Ab initio chemical kinetics for the N2H4 + NOx (x=1-3) reactions and related reverse processes

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Ab initio chemical kinetics for the N

2

H

4

+ NO

x

(x = 1–3) reactions

and related reverse processes

P. Raghunath, Y.H. Lin, M.C. Lin

⇑

Center for Interdisciplinary Molecular Science, Department of Applied Chemistry, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history: Received 26 May 2014

Received in revised form 14 July 2014 Accepted 19 July 2014

Available online 29 July 2014 Keywords:

N2H4–NOxreactions

Thermochemistry Kinetics and mechanisms

a b s t r a c t

The kinetics and mechanisms for N2H4+ NOx(x = 1–3) reactions and the related reverse reactions have been investigated by ab initio molecular orbital theory based on the CCSD(T)/CBS//CCSD/6-31G(d,p), CCSD(T)//B3LYP and CCSD(T)//BH&HLYP methods with the 6-311++G(3df,2p) basis set. These reactions are important to the propulsion chemistry of the N2H4–N2O4propellant system. The results show that the reactions of N2H4with NO and NO2producing N2H3+ HNO and N2H3+ c-HONO by H-abstraction with 33.7 and 10.3 kcal/mol barriers, respectively, are dominant. For the N2H4+ NO3(D3h) reaction via two pre-reaction van der Waals complexes with 0.5 kcal/mol and 1.6 kcal/mol binding energies produces HNO3+ N2H3by H-abstraction and t-HONO + N2H3O by concerted O- and H-atom transfers, respectively. The predicted enthalpies of formation of various products at 0 K are in good agreement with available experimental data within reported errors. Furthermore, the rate constants for the forward and some key reverse reactions have been predicted in the temperature range 300–2000 K with tunneling correc-tions using transition state theory (for direct abstraction) and variational Rice–Ramsperger–Kassel–Mar-cus theory (for association/decomposition) by solving the master equation.

1. Introduction

N2H4and NOx(x = 1–3) co-exist in the early stages of the

hydra-zine-NTO (N2O4nitrogen tetroxide) combustion reaction. The NOx

species can be produced by the rapid dissociation N2O4?2NO2

followed by the disproportionation reaction, 2NO2?NO + NO3.

In addition, a large amount of NO can be generated by the very fast metathetical reaction of NO2with radicals such as H and NH2and

N2H3, among others. Kinetics and mechanisms for the N2H4+ NOx

reactions have not been experimentally investigated before by direct reagent or product detection[1]. A reliable characterization of these processes by a high-level ab initio chemical kinetic study is therefore called for.

Among the 3 reactions studied in the present work, the NO2+

N2H4reaction has been investigated by 3 research groups before.

Sawyer and Glassman[2]studied the reaction in 1967 in an adia-batic ﬂow reactor using a thermocouple to measure the rate of temperature rise. They attributed the T-rise to the initiation pro-cess which obeyed the kinetics: d[N2H4]/dt = k2[NO2][N2H4] with

k2= 1015.83 exp [26,700/RT] cm3mol1s1 (where R = 1.987

cal mol1_K1_), _suggesting _that _the _reaction _NO

2+ N2H4?

HONO + N2H3 has a very high barrier of 26.7 kcal mol1. More

recently Lai et al.[3]reported the rate constant for the bimolecular reaction to be k2= 3.23 T3.56exp(384/T) cm3mol1s1from 250

to 2500 K temperature range based on the transition state theory (TST) calculation using the potential energy surface predicted with the G2M (CC3) method[4]. A similar ab initio chemical kinetic cal-culation employing the CBS-QB3 method by Koshi and coworkers

[5], in their attempt to model the hypergolic reaction of NTO and hydrazine, gave rise to the expression k2= 4.89 101 T3.43

exp(5566/T) cm3_mol1_s1 _{from 300 to 3000 K temperature.}

These data will be compared with our present results using a higher level of theory in conjunction with TST calculations.

The kinetics and mechanisms for both N2H4reactions involving

NO and NO3 have not been studied before. The processes are

believed to be intimately involved in the early stages of the hyper-golic combustion of N2H4and NTO because of the expected high

concentrations of both NOxspecies, directly or indirectly through

their reverse processes (such as N2H3+ HNO ? N2H4+ NO,

N2H3+ HONO ? N2H4+ NO2 and N2H3+ HONO2?N2H4+ NO3 at

high temperatures). Parenthetically it should be mentioned that HNO3is one of the major products formed in the exothermic

bimo-lecular initiation reaction, N2H4+ ONONO2?HNO3+ H2NHNO

proposed by Lai et al.[3]. The results of the present calculations on the title reactions will be discussed in detail below.

⇑ Corresponding author.

E-mail address:[email protected](M.C. Lin).

Contents lists available atScienceDirect

Computational and Theoretical Chemistry

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2. Computational methods

The mapping of the potential energy surfaces (PESs) of N2H4+

NOx(x = 1–3) and some key reverse processes was carried out by

using the Gaussian 09 program [6]. The geometries of the reactants, intermediates, transition states, and products of all the reactions were optimized with the CCSD/6-31G(d,p) [7], BH&HLYP/6-311++G(3df,2p) [8] and B3LYP/6-311++G(3df,2p) [9]

methods. It should be mentioned that the ground state structure of the NO3 radical predicted by some DFT methods such as

BH&HLYP had C2vsymmetry, which is inconsistent with the

exper-imentally determined D3hstructure[10]; accordingly, for the NO3

reaction, we only used the geometries predicted by the CCSD method. Vibrational frequencies employed to characterize station-ary points and zero-point vibrational energy (ZPVE) corrections were also calculated at the same levels of theory. For obtaining more accurate energies, we carried out higher level single-point energy calculations with extrapolation to the complete basis set (CBS) limit[11]using the CCSD/6-31G(d,p) optimized geometries for all 3 reactions. The CBS energies were evaluated with these geometries as follows. The total energies E(X) computed with the cc-PVXZ basis sets (X = 2, 3, 4) extrapolated to the CBS limits ECBS

employing a three-point extrapolation scheme[11],

EðXÞ ¼ ECBSþ b exp½ðX 1Þ þ c exp½ðX 1Þ

2

where E(X) is the single point energy calculated by CCSD(T)/ cc-pVXZ method [12], X is the cardinal number of the basis sets connected with X = 2 (DZ), 3 (TZ), 4 (QZ) and ECBS, b, and c are

parameters to be ﬁtted. In addition, we also carried out single point energy calculations by the CCSD(T)/6-311++G(3df,2p) method [7]

using the DFT optimized geometries for NO and NO2reactions. Rate

constant predictions for all forward reactions were based on the CBS-limit values using the CCSD/6-31G(d,p) optimized geometries; whereas for some key reverse processes of the NO and NO2

reac-tions, their rate constants were predicted with the CCSD(T)/ 6-311++G(3df,2p)//BH&HLYP/6-311++G(3df,2p) energies.

The rate constants were calculated using the microcanonical transition-state theory (TST) and/or the Rice–Ramsperger– Kassel–Marcus (RRKM) theory by solving the one-dimensional master equation to derive the nonequilibrium distribution function for each channel with the VARIFLEX program suite[13]. For a bar-rierless dissociation process, the Morse function, V(R) = De{1–

exp[b(R–Re)]}2, was employed to ﬁt the dissociation energy curve

and approximate the minimum energy path (MEP) for the rate con-stant calculation. In the Morse function, R is the reaction coordi-nate (i.e., the distance between two bonding atoms), Re is the

equilibrium value of R, and De is the binding energy excluding

zero-point energy. For tight transition states, the numbers of states were evaluated according to the rigid-rotor harmonic-oscillator assumption[13].

3. Results and discussions

3.1. Potential energy surfaces and reaction mechanisms

The equilibrium geometries of various species involved in the reactions computed at the CCSD/6-31G(d,p) level are summarized inFig. 1. For the NO and NO2reactions, the structures predicted

with the BH&HLYP/6-311++G(3df,2p) method are very similar. The potential energy surfaces (PES’s) of the 3 N2H4reactions were

predicted at the CBS//CCSD/6-31G(d,p) + ZPVE level of theory; the results are presented inFig. 2. Additionally, the results for N2H4

reactions with NO and NO2 obtained by different methods have

been compared inTable 1 and 2. In this work, we also study the reverse reaction mechanisms of N2H3 with HNO and HONO at

the CCSD(T)//BH&HLYP/6-311++G(3df,2p) level as shown in

Fig. 3. The moments of inertia and the vibrational frequencies of all the species involved in these reactions are listed inTables S1 and S2for the kinetic calculations. The calculated heats of forma-tion of major species at 0 K are compared with experimental data inTable 3. The following discussion will be based on the energies computed at the CBS//CCSD/6-31G(d,p) level unless speciﬁed otherwise, and all the energies of TS’s and intermediates cited are relative to the reactants.

Fig. 1. The optimized geometries of the reactants, intermediates, transition states and products computed at the CCSD/6-31G(d,p) level (length in Å and angle in degree).

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3.1.1. Reactions of N2H4+ NO

We have carried out an exhaustive search for the potential energy paths and mechanism of the N2H4+ NO reaction as shown

inFig. 2A. As revealed by the PES, there are two possible reaction paths: H abstraction and substitution reactions. Both the product

channels proceed via a loose pre-reaction complex formed by the nitrogen of NO bonding with one of the nitrogen of the N2H4

form-ing the LM1 complex with 1.6 kcal/mol bindform-ing energy. The bond length of N N complex is 2.938 Å. Firstly, the intermediate LM1 can react via the lowest energy channel by NO attacking on one of the hydrogen atoms in N2H4through TS1 to yield the HNO + N2

H3via the post-reaction complex HNO:N2H3, LM2, with a barrier of

33.7 kcal/mol; the CBS/CCSD value is very close to the values obtained by the CCSD(T)//BH&HLYP and CCSD(T)//B3LYP methods with the 6-311++G(3df,2p) basis set, 33.7 and 35.1, respectively (see Table 1). The endothermicity of this process predicted by CCSD(T)/BH&HLYP and that by the CBS extrapolation, 33.7 and 33.6 kcal/mol, respectively, are in close agreement with the exper-imental value, 33.0 kcal/mol[14,15]. In the second mechanism, the reaction takes place by the attack of the nitrogen of NO at one of the nitrogen atoms of N2H4via TS2 to give NH2+ H2NNO. The

reac-tion barrier is 38.5 kcal/mol, which is 4.8 kcal/mol higher than TS1. Furthermore, the endothermicity of this process is computed to be 17.0 kcal/mol. The reaction barriers and the heats of reaction

Fig. 2. Schematic energy diagram for N2H4reactions with NOx(x = 1–3) computed at the CCSD(T)/CBS//CCSD/6-31G(d,p) level with ZPVE corrections. Relative energies are

given in kcal/mol at 0 K.

Table 1

The energies of species for N2H4+ NO computed at the various levels of theory with

ZPVE corrections. Relative energies are given in kcal/mol.

Species CCSD(T)/6-311++G(3df,2p) CCSD(T)/CBS //CCSD BH&HLYP B3LYP N2H4+ NO 0.0 0.0 0.0 LM1 1.6 1.4 1.4 LM2 28.0 27.9 28.0 TS1 33.7 35.1 33.7 TS2 39.1 38.9 38.5 HNO + N2H3 33.7 33.7 33.6 NH2+ H2NNO 17.5 17.7 17.0

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predicted by the 3 methods agree within 1.5 kcal/mol for the worst case.

3.1.2. Reactions of N2H4+ NO2

The potential energy diagram obtained at the CBS//CCSD/6-31G(d,p) level and their optimized geometries computed with the CCSD/6-31G(d,p) method are presented in Figs. 2B and 1, respectively. The relative energies of all the H-abstraction reaction

channels obtained at different levels of theory are listed inTable 2

including the G2M(CC1) and CBS-QB3 results obtained by Lai et al.

[3]and Daimon et al.[5], respectively. Comparison of these data will be made below for the individual channels discussed. As shown in the PES, there are three H-abstraction and one associa-tion channel. All the H-abstracassocia-tion channels take place via the pre-reaction complex, LM3 (O2N NH2–NH2), at a separation of

2.862 Å between the NO2 and N2H4 with 2.0 kcal/mol binding

Table 2

The energies of species for N2H4+ NO2computed at the various levels of theory with ZPVE corrections. Relative energies are given in kcal/mol.

Species CCSD(T)/6-311++G(3df,2p) G2M(CC1)3 CBS-QB35 CCSD(T)/CBS//CCSD BH&HLYP B3LYP N2H4+ NO2 0.0 0.0 0.0 0.0 0.0 LM3 2.6 3.6 2.5 1.9 2.0 LM4 5.1 6.1 4.9 – 4.5 LM5 7.5 8.3 7.6 6.9 7.3 LM6 0.4 1.1 0.5 0.3 0.0 TS3 10.5 7.7 7.6 – 10.3 TS4 16.4 15.6 13.0 12.8 16.6 TS5 12.8 10.4 9.4 9.3 12.0 N2H3+ c-HONO 3.5 3.4 3.4 – 3.6 N2H3+ t-HONO 3.2 3.9 2.9 4.0 3.2 N2H3+ HNO2 11.5 12.6 11.3 12.0 11.7

Fig. 3. Schematic energy diagram for the reverse processes of N2H3with HNO and HONO computed with the CCSD(T)//BH&HLYP method employing the 6-311++G(3df,2p)

basis set with ZPVE corrections. Relative energies are given in kcal/mol at 0 K.

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energy. The ﬁrst low energy barrier products are N2H3+ cis-HONO,

formed by one of the oxygen atoms in the NO2abstracting an H

atom from N2H4 via transition state TS3 with a barrier of

10.3 kcal/mol, yielding a product complex, LM4 (ONOH NHNH2),

with the exothermicity of 4.5 kcal/mol. LM4 dissociates further to N2H3+ cis-HONO with the dissociation energy of 8.1 kcal/mol.

The barrier height of this reaction is 2.7 kcal/mol higher than the value 7.6 kcal/mol calculated at the G2M(CC1)//B3LYP/6-311++G(3df,2p) level by Lai et al.[3]. The latter is, however, close to the value, 7.5 kcal/mol, predicted by the CCSD(T)/6-311++G(3df,2p) //B3LYP/6-311++G(3df,2p) method. At the CCSD(T)//BH&HLYP level, the barrier was found to be 10.5 kcal/ mol, in close agreement with the CBS//CCSD/6-31G(d,p) result. In the second product channel producing N2H3+ trans-HONO, one

of the oxygen atoms in NO2abstracts one of H atoms of N2H4via

a trans-position through TS4 which requires a 16.6 kcal/mol bar-rier energy via the LM5 complex. This value is 3.6 and 3.8 kcal/ mol higher than those predicted by Lai et al. [3] and Daimon et al.[5], respectively. To summarize both abstraction results, the formation of cis-HONO requires 6.3 kcal/mol lower energy barrier, when compared to the production of trans-HONO. As shown in

Fig. 2B, the next low-energy H-abstraction product channel pro-duces N2H3+ HNO2occurring by the abstraction by the central N

atom of the NO2one of H atoms in N2H4via TS5 with 12.0 kcal/

mol energy.

Lastly, we studied the production of H2NN(O)H2+ NO which

occurs by the association of one of O atoms in NO2with one of N

atoms in N2H4via TS6 with a high barrier of 44.8 kcal/mol, forming

a product complex, LM7, with the endothermicity of 16.6 kcal/mol calculated by the CCSD(T)//BH&HLYP method; seeFig. S2 of Sup-porting Information. The intermediate LM7 dissociates to give H2NN(O)H2+ NO with the dissociation energy of 2.4 kcal/mol.

The H2NN(O)H2radical can undergo further H migration involving

one of the vicinal H atoms attaching to the N atom to the O atom giving H2NN(H)OH. This process has a high energy barrier

(47.8 kcal/mol) via TS7 with an overall exothermicity computed to be 1.2 kcal/mol. As shown inTable 2, the transition state ener-gies predicted at the CBS//CCSD level are close to those computed by the CCSD//BH&HLYP method. The results obtained by CCSD// B3LYP, G2M(CC1) and CBS-QB3 methods are about 3.0 kcal/mol lower, essentially resulted from the looser TS structures predicted by the B3LYP method. The barrier height at TS3 predicted by B3LYP/6-311++G(3df,2p) was found to be as low as 2.4 kcal/mol, which may be compared with the values predicted by BH&HLYP/ 6-311++G(3df,2p) and CCSD/6-31G(d,p), 8.6 and 9.5 kcal/mol, respectively, which are much closer to the CBS result of 10.3 kcal/mol.

3.1.3. Reactions of N2H4+ NO3

For this reaction system, we employed the CCSD(T)/CBS//CCSD/ 6-31G(d,p) method for PES mapping as some DFT methods such as BH&HLYP and others alluded to below failed to predict the ground state structure of the NO3radical as aforementioned. Previously,

Wille et al.[16]studied NO3reactions with alkynes using various

DFT methods: B3LYP, BH&HLYP, mPW1PW91 and mPW1K. They reported that NO3with the C2vsymmetry geometry is more stable

by the BH&HLYP, mPW1PW91 and mPW1K methods, whereas the D3hsymmetry geometry is more stable with B3LYP. The results by

BH&HLYP/cc-pVDZ were found to be in good agreement with those of QCISD and CCSD(T) methods. In the present case, however, the B3LYP method failed to predict the existence of the pre-reaction complexes for the N2H4+ NO3 reaction. In the present reaction,

D3hsymmetry has been considered for the NO3radical, to be

con-sistent with experimental data. The N–O bond length in NO3with

the D3hsymmetry is 1.238 Å calculated by CCSD/6-31G(d,p) which

is in close agreement with the experimental value of 1.240 Å[10]. In this reaction, there are two possible low energy channels via pre-reaction complexes computed with the CCSD(T)/CBS//CCSD/6-31G(d,p) method as shown in Fig. 2C; their related optimized structures are given in Fig. 1. In the ﬁrst channel producing t-HONO + N2H3O, the reaction proceeds via the van der Waals

complex, N2H4:NO3(LM8) with 0.5 kcal/mol binding energy. The

pre-reaction complex can further readily isomerize by H-abstraction from one of the N–H bonds in N2H4 via TS8, with 1.4 kcal/mol

barrier above the reactants to produce the post-reaction complex, N2H3+ HNO3 (LM9). The process is exothermic by 35.9 kcal/mol

at the CBS//CCSD level; LM9 can easily decompose to produce N2H3+ HNO3(at 21.0 kcal/mol) with no intrinsic TS. As shown

inFig. 2C, the second reaction occurs via the pre-reaction complex LM10 with 1.6 kcal/mol binding energy in which one of the O atoms of NO3interacting with one of the N atoms in N2H4by a

bar-rierless process. As the reaction proceeds, the LM10 can dissociate further by the transfer of an H atom from N2H4to a neighboring O

atom in NO3with the concerted transfer of another O atom to N2H3

via the ﬁve-membered ring low-energy TS9 with the negative acti-vation barrier of 3.1 kcal/mol to form a product complex, LM11. Further dissociation the reaction produces N2H3O + t-HONO with

26.3 kcal/mol exothermicity. In this potential energy surface, the reactions LM8 ? N2H4+ NO3, LM9 ? N2H3+ HNO3, LM10 ? N2

H4+ NO3 and LM11 ? t-HONO + N2H3O have no well-deﬁned

intrinsic transition states; their association/dissociation potential functions, computed variationally to their separate products at the DFT level, are ﬁtted to the Morse functions with b = 1.40 Å1_,

b= 1.35 Å1_{, b = 1.82 Å}1_{and b = 1.13 Å}1_{respectively. These}

val-ues will be used in the rate constant calculations to be discussed below. It is worth noting that the PES for the exothermic reaction of the nitrate radical is quite similar to those of analogous exother-mic reactions of NH3with chlorate (ClO3) and perchlorate (ClO4)

radicals[17].

3.2. Reverse reaction mechanisms of N2H3with HNO and HONO

The low-energy products, N2H3, HNO and HONO formed in the

N2H4reactions with NO and NO2may co-exist in the

hydrazine-NTO combustion system. Considering the important role of these species, we studied the reverse reactions in which many new reac-tion channels inaccessible to the forward processes are open with relatively low reverse barriers. As presented in previous sections, the results from both DFT and CCSD(T)/CBS//CCSD methods appear to agree closely in geometries and relative energies for the reac-tions of N2H4with NO and NO2. The PES’s of the reverse reactions

of N2H3with HNO and HONO were predicted only by using the

CCSD(T)//BH&HLYP method with the 6-311++G(3df,2p) basis set as shown inFig. 3 and their geometrical parameters are shown

Table 3

Heats of formation (DfH0°) of product species at 0 K predicted by CCSD(T)/CBS//CCSD/

6-31G(d,p) given in kcal/mol. Values in parentheses are predicted by CCSD(T)// BH&HLYP/6-311++G(3df/2p).

Species Reactions Heat of formation DfH0°

Calculated Literaturea N2H3 N2H4+ NO ? HNO + N2H3 56.8 (56.9) 56.2 H2NNO N2H4+ NO ? NH2+ H2NNO 19.5 (20.0) 20.0 HNO2 N2H4+ NO2?N2H3+ HNO2 9.7 (9.9) – c-HONO N2H4+ NO2?N2H3+ c-HONO 17.8 (17.9) 16.9 ± 0.3 t-HONO N2H4+ NO2?N2H3+ t-HONO 18.2 (18.2) 17.4 ± 0.3 HNO3 N2H4+ NO3?N2H3+ HNO3 32.1 29.8 ± 0.1 N2H3O N2H4+ NO3?t-HONO + N2H3O 37.8 37.7 ± 0.2 a _{The experimental values utilized in the calculations are obtained based on the}

enthalpies of formation at 0 K for N2H4= 26.2 kcal/mol; NO = 21.5 ± 0.1 kcal/mol;

NO2= 8.6 ± 0.2 kcal/mol; HNO = 24.5 kcal/mol; NO3= 18.9 kcal/mol; HNO3=

29.8 ± 0.1 kcal/mol; t-HONO = 17.4 ± 0.3 kcal/mol; c-HONO = 16.9 ± 0.3 kcal/ mol (Ref.[14]); N2H3= 56.2 kcal/mol (Ref.[15]); NH2= 45.2 ± 0.24 kcal/mol (Ref.

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inFig. S2. As discussed inFig. 2A, the N2H4+ NO reaction produces

N2H3+ HNO with the lowest energy barrier. Similarly in the

reverse reaction of N2H3+ HNO, the N2H4+ NO reaction is also

the lowest energy process. Another product pair, N2H3NO and H,

can be formed from the N2H3+ HNO reaction via association

pro-cess yielding the N2H3N(H)O intermediate with the energy barrier

of 3.7 kcal/mol(TS-r1). However, the elimination of the H atom requires a high energy barrier of 20.6 kcal/mol at TS-r2. On this PES, the next low energy product channel gives t-N2H2+ t-HONH

occurring by direct H-abstraction via TS-r3 with 15.5 kcal/mol bar-rier energy.

3.2.1. N2H3+ HONO

As discussed above, N2H3+ HONO are the low-energy products

in the N2H4+ NO2reaction. HONO molecule possesses two stable

geometrical conformers in the gas phase: cis and trans. The trans-conformer is 0.3 kcal/mol more stable than the cis-trans-conformer. The barrier energy for trans to cis-conformation is 10.5 kcal/mol (TS-r7) calculated by the CBS//CCSD method (seeFig. 3B). For the reverse processes, we calculated major low energy product chan-nels from N2H3+ c-/t-HONO by CCSD(T)//BH&HLYP; the PES is

pre-sented inFig. 3B. The calculation of N2H3+ c-HONO reaction shows

that the direct hydrogen abstraction occurred by the attack of H atom of c-HONO at the N atom of the NH group in N2H3produces

the N2H4+ NO2via TS3 with a 6.7 kcal/mol barrier energy at the

CBS//CCSD level. Similarly, we also calculated for the barrier energy of the N2H3+ t-HONO reaction channel producing N2H4+ NO2to be

13.4 kcal/mol. Based on our results the reverse reaction of N2H3

with cis- and trans-HONO producing N2H4+ NO2have the lowest

energy barriers. The next pathway of the lower energy reaction occurs by the attack of the N atom of the NH group of N2H3 at

the N atom of HONO via TS-r4 with 17.2 kcal/mol barrier energy, according to CCSD(T)//BH&HLYP, producing N2H3NO + OH

prod-ucts. The next low-energy channel involves the interaction of the HO group of HONO with one of the H atoms of the NH2group in

N2H3 leading to the formation of c-N2H2+ H2O + NO with

18.6 kcal/mol barrier at TS-r5. Finally, H-abstraction from the NH2 group of N2H3 by the terminal O atom of HONO via TS-r6

with 20.1 kcal/mol barrier energy producing t-N2H2+ HONOH

with an overall endothermicity of 11.2 kcal/mol. The rate constants for these reverse processes have been computed and reported below.

3.3. Enthalpies of formation

The predicted heats of formation of most product species involved in the N2H4+ NOx (x = 1–3) reactions are presented in

Table 3 based on the energies computed at the

CBS//CCSD/6-31G(d,p) and CCSD(T)//BH&HLYP/6-311++G(3df,2p) levels. The heats of formation were determined by combining the computed heats of reaction (DrH0°) based on the CBS-limit values and

CCSD(T)//BH&HLYP with the experimental heats of formation (DfH0°) of other known species involved in the reaction at 0 K

[14,15,18]. With the experimental heats of formation referenced in the footnote of Table 3, we obtained the values at 0 K for N2H3, c-HONO and t-HONO to be 56.8, 17.8 and 18.2 kcal/mol,

respectively, with an estimated error of ±1.2 kcal/mol. The pre-dicted heats of formation of these and other products are listed inTable 3for comparison with available data in the literature; in general, the agreement is quite good.

3.4. Rate constant calculations

The predicted rate constants for all the low-energy product channels of N2H4+ NOx(x = 1–3) are summarized below:

N2H4þ NO ! N2H3þ HNO ð1Þ ! NH2þ H2NNO ð2Þ N2H4þ NO2! N2H3þ c-HONO ð3Þ ! N2H3þ HNO2 ð4Þ ! N2H3þ t-HONO ð5Þ N2H4þ NO3! N2H3þ HNO3 ð6Þ ! t-HONO þ N2H3O ð7Þ

The rate constants for the forward reactions of N2H4+ NOx

(x = 1–2) channels using the transition state theory (TST)[19]with Eckart tunneling corrections[20]have been computed in the tem-perature range of 300–2000 K with the Variﬂex code[13], whereas the higher energy H-production channels are neglected. Rate con-stants are calculated according to the predicted PES’s as shown in

Fig. 2using energies obtained at the CCSD(T)/CBS//CCSD level and the moment of inertia and harmonic vibrational frequencies obtained by the CCSD/6-31G(d,p) presented inTable S1 in Support-ing Information.

The Arrhenius plots for the N2H4+ NO reaction product rate

constants of reactions (1) and (2) are presented in Fig. 4. The three-parameter ﬁts in the 300–2000 K temperature range give the following expression in cm3_molecule1_s1_:

k1¼ 1:07 1022T3:16 expð15342=TÞ

k2¼ 8:35 1023 T2:98 expð17919=TÞ

The rate constants for the H-abstraction reactions of N2H4with

NO2forming various products k3–k5are predicted in the

tempera-ture range of 300–2000 K and compared with the available com-puted results as shown inFig. 5. It is evident that H-abstraction producing c-HONO via TS3 is predominant comparing with other two abstraction reactions. The three-parameter ﬁts in the 300– 2000 K temperature range given in the unit of cm3molecule1_s1

for reactions (3)–(5) are given below along with previous com-puted results: k3¼ 1:37 1022T3:13expð4460=TÞ k3¼ 3:20 1025T3:74 expð1663=TÞ (Ref.[3]) k4¼ 4:00 1026 T4:14 expð3999=TÞ k5¼ 5:45 1026T4:00 expð6500=TÞ 0.5 1.0 1.5 2.0 2.5 3.0 10-42 10-38 10-34 10-30 10-26 10-22 10-18 10-14 NH₂ + H₂NNO (k₂) N₂H₃ + HNO (k₁) k (c m 3 m ol ecul e -1 s -1 ) 1000/T (K-1)

Fig. 4. The predicted rate constants for the N2H4+ NO reaction forming N2H3+ HNO

(k1) and NH2+ H2NNO (k2) based on the CCSD(T)/CBS//CCSD/6-31G(d,p) results.

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k5¼ 8:12 1023T3:43 expð5566=TÞ (Ref.[5])

The recent result of N2H4+ NO2?c-HONO + N2H3 (k3)

calcu-lated by Lai et al.[3]with the G2M//B3LYP method is also shown inFig. 5. The result is found to be 3 order magnitudes higher at low temperatures because of their smaller barrier energy at TS3 (2.7 kcal/mol lower than that by the CCSD(T)/CBS//CCSD method). Our predicted rate constant k5for t-HONO formation is lower than

the value reported by Daimon et al.[5], attributable to our higher barrier energy at TS4, 16.6 kcal/mol, than their 12.8 kcal/mol. We have also carried out rate constant calculations using the CCSD(T)/BH&HLYP values for the N2H4+ NOx (x = 1–2) and

com-pared with those by CBS//CCSD/6-31G(d,p) as shown inSupporting Information Figs. S3 and S4. All the calculations at the CCSD(T)/ BH&HLYP level of theory are in good accord with those by the CBS//CCSD/6-31G(d,p) method.

The rate constants for the N2H4+ NO3 reaction product

chan-nels are calculated by variational TST and RRKM rate theory using the energetics presented in Fig. 2C along with Morse potential energies and the vibrational frequencies and rotational constants are displayed inTable S1. The VTST calculations were carried out with the uniﬁed statistical formulation of Miller

[21] including multiple reﬂection corrections [22] above the shallow wells of the pre-reaction and post-reaction complexes. The Lennard-Jones parameters for collision rate estimates are obtained by using

r

= 3.85 Å and

e

= 153.4 K for N2H4:NO3,

derived from those of N2H4 (

r

= 4.23 Å and

e

= 205 K) and

NO3 (

r

= 3.462 Å and

e

= 114.8 K) and N2H4 buffer gas [23].

The predicted rate constant for the both product channels N2H3+ HNO3 (k6) and t-HONO + N2H3O (k7) are

pressure-independent at <100 atm, as shown in Fig. 6 given by the following expressions in the units of cm3_molecule1_s1 _in

the different temperature ranges:

k6¼ 1:03 1020T2:64 expð299=TÞ (300–2000 K)

k7¼ 2:70 1013T0:20 expð1097=TÞ (300–1000 K)

¼ 3:38 1021T2:59 _{expð2824=TÞ (1000–2000 K)}

3.4.1. Reverse reaction rate constants

As shown in the PES (Fig. 3), the reverse reactions of N2H3with

HNO and HONO can occur through many product channels as shown below. N2H3þ HNO ! N2H4þ NO ð8Þ ! N2H3NO þ H ð9Þ ! t-N2H2þ t-HONH ð10Þ N2H3þ t-HONO ! N2H3NO þ OH ð11Þ ! c-N2H2þ H2O þ NO ð12Þ N2H3þ c-HONO ! N2H4þ NO2 ð13Þ N2H3þ t HONO ! N2H4þ NO2 ð14Þ

Rate constants for all these product channels have been calcu-lated by the transition state theory (TST) with Eckart tunneling cor-rections, employed in the Variﬂex [13]. For the rate constant calculations, we used the CCSD(T)//BH&HLYP barrier heights and the BH&HLYP/ 6-311++G(3df,2p) molecular parameters of the reac-tants and transition states which are presented in Table S2. The calculated rate constant expressions for all the reaction channels R8-R13 obtained by three-parameter ﬁtting for the 300–2000 K temperature range are given below,

k8¼ 1:05 1020T2:14expð1254=TÞ (300–600 K) ¼ 1:61 1025T3:67expð2017=TÞ (600–2000 K) k9¼ 2:74 1026T3:82 expð8947=TÞ (300–2000 K) k10¼ 8:06 1041T8:15 expð455=TÞ (300–2000 K) k11¼ 7:78 1024T2:94 expð7739=TÞ (400–2000 K) k12¼ 4:64 1032T5:51 expð5592=TÞ (400–2000 K) k13¼ 1:15 1021T2:57 expð2838=TÞ (300–2000 K) k14¼ 8:95 1026T3:64 expð5088=TÞ (300–2000 K). 4. Conclusion

The kinetics and mechanisms for a series of reactions of N2H4

with NOx (x = 1–3) radicals and its reverse reactions have been

studied at the CCSD(T)/CBS//CCSD/6-31G(d,p) level of theory for all 3 systems and with additional two DFT based methods, CCSD(T)//B3LYP/6-311++G(3df,2p) and CCSD(T)//BH&HLYP/6-311++G(3df,2p) for NO and NO2reactions and some of their key

reverse processes, in conjunction with RRKM and TST calculations. The results of these calculations show that the H-abstraction pro-cess is the most favorable low-energy reaction channel in each of these reactions. The predicted barriers for N2H4with NO and NO2

reactions are 33.7 kcal/mol and 10.3 kcal/mol, respectively, for the formation of N2H3+ HNO and N2H3+ c-HONO products. In

the case of N2H4+ NO3, the results show that the reaction can

0.5 1.0 1.5 2.0 2.5 3.0 10-25 10-23 10-21 10-19 10-17 10-15 10-13 10-11 k (c m 3 m ole cu le -1s -1) 1000/T (K-1) k3 (Ref 3) k5 (Ref 5) k3 k4 k5

Fig. 5. The predicted rate constants for the N2H4+ NO2reaction forming N2H3+

c-HONO (k3), N2H3+ HNO2(k4) and N2H3+ t-HONO (k5) based on the CCSD(T)/CBS//

CCSD/6-31G(d,p) results, comparing with the data of Lai et al.[3]and Daimon et al. [5]. 0.5 1.0 1.5 2.0 2.5 3.0 10-14 10-13 10-12 10-11 10-10

k

₇

k

₆ k (c m 3 molecule -1s -1) 1000/T (K-1)

Fig. 6. Predicted rate constants for the N2H4+ NO3reaction forming N2H3+ HNO3

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produce two key low energy products HNO3+ N2H3via the direct

H-abstraction path with 1.2 kcal/mol barrier and t-HONO + N2H3O

via the concerted transfer of O and H atoms via a 5-member-ring intermediate with 3.1 kcal/mol negative barrier lying below the intermediate. The computed heats of formationDfH0° at 0 K for

N2H3, H2NNO, N2H3O, 56.8, 19.5 and 37.8 kcal/mol, respectively,

are in good agreement with experimental results. The rate con-stants for reactions occurring without intrinsic barriers (such as those in the NO3reactions and some reverse processes involving

HNO and HONO) were evaluated with the variational transition state theory. Their rate constants have been calculated in the tem-perature range 300–2000 K by TST/VTST and/or RRKM theory with Eckart tunneling and multiple-reﬂection corrections.

Acknowledgments

The authors deeply appreciate the support by Taiwan’s National Science Council (NSC) under contract No. NSC100-2113-M-009-013 and by the Ministry of Education’s ATU program. MCL also acknowledges the support from the NSC for the distinguished vis-iting professorship at National Chiao Tung University in Hsinchu, Taiwan. We are also grateful to the National Center for High-per-formance Computing for computer time and the use of its facilities.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.comptc. 2014.07.011.

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