Ab Initio Chemical Kinetics for Singlet CH2 Reaction with N2 and the

Mar 26, 2010 - Rate constants for various product channels in the temperature range of 300−3000 K are predicted by the variational transition state ...
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J. Phys. Chem. A 2010, 114, 5195–5204

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Ab Initio Chemical Kinetics for Singlet CH2 Reaction with N2 and the Related Decomposition of Diazomethane Shucheng Xu† and M. C. Lin*,†,‡ Department of Chemistry, Emory UniVersity, Atlanta, Georgia 30322, and Institute of Molecular Science, Department of Applied Chemistry, National Chiao Tung UniVersity, Hsichu, Taiwan 300 ReceiVed: NoVember 20, 2009; ReVised Manuscript ReceiVed: February 24, 2010

The kinetics and mechanism for the reaction of singlet state CH2 with N2 have been investigated by ab initio calculations with rate constant prediction. The potential energy surface of the reactions has been calculated by single-point calculations at the CCSD(T)/6-311+G(3df,2p) level based on geometries optimized at the B3LYP/6-311+G(3df,2p) level. By comparing the differences in the predicted heats of reaction with the available experimental values, we estimate the uncertainties in the calculated heats of reactions are (1.4 kcal/mol. Rate constants for various product channels in the temperature range of 300-3000 K are predicted by the variational transition state and RRKM theories. The predicted total rate constants for 1CH2 + N2 at 760 Torr Ar pressure can be represented by the expressions s-ktotal ) 9.67 × 10+7 × T -6.88 exp (-1345/T) cm3 molecule-1 s-1 at T ) 300-2400 K and 3.15 × 10-229 × T +56.18 exp (128 000/T) cm3 molecule-1 s-1 at T ) 2400-3000 K. The branching ratios of the primary channels for 1CH2 + N2 are predicted: k1 for forming singlet s-CH2N2-a (diazomethane) accounts for 0.97-0.01, k2 + k4 for producing HCNN-a + H accounts for 0.00-0.69, k3 for forming singlet s-CH2N2-b (3H-diazirine) accounts for 0.03-0.00, k5 for producing HCN + NH accounts for 0.00-0.18, and k6 for producing CNNH + H accounts for 0.00-0.11 in the temperature range of 300-3000 K. The rate constant predicted for the unimoclecular decomposition of diazomethane producing 1CH2 + N2 agrees closely with experimental results. Because of the low stability of the two isomeric CH2N2 adducts and the high barriers for production of CN-containing products, the contribution of the CH2 + N2 reaction to NO formation becomes very small. Introduction Prompt NO formation is an important process in the primary reaction zone of a hydrocarbon flame. The reaction of N2 with methylene (CH2) has long been considered, along with CH, as a potential precursor for the prompt NO formation.1 In general, the reaction of CH2 with molecular nitrogen is more endothermic than the analogous reaction with CH; thus, CH2 has been thought to be a minor contributor to the overall prompt NO formation. The reaction of CH with N2 has been studied by many groups.2-8 In our laboratory, HNCN was first proposed to be the key stable intermediate of the CH + N2 reaction producing NCN + H at high temperatures, instead of HCN + N, along a spin-allowed doublet electronic state path.3 In a previous study on the reaction of CH2 with N2, the rate constant for CH2 + N2 f HCN + NH was modeled using the RRKM theory based on assumed singlet and triplet transition states by Sanders and Lin.9 In that study, the rate constant predicted for the singlet state could be represented by k ) 8 × 10-12 exp (-18,000/T) cm3 molecule-1 s-1. In addition, the minimum energy paths for 1CH2 + N2 f HCN2 + H and 1CH2 + N2 f CH2N + N were calculated by Walch using the CASSCF method,10 where two primary singlet intermediates for CH2N2, diazomethane (DM), and 3H-diazirine were found. Recently, a new experimental and modeling study of NO formation in 10 Torr methane and propane flames was reported by Williams and Fleming.11 In this paper, singlet and triplet CH2 with N2, 1,3CH2 + N2 f HNCN +H, 1,3CH2 + N2 * To whom correspondence should be addressed. E-mail: chemmcl@ emory.edu. † Emory University. ‡ National Chiao Tung University.

f1,3NCN +H2, and 1,3CH2 + N2 + M f,13H2NCN + M were assumed in a modified kinetic mechanism for NO formation. Following the conference presentation of a preliminary version of the present work,12 Williams, Sutton, and Fleming continued to investigate the plausibility of 1CH2 + N2 in the prompt NO formation mechanism via examination of experimental species profiles with kinetic modeling of several low-pressure methane-oxygen-nitrogen flames.13 In this report, they have constructed a kinetic pathway initiated by the recombination of 1CH2 with N2 to form CH2N2 early in the flame. The related thermal decomposition of DM producing 1CH2 and N2 has been experimentally measured by a number of investigators14-18 because it is an important source of the CH2 radical. In an early static photolysis experiment in the pressure range 40-200 Torr at temperatures between 410 and 490 K, Steacie14 observed that the decomposition of DM was homogeneous and bimolecular with an activation energy of approximately 36 kcal/mol. In addition, Shantarovich19 studied the decomposition of DM in a flow tube with 20 Torr N2 in the temperature range 600-700 K and obtained k1 ) 0.8 × 1011 exp(-31 750/RT) s-1 (R ) 1.987 cal/mol · deg). Rabinovitch and Setser15 studied the decomposition of DM in the temperature range 453-653 K, and their results on the thermal decomposition of DM supported the singlet 1CH2 product. Setser and Rabinovitch16 continued to measure this reaction in different conditions and reported k1 ) 1.2 × 1012 exp(-34 000/RT) s-1 in 1:20 mixtures of diazomethane:cis-butene (at a total pressure of 25 Torr in the temperature range 505-530 K). Dunning and McCain17 obtained k1 ) 0.9 × 1012 exp(-32 000/RT) s-1 in the mixtures of 0.42 Torr DM and 110 Torr hydrogen in the

10.1021/jp911048p  2010 American Chemical Society Published on Web 03/26/2010

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temperature range 566-546 K. Dove and Riddick18 studied the kinetics and mechanism for the decomposition of DM diluted with krypton in shock waves using the time-of-flight spectrometer and obtained the second-order rate constants, k1′,Kr ) 109.61(0.21 exp(-15 800 ( 1000/RT) 1 mole-1 s-1 at pressures 45-95 Torr in the temperatures 820-1200 K. Here we corrected the units in this expression as l mole-1 s-1 according to the experimental data plotted in their Figures 6 and 7 in l mole s units, not cm3 mole-1 s-1 given by the authors in the abstract. Otherwise the second-order data can not match the first-order experimental data plotted in their Figure 9.18 Furthermore, the triplet state 3CH2 was long confirmed to be the ground electronic state of methylene.20 The question arose whether the thermal decomposition of DM proceeded with conservation of spinangular momentum to give the singlet radical or nonadiabatically to give triplet methylene. The latter possibility is analogous to the decomposition of isoelectronic N2O, which yields the O(3P) atom, with a rate characterized by a low frequency factor.21 However, Setser and Rabinovitch’s experimental results supported the singlet nature of the radical product.15,16 In a previous theoretical study, the thermal decomposition of DM was theoretically estimated using the ab initio SCF method and RRKM theory by Yano et al.22 In addition, the electronic ground-state structure and dissociation energies of CH2N2 were calculated by Papakondylis and Mavridis23 using the ab initio CASPT2 and CASPT3 methods with the correlation consistent cc-pVTZ basis set. In that work, the calculated dissociation energies for CH2N2 f 1CH2 + N2 and 3CH2 + N2 were 38.2 and 27.2 kcal/mol, respectively. In present study, the reaction of the singlet 1CH2 with N2 has been calculated in detail at the CCSD(T)/6-311+G(3df, 2p)// B3LYP/6-311+G(3df,2p) level of theory. In addition, the rate constants and branching ratios for the primary product channels of the 1CH2 + N2 reaction and the decomposition of DM in the temperature range of 300-3000 K have been predicted for combustion modeling applications. Computational Methods The optimized geometries of the reactants, transition states, intermediates, and products for the 1CH2 + N2 reaction have been calculated at the B3LYP/6-311+G(3df,2p) level. The energies for the PES are improved by single point calculations at the CCSD(T)/6-311+G(3df,2p) level of theory based on the optimized geometries at the B3LYP/6-311+G(3df, 2p) level. The rate constants for the key product channels have been computed with variational TST and RRKM theory using the Variflex code.24 All quantum chemistry calculations have been carried out by the Gaussian 03 program25 using a PC cluster and the computers at Cherry L. Emerson Center for Scientific Computation at Emory University. Results and Discussion The singlet 1CH2 radical is the first excited state of CH2.20 The predicted energy difference between 1CH2 and 3CH2 at the CCSD(T)/6-311+G(3df,2p)//B3LYP/6-311+G(3df, 2p) level is 9.8 kcal/mol, which is in reasonable agreement with the experimental values, 9.292 ( 0.57,26 9.215 ( 0.0036,27 and 9.0 ( 0.8 kcal/mol, based on their heats of formation at 0 K, ∆fH0(1CH2) ) 102.37 ( 0.38 kcal/mol28 and ∆fH0(3CH2) ) 93.38 ( 0.38 kcal/mol.28 In the following section, the PES and reaction mechanism for the reaction of 1CH2 with N2 are discussed. For convenience of discussion, the energies of all the species obtained by both methods are listed in Table 1, and the optimized geometries for the species involved in the reaction

Xu and Lin are shown in Figure 1. The potential energy diagram of the reaction obtained by the CCSD(T)/ 6-311+G(3df,2p)//B3LYP/ 6-311+G(3df,2p) method is shown in Figure 2, panels a and b. 1. Singlet State Potential Energy Surface and Reaction Mechanism. As shown in Figure 2a, the reaction of 1CH with N2 forms first primary intermediates s-CH2N2-a (diazomethane) with the linear structure of C-N-N and s-CH2N2-b (3Hdiazirine) with ∠N-C-N ) 67.5° by barrierless association processes. The predicted binding energies of 1CH2 and N2 giving s-CH2N2-a and s-CH2N2-b are 35.0 and 25.5 kcal/mol, respectively. The calculated dissociation energy of diazomethane is in reasonable agreement with the experimental activation energies for diazomethane decomposition, 32-36 kcal/mol,14-16 and the value, 31.1 ( 5.0 kcal/mol, based on the heats of formation at 0 K, ∆fH0(1CH2) ) 102.37 ( 0.38 kcal/mol,28 ∆fH0(N2) ) 0.00 kcal/mol,28 ∆fH0(CH2N2) ) 71.26 ( 4.61 kcal/ mol.29 However, using the predicted heat of formation of CH2N2, ∆fH0(CH2N2) ) 64.4 ( 1.8 kcal/mol, the average of the values evaluated by the isodesmic reactions, CH2CO + N2 f CH2N2 + CO and CH2CO + N2O f CH2N2 + CO2, we obtain D(CH2-N2) ) 38.0 ( 2.2 kcal/mol. The former reaction with ∆rH0 ) 47.8 ( 1.4 kcal/mol gives ∆fH0(CH2N2) ) 64.1 ( 1.8 kcal/mol, and the latter reaction with ∆rH0 ) 39.0 ( 1.4 kcal/ mol leads to ∆fH0(CH2N2) ) 64.7 ( 1.8 kcal/mol. These calculations were made with available experimental heats of formation, ∆fH0(CH2CO) ) -10.88 ( 0.38 kcal/mol,28 ∆fH0(CO) ) -27.20 ( 0.04 kcal/mol,28 ∆fH0(CO2) ) -94.0 ( 0.1 kcal/mol,30 and ∆fH0(N2O) ) 20.4 ( 0.1 kcal/mol.30 In addition, the calculated dissociation energy of 3H-diazirine in this work is in reasonable agreement with the activation energy for its decomposition reaction, 25.4 ( 3.4 kcal/mol, based on calculated ∆fH0(c-CH2N2) ) 77.0 ( 3.0 kcal/mol.31 s-CH2N2-a and s-CH2N2-b can isomerize to seven other intermediates. For example, s-CH2N2-a can transform to sHNC(H)N-b via s-TS4 with a barrier of 104.0 kcal/mol. Similarly, s-CH2N2-b can transform to s-HCNNH via s-TS1 with a barrier of 69.2 kcal/mol or to s-HNC(H)N-a via s-TS2 with a barrier of 103.5 kcal/mol. In addition, s-HNC(H)N-a can isomerize to s-CNNH2 through s-TS6 with a barrier of 40.5 kcal/ mol or to s-cyc-HNCNH via s-TS10 with a barrier of 57.6 kcal/ mol and s-HCNNH to s-HNC(H)N-b through s-TS5 with a barrier of 65.9 kcal/mol. Furthermore, s-CNNH2 can undergo isomerization to s-NCNH2 (cyanamide) through s-TS7 with a barrier of 28.3 kcal/mol, and s-NCNH2 can isomerize to s-HNCNH through s-TS11 with a barrier of 79.1 kcal/mol. The predicted exothermicity for formation of the most stable intermediate, s-NCNH2, 66.5 kcal/mol, is in reasonable agreement with the available experimental value of 68.7 ( 1.1 kcal/ mol based on ∆fH0(1CH2) ) 102.37 ( 0.38 kcal/mol,28 ∆fH0(N2) ) 0.00 kcal/mol,28 and ∆fH0(NCNH2) ) 33.67 ( 0.66 kcal/ mol.29 These isomerization reactions can also occur reversely as one would expect. As shown in Figure 2b, the 1CH2 + N2 reaction may produce the following products via the above-mentioned intermediates with the predicted enthalpy changes: HNCN + H, 25.6 kcal/ mol; 1NCN + H2, 34.1 kcal/mol; 1CNN + H2, 57.3 kcal/mol; HCN + 1NH, 56.7 kcal/mol; HCNN-a + H, 60.7 kcal/mol; CH2N + N, 63.4 kcal/mol; CNNH + H, 64.9 kcal/mol; HCNN-c + H, 65.0 kcal/mol; HCNN-b + H, 80.2 kcal/mol; CH + NH2, 97.9 kcal/mol. In principle, the 1NCN + H2 products may be formed by the dissociation of the intermediates s-NCNH2 by overcoming a high barrier of 108.2 kcal/mol at s-TS8 or s-CH2NN-b by overcoming an even higher barrier of 125.5 kcal/ mol at s-TS3. Similarly, the 1CNN + H2 products may be formed

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TABLE 1: Total and Relative Energiesa of Reactants, Transition States, and Products of the Reaction 1CH2 + N2 B3LYP/6-311+G(3df, 2p) species or reactions 1 CH2 + N2 s-CH2N2-a

ZPE 0.022176 6.0

CCSD(T)b/6-311+G(3df,2p)

energies

B3LYP/6-311+G(3df, 2p)

-148.695210 -44.3

-148.432050 -35.0

s-CH2N2-b s-HCNNH s-HNC(H)N-a s-HNC(H)N-b s-CNNH2 s-NCNH2 s-HNCNH s-TS1 s-TS2 s-TS3 s-TS4 s-TS5 s-TS6 s-TS7 s-TS8 s-TS9 s-TS10 CP 3 CH2 + N2

7.0 6.0 6.5 5.2 7.2 7.4 6.6 2.6 1.3 0.1 1.6 4.8 3.7 6.0 -0.9 -1.1 3.3 2.7 0.4

-27.6 -20.9 -8.2 49.8 -28.3 -72.7 -74.3 40.1 76.5 107.8 75.3 52.3 34.4 0.9 29.7 62.0 50.0 -1.9 -10.9

-25.5 -10.6 -4.8 53.3 -22.5 -66.5 -63.6 43.8 78.1 100.0 78.0 55.3 35.7 5.8 41.7 73.8 52.8 -1.6 -9.8

t-TS1 t-CH2N2-a 1 NCN + H2 1 CNN + H2 HNCN + H CNNH + H HCN + 1NH HCNN-a + H HCNN-b + H HCNN-c + H CH2N + N CH + HN2

2.1 5.0 -1.8 -3.1 -1.6 -1.7 1.1 -2.2 -1.8 -1.7 1.8 -1.6

2.8 -12.4 27.3 51.6 16.7 57.0 59.7 50.4 77.9 60.7 63.1 90.5

10.4 -1.6 34.1 57.3 25.6 64.9 56.8 60.7 80.2 65.0 63.4 97.9

∆H0,exptc (0.0) -31.1 ( 5.0 -38.0 ( 2.2d -25.4 ( 3.4e

-68.7 ( 1.1

-9.292 ( 0.5726 -9.215 ( 0.003627 -9.0 ( 0.8c

26.5 ( 1.1 65.0 ( 4.6 65.0 ( 4.6

a Total energies for 1CH2 + N2 are in au and relative energies for others are in kcal mol-1. b Single point energies based on optimized geometries calculated at the B3LYP/6-311+G(3df, 2p) level. c At 0 K, ∆fH0 are as follows: ∆fH0(1CH2) ) 102.37 ( 0.38 kcal/mol;28 ∆fH0(3CH2) ) 93.38 ( 0.38 kcal/mol;28 ∆fH0(N2) ) 0.00 kcal/mol;28 ∆fH0(HNCN) ) 72.3 ( 0.7 kcal/mol;29 ∆fH0(H) ) 51.63 kcal/mol;28 ∆fH0(H2) ) 0.00 kcal/mol;28 ∆fH0(HCN) ) 30.9 ( 0.7 kcal/mol;29 ∆fH0(HCNN) ) 115.76 ( 4.15 kcal/mol;29 ∆fH0(CH2N2) ) 71.26 ( 4.61 kcal/mol;29 ∆fH0(NCNH2) ) 33.67 ( 0.66 kcal/mol.29 d At 0 K, ∆fH0(CH2N2) ) 64.4 ( 1.8 kcal/mol derived from CH2CO + N2 f CH2N2 + CO and CH2CO + N2O f CH2N2 + CO2, where the heats of reaction ) 47.8 ( 1.4 and 39.0 ( 1.4 kcal/mol, respectively, calculated at the CCSD(T)//B3LYP level, ∆fH0(CH2CO) ) -10.88 ( 0.38 kcal/mol,28 ∆fH0(CO) ) -27.20 ( 0.04 kcal/mol,28 ∆fH0(CO2) ) -94.0 ( 0.1 kcal/ mol,30 and ∆fH0(N2O) ) 20.4 ( 0.1 kcal/mol,30 e Calculated ∆fH0(c-CH2N2) ) 77.0 ( 3.0 kcal/mol.31

by the dissociation of the intermediate s-CNNH2 by overcoming the barrier of 96.3 kcal/mol at s-TS9. The HNCN + H products may be produced by the direct dissociation of the intermediates with predicted dissociation energies: s-NCNH2, 92.1 kcal/mol; or s-HNCNH, 89.2 kcal/mol; or s-cyc-HNCNH, 2.8 kcal/mol. Similarly, the production of other radical product pairs may take place by the direct dissociation processes with the predicted dissociation energies: HCN + 1NH from s-HCNNH, 67.3 kcal/ mol; HCNN-a + H from s-HCNNH, 71.3 kcal/mol; s-CH2NNa, 95.7 kcal/mol; and s-HNC(H)N-b, 6.4 kcal/mol, respectively; HCNN-c + H from s-HNC(H)N-a, 69.8 kcal/mol; CNNH + H from s-HCNNH, 75.5 kcal/mol and s-CNNH2, 87.4 kcal/mol, respectively; CH2N + N from s-CH2N2-a, 98.4 kcal/mol; HCNN-b + H from s-CH2N2-b, 105.7 kcal/mol, and finally, CH + NH2, from s-HCNNH, 108.5 kcal/mol and s-HNC(H)N-b, 44.6 kcal/mol, respectively. By comparing the differences in the calculated heats of reaction with the available experimental values listed in Table 1, we estimate the uncertainties in the predicted heats of reaction at the CCSD(T)/6-311+G(3df,2p)// B3LYP/6-311+G(3df, 2p) level are (1.4 kcal/mol. The singlet 1CH2 may transform to its ground state triplet 3 CH2 by collisional quenching during the reaction. The 3CH2

can also react with N2. As shown in Figure 2a, the reaction of 3 CH2 with N2 can form triplet intermediate t-CH2N2-a with a 124.6° ∠C-N-N by overcoming a 20.2 kcal/mol barrier at t-TS1. There should be a crossing point (CP) between the triplet and singlet association curves. To search for the CP, the minimum energy path (MEP) for 1CH2 + N2 f s-CH2N2-a was calculated along the reaction coordinate of C-N from 1.3 to 5.0 Å with a step size of 0.1 Å at the B3LYP/6-311+G(3df,2p) level. Meanwhile, the IRC for t-TS1 f 3CH2 + N2 were also calculated along the reaction coordinate of C-N at the same level. By drawing the MEP and IRC curves shown in Figure 3, the reaction coordinate for the CP was obtained at C-N ) 2.313 Å. The CP with the fixed C-N ) 2.313 Å was reoptimized at the CASSAF (6,6,Slaterdet)/6-311+G(3df,2p) level. Finally, the single point energy for the CP with a 11.4 kcal/mol barrier from the triplet state reactants was obtained at the CCSD(T)/ 6-311+G(3df,2p) level based on the optimized geometry using the CASSAF method. Prediction of Rate Constants a. Methods Employed for Rate Constant Calculation. The rate constants for the following primary channels of the

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reaction of 1CH2 with N2 have been predicted by statistical calculations:

CH2 + N2 f s-CH2N2-a* f s-CH2N2-a (+M)

1

f HCNN-a + H 1

CH2 + N2 f s-CH2N2-b* f s-CH2N2-b (+M)

(1) (2) (3)

f HCNN-a + H

(4)

f HCN + 1NH

(5)

f CNNH + H

(6)

The rate constants for the primary six singlet state channels of the reaction as discussed above have been calculated using variational TST and RRKM theory by the Variflex Code24 in the temperature range 300-3000 K with Ar as bath gas. In principle, the quenching process of 1CH2 to 3CH2 by N2 may take place by the collision inducing curve-crossing as discussed

in the above section. However, this process may not compete with the formation of s-CH2N2-a as its reverse dissociation reaction occurs primarily via the singlet path producing 1CH2 + N2 (vide infra). Therefore, comparing with the barrierless association 1CH2 + N2 f s-CH2N2-a, the quenching process will be neglected in the association rate constant calculation. Channels 1 and 3 are association reactions forming the intermediates s-CH2N2-a and s-CH2N2-b, respectively. Channel 2 is a dissociation reaction via the intermediate CH2NN-a, directly giving the products HCNN-a + H. Channels 4, 5, and 6 are treated as dissociation reactions via the transition state s-TS1 and the intermediate s-HCNNH, producing HCNN-a + H, HCN + NH, and CNNH + H, respectively. For the barrierless association processes of 1CH2 + N2 f s-CH2N2-a via MEP1 and 1CH2 + N2 f s-CH2N2-b via MEP2 and direct dissociation process of s-CH2N2-a f HCNN-a +H via MEP3, s-HCNNH f HCNN-a +H via MEP4, s-HCNNH f HCN + 1NH via MEP5, and s-HCNNH f CNNH +H via MEP6, they were obtained by computing the potential energy curves along the reaction coordinate from their equilibrium distance R0 to 5.0 Å with a step size of 0.1 Å calculated at the

Figure 1. The optimized geometries of the reaction 1CH2 +N2 calculated at the B3LYP/6-311+G(3df,2p) level, where the crossing point was optimized at the CASS (6, 6, Slaterdet)/6-311+G(3df,2p) level with fixed reaction coordinate C-N ) 2.313 Å obtained from the crossing point searching.

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Figure 2. The PES of the reaction 1CH2 + N2 calculated at the CCSD(T)/6-311+G(3df,2p)//B3LYP/6-311+G(3df,2p) level, where the energy of the crossing point was obtained at the CCSD(T)/ 6-311+G(3df,2p)//CASS (6,6,Slaterdet)/6-311+G(3df,2p) level with ZPE correction at the B3LYP/ 6-311+G(3df,2p) level; (a), formation of intermediates; (b), production of products.

Figure 3. The crossing point searching for MEP of 1CH2 +N2 f s-CH2N2-a with IRC of 3CH2 +N2 f t-TS1 calculated at the B3LYP/6-311+G(3df,2p) level without ZPE correction.

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Figure 4. The predicted rate constants of k1 (a), k2 (b), s-ktotal (c) at the Ar pressures of 1, 10, 100, 300, and 760 Torr and 10 atm in the temperature range of 300-3000 K.

UB3LYP/6-311+G(3df,2p) level. The calculated MEPs could be fitted to the Morse potential function. The obtained parameters of the Morse potential function are the following, β ) 3.097 Å-1 with R0 ) 1.292 Å, and De ) 40.0 kcal/mol for MEP1; β ) 3.907 Å-1 with R0 ) 1.478 Å, and De ) 32.5 kcal/ mol for MEP2; β ) 1.439 Å-1 with R0 ) 1.076 Å, and De ) 103.9 kcal/mol for MEP3; β ) 1.678 Å-1 with R0 ) 1.023 Å, and De ) 79.5 kcal/mol for MEP4; β ) 2.668 Å-1 with R0 ) 1.239 Å, and De ) 72.3 kcal/mol for MEP5; and β ) 1.561 Å-1 with R0 ) 1.075 Å, and De ) 83.2 kcal/mol for MEP6; where the energies for De were slightly scaled to the CCSD(T) correction energy limit without ZPE corrections. For the variational rate constant calculations by the Variflex code, a statistical treatment of the transitional-mode contributions to the transition-state partition functions was performed variationally. The numbers of states are evaluated according to the variable reaction coordinate flexible transition state theory.24,32 The energy grain size of 1.00 cm-1 was used for the convolution of the conserved mode-vibrations, and the grain size of 50.00 cm-1 was used for the generation of the transitional-mode numbers of states. The estimate of the transitional mode contribution to the transition state number of states for a given energy is evaluated via Monte Carlo integration with 10 000 configuration

numbers. The energy-transfer process was computed on the basis of the exponential down model with the 〈∆E〉down values (the mean energy transferred per collision) of 400 cm-1 for Ar, 150 cm-1 for H2, 230 cm-1 for N2, 600 cm-1 for Kr, and 1000 cm-1 for cis-butene, respectively. The Lennard-Jones (L-J) parameters for different moelcules are employed: Ar, σ ) 3.47 Å, ε/k ) 114.0 K;33 N2, σ ) 3.74 Å, ε/k ) 82.0 K;33 Kr, σ ) 3.66 Å, ε/k ) 178.0 K;33 H2, σ ) 2.04 Å, ε/k ) 282.0 K derived from its critical temperature (Tc ) 314.0 K) and volume (Vc ) 17.6 cm3/mol); cis-butene, σ ) 4.87 Å, ε/k ) 426.4 K derived from its Tc ) 475.4 K and Vc ) 238.8 cm3/mol; s-CH2N2-a, σ ) 3.90 Å, ε/k ) 528.0 K derived from its Tc ) 588.6 K and Vc ) 122.6 cm3/mol; and s-CH2N2-b, σ ) 3.91 Å, ε/k ) 525.4 K derived from its Tc ) 585.7 K and Vc ) 123.5 cm3/mol), using the formulas (ε/k ) 0.897 Tc, σ ) 0.785 Vc1/3),33 where the critical temperatures and volumes for the species are calculated based on their optimized geometries. To achieve convergence in the integration over the energy range, an energy grain size of 100 cm-1 was used. The total angular momentum J covered the range from 1 to 250 in steps of 10 for the E, J-resolved calculation. The Morse potentials with the above-mentioned parameters, the L-J pairwise potential and the anisotropic potential are added together to form the final potential, similar

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TABLE 2: The Predicted Rate Expressionsa of k1, k2, k3, k4, k5, k6,and ktotalb at the Ar Pressures of 1, 10, 100, 300, 760, and 7600 Torr in the Temperature Range of 300-3000 K reaction k1

k3

k5

s-ktotalb

P (Torr) 1 10 100 300 760 7600 1 10 100 300 760 7600 1 10 100 300 760 7600 1 10 100 300 760 7600

T1 T2 T1 T2 T1 T2 T1 T2 T1 T2 T1 T2

A

n

B

reaction

P (Torr)

A

n

B

2.03 × 106 1.57 × 107 7.55 × 107 2.42 × 108 1.23 × 109 2.30 × 1010 4.59 × 103 2.66 × 104 3.96 × 105 1.37 × 106 3.84 × 106 7.27 × 107 5.75 × 10-6 5.75 × 10-6 5.75 × 10-6 5.75 × 10-6 5.75 × 10-6 5.75 × 10-6 3.12 × 104 1.17 × 10-42 1.23 × 106 1.34 × 10-131 2.06 × 106 1.66 × 10-146 1.21 × 107 2.89 × 10-143 9.67 × 107 3.15 × 10-229 8.23 × 109 0.00 × 10°

-7.25 -7.22 -7.13 -7.13 -7.21 -7.28 -6.92 -6.86 -6.90 -6.92 -6.92 -6.99 -1.52 -1.52 -1.52 -1.52 -1.52 -1.52 -6.69 8.38 -6.88 31.46 -6.65 35.12 -6.73 34.15 -6.88 56.18 -7.14 79.97

-1374 -1376 -1335 -1396 -1554 -1914 -1316 -1279 -1404 -1480 -1557 -1871 -33 400 -33 400 -33 400 -33 400 -33 400 -33 400 -1053 -12 400 -1175 47 700 -1044 62 700 -1152 63 400 -1345 128 000 -1828 214800

k2

1 10 100 300 760 7600 1 10 100 300 760 7600 1 10 100 300 760 7600

7.43 × 10-49 6.07 × 10-46 3.94 × 10-39 1.07 × 10-37 8.84 × 10-36 1.58 × 10-35 1.47 × 10-17 1.47 × 10-17 1.47 × 10-17 1.47 × 10-17 1.47 × 10-17 1.47 × 10-17 1.11 × 10-14 1.11 × 10-14 1.11 × 10-14 1.11 × 10-14 1.11 × 10-14 1.11 × 10-14

10.63 9.77 7.87 7.47 6.94 6.87 1.57 1.57 1.57 1.57 1.57 1.57 0.92 0.92 0.92 0.92 0.92 0.92

-19 900 -21 010 -23 360 -23 850 -24 500 -24 610 -30 400 -30 400 -30 400 -30 400 -30 400 -30 400 -32 800 -32 800 -32 800 -32 800 -32 800 -32 800

k4

k6

a Rate constants are represented by k ) ATn exp(B/T) in units of cm3 molecule-1 s-1. b ktotal ) k1 + k2 + k3 + k4 + k5 + k6. T1, for the temperature range of 300-2400 K; T2, for the temperature range of 2400-3000 K.

Figure 5. The predicted branching ratios for the primary channels of the 1CH2 +N2 reaction at the 760 Torr Ar pressure in the temperature range of 300-3000 K.

to that employed in the OH + CH2O,34 OH + HNCN,35 OH + CH3OH, OH + C2H5OH reactions.36 The tunneling effect on the transition states, s-TS1 is considered because its barrier involving H-transfer is much higher than the reactants.34-36 In this study, the tunneling effects are treated using the Eckart’s tunneling corrections. In addition, s-TS1 involved in channels 4, 5, and 6 has two optical isomers, s-CH2N2-a in channel 2 has two H atoms for dissociation, so a statistical factor of 2 is employed in the rateconstant calculation.

Finally, the rate constants for the unimolecular decomposition of DM producing singlet 1CH2 have also been predicted:

s-CH2N2-a f 1CH2 + N2

(7)

b. Predicted Rate Constants of 1CH2 + N2. The predicted values for k1 forming singlet s-CH2N2-a at 6 specific pressures between 1 and 7600 Torr in the temperature range of 300-3000 K are shown in Figure 4a and are also listed in Table 2. The

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value of k1, as expected, has a strong pressure dependence in the whole temperature range; it decreases with increasing temperature from 300 to 3000 K. When the pressure increases from 1 Torr to 10 atm, k1 increases linearly. The predicted results for k2 producing HCNN-a + H at 6 specific pressures between 1 and 7600 Torr in the temperature range of 300-3000 K are shown in Figure 4b and are also listed in Table 2. The properties of k2 are completely different from those of k1 because this pathway has to overcome a dissociation energy of 95.7 kcal/mol. k2 increases with increasing temperature from 300 to 3000 K; it has a weak pressure dependence only at temperatures below 500 K due to competition with k1 through collisional deactivation of the excited s-CH2N2-a. The predicted values for k3 forming singlet s-CH2N2-b at 6 specific pressures between 1 and 7600 Torr in the temperature range of 300-3000 K are listed in Table 2. k3 has a strong pressure dependence in the entire temperature range as k1. When the pressure increases from 1 Torr to 10 atm, k3 also increases nearly linearly. In addition, k3 also decreases with increasing temperature from 300 to 3000 K. The predicted results for k4 producing HCNN-a + H, k5 producing HCN + NH, and k6 producing CNNH + H at 6 specific pressures between 1 and 7600 Torr in the temperature range of 300-3000 K are listed in Table 2. The k4, k5, and k6 have similar P,T-dependence properties as k2. They increase with increasing temperature from 300 to 3000 K and are not pressuredependent in the whole temperature range. The predicted total rate constant for ktotal ) ∑6i ) 1ki at 6 specific pressures between 1 and 7600 Torr in the temperature range 300-3000 K are shown in Figure 4c and are also listed in Table 2. The P,T-dependences of ktotal are closely parallel with those of k1 and k3 in the temperature range below about 2000 K. However, when temperature is over 2000 K, the property of ktotal is close to that of k2; it increases with increasing temperature from 2000 to 3000 K. The predicted individual rate constants given in units of cm3 molecule-1 s-1 at the 760 Torr Ar pressure in the temperature range 300-3000 K can be represented by:

Xu and Lin

k1 ) 1.23 × 10+9 × T-7.21 exp(-1554/T) k2 ) 8.84 × 10-36 × T+6.94 exp(-24500/T) k3 ) 3.84 × 10+6 × T-6.92 exp(-1557/T) k4 ) 1.47 × 10-17 × T+1.57 exp(-30400/T) k5 ) 5.75 × 10-6 × T-1.52 exp(-33400/T) k6 ) 1.11 × 10-14 × T0.92 exp(-32800/T) The total rate constant for the reaction 1CH2 + N2 at the 760 Torr Ar pressure can be represented by the expression ktotal ) 9.67 × 10+7 × T -6.88 exp (-1345/T) cm3 molecule-1 s-1 at T ) 300-2400 K and 3.15 × 10-229 × T +56.18 exp (128 000/T) cm3 molecule-1 s-1 at T ) 2400-3000 K. Based on the estimated uncertainties in the calculated heats of reaction (1.4 kcal/mol, the uncertainties in the k1-k6 and ktotal are estimated to be (3%. c. Predicted Branching Ratios of 1CH2 + N2. The branching ratios of the rate constants k1-k6 for the reaction of 1CH2 + N2 at the Ar-pressure of 760 Torr in the temperature range of 300-3000 K are shown in Figure 5a. k1 for forming singlet s-CH2N2-a (diazomethane) accounts for 0.97-0.01, k2 + k4 for producing HCNN-a + H accounts for 0.00-0.69, k3 for forming singlet s-CH2N2-b (3H-diazirine) accounts for 0.03-0.00, k5 for producing HCN + NH accounts for 0.00-0.18, and k6 for producing CNNH + H accounts for 0.00-0.11 in the temperature range of 300-3000 K. d. Predicted Rate Constant for Decomposition of DM. The predicted high-pressure, first-order rate constant for the decomposition reaction of DM producing 1CH2 + N2 in the temperature range 300-3000 K can be given by

Figure 6. The decomposition rate constants of DM f 1CH2 + N2. Experimental data: O, data obtained by Shantarovich with 20 Torr N2;19 0, data obtained by Setser and Rabinovitch with 25 Torr cis-butene-2;16 9, data obtained by Dunning and McCain with 110 Torr H2.17 2, data obtained by Dove and Riddick with 45-95 Torr Kr.18 Predicted values in the temperature range of 300-3000 K: dashed curve, rate constants with N2 as bath gas at 20 Torr; solid curve is the rate constants with cis-butene as bath gas at 25 Torr; dotted curve, rate constants with H2 as bath gas at 110 Torr; dash dotted curve, rate constants with Kr as bath gas at 70 Torr.

Decomposition of Diazomethane ∞ k-1 ) 5.22 × 1015 exp[-17600/T] s-1

At 760 Torr, a variety of colliders including N2, O2, Ar, and CO2 have been applied to the decomposition of DM in order to compare the third-body efficiencies for this reaction. We conclude that CO2 has the largest third-body effect among the selected colliders. The rate expression using CO2 at 760 Torr is k-1 ) 2.58 × 1036 T-7.59 exp(-19 700/T) s-1. In addition, in order to compare the theory with experiments, we have calculated the rate constants in the temperature range of 300-3000 K under different experimental conditions. The predicted rate expressions can be represented as: k-1 ) 3.73 × 1033 × T-7.36 exp(-19000/T) s-1 using N2 as bath gas at 20 Torr pressure employed by Shantarovich;19 k-1 ) 7.31 × 1033 × T-7.37 exp(-19 000/T) s-1 using cis-butene as bath gas at 25 Torr pressure employed by Setser and Rabinovitch;16 k-1 ) 1.88 × 1035 × T-7.64 exp(-19 100/T) s-1 using H2 as bath gas at 110 Torr pressure employed by Dunning and McCain;17 and k-1 ) 2.25 × 1033 × T-7.19 exp(-19 400/T) s-1 using Kr as bath gas at 70 Torr pressure employed by Dove and Riddick.18 The predicted values shown in Figure 6 for the conditions close to those employed experimentally compare reasonably with experimental results cited above. The good agreement between theory and experiment supports the experimental finding that in the thermal decomposition of DM, 1CH2, instead of 3CH2, is the radical product. This result is understandable because of the small singlet-triplet energy difference and the barrierless association reaction involving 1CH2, comparing with the tight transition state for 3CH2 + N2 lying merely 1 kcal/mol below 1 CH2 + N2. Furthermore, based on the estimated uncertainties of (1.4 kcal/mol in the calculated D(CH2-N2) ) 35.0 kcal/ mol, the predicted rate constant for the DM decomposition under the condition employed by Setser and Rabinovitch16 was found to increase to 2 times that of the experimental result when the D(CH2-N2) was decreased by 1.4 kcal/mol and to decrease by 50% when the D(CH2-N2) was increased by 1.4 kcal/mol. The experimental value could be matched exactly with D(CH2-N2) )36.4 kcal/mol. We also tested the decomposition of DM under other conditions; we found that the predicted rate constant with D(CH2-N2) ) 36.0 kcal/mol could match the experimental data by Shantarovich,19 and that with D(CH2-N2) ) 34.9 kcal/mol could match the experimental data by Dunning and McCain.17 In addition, we also used the value of D(CH2-N2) ) 38.0 ( 2.2 kcal/mol based ∆fH0(CH2N2) ) 64.4 ( 1.8 kcal/mol to predict the rate constant for the decomposition of DM under the condition employed by Setser and Rabinovitch;16 it was found the predicted value was 5 times smaller than the experimental result and 10 times smaller than that calculated with D(CH2-N2) ) 35.0 kcal/mol. Because of the high sensitivity of the unimolecular decomposition rate constant to the dissociation energy, the above tests with different experimental data allow us to arrive at a more reliable bond energy, D(CH2-N2) ) 36.0 ( 2.0 kcal/mol, which effectively overlaps with those from the direct dissociation calculation (35.0 ( 1.4 kcal/mol) and the isodesmic estimation (38.0 ( 2.2 kcal/mol). Conclusions The kinetics and mechanism for the reaction of 1CH2 with N2 has been studied at the CCSD(T)/6-311+G(3df,2p)//B3LYP/6311+G(3df,2p) level of theory. By comparing the differences in the calculated heats of reactions with the available experimental values, we estimate the uncertainties in the calculated heats of reactions at the present level are (1.4 kcal/mol. The rate constants

J. Phys. Chem. A, Vol. 114, No. 15, 2010 5203 for the primary channels of the reaction in the temperature range of 300-3000 K are predicted. The primary intermediates formed, s-CH2N2-a (DM) and s-CH2N2-b (3H-diazirine) are stable at temperatures below 1000 K and begin to dissociate when temperature is higher than 2000 K, giving rise to the high-energy radical products HCNN-a + H, CNNH + H, and HCN + 1NH. The low stability of the CH2N2 isomers under combustion conditions and the high barriers for the formation of these products reduce the contribution of this reaction to NO formation by about a factor of 40, which was confirmed in ref 13. It is worth noting that the predicted rate constant for the unimoclecular decomposition of DM is in good agreements with existing experimental results under the conditions employed. Acknowledgment. The authors are grateful for the support of this work from the Basic Energy Sciences of DOE. MCL acknowledges the support from the National Science Council of Taiwan for a distinguished visiting professorship and the Taiwan Semiconductor Manufacturing Company for the TSMC Distinguished Professorship. Supporting Information Available: Table S1: Cartesian Coordinates (in Å) of the Optimized Geometries of Intermediates and Transition Sates of the Reaction of 1CH + N2 at theB3LYP/ 6-311+G(3df, 2p) level. Table S2: Frequencies and Moments of Inertia Ii of the Reaction of 1CH + N2 Calculated at the B3LYP/6-311+G(3df, 2p) Level. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Blauwens, J.; Smets, B.; Peeters, J. Proc. Combust. Inst 1976, 16, 1055. (2) Wada, A.; Takayanagi, T. Chem. Phys. 2002, 116, 7065. (3) Moskaleva, L. V.; Xia, W. S.; Lin, M. C. Chem. Phys. Lett. 2000, 331, 269. (4) Cui, Q.; Morokuma, K.; Bowman, J. M.; Klippenstein, S. J. J. Chem. Phys. 1999, 110, 9469. (5) Rogers, A. S.; Smith, G. P. Chem. Phys. Lett. 1996, 253, 313. (6) Seidman, T. J. Chem. Phys. 1994, 101, 3662. (7) Manaa, M. R.; Yarkony, D. R. Chem. Phys. Lett. 1992, 188, 352. (8) Berman, M. R.; Lin, M. C. J. Phys. Chem. 1983, 87, 3933. (9) Sanders, W. A.; Lin, C. Y.; Lin, M. C. Combust. Sci. Technol. 1987, 51, 103. (10) Walch, S. P. J. Chem. Phys. 1995, 103, 4930. (11) Williams, B. A.; Fleming, J. W. Proc. Combust. Inst. 2007, 31, 1109. (12) Xu, S. C.; Lin, M. C. Fall Technical Meeting, Eastern States Section; Combustion Institute: Charlottesville, VA, October 21-24, 2007. (13) Williams, B. A.; Sutton, J. A.; Fleming, J. W. Proc. Combust. Inst. 2009, 32, 343. (14) Steacie, E. W. R. J. Phys. Chem. 1931, 35, 1493. (15) Rabinovitch, B. S.; Setser, D. W. J. Am. Chem. Soc. 1961, 83, 750. (16) Setser, D. W.; Rabinovitch, B. S. Can. J. Chem. 1962, 40, 1425. (17) Dunning, W. J.; McCain, C. C. J. Chem. Soc. Phys. Org. 1966, 68. (18) Dove, J. E.; Riddick, J. Can. J. Chem. 1970, 48, 3623. (19) Shantarovich, P. S. Dokl. Akad. Nauk. SSSR 1957, 116, 255. (20) Herzberg, G.; Shoosmith, J. Nature 1959, 183, 1801. (21) Gill, E. K.; Laidler, K. J. Can. J. Chem. 1958, 36, 1570. (22) Yano, K.; Itoh, R. Rikogaku Kenkyusho Hokoku, Waseda Daigaku 1981, 94, 48. (23) Papakondylis, A.; Mavridis, A. J. Phys. Chem. A 1999, 103, 1255. (24) Klippenstein, S. J.; Wagner, A. F.; Dunbar, R. C.; Wardlaw, D. M.; Robertson, S. H. Variflex 1999. (25) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill,

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