Theoretical study on the mechanism and kinetics of atmospheric reactions NH2OH + OOH and NH2CH3 + OOH

Physics Letters A 378 (2014) 777–784

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Physics Letters A www.elsevier.com/locate/pla

Theoretical study on the mechanism and kinetics of atmospheric reactions NH2 OH + OOH and NH2 CH3 + OOH Younes Valadbeigi, Hossein Farrokhpour ∗ , Mahmoud Tabrizchi Department of Chemistry, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e

i n f o

Article history: Received 28 October 2013 Received in revised form 2 January 2014 Accepted 3 January 2014 Available online 17 January 2014 Communicated by Z. Siwy Keywords: Reaction mechanisms Transition State Theory Atmospheric reaction Rate constant Density functional theory

a b s t r a c t Mechanism and kinetics of NH2 OH + OOH and NH2 CH3 + OOH reactions were studied at the B3LYP and M062X levels of theory using the 6-311++G(3df, 3pd) basis set. The NH2 OH + OOH and NH2 CH3 + OOH reactions proceed through different paths which lead to different products. Transition state structure and activation energy of each path were calculated. The calculated activation energies of hydrogen abstraction reactions were smaller than 25 kcal/mol and of substitution reactions are in the range of 50–70 kcal/mol. The rate constants were calculated using transition state theory (TST) modified for tunneling effect at 273–2000 K. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Ammonia, NH3 , and hydroperoxyl radical, OOH, are important atmospheric species that participate in atmospheric reactions. Most of the NH3 reactions are pertinent to hydrogen abstraction to produce NH2 radical [1–5]. Kinetics of NH3 + OH/H/C2 H2 reactions which leads to NH2 radical have been well studied experimentally and theoretically [3–5]. The reaction of NH2 radical with NO2 or O produces H2 NO [6–8]. In addition, NH2 radical reacts with CH3 to form NH2 CH3 [9]. The reactions of OOH radical have been studied theoretically and experimentally by several researchers [10–14]. Kinetic and mechanism of OOH reactions with itself and other molecules result in producing of several important intermediates and products [10,11,13,15]. Farnia et al. [10] studied kinetics and mechanism of reactions of hydroperoxyl and acetaldehyde and showed that these reactions leads to products such as CH3 OOH and CH3 CH(OO)OH. Zhu and Lin [15] investigated reaction mechanism of OOH + OOH and showed that the most favored product channel is producing H2 O2 + O2 . Garcia and Gil [14] studied the reaction mechanism of OOH formation due to reaction between ozone, O3 , and OH. Their calculations at the QCISD level of theory showed that the reaction proceeds through only one transition state between reactants and products, while at the MP2 level, the reaction proceeds through a two-step mechanism with two transition state structures. Mebel

*

Corresponding author. E-mail address: [email protected] (H. Farrokhpour).

0375-9601/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2014.01.006

and Lin [16] have calculated activation barrier for NH2 + CH4 reaction as 15.2 kcal/mol using the G2M method. It is clear that the OOH, NH3 , NH2 and its derivatives such as NH2 CH3 and NH2 OH are important species in atmospheric and combustion systems and study of their reactions is interesting. In this work, we perform a detailed theoretical study on the different paths for producing NH2 OH and NH2 CH3 from NH2 and NH3 and mechanism of OOH + NH2 CH3 and OOH + NH2 OH reactions. The rate constants of these reactions are calculated in the 298–2000 K temperature range. 2. Computational details The structures of reactants, pre-reactive complexes, transition states, products and pre-product complexes were optimized at the B3LYP and M062X [17] levels of theory using 6-311++G(3df, 3pd) [18] basis set. DFT methods have been used to study kinetic and mechanism of radicalic systems [10,15]. Zhu and Lin [15] studied OOH + OOH reactions using B3LYP, MP2, G2M and CCSD(T). Their results showed that calculated data using CCSD(T) and B3LYP are in agreement and are different from those calculated by MP2. Zhao and Truhlar [17] assessed the performance of M062X for three databases containing 40 bond lengths, 38 vibrational frequencies and 15 vibrational zero point energies. They recommended the M062X as an accurate method for applications involving maingroup thermo-chemistry, kinetics, non-covalent interactions, and electronic excitation energies to valence and Rydberg states [17]. Moreover, there are many studies on the assessment of the performance of the M062X and B3LYP methods for interactions in

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Fig. 1. Structures of transition states for NH2 formation reactions optimized at M062X/6-311++G(3df, 3pd) level of theory.

bio-molecules and clusters which confirm the accuracy of these methods [19–21]. Also, the frequency calculations were carried out by the same levels of theories and basis set at 298.15 K. The QST2 method was used to determine the transition state structures. Intrinsic reaction coordinate (IRC) [22] calculations were performed to confirm the correct transition states structures. The basis set superposition error (BSSE) calculations was also performed at the B3LYP level of theory according to the counterpoise method proposed by Boys and Bernardi [23] for pre-reactive and pre-product complexes. All calculations were performed using Gaussian 09 quantum chemistry package [24]. Rate constants were calculated using the transition state theory [25]

 k( T ) = κ

kB T h



 exp −

 G =

(i)

RT

where k B is Boltzmann constant, h is Planck constant and G # is Gibbs free energy difference between reactants and their transition state structures. The calculated values of G # at 298.15 K was used to obtain the rate constants. The Wigner coefficient, κ , is inserted in order to take into account tunneling effect [26,27].

κ =1+

1 24



hcv kB T

2  1+

RT



 E B ,0

(ii)

where v (cm−1 ) is imaginary frequency of transition state structure and  E B ,0 is electronic barrier height of considered reaction. 3. Results and discussion 3.1. Mechanism of NH2 OH and NH2 CH3 formation All of the elementary reactions considered in this work begin with the formation of a pre-reactive complex from the reactants. The pre-reactive complex passes through a transition state to form a pre-product complex. We study twenty two reactions whose general forms can be written as

Reactants → An → TSn → Bn → Products where n is the number of the reaction (from 1 to 22), An and Bn are the pre-reactive and pre-product complexes of reaction n, respectively. The activation energies, E a , are calculated from the electronic energies difference between pre-reactant/pre-product complexes and transition state structures. NH2 OH and NH2 CH3 can be produced directly through the reaction of NH2 with OH and CH3 radicals [6–9]. It is known that these radically reactions do not have activation energy; therefore we do not investigate the thermodynamic and kinetic of these two reactions. The NH2 radical is produced during the atmospheric reactions [1–5]. Reactions (1)–(6) are some of the reactions that pro-

duce NH2 radical from NH3 molecules via the hydrogen abstraction reaction

NH3 + H → TS1 → NH2 + H2

(1)

NH3 + OH → TS2 → NH2 + H2 O

(2)

NH3 + OCl → TS3 → NH2 + HOCl

(3)

NH3 + OOH → TS4 → NH2 + H2 O2

(4)

NH3 + CH3 → TS5 → NH2 + CH4

(5)

NH3 + Cl → TS6 → NH2 + HCl

(6)

where TSs are the transition states. Fig. 1 shows the transition state structures of reaction (1)–(6) (TS1–6) optimized at the M062X/6-311++G(3df, 3pd) level of theory. Geometrical parameters of the transition state structures are reported in the supplementary information, Section A. Table 1 presents the calculated activation energies, E a , and Gibbs free energies, G # , of forward and backward paths of reactions (1)–(6), at 298.15 K. Also, imaginary frequencies of transition state structures are tabulated in Table 1. Since the imaginary frequency is a property of the transition state and therefore it is the same for forward and backward reactions, only one quantity is reported in Tables 1–4 for both processes. The activation energies calculated by B3LYP method are slightly less than those obtained using M062X method. This trend is observed for all reactions presented in Tables 1–4. Zhao and Truhlar [19] showed that the B3LYP method underestimates the reaction barrier heights which is in agreement with our results. Forward and backward reactions (1)–(6) are hydrogen abstraction whose activation energies are less than 20 kcal/mol. In addition, the activation energies of forward reactions, formation of NH2 , are more than those of backward reactions. NH2 OH and NH2 CH3 may be formed via following substitution reactions

NH2 + H2 O → TS7 → NH2 OH + H

(7)

NH3 + OH → TS8 → NH2 OH + H

(8)

NH2 + H2 O2 → TS9 → NH2 OH + OH

(9)

NH2 + CH4 → TS10 → NH2 CH3 + H

(10)

NH3 + CH3 → TS11 → NH2 CH3 + H

(11)

Fig. 2 shows the transition state structures of reactions (7)–(11) (TS7–11) optimized at the M062X/6-311++G(3df, 3pd) level of theory. Geometrical parameters of the transition state structures are collected in supplementary information, Section A. The reactions (7)–(11) are substitution reactions. In reaction (9), a hydroxyl group of H2 O2 molecule is replaced with the NH2 to form a hydroxylamine molecule, NH2 OH. In the case of reactions (8) and

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Table 1 Activation energies, E a , Gibbs free energies, G # , and imaginary frequencies, v, of the forward and backward paths of reactions (1)–(6) calculated at 298.15 K. Reaction

B3LYP/6-311++G(3df, 3pd)

M062X/6-311++G(3df, 3pd)

Ea (kcal/mol)

G #

v /i (cm−1 )

Ea (kcal/mol)

G #

1-(forward) 1-(backward) 2-(forward) 2-(backward) 3-(forward) 3-(backward) 4-(forward) 4-(backward) 5-(forward) 5-(backward)

7.26 7.24 2.27 10.58 10.57 2.69 18.95 1.27 12.68 10.65

8.81 9.50 2.10 12.76 12.70 3.04 22.23 2.89 14.83 15.31

1141.61

13.12 9.40 4.35 17.92 12.91 9.74 24.51 6.70 19.56 13.83

13.03 1.62

292.73

12.17 7.98 3.34 16.01 12.21 8.39 22.70 6.01 15.56 11.99 15.2 [16] 12.71 1.81

6-(forward) 6-(backward)

10.84 0.71

(kcal/mol)

1188.57 1524.89 1716.94 1667.91

(kcal/mol)

12.89 2.45

v /i (cm−1 ) 1640.13 1199.25 1978.36 2057.01 1650.70

806.02

Table 2 Activation energies, E a , Gibbs free energies, G # , and imaginary frequencies, v, of the forward and backward paths of reactions (7)–(11) calculated at 298.15 K. Reaction

7-(forward) 7-(backward) 8-(forward) 8-(backward) 9-(forward) 9-(backward) 10-(forward) 10-(backward) 11-(forward) 11-(backward)

B3LYP/6-311++G(3df, 3pd)

M062X/6-311++G(3df, 3pd)

Ea (kcal/mol)

G #

67.02 7.01 51.93 2.41 16.76 28.71 52.76 28.36 42.57 16.13

69.45 10.34 53.32 3.19 18.53 29.91 60.05 33.09 46.07 17.81

(kcal/mol)

v /i (cm−1 )

Ea (kcal/mol)

G #

1171.27

70.27 13.47 53.05 6.72 25.77 40.79 52.59 34.13 44.24 22.28

71.95 15.23 54.33 7.38 26.82 41.55 55.35 37.06 48.87 23.90

984.51 863.68 1396.50 763.56

(kcal/mol)

v /i (cm−1 ) 1466.48 1228.84 1075.40 1500.22 701.28

Table 3 Activation energies, E a , Gibbs free energies, G # , and imaginary frequencies, v, of the forward and backward paths of reactions (12)–(16) calculated at 298.15 K. Values in parenthesis include BSSE corrections. Reaction

12-(forward) 12-(backward) 13-(forward) 13-(backward) 14-(forward) 14-(backward) 15-(forward) 15-(backward) 16-(forward) 16-(backward)

B3LYP/6-311++G(3df, 3pd)

M062X/6-311++G(3df, 3pd)

Ea (kcal/mol)

G #

11.23 18.27 13.99 9.75 68.86 8.44 8.33 −9.47 11.87 19.70

11.39 19.17 15.10 11.56 71.08 9.41 8.87 −6.44 11.42 21.53

(10.64) (17.76) (13.55) (9.46) (68.42) (8.15) (7.43) (−9.89) (11.49) (19.28)

(kcal/mol)

(10.84) (18.71) (14.82) (11.42) (70.08) (8.99) (8.01) (−7.30) (11.28) (20.66)

v /i (cm−1 )

Ea (kcal/mol)

G #

1228.80

16.19 23.03 19.52 15.30 71.57 14.81 12.01 −2.61 18.27 36.35

16.68 24.00 20.01 16.25 73.11 17.68 12.14 −0.68 17.97 37.33

1558.87 1094.35 658.25 755.57

(kcal/mol)

v /i (cm−1 ) 1944.61 1945.58 1439.94 689.85 1110.78

Table 4 Activation energies, E a , Gibbs free energies, G # , and imaginary frequencies, v, of the forward and backward paths of reactions (17)–(22) calculated at 298.15 K. Values in parenthesis include BSSE corrections. Reaction

17-(forward) 17-(backward) 18-(forward) 18-(backward) 19-(forward) 19-(backward) 20-(forward) 20-(backward) 21-(forward) 21-(backward) 22-(forward) 22-(backward)

B3LYP/6-311++G(3df, 3pd)

M062X/6-311++G(3df, 3pd)

Ea (kcal/mol)

G #

8.07 5.73 11.41 4.08 68.64 6.82 58.11 28.27 10.77 11.70 11.34 29.80

11.50 6.38 14.60 5.41 73.14 8.64 62.05 28.75 12.55 13.05 10.64 30.93

(7.96) (5.33) (11.29) (3.76) (68.53) (6.47) (58.00) (28.18) (9.96) (11.24) (10.35) (29.34)

(kcal/mol)

(12.58) (6.02) (15.68) (5.17) (74.22) (7.87) (63.13) (29.80) (12.07) (12.63) (9.67) (30.51)

v /i (cm−1 )

Ea (kcal/mol)

G #

1467.67

10.95 10.60 15.59 8.083 69.47 10.08 60.53 36.78 11.59 19.96 25.07 40.66

14.59 11.37 18.70 8.14 73.88 10.10 64.69 39.61 13.42 21.61 24.65 42.42

1664.38 1061.46 1331.49 710.69 946.32

(kcal/mol)

v /i (cm−1 ) 1554.42 2035.64 1385.35 1583.60 759.49 1348.94

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Fig. 2. Structures of transition states for NH2 OH and NH2 CH3 formation (reactions (7)–(11)) at the M062X/6-311++G(3df, 3pdp) level of theory.

Fig. 3. Structures of transition states for NH2 OH + OOH (reactions (12)–(16)) optimized at M062X/6-311++G(3df, 3pd) level of theory.

(11), a hydrogen atom of NH3 is replaced with OH and CH3 species, respectively. Table 2 presents the activation energies, E a , Gibbs free energies, G # , and imaginary frequencies, ν , of forward and backward reactions (7)–(11) calculated at 298.15 K. The activation energies calculated by the two methods have the same trend. The activation energies of the forward reactions (7)–(11) are in the range of 20–50 kcal/mol which are greater than those of hydrogen abstractions (reactions (1)–(6)). Table 2 shows that formation of NH2 OH via reaction (9) (NH2 + H2 O2 ) is, energetically, the most favored path among the reactions (7)–(9). The activation energies of the backward reactions are less than 30 kcal/mol. 3.2. Reaction between OOH and NH2 OH The hydroxylamine, NH2 OH, reacts with OOH to produce some products and intermediates. We investigate five forward and five backward reactions for the NH2 OH + OOH system (reactions (12)–(16))

NH2 OH + OOH → TS12 → NH2 O + H2 O2

(12)

NH2 OH + OOH → TS13 → NHOH + H2 O2

(13)

NH2 OH + OOH → TS14 → NH(OOH)OH + H

(14)

NH2 O + OOH → TS15 → NH2 OH + O2

(15)

NHOH + OOH → TS16 → NH2 OH + O2

(16)

The transition state structures of reactions (12)–(16) (TS12–16) optimized at the M062X/6-311++G(3df, 3pd) level of theory are

shown in Fig. 3. Geometrical parameters of the transition state structures are collected in supplementary information, Section A. Forward paths of the reactions (12) and (13) are hydrogen abstraction in which the OOH radical separates a hydrogen atom from oxygen and nitrogen atoms of NH2 OH, respectively. Two intermediates, NH2 O and NHOH, are produced in reaction (12) and (13). In reaction (14) a new compound, NH(OOH)OH is formed due to replacement of OOH with a hydrogen atom of NH2 OH. An O2 molecule is produced from the reaction of NH2 O or NHOH intermediates with OOH (reactions (15), (16)). Table 3 tabulates the activation energies, E a , Gibbs free energies, G # , and imaginary frequencies, ν , of the forward and backward reactions (12)–(16) calculated at 298.15 K. All activation energies of reactions (12)–(16) are almost less than 20 kcal/mol except that of reaction (14) that is a replacement reaction with a high activation energy. Formation of NH(OOH)OH via replacement reaction between NH2 OH and OOH (reaction (14)) is not energetically favored. However, this molecule may be produced through direct reaction of NHOH and OOH radicals. A negative activation energy was observed for backward reaction (15), i.e., reaction of NH2 OH + O2 . In the reactions with negative energy barriers the molecules are captured in a potential well, during the reaction proceeding. Also, the negative energy barriers in some reactions is due to the formation of a pre-reactive complex which its formation has not any activation energy, then a stable the transition state structure is produced whose energy is less than the energy of the separated reactant molecules [28]. In the reaction (15), O2 and NH2 OH produce a stable transition state structure, TS15, that is resemble of cage complex structure in which oxygen and nitrogen

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Fig. 4. Optimum structures of transition states for NH2 CH3 + OOH reactions optimized at M062X/6-311++G(3df, 3pd) level of theory.

atoms of the two molecules interact with each other to stabilize the structure (Fig. 3). 3.3. Reactions between OOH and NH2 CH3 The methylamine, NH2 CH3 , reacts with OOH to produce some products and intermediates. 6 forward and 6 backward reaction for NH2 CH3 + OOH system are studied (reactions (17)–(22))

NH2 CH3 + OOH → TS17 → NH2 CH2 + H2 O2

(17)

NH2 CH3 + OOH → TS18 → NHCH3 + H2 O2

(18)

NH2 CH3 + OOH → TS19 → NH(OOH)CH3 + H

(19)

NH2 CH3 + OOH → TS20 → NH2 CH2 OOH + H

(20)

NH2 CH2 + OOH → TS21 → NH2 CH3 + O2

(21)

NHCH3 + OOH → TS22 → NH2 CH3 + O2

(22)

The transition state structures of reactions (17)–(22) (TS12–22) optimized at the M062X/6-311++G(3df, 3pd) level of theory are shown in Fig. 4. Geometrical parameters of the transition state structures are collected in supplementary information, Section A. In reactions (17) and (18), a hydrogen atom separated from carbon and nitrogen atoms of NH2 CH3 by OOH radical, respectively. Two new compounds, NH(OOH)CH3 and NH2 CH2 OOH, are produced in the reactions (19) and (20) by replacement of OOH with a hydrogen atom, respectively. Intermediates NH2 CH2 and NHCH3 , that are produced in the reactions (17) and (18), take part in a hydrogen abstraction with OOH and therefore an oxygen molecule, O2 , is produced (reactions (21), (22)). Table 4 reports the activation energies, E a , and Gibbs free energies, G # , of the forward and backward reactions (17)–(22) calculated at 298.15 K. Right column of Table 4 shows imaginary frequencies of transition state structures calculated using the M062X/6-311++G(3df, 3pdp) level of theory. The reactions (17), (18), (21) and (22) are hydrogen abstraction reactions and their activation energies are almost less than 20 kcal/mol. For the forward reactions (19) and (20) activation energies are very high and in the range of 60–75 kcal/mol. Although there is a good agreement between the results obtained using the B3LYP and M062X methods, the differences between the activation energies calculated with the two DFT methods are 10–15 kcal/mol in reactions (9), (16)-backward, and (22). These differences in the activation energies lead to huge differences in the rate coefficient. Since many researchers [17,19–21] confirmed that the M062X is more accurate method than the B3LYP, it is reasonable to accept the results obtained with the M062X. Moreover, it is proved that the B3LYP method underestimates the activation energies of reactions [19], therefore, real

Fig. 5. The schematic potential energy surface (PES) of the NH2 CH3 + OOH reaction.

values of activation energies are larger than those obtained with the B3LYP method. Fig. 5 shows the schematic potential energy surface (PES) of the NH2 CH3 + OOH reaction. As seen, the intermediate NH2 CH2 is more stable than NHCH3 . Also, the activation energy of NH2 CH2 formation (reaction (17)) is less than that for NHCH3 formation (reaction (18)). The reactions (19) and (20) lead to products NH(OOH)CH3 and NH2 CH2 OOH, respectively, which the latter is more stable. Direct formation of NH2 CH2 OOH from NH2 CH3 and OOH (reaction (20)) needs an activation energy which is less than that for formation of NH(OOH)CH3 (reaction (19)). Another path to NH2 CH2 OOH formation is NH2 CH2 + OOH reaction in which NH2 CH2 is formed via reaction (17). Therefore, the NH2 CH2 OOH formation by the two path is more favored than NH(OOH)CH3 formation. The reactions (21) and (22) produce the same products, however, the products are rapidly formed through reaction (21). The same potential energy surface (PES) can be constructed for reactions (12)–(16) using energy values reported in Table 3. 3.4. Thermodynamic study As mentioned, An and Bn are the pre-reactive and pre-product complexes of reaction n, respectively. Fig. 6 shows the structures of pre-reactive and pre-product complexes optimized at the M062X level of theory. Geometrical parameters of the pre-reactant and

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Fig. 6. The structures of pre-reactive and pre-product complexes optimized at M062X level of theory.

pre-product complex structures are collected in supplementary information, Section B. The electronic energies, E, enthalpies, H , and Gibbs free energies, G, of the pre-reactive and pre-product complexes were calculated using the B3LYP and M062X methods at 298.15 K. These thermodynamic properties for the pre-reactive and pre-product complexes are tabulated in Table A and Table B as supplementary information section, respectively. All calculated energies include only thermal corrections, and ZPEs are presented separately. The results of the two methods are in good agreement with each other. The entropies, S, and Helmholtz energies, A, can be directly calculated using G = H − T S and A = E − T S, respectively. To complete the thermodynamic study of the overall paths of the reactions, the electronic energies, E, enthalpies, H , and Gibbs free energies, G, of the reactants and products were also calculated using the B3LYP and M062X methods at 298.15 K. These thermodynamic properties are tabulated in Table C in Supplementary information section. All calculated energies include only thermal corrections, and ZPEs are presented separately. Thermodynamic properties of the reactant ↔ pre-reactive complex and product ↔ pre-product complex reactions are calculated using data reported

in Tables A–C (supplementary information). By making use of data reported in Tables 1–4 and Tables A, B and C, we can obtain the energies for each step in each reaction. Moreover, the reported data in Table C (supplementary information section) can be used to calculate thermodynamic properties of atmospheric reactions other than these 22 reactions. For example, enthalpy,  H , of any reaction with general form of A + B → C + D, can be calculated as ( H C + H D ) − ( H A + H B ) that the values of H A , H B , H C and H D have been presented in Table C. Other thermodynamic data,  E and G, are calculated using the same method. We believe that reported data in Table C are general and very useful, because the given molecules and radicals in Table C take part in many atmospheric reactions. However,  H and G of the reactions (1)–(22) were calculated using thermodynamic data reported in Table C. Table 5 summarizes the values of  H and G calculated at B3LYP and M062X levels of theory. Reactions (2), (13), (21) and (22) are exothermic and others are endothermic. In reactions (21) and (22) two stable molecules are produced from two radicals, therefore, these reactions are thermodynamically favored.

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Table 5 Calculated  H and G of reactions (1)–(22) at 298.15 K. Reaction

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

B3LYP

M062X

H

G

H

(kcal/mol)

(kcal/mol)

(kcal/mol)

G (kcal/mol)

+0.26 −11.12 +13.79 +22.45 +1.80 +2.65 +56.52 +45.39 −13.91 +23.94 +25.75 −10.12 −0.75 +55.35 +14.99 +5.62 +6.92 +13.63 +61.55 +29.65 −2.05 −8.76

−0.76 −12.20 +12.77 +22.53 +2.88 +0.76 +59.78 +47.58 −12.74 +26.15 +29.04 −10.33 −0.67 +59.56 +17.54 +7.89 +6.74 +13.34 +65.51 +34.34 +0.47 −6.12

+3.97 −10.85 +11.52 +21.50 +2.77 +3.72 +52.40 +41.56 −16.13 +17.92 +20.69 −9.79 −0.86 +50.25 +9.32 +0.39 +6.74 +13.74 +56.13 +22.89 −7.21 −14.21

+2.95 −11.91 +10.53 +21.60 +3.94 +1.84 +55.72 +43.80 −14.92 +20.14 +24.08 −10.17 −0.82 +54.63 +12.06 +2.70 +6.56 +13.50 +60.27 +27.71 −4.67 −11.61

3.5. The rate constants The rate constants of the forward reactions (12)–(22) were calculated using Eq. (i), G # values and imaginary frequencies reported in Tables 3 and 4. Fig. 7a and b show the calculated rate constants of the NH2 OH + OOH and NH2 CH3 + OOH reactions as a function of temperature, respectively. Since the values of Gibbs free energy (G = H − T S) depend on temperature we also calculated Gibbs free energies of some pre-reactant complexes and TS structures at 1000 K. These data are not reported here. It is observed that Gibbs free energies, G, of both reactants and T S increase with increase in temperature. However, G 1000 − G 298 for a T S and its corresponding reactant was almost the same. Therefore, although G depends on temperature, G # can be considered as a temperature independent property. For example, the value of G # (1000) − G # (298) was calculated as 1.9 kcal/mol for reaction (12). For this reason, it was assumed that variations of Gibbs free energies are negligible in the temperature range, therefore, we used G # calculated using M062X method, at 298.15 K to obtain the rate constants. Since some of reactions (12)–(22) may participate in combustion systems we calculated the rate constants in temperatures range of 273–2000 K. Table 6 presents the rate constants of the reactions (12)–(22) at 298, 500, 1000, 1500 and 2000 K. However, the rate constants of any temperature can be calculated from Fig. 7. The plots shown in Fig. 7(a)–(b) were fitted in Arrhenius equation

ln(k) = ln A − ( E Arr / R T )

(iv)

where A and E Arr are Arrhenius constant and activation energy, respectively. The obtained E Arr ’s are tabulated in Table 6. The Arrhenius activation energies are in good agreement with the trend observed for the calculated activation energies which indicates that the tunneling effect can be negligible. Wigner coefficient, κ , is 1.01–2.03 in the considered temperature range. Furthermore, since the Arrhenius plot is linear in the considered temperature range (273–2000 K), it implies that E a and G # are almost temperature independent and can be used for the other temperatures. Amount of variations in the rate constants of the reactions (14), (19) and (20) is large and their values change as 50 order in the range of 298–2000 K (Table 6). According to Arrhenius equation, reaction

Fig. 7. The rate constants calculated using M062X method for (a) NH2 OH + OOH and (b) NH2 CH3 + OOH reactions as a function of temperature.

with high activation energy values have a greater temperature dependency. In summary, a view of the whole process studied in this work can be shown as following: I. Reactions (1)–(6): NH3 → NH2 ; II. Reactions (7)–(11): [NH2 and NH3 ] → [NH2 CH3 and NH2 OH]; III. Reactions (12), (13), (17), (18): [NH2 CH3 and NH2 OH] → Intermediates; IV. Reactions (14), (19), (20): Intermediates → [NH2 CH2 OOH, NH(OOH)CH3 and NH(OOH)OH]. The calculated activation energies of reactions involved in step I are smaller than 25 kcal/mol. These values for reactions of step II are almost in the range of 50–70 kcal/mol. The activation energies of intermediates formation reactions (step III) are smaller than 20 kcal/mol and of reactions involved in step IV are about 70 kcal/mol.

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Table 6 The rate constants of the forward reactions at different temperatures and comparison of Arrhenius activation energies, E Arr , and calculate activation energies using M062X/6-311++G(3df, 3pd). Forward reaction

k (cm3 /s) 298 K

k (cm3 /s) 500 K

k (cm3 /s) 1000 K

k (cm3 /s) 1500 K

k (cm3 /s) 2000 K

E Arr (kcal/mol)

Ea (kcal/mol)

12 13 14 15 16 17 18 19 20 21 22

7.15 × 10−19 2.53 × 10−21 1.91 × 10−60 4.67 × 10−16 3.73 × 10−20 1.72 × 10−17 2.48 × 10−20 4.74 × 10−61 3.10 × 10−54 5.66 × 10−17 5.88 × 10−25

8.72 × 10−14 3.00 × 10−15 1.38 × 10−38 4.12 × 10−12 1.43 × 10−14 5.63 × 10−13 1.18 × 10−14 6.02 × 10−39 7.01 × 10−35 1.16 × 10−12 1.96 × 10−17

8.80 × 10−10 1.63 × 10−10 3.56 × 10−22 6.60 × 10−9 3.75 × 10−10 2.28 × 10−9 3.24 × 10−10 2.36 × 10−22 2.51 × 10−20 3.48 × 10−9 1.35 × 10−11

2.78 × 10−8 9.04 × 10−9 1.54 × 10−16 1.11 × 10−7 1.62 × 10−8 5.32 × 10−8 1.42 × 10−8 1.18 × 10−16 2.63 × 10−15 7.25 × 10−8 1.76 × 10−9

3.06 × 10−7 1.03 × 10−7 9.35 × 10−14 8.31 × 10−7 2.31 × 10−7 3.08 × 10−7 1.14 × 10−7 1.00 × 10−13 1.02 × 10−12 3.93 × 10−7 2.40 × 10−8

19.47 22.61 75.38 15.07 20.74 16.95 20.73 75.38 65.33 15.7 26.50

16.19 19.52 71.57 12.01 18.27 10.95 15.59 69.74 60.53 19.96 25.07

4. Conclusion NH2 OH, NH2 CH3 and OOH take part in both hydrogen abstraction and replacement reactions. The calculated activation energies of hydrogen abstractions are less than those of replacement reactions. Two products, NH2 CH2 OOH and NH(OOH)CH3 , are produced due to reactions occur between NH2 CH3 and OOH species, which NH2 CH2 OOH has greater stability. In addition, the thermodynamic properties of all paths of the reactions were calculated using the B3LYP and M062X computational methods which can be used to construct potential energy surfaces (PES) of the reactions. Our results showed that B3LYP underestimated the activation energies of the reactions that is in agreement with previous finding [19]. We assumed that temperature dependency of G # values is little and therefore calculated G # values at 298.15 K were used to obtain the rate constants. Comparison between the obtained plots and Arrhenius ones showed that our supposition is reasonable. Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.physleta.2014.01.006. References [1] L.V. Moskaleva, M.C. Lin, J. Phys. Chem. A 102 (1998) 4687–4693. [2] J.C. Corchado, J. Espinosa-Garcia, M. Yang, J. Chem. Phys. 135 (2011) 014303–014311. [3] M. Monge-Palacios, C. Rangel, J. Espinosa-Garcia, J. Chem. Phys. 138 (2013) 084305. [4] J.B. Jeffries, G.P. Smith, J. Phys. Chem. 90 (1986) 487. [5] R. Atkinson, D.L. Baulch, R.A. Cox, J.N. Crowley, R.F. Hampson, R.G. Hynes, M.E. Jenkin, M.J. Rossi, J. Troe, Atmos. Chem. Phys. 4 (2004) 1461. [6] H. Meunier, P. Pagsberg, A. Sillesen, Chem. Phys. Lett. 261 (1996) 277–282. [7] D.L. Yang, M.L. Koszykowski, J.L. Durant, J. Chem. Phys. 101 (1994) 1361–1368. [8] R. Sumathi, D. Sengupta, M. Tho Nguyen, J. Phys. Chem. A 102 (1998) 3175–3183.

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