-1. Besides, the whole reaction process needed energy of 11.62 kcal·mol^-1. In view of above data, the reaction of NO₃ with N₂ was difficult to occur. The similar conclusion can be obtained from the calculated rate constants. As shown in the Fig. 11 (point), the rate constants were relatively slow especially at low temperature range. It seemed that NO₃ cannot react with N₂ at low 1000 K. However, the high-temperature research of this reaction was necessary. Clearly, as the temperature increased, the rate constants were rising, and they were more than 10 orders of magnitude at above 2000 K.

The Reaction of NO₃+HO₂. In term of the reaction, the PES was shown in the Fig. 10, which included the two reactants, one intermediate (IM6), two products and two transition states. The barriers respectively were 53.04 kcal·mol^-1 and 24.66 kcal·mol^-1 for the reaction channel of TS9a and TS9b. Similarly, the rate constant of R9 can be expressed by that TS9a. Therefore, the rate constants were obtained by related parameters in the Fig. 11 (point). It was clear that the anharmonic effects and rate constants were up as the temperature increased. Specifically, the anharmonic (1.35×10^-26 cm³·mol^-1·s^-1) and harmonic (2.15×10^-25 cm³·mol^-1·s^-1) rate constants were close at 298 K, but there had visible distinctions (3 orders of magnitude) between harmonic (7.73×10¹⁶ cm³·mol^-1·s^-1) and anharmonic (9.78×10¹³ cm³·mol^-1·s^-1) rate constants at 4000 K. Therefore, for this reaction, the anharmonic correction was noteworthy especially at high temperature range.

The Reaction of NO₃+H₂O₂. For the reaction of NO₃ with H₂O₂, the reactants directly reacted to generate the products of HNO₃ and HO₂ via TS10. The reaction process, related energies and potential energy surface were shown as the Fig. 10. The 14.59 kcal·mol^-1 of energy can be released to help others occurring on the reaction process. Meanwhile, the barrier of this exothermic reaction was 10.82 kcal·mol^-1. According to the calculated barrier and frequencies, anharmonic and harmonic rate constants were displayed in the Fig. 12 (point). Clearly, as the temperature increased, the difference between anharmonic and harmonic rate constants was up. In other words, the anharmonic effect was clear. On the basis of calculated data, the difference between harmonic (8.15×10¹⁸ cm³·mol^-1·s^-1) and anharmonic (6.44×10¹⁴ cm³·mol^-1·s^-1) rate constant reached 4 orders of magnitude at 4000 K. Even when the temperature was 298 K, the harmonic (1.49×10⁶ cm³·mol^-1·s^-1) rate constant differed from that anharmonic (1.10×10⁵ cm³·mol^-1·s^-1) one by 1 order The Reaction of NO₃+H₂O₂. For the reaction of NO₃ with H₂O₂, the reactants directly reacted to generate the products of HNO₃ and HO₂ via TS10. The reaction process, related energies and potential energy surface were shown as the Fig. 10. The 14.59 kcal·mol^-1 of energy can be released to help others occurring on the reaction process. Meanwhile, the barrier of this exothermic reaction was 10.82 kcal·mol^-1. According to the calculated barrier and frequencies, anharmonic and harmonic rate constants were displayed in the Fig. 12 (point). Clearly, as the temperature increased, the difference between anharmonic and harmonic rate constants was up. In other words, the anharmonic effect was clear. On the basis of calculated data, the difference between harmonic (8.15×10¹⁸ cm³·mol^-1·s^-1) and anharmonic (6.44×10¹⁴ cm³·mol^-1·s^-1) rate constant reached 4 orders of magnitude at 4000 K. Even when the temperature was 298 K, the harmonic (1.49×10⁶ cm³·mol^-1·s^-1) rate constant differed from that anharmonic (1.10×10⁵ cm³·mol^-1·s^-1) one by 1 order of magnitude. Therefore, the anharmonic correction cannot be ignored for this reaction. Besides, owing to the relatively low barrier for the exothermic reaction, the rate constant was relatively high. Therefore, the reaction of NO₃ with H₂O₂ had a significant influence on the whole conversion process of NO₃ and its anharmonic correction cannot be neglected. Besides, the rate constant of Burrows was drawn in the Fig. 15.⁴⁸

Figure10.

The potential energy surfaces for R8-R10 by QCISD (T) method.

Figure11.

The rate constants of anharmonic and harmonic for R8 and R9 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

Figure12.

The rate constants of anharmonic and harmonic for R10 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

The Reaction of NO₃+OH. Concerning this reaction, the NO₃ combined with OH to form NO₂+HO₂ (R11-1) and HNO₃+O (R11-2) via TS11a and TS11b, respectively. The detailed reaction process, related energies and PES was shown and drawn in the Fig. 13. For R11-1, the barrier was 26.26 kcal·mol^-1 and the energy of 15.01 kcal·mol^-1 can be released. But for R11-2, the barrier was 12.37 kcal·mol^-1 and the energy of 1.92 kcal·mol^-1 can be released on this reaction process. Therefore, R11-2 was relatively easy to occur and it was the main reaction for R11. According to these obtained data, anharmonic and harmonic rate constants of the two reaction channels were worked out and shown in the Fig. 14 (point). With the increasing temperature, the rate constants all were up. And, the anharmonic rate constant of R11-2 was greater than that of R11-1 at arbitrary calculated temperature. It was to say that R11-2 was the main reaction for R11. Besides, the anharmonic effect of R11-2 was more and more clear as the temperature increased. The anharmonic (8.17×10¹⁷ cm³·mol^-1·s^-1) and harmonic (1.92×10¹⁶ cm³·mol^-1·s^-1) rate constants differed about 43 times at 4000 K. Therefore, the anharmonic correction was necessary for R11.

Figure13.

The potential energy surfaces for R11 by QCISD(T) method.

Figure14.

The rate constants of anharmonic and harmonic for R11 at 300-4000 K. And comparing of calculating rate constants and fitting Arrhenius equation.

According to above results, as shown in the Table 1, A, n, and E of related reactions were respectively fitted through anharmonic and harmonic rate constants. As shown in related figures (2, 4, 6, 7, 9, 11, 12, 14) of rate constants, fitting kinetic equations (line) were drawn on related figures and compared with corresponding calculated rate constant (point), the results showed that the fitting Arrhenius equations were well agreement with the calculated results. Therefore, the fitted method of kinetic was reliable in this paper. Notably, the fitted E was close to the calculated barrier of related reaction by ab initio calculations. Such as the reaction of R2-1, its calculated barrier was 4.90 kcal·mol^-1 by QCISD(T) method and fitted E respectively was 5.12 kcal·mol^-1 and 4.43 kcal·mol^-1 through anharmonic and harmonic rate constants. Therefore, the fitting result can well express the physical meaning of E in Arrhenius equation. Besides, owing to the existing of anharmonic effect, the harmonic and anharmonic kinetic parameters had some difference. In order to ensure the fitting accuracy, the fitted A and n by anharmonic and harmonic rate constants were different for same reaction. It meant that the anharmonic correction on rate constant cannot be ignored.

C^o_p and S^o can be directly got by Gaussian software, however, Δ_fH^o were worked out by related equation and the obtained H. According to the equation (16), and the Δ_fH^o of stable elementary substance were zero. Therefore, for H₂, N₂ and O₂, Δ_fH^o₂₉₈ (enthalpies of formation at 298 K) all were zero. Therefore, Δ_fH^o₂₉₈ for H(g), N(g), O(g) can be worked out. In this paper, several methods (B3LYP, M062X, CBS-QB3 and G4) were used to calculate and compare Δ_fH^o₂₉₈ for an elementary substance in the Table 2. Meanwhile, the calculating results were compared with other data.^49,50 According to the comparing, the results of B3LYP were used to calculate Δ_fH^o. Therefore, Δ_fH^o₂₉₈ for H(g), N(g), O(g) were 52.2035, 111.0977 and 59.0954 kcal·mol^-1. Then, Δ_fH^o of reactant at different temperatures was worked out by the corresponding expressions. Therefore, C^o_p, S^o and Δ_fH^o were calculated at temperatures 300 K-4000 K by Gaussian software and related equations in the Figs. 15-17, respectively.

Then, in order to verify the obtained value of C^o_p, S^o and Δ_fH^o, related calculated results were compared with other researches⁵⁰⁻⁵⁶ in the Figs. 15-17, respectively. Such as C^o_p, related value of O, O₂, O₃, N, N₂, NO₃, H₂O₂, HO₂, H and OH, were listed in Fig. 15(a-g) and compared with other references, respectively. From these figures, it can be seen that corresponding results of C^o_p were different in different mechanisms for the same species. The accuracy of related thermodynamic data in other mechanisms was not discernable. Therefore, related thermodynamic data were worked out in this paper, and the calculating results were basically in agreement for most species.

Figure15.

The comparing of calculating C^o_p, fitting results of thermodynamic equation and related data of other researches for O, O₂, O₃, N, N₂, NO₃, H, OH, HO₂ and H₂O₂, respectively.

Figure16.

The comparing of calculating S^o, fitting results of thermodynamic equation and related data of other researches for O, O₂, O₃, N, N₂, NO₃, H, OH, HO₂ and H₂O₂, respectively.

Figure17.

The comparing of calculating Δ_fH^o, fitting results of thermodynamic equation and related data of other researches for O, O₂, O₃, N, N₂, NO₃, H, OH, HO₂ and H₂O₂, respectively.

Next, according to the obtained thermodynamics data, a₁-a₇ of related species were fitted through thermodynamics equations and least square method in the Table 3. Notably, owing to the temperature partition of input thermodynamics files by CHEMKIN, the fitted temperatures were divided into 300-1000 K and 1000-4000 K.

Finally, in order to verify fitted thermodynamic equations, related fitted equations (line) were compared with calculated data (point) and drawn on the Figs. 15-17, respectively. From these figures, it was clear that the fitting and calculating results were basically in agreement. Therefore, the fitted method of thermodynamics parameters was reliable in this paper.