Kinetic Modeling of the Free-Radical Process during the Initiated

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Kinetic Modeling of the Free-Radical Process during the Initiated Thermal Cracking of Normal Alkanes with 1-Nitropropane as an Initiator Yulei Guan, Bolun Yang,* Suitao Qi, and Chunhai Yi Department of Chemical Engineering, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China ABSTRACT: To elucidate the accelerating effect of 1-nitropropane on the thermal cracking of normal alkanes, ab initio and density functional theory calculations were performed to investigate the elementary reactions involved in the initiated thermal cracking of a 1-nitropropane/n-heptane mixture. The kinetic parameters were evaluated on the basis of standard transition state theory (TST) or variational transition state theory (VTST) with Wigner tunneling correction. The activation energy for the C
N bond rupture of 1-nitropropane to produce primary n-propyl and nitro radicals is calculated to be 234
269 kJ/mol, while the bond dissociation energy of the C
C bond within the n-heptane molecule is predicted to be at least 335 kJ/mol. These calculated results demonstrate that the presence of 1-nitropropane makes the free radical formation become relatively easier compared with single n-heptane cracking. Furthermore, compared with the C
H bond cleavage and 1,3-H intramolecular transfer, the unstable n-propyl radical _ 3 radical. After formation of these free mainly follows the C
C bond cleavage pathway to produce ethylene and secondary CH radicals, the H-abstraction of n-heptane with radicals occurs readily with considerably lower activation energy than the radical formation step to initiate the chain reaction. The analysis result indicates that the thermal cracking of n-heptane is accelerated mainly due to the change of the initial step from the C
C bond cleavage of n-heptane to the C
N bond rupture of 1-nitropropane.

1. INTRODUCTION In aeronautics research, great emphasis has been given to increasing the cruising speed of aircraft to the hypersonic range. At high Mach numbers, the heat load of whole vehicle increases dramatically so that the sensible heat sink of fuel is not sufficient to meet cooling requirements. Therefore, hypersonic aircraft has a large demand for an extra heat sink to transfer the excess heat. A potential candidate for this is endothermic cracking of the hydrocarbon propellant fuels prior to combustion to increase the cooling capacity of fuels.1
4 The more hydrocarbon fuels crack, the more heat can be absorbed. However, these chemical reactions require very high temperatures to occur at useful reaction rates and hence reduce the allowable stress of the heat exchanger material. To avoid a mechanical fault, the walls must be thickened. This will add size, weight, and complexity to the heat exchanger, which, in turn, significantly increases the launching cost and degrades the aircraft performance. An attractive solution is to introduce an initiator as a soluble additive into the endothermic hydrocarbon fuels, through which the thermal cracking can be carried out at a lower temperature with the same cracking conversion level and the overall fuel heat sink can be significantly augmented.5
7 In general, the accepted mechanism for hydrocarbon cracking is the free radical chain reaction.5,8
13 The initial step, that is, the formation of free radicals, is considered to be the rate-determining step in the reaction sequence. Once a free radical is generated, it can initiate an unlimited amount of hydrocarbon molecules to decompose into smaller fragments. Therefore, the additive in very small quantities that produces the reactive free radicals prior to the initial step of single hydrocarbon cracking would lead to a remarkable increase in the overall cracking rate. r 2011 American Chemical Society

A few experimental studies have been reported on the initiated thermal cracking of hydrocarbons. Chang et al.14,15 found that the addition of di-tert-butyl peroxide and elemental sulfur could produce free radicals at lower temperatures, and hence enhance the hydrothermal cracking of heavy oil remarkably. They also proposed that thermal cracking of heavy oil was promoted by oxidation pretreatment by air, and the accelerating effect was attributed to the formation and subsequent decomposition of peroxides serving as initiators in the chain reaction.16 However, these initiators were not applicable for the endothermic fuels due to their low solubility and thermal instability. Yu and Eser17 pyrolyzed binary mixtures of jet fuel model compounds under supercritical conditions and observed that the thermal cracking of n-dodecane was promoted in the presence of n-butylbenzene, which was due to the much easier cleavage of CR
Cβ in the side chain of n-butylbenzene than the breaking of any other bond in the two compounds. Nevertheless, alkylbenzenes could not be employed as initiators, because the aromatic hydrocarbons were restricted to a low content in the endothermic fuels: adding alkylbenzenes would cause a significant increase in coke deposition during the cracking process.18 Wickham and co-workers5
7 made a great effort to study the role of initiators in accelerating the thermal cracking of endothermic fuels under supercritical conditions. Their experimental results clearly demonstrated that the adopted initiators were very effective in increasing the fuel heat sink capacities and lowering the thermal cracking temperatures Received: March 16, 2011 Accepted: June 22, 2011 Revised: June 16, 2011 Published: June 23, 2011 9054

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Industrial & Engineering Chemistry Research of n-heptane and JP-7. Meanwhile, they also confirmed that the initiators only promoted the cracking rate and did not change the overall thermal cracking mechanism by analyzing the product distributions obtained with and without initiators. However, the identities of these initiated additives were not reported. Up to now, some chemicals have been disclosed to be effective as initiators to promote the thermal cracking of normal alkanes. In the work of Wang et al.,19
21 the initiated thermal cracking of nheptane was studied in the presence of triethylamine and tributylamine in the temperature range 823
923 K. It was found that triethylamine and tributylamine could substantially increase the cracking rate of n-heptane, and higher conversions were obtained compared with pure n-heptane cracking at all testing temperatures. Yanovskiy et al.22 reported that adding 0.8
1.5% nitromethane could increase the conversion of fuel and gas formation by 28% at 893
913 K. From the thermal stability point of view, 1-nitropropane is more suitable to serve as an initiator than nitromethane due to its higher flash point and stability. Consequently, Liu et al.23,24 studied the conversion and reaction kinetics of n-dodecane cracking with and without 1-nitropropane at temperatures ranging from 673 to 728 K. They observed that the conversion of n-dodecane increased by 32.5% at 693 K and by 150% at 728 K with only 2 wt % 1-nitropropane added, and the activation energy for the thermal cracking of n-dodecane was decreased by 27.6% in the presence of 1-nitropropane. In their recent study, they revealed that the addition of 1-nitropropane was also capable of inhibiting the formation of pyrolytic deposition.25 These researchers attributed the accelerating effect to a somewhat weaker C
N bond existing in these compounds, and the C
N bond could dissociate more easily to produce reactive free radicals compared with C
C and C
H bonds within normal alkane molecules. However, the accelerating mechanism was discussed without touching on the fundamental nature of the model in these works, and few studies concerned the action mode of initiator on the fuel molecule. To elucidate the structure
function relationship, as well as the interaction of initiator with fuel, it is essential to investigate the elementary reactions involved in the initiated thermal cracking in detail, and the relevant transition states and possible intermediates linking the reactants and products need to be explored. However, it would be very difficult to study by means of experimental methods due to various simultaneous reaction pathways and very short lifetimes of radicals. With the fast development of computer technology and numerical algorithms, computational quantum chemistry provides a feasible way to investigate reaction mechanisms at the atomic level. Meanwhile, to the best of our knowledge, there have been few theoretical investigations reported on these initiated thermal cracking systems. In the present work, n-heptane and 1-nitropropane are chosen as the model fuel and initiator. Ab initio and density functional theories are applied to study the initiated thermal cracking process of the 1-nitropropane/n-heptane mixture. Through quantum chemical calculations, the reasonable elementary reaction pathways and relevant transition states involved in the reaction mechanism are identified, and the accelerating effect on the thermal cracking of normal alkane is well analyzed by interpretation of the action mode of 1-nitropropane. Comparison of the calculated results with available reported data allows us to validate the theoretical approach. Hopefully, this investigation will provide some insight into the underlying chemistry for developing more effective initiators.

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2. COMPUTATIONAL METHOD 2.1. Electronic Energy Calculation. All the theoretical calculations were carried out using the Gaussian 09 computational package.26 Geometries of reactants, products, and transition states were optimized with “tight” convergence criteria at the B3LYP,27,28 B3PW91,27,29,30 and MP231
33 levels, in combination with the 6-31+G(d)34 basis set. In this work, the B3LYP/ 6-31+G(d), B3PW91/6-31+G(d), and MP2/6-31+G(d) levels were entitled “B3LYP”, “B3PW91”, and “MP2” for simplification. Calculations on radicals and transition states were performed with an unrestricted open-shell wave function, while restricted theory was applied to the normal alkanes, alkenes, and initiator closed-shell systems.35 Vibrational frequencies were calculated at the same levels as the optimizations to confirm the nature of the stationary points: minima with all real frequencies and first-order saddle point with only one imaginary frequency. Additionally, the intrinsic reaction coordinate (IRC) calculations36
38 were performed to verify that the transition states lead to the desired products and reactants on the minimum energy path (MEP). For the 1-nitropropane/n-heptane binary mixture, the bond dissociation energies (BDEs) determine which one will first dissociate to form free radicals. In view of this, we also applied high-level composite methods (G2,39 CBS-4M,40,41 and CBS-Q40,42) to obtain more reliable BDE(C
C) of n-heptane and BDE(C
N) of 1-nitropropane. In the kinetics study of 1-nitropropane decomposition, the thermodynamic data were derived from the rotational, translational, vibrational, and internal rotational partition functions of reactant and transition state based on statistical thermodynamics. The first three were determined by the standard harmonic oscillator analysis, and the internal rotational partition function was treated using the McClurg method.43
45 2.2. Rate Constant Calculation. When a transition state was located on the MEP that connected the desired reactant and product, the reaction rate constant was estimated by means of the standard transition state theory (TST) using eq 1:46
49 ! kB T
Δq G TST exp k ðTÞ ¼ ð1Þ h RT

where ΔqG is the zero point corrected activation free energy change between the reactant and the transition state, kB and h are the Boltzmann and Planck constants, respectively, T is the absolute temperature, and R is the ideal gas constant. By introducing the relationship ΔqG = ΔqH
TΔqS into eq 1, this leads to ! ! kB T Δq S
Δq H TST exp k ðTÞ ¼ exp ð2Þ h R RT If a well-defined transition state was not present on the MEP, the variational transition state theory50,51 (VTST) was applied to calculate the reaction rate constant using eq 3. According to VTST, the rate constant kVTST at T was obtained through determining the maximum ΔG(T,s) along the reaction coordinate (s).   kB T
ΔGðT, sÞ VTST exp ðTÞ ¼ min ð3Þ k s h RT To include the quantum mechanical tunneling and nonclassical effects, the TST and VTST rate constants were corrected by the Wigner tunneling correction k(T).52,53 The final rate 9055

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Table 1. Calculated BDEs(C
C) (kJ/mol) of n-Heptane at the B3LYP, B3PW91, MP2, G2, CBS-4M, and CBS-Q Levels level

C7 f C1 + C6

C7 f C2 + C5

C7 f C3 + C4

B3LYP

347.94

335.09

337.07

B3PW91 MP2

354.51 374.22

341.53 373.33

343.72 377.27

G2

373.82

372.76

377.04

CBS-4M

373.54

369.61

371.97

CBS-Q

375.13

370.82

372.05

expt55

368.19 ( 6.28

361.08 ( 6.28

365.26 ( 3.76

constant was given by kF ðTÞ ¼ kðTÞ kTST ðTÞ or kF ðTÞ ¼ kðTÞ kVTST ðTÞ

ð4Þ

In the present work, the kinetic parameters were evaluated at the absolute temperature T = 698 K, since the experiment was conducted in the temperature range 673
728 K in refs 23 and 24.

3. RESULTS AND DISCUSSION Analysis of the elementary reactions during the initiated thermal cracking process will be presented in the following sections (Primary Radical Formation, Further Reaction of the n-Propyl Radical, and Hydrogen-Abstraction Reaction) to elucidate the substantial reason that the presence of 1-nitropropane might accelerate the thermal cracking of n-heptane at lower temperature. 3.1. Primary Radical Formation. For the 1-nitropropane/ n-heptane binary mixture, the electronic structures and decomposition energetics of n-heptane and 1-nitropropane need to be studied in detail respectively to predict the free radical formation pathway. 3.1.1. Homolytic Cleavage of n-Heptane. n-Heptane is a saturated alkane, and the initial cracking step is the homolytic cleavage of a C
C bond within the n-heptane molecule to produce two free radicals, in which there is no classical transition state.54 Table 1 lists the calculated BDEs(C
C) for n-heptane cracking to form the corresponding radical pairs: methyl and hexyl; ethyl and pentyl; propyl and butyl. The calculated BDEs(C
C) at the B3LYP and B3PW91 levels are 14
28 kJ/mol lower than the reference values, and the MP2 and G2 results are 6
12 kJ/mol higher than the reference values, but giving the incorrect energetic order for the BDEs(C
C). The CBS-4M and CBS-Q levels predict the most accurate results with the gap less than 10 kJ/mol and yield the same energetic order as the experimental results. As shown in Table 1, the calculated results indicate that the weakest C
C bond within the n-heptane molecule is the β C
C bond, which is in accordance with the common feature of nalkane cracking.56 Therefore, one can expect that the homolytic cleavage of the β C
C bond is the fastest initial step and the ethyl radical might have slightly more stability than other primary radicals. In view of no transition state during the C
C bond cleavage process, the initial cracking step of n-heptane is mainly controlled by the reaction temperature, and the needed energy is no less than 335 kJ/mol to break the C
C bond to form free radicals. 3.1.2. Decomposition of 1-Nitropropane. According to the literature, there are two principal mechanisms for nitroalkane

Figure 1. Optimized structures of (a) equilibrium state of 1-nitropropane and (b) transition state (CME_TS) for CME reaction at the B3LYP, B3PW91, and MP2 levels. The italic values are obtained at the B3PW91 level. The values in parentheses are the MP2 results. All distances are in angstroms.

decomposition.57
60 One is the concerted molecular elimination (CME) mechanism: when the temperature is below 573 K, the reaction will proceed through a planar five-membered transition structure to give the corresponding alkene and HONO. The other is the radical decomposition mechanism: at higher temperature, due to a larger pre-exponential factor, the C
N bond dissociates without an apparent transition structure to produce two free radicals. Accordingly, the two possible decomposition pathways of 1-nitropropane are shown as 1R and 2R in eq 5.

3.1.3. CME Mechanism. Figure 1 shows the fully optimized structures of 1-nitropropane and the transition state (CME_TS) for HONO elimination from 1-nitropropane. As can be seen, a five-membered ring is formed in the CME_TS structure. The (Φ(C1NO1H),Φ(O1NC1C2)) dihedral angles are predicted to be (0.2
,
2.9
) at the B3LYP level, (0.3
,
3.0
) at the B3PW91 level, and (
0.4
,
3.2
) at the MP2 level. These values are quite close to zero, indicating that the five-membered ring is nearly planar. Figure 1 also lists the calculated structural parameters of 1-nitropropane and CME_TS. Taking the B3LYP results as an example to elucidate the CME reaction mechanism, the most significant change is the C1
N bond length during the reaction course, which is strongly elongated to reach the CME_TS structure (2.278 Å) from 1.508 Å of 1-nitropropane. The distance between atoms C2 and H is also elongated to 1.339 Å in the CME_TS structure from 1.096 Å of 1-nitropropane, implying the breaking of the C2
H bond. The formation of the O1
H bond is predicted by the significant decrease of the distance between the two atoms. In addition, the distance of the C1
C2 bond shortens as the bond changes from single bond (1.533 Å) to double bond (1.394 Å). The calculated kinetic and thermodynamic parameters of CME reaction are presented in Table 2. In addition, the experimental data reported in refs 57 and 59 are also included in 9056

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Table 2. Kinetic and Thermodynamic Parameters of CME Reaction of 1-Nitropropane at 698 K Calculated at the B3LYP, B3PW91, and MP2 Levels B3LYP

B3PW91

expt57,59

MP2

ΔqH (kJ/mol)

186.41

191.95

219.80

188.1

ΔqS (J/(mol 3 K))


9.12


7.14


5.58




log A (s
1)

12.69

12.79

12.87

12.15

k (s
1)

5.48  10
2 2.68  10
2 2.66  10
4


ΔqSInt (J/(mol 3 K))
14.79 12.39 log AInt (s
1) kInt (s
1)


11.76 12.55





2.77  10
2 9.97  10
3 1.26  10
4


k (s )

3.62  10
2 1.29  10
2 1.67  10
4 1.19  10
2

ΔE (kJ/mol)

74.87

F


1


15.36 12.36

92.10

99.54




Table 2. As can be seen, compared with the MP2 level, the calculated activation enthalpies (ΔqH) at the B3LYP and B3PW91 levels are in better agreement with the experimental results. The absolute average deviation from experiment is less than 3 kJ/mol. It is obvious that the activation enthalpy of the CME reaction of 1-nitropropane is lower than the BDE(C
C) of n-heptane. Therefore, the CME reaction of 1-nitropropane should proceed prior to the C
C bond cleavage of n-heptane. Based on the assumption of complete hindered rotation in the equilibrium state and CME_TS, the pre-exponential factor (log A) and reaction rate constant (k) calculated at the density functional theory (DFT) methods are considerably larger than the experimental results. The internal rotations of the
NO2 group around the C1
N bond, the
CH3 group around the C2
C3 bond, and the
C1NO2 group around the C1
C2 bond exist in the equilibrium state, and the internal rotation of the
CH3 group around the C2
C3 bond is present in the CME_TS. By inclusion of all internal rotations, one gets the kinetic results (log AInt and kInt). The deviation at the DFT methods from experiment decreases and is less than 0.24 log unit, indicating that the internal rotations practically affect the change of the pre-exponential factor, which was testified to previously by Shamsutdinov et al.61 The kF value is obtained by mutiplying kInt and the Wigner tunneling correction evaluated using the imaginary frequency (1312i, 1292i, and 1356i cm
1 at the B3LYP, B3PW91, and MP2 levels, respectively); the calculated kF (1.29  10
2 s
1) at the B3PW91 level agrees very well with the experimental result of 1.19  10
2 s
1. It can be also seen from Table 2 that this reaction, which has a chemical endothermicity (ΔE) of 75
100 kJ/mol, can be identified as a potential source of cooling to increase the fuel heat sink. 3.1.4. Radical Decomposition Mechanism. The second pathway is the rupture of the C
N bond in the 1-nitropropane molecule, which is often suggested as an initial step in the thermal decomposition of nitroalkanes, because the attachment of the nitro group leads to relatively weaker bond strength with BDE(C
N) equal to 256 kJ/mol.55 To model the C
N bond rupture process of 1-nitropropane, the potential energy surface scan is performed with R(C
N) in the range 1.5
4.5 Å at a 0.1 Å interval at the B3LYP and B3PW91 levels. In this case, calculations are carried out using unrestricted theory designated with “UB3LYP” and “UB3PW91”. For each studied point, the bond length of C
N is fixed at a certain value, while the rest of the electronic structure is allowed to relax to a minimum energy configuration. Figure 2 shows the calculated results at the B3LYP and B3PW91 levels. As the C
N

Figure 2. Fully optimized B3LYP and B3PW91 potential energy curves for the C
N bond rupture of 1-nitropropane.

bond is stretched, the predicted energy increases with the C
N bond length. When R(C
N) is larger than 3.5 Å, the energy approaches a constant value (253 kJ/mol at the B3LYP level and 257 kJ/mol at the B3PW91 level). This value is slightly higher than the calculated BDE(C
N) at the B3LYP (234 kJ/mol) and B3PW91 (238 kJ/mol) levels. As shown in Figure 2, there is no transition state on the potential energy curve. Therefore, the radical decomposition pathway of 1-nitropropane is controlled by the strength of the C
N bond. To obtain more reliable values, the BDE(C
N) calculated at the G2, CBS-4M, and CBS-Q high-level composite methods are 269, 257, and 262 kJ/mol, respectively. Because the BDE(C
N) of 1-nitropropane is almost 100 kJ/mol lower than the BDE(C
C) of n-heptane, 1-nitropropane should have the priority to produce free radicals by the rupture of the C
N bond. To determine the branching ratio of 1-nitropropane following the radical decomposition pathway, a reliable estimation of the rate constant for C
N bond rupture is necessary. According to VTST, for each optimized point, frequency analysis is performed to search for the maximum ΔG(T,s). Figure 3 shows the dependence of the activation free energy on the reaction coordinate (the distance between C1 and N atoms) at 698 K. The zone around the highest point is depicted by the enlarged picture. One can see that ΔG(T,s) arrives at a maximum point at the C
N bond length of 3.16 Å at the B3LYP level and 3.13 Å at the B3PW91 level in the C
N bond rupture process. The structure corresponding to the maximum of the activation free energy is proposed to be a transition state at this temperature. According to eq 3, kVTST is estimated to be 5.25  10
3 s
1 (B3LYP) and 2.97  10
3 s
1 (B3PW91). By taking into account all internal rotations in the equilibrium state of 1-nitropropane, kIntVTST is calculated to be 1.62  10
3 s
1 (B3LYP) and 8.36  10
4 s
1 (B3PW91), respectively, in good agreement with the experimental result of 1.05  10
3 s
1. Due to small imaginary frequency values (116i and 118i cm
1 at the B3LYP and B3PW91 levels, respectively) of the transition state, the Wigner tunneling correction k(T) is approximately equal to 1 at 698 K, having very little effect on the change of the reaction rate constant. It can be concluded that 1-nitropropane reacts through two pathways simultaneously at 698 K, complicated by the competitive elimination of HONO and radical decomposition. The elimination of HONO occurs at rates almost 10 times faster than the radical decomposition pathway at 698 K. The CME pathway 9057

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Figure 3. Fully optimized B3LYP and B3PW91 activation free energy curves for the C
N bond rupture of 1-nitropropane at 698 K.

can increase the yield of propylene and the fuel heat sink; more importantly, 1-nitropropane reacts through the radical decomposition pathway to provide the reactive free radicals at lower temperature compared with the C
C bond cleavage of n-heptane. 3.2. Further Reaction of n-Propyl Radical. The primary n-propyl radical generated from the C
N bond rupture of 1-nitropropane is unstable in the reaction system. Besides undergoing hydrogen-abstraction reaction, other reaction pathways56,62 from n-propyl are also studied.

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Figure 4. Main geometric parameters of n-propyl and transition structures for the three possible reactions of n-propyl optimized at the B3PW91 level. All distances are in angstroms; bond angles are in degrees.

Table 3. Calculated Activation Energies (ΔEa, kJ/mol) and Reaction Energies (ΔE, kJ/mol) of n-Propyl Further Reactions at the B3LYP, B3PW91, and MP2 Levelsa 1T

a

The first pathway (1T) in eq 6 is that n-propyl radical reacts through the cleavage of the β C
C bond to the radical center to _ 3 radical, the second one produce ethylene and the secondary CH (2T) is the cleavage of a C
H bond to produce one H radical and propylene, and the third one (3T) is the isomerization reaction of n-propyl radical, completing 1,3-H atom intramolecular transfer via a four-membered-ring transition structure. Figure 4 shows the optimized structures of n-propyl radical and the corresponding transition states for the three possible reaction pathways of n-propyl at the B3PW91 level. In the C
C bond cleavage pathway, the breaking C2
C3 bond has increased in length by 54.34% to reach the TS1 structure compared with that (1.542 Å) of the parent n-propyl radical. For the C
H bond cleavage, the lengthening of the C2
H2 bond is combined with a shortening of the C1
C2 bond from 1.490 Å of n-propyl to 1.348 Å in the TS2 structure, implying the formation of a C1dC2 double bond. With regard to 1,3-H isomerization reaction, the distances between the transferring H3 atom and the two bridgehead C atoms are both 1.401 Å symmetrically in the TS3 structure, and the — C1 3 3 3 H3 3 3 3 C3 angle is calculated to be 103.3
, indicating that this ring is highly strained and difficult to be formed. To determine the preferred reaction route of n-propyl, theoretical calculations were performed to obtain the activation energies of the three possible reaction pathways in eq 6, and the calculated results are presented in Table 3. As can be seen, the best agreement with reported values is achieved at the B3LYP level. The absolute average deviation relative to reported values is less than 4 kJ/mol. Both B3LYP and B3PW91 results indicate

ΔEa

level

2T

3T

ΔE

ΔEa

ΔE

ΔEa

ΔE 0

B3LYP

127.55

97.86

155.53

145.35

170.87

B3PW91

140.76

112.69

156.69

143.57

166.28

0

MP2

172.16

99.84

166.95

99.63

193.29

0

refs 62
64

130.80




148.29




171.54




Pathways 1T, 2T, and 3T in eq 6.

Table 4. Calculated Activation Energies (kJ/mol) of Forward _ 3 Radical and n-Heptane and Reverse Reactions between CH at the B3LYP, B3PW91, and MP2 Levelsa 1C

a

2C 0

3C

ΔEa

ΔEa0

ΔEa

ΔEa

level

ΔEa

ΔEa

B3LYP

49.58

67.66

43.52

76.71

44.16

B3PW91

49.90

67.93

42.23

76.22

44.25

MP2

80.91

91.84

70.32

91.10

70.11

4C 0

ΔEa

ΔEa0

78.02

46.07

79.46

77.51

44.31

77.79

90.05

69.77

87.31

Pathways 1C
4C in eq 7.

that the C
C bond cleavage pathway is energetically favored over the C
H bond cleavage and 1,3-H isomerization reaction, which is in accordance with the reactive order (1T > 2T > 3T; see eq 6) derived from refs 62
64. The worst agreement with reported values is produced by the MP2 level, overestimating the activation energies by an absolute average value of 27 kJ/mol. What is worse, the MP2 results indicate that n-propyl should mainly react through the 2T pathway, which is clearly in contradiction to the reported reactive order. As shown in Table 4, the β C
C bond cleavage pathway has a reaction endothermicity of 98
113 kJ/mol and the endothermic reaction of this type for alkyl radical is considered to be the main contribution to the fuel heat sink.5 According to the quantitative characteristics described above, the route for the radical formation during the initiated thermal 9058

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_ 3 + n-heptane optimized at the B3LYP, B3PW91, and MP2 levels. R1 is the length of the Figure 5. Transition structural parameters for the reaction of CH forming bond; R2 is the length of the breaking bond. The italic values are obtained at the B3PW91 level. The values in parentheses are the MP2 results. All distances are in angstroms; bond angles are in degrees.

cracking process is determined: the first condition favors the C
N bond rupture of 1-nitropropane to produce the primary radicals (n-propyl and nitro) with lower activation energy compared with the C
C bond cleavage of n-heptane; then _ 3 radical is generated from the C
C bond cleavage secondary CH of n-propyl radical. 3.3. Hydrogen-Abstraction Reaction. Hydrogen-abstraction reaction, in which a radical captures an H atom from one molecule to form a new molecule and a new radical, plays an important role in the chain reaction sequence, as it has a profound effect on the product distribution. The main goal of this section is to validate that the H-abstraction can proceed with lower activation energy compared with the radical formation step. In the binary mixture, n-heptane is considered to be the attacked model fuel, and the free radicals methyl, n-propyl, and nitro generated from the decomposition of 1-nitropropane work as chain-propagating carriers to initiate the chain reaction. The four reactions in eq 7 describe the methyl radical attacking the H atoms attached to the first (C1), second (C2), third (C3), and fourth (C4) C atoms from the terminal
CH3 group within n-heptane molecule, respectively.

Figure 5 shows the four optimized transition structures of _ 3 + n-heptane reaction. The identified transition states are CH characterized by roughly a half-transfer of the H atom. Take the 2C reaction pathway in eq 7 calculated at the B3PW91 level, for example; the breaking C 3 3 3 H bond is lengthened by 18.29% to reach the transition structure (1.300 Å) from the reactant (1.099 Å) and the forming C 3 3 3 H bond is stretched by 28.98%

to reach the transition structure (1.411 Å) compared with that (1.094 Å) of CH4. The — C 3 3 3 H 3 3 3 C bond angles between the forming (C 3 3 3 H) and breaking (C 3 3 3 H) bonds in the four transition structures are predicted to be nearly linear. Examination of the C 3 3 3 H semibond distances reveals that R1 is modestly longer than R2 (average difference approximately 0.098 Å) at the B3PW91 level, which implies an “early”, reactant-like transition state (in the Hammond sense65) and is in accordance with an exothermic reaction process. The forming and breaking C 3 3 3 H bonds at the secondary site (
CH2
group) are averagely 2.37% longer and 1.49% shorter than those at the primary site (
CH3 group), respectively. This predicts that the transition state of H-abstraction at the secondary site is more reactant-like than that at primary site, indicating that the secondary H-abstraction channel has the lower activation energy and higher exothermicity according to Hammond’s postulate.65 _ 3 radical to abstract an The required activation energies for the CH H atom from each of the four C atoms are presented in Table 4. The activation energies are predicted to be 42
50 kJ/mol at the B3PW91 level, close to the classical barrier height of 42 kJ/mol.66
68 As can be seen, it requires less energy to abstract an H atom from the secondary site than from the primary site, which is in accordance with the structural implication discussed above. As a second point, pathway 2C in eq 7 with the lowest activation energy should be the preferred reaction pathway, which is similar to the _ 3 radical and ncalculated result of H-abstraction between CH pentane by Xiao et al.66 This reaction selectivity is presumably correlated with the bond dissociation energies of the C
H bonds, because a weaker C
H bond is present at the C2 site,55 and the _ 3 attached H atom should be first bound and captured by the CH radical. The calculated results at the B3LYP level leads to the _ 3 radical should mainly capture the same conclusion that the CH H atom attached to the C2 atom with the lowest activation energy. However, by changing the level of theory from DFT to MP2, the discrepancy between calculated and classic values is 9059

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Industrial & Engineering Chemistry Research

ARTICLE

Figure 7. Optimized structures of the three possible transition states for the reaction of nitro with n-heptane at the B3LYP and B3PW91 levels. The italic values are obtained at the B3PW91 level. All distances are in angstroms. Figure 6. Reaction profile of H-abstraction between n-propyl radical and n-heptane calculated at the B3PW91 level.

_ 3 radical is predicted to capture the H beyond 28 kJ/mol, and the CH atom attached to the C4 atom, failing to simulate the overall energetic trend. The reverse reactions are also the interaction of free radical with normal alkane. We define the ratio of the C atom number between the radical and the normal alkane as the factor ξ. Through comparison of the activation energies of the forward and reverse reactions, the calculated results shown in Table 4 apparently indicate the same reaction characteristics at three adopted levels; when ξ > 1, the reaction probability is lower with higher activation energy. This reactivity is presumably due in large part to its greater steric interaction. The remaining CH groups of a larger radical might hinder the accessibility of CH4 to the reactive radical center, which would decrease its reactivity. To clarify the influence of size and variety of free radicals on the H-abstraction reaction, which will be useful for developing initiators, the following reactions are considered at the B3LYP and B3PW91 levels, taking the H atom attached to the C2 atom as the attacking center. _ 2 CH2 CH3 CH3 ðCH2 Þ5 CH3 þ CH 5C

_ sf CH3 ðCH2 Þ4 CHCH 3 þ CH3 CH2 CH3

ð5CÞ

_ 2 CH3 ðCH2 Þ5 CH3 þ NO 6C

_ sf CH3 ðCH2 ÞCHCH 3 þ HNO2

ð6CÞ

3.3.1. Effect of Radical Size on Barrier Height. Figure 6 depicts the reaction profile of eq 5C calculated at the B3PW91 level. The lengths of the two C 3 3 3 H semibonds are predicted to be 1.384 and 1.335 Å, respectively. The transferring H atom lies much closer to the midpoint of the two semibonding C atoms compared with the TS2H structure. The forming and breaking C 3 3 3 H bonds in the TS5H structure are 1.91% shorter and 2.69% longer than those in the TS2H structure, which implies that TS5H is less reactant-like than TS2H and the activation energy of reaction 5C is higher than that of reaction 2C in eq 7 according to Hammond’s postulate.65 The activation energy of reaction 5C is calculated to be 49 kJ/mol at the B3LYP level and 47 kJ/mol at the B3PW91 level, respectively. Both values are 4
6 kJ/mol higher than that of reaction 2C in eq 7. Such a small difference indicates that the activation energy of H-abstraction is not very sensitive to the size of alkyl radicals. 3.3.2. Effect of Radical Variety on Barrier Height. There are three possible reaction pathways for the reaction of nitro radical

Table 5. Calculated Activation Energies (ΔEa, kJ/mol) for the Reaction of Nitro with n-Heptane at the B3LYP and B3PW91 Levels level

6C1

6C2

6C3

B3LYP

112.28

93.77

98.03

B3PW91

118.64

97.39

104.81

ref 72a

120.50

97.07

107.11

a

Activation energies of nitro abstracting the secondary H atom from propane calculated at the BHandHLYP/6-311G** level derived from ref 72.

with n-heptane to form trans-HONO, cis-HONO, and HNO2, respectively. Figure 7 shows the optimized structures of the three possible transition states. The calculated activation energies are presented in Table 5, where 6C1, 6C2, and 6C3 represent the formation pathways of trans-HONO, cis-HONO, and HNO2, respectively. The activation energy of reaction 6C is calculated to be 94
112 kJ/mol at the B3LYP level and 97
119 kJ/mol at the B3PW91 level. While there is no experimental value available for the calculated activation energy to compare with directly, the values of related systems can be obtained. The activation energy of H-abstraction of nitro with CH4 was 120
126 kJ/mol,69,70 and for the reaction of nitro with propane, the activation energy was reported to be 95
126 kJ/mol.71,72 It is found that the DFT results are close to experimental analogues. As indicated in Table 5, the activation energies for the three possible reaction pathways are predicted to be in the order 6C1 > 6C3 > 6C2, which agrees with the calculated results of Chan et al.72 By comparing alkyl radicals acting on n-heptane, nitro abstracting an H atom from n-heptane is characterized with higher activation energy by about 50 kJ/mol. This tells that the radical variety would have a significant effect on the H-abstraction reaction. Through the H-abstraction of free radicals with n-heptane discussed above, the heptyl radical is formed, and then it can continue to undergo the C
C bond cleavage, each time producing an alkene molecule until a methyl radical is ultimately formed. If the chain reaction is not close to termination, the methyl radical would likely react with another n-heptane molecule to make the chain reaction proceed continuously.

4. CONCLUSIONS In this work, quantum chemical calculations were performed to study the detailed kinetics and mechanism of the initiated thermal cracking of n-heptane with 1-nitropropane as an initiator. 9060

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Industrial & Engineering Chemistry Research The calculated results can succeed in describing the free radical process and elucidating the accelerating effect of 1-nitropropane. On the basis of kinetics analysis, 1-nitropropane reacts through two pathways simultaneously at the investigated temperature: one is the elimination of HONO, which is an endothermic reaction to increase the fuel heat sink; the other is the rupture of the C
N bond to produce the primary n-propyl and nitro radicals. The rupture of the C
N bond with lower activation energy can proceed prior to the C
C bond cleavage of nheptane. Compared with the C
H bond cleavage and 1,3-H intramolecular isomerization, the unstable n-propyl radical mainly follows the C
C bond cleavage pathway to produce ethylene _ 3 radical. The analysis of H-abstraction of and the secondary CH radicals with n-heptane at the B3LYP and B3PW91 levels validates that the H-abstraction can proceed with lower activation energy compared with the radical formation step. Alkyl radicals of different sizes acting on normal alkanes have almost identical barrier heights of 50 kJ/mol, while the nitro radical acting on normal alkane shows an activation energy value about 50 kJ/mol higher than those of alkyl radicals, indicating that the radical variety, rather than radical size, significantly affects the H-abstraction reaction. Taking all analytical results together, the mechanism investigation supports the assumption that the process of free radicals obtained by the decomposition of 1-nitropropane, capturing an H atom from a normal alkane to initiate the chain reaction, should be the main reason for the accelerating effect.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: +86-29-82663189. Fax: +86-29-82668789. E-mail:blunyang@ mail.xjtu.edu.cn.

’ REFERENCES (1) Maurice, L. Q.; Lander, H.; Edwards, T.; Harrison, W. E., III. Advanced Aviation Fuels: A Look Ahead via a Historical Perspective. Fuel 2001, 80, 747. (2) Huang, H.; Sobel, D. R.; Spadaccini, L. J. Endothermic Heat-Sink of Hydrocarbon Fuels for Scramjet Cooling. Presented at the AIAA/ ASME/SAE/ASEE Joint Propulsion Conference Exhibit, Indianapolis, IN, July 2002; Paper AIAA 2002-3871. (3) Huang, H.; Spadaccini, L. J.; Sobel, D. R. Fuel-Cooled Thermal Management for Advanced Aeroengines. J. Eng. Gas Turbines Power 2004, 126, 284. (4) Edwards, T. Advancements in Gas Turbine Fuels from 1943 to 2005. J. Eng. Gas Turbines Power 2007, 129, 13. (5) Wickham, D. T.; Engel, J. R.; Hitch, B. D.; Karpuk, M. E. Initiators for Endothermic Fuels. J. Propul. Power 2001, 17, 1253. (6) Wickham, D. T.; Engel, J. R.; Hitch, B. D. Additives to Increase Fuel Heat Sink Capacity. Presented at the AIAA/ASME/SAE/ASEE Joint Propulsion Conference Exhibit, Indianapolis, IN, July 2002; Paper AIAA 2002-3872. (7) Wickham, D. T.; Engel, J. R.; Rooney, S.; Hitch, B. D. Additives to Improve Fuel Heat Sink Capacity in Air/Fuel Heat Exchangers. J. Propul. Power 2008, 24, 55. (8) Rice, F. O. The Thermal Decomposition of Organic Compounds from the Standpoint of Free Radicals. III. The Calculation of the Products Formed from Paraffin Hydrocarbons. J. Am. Chem. Soc. 1933, 55, 3035. (9) Rice, F. O.; Herzfeld, K. F. The Thermal Decomposition of Organic Compounds from the Standpoint of Free Radicals. VI. The Mechanism of Some Chain Reactions. J. Am. Chem. Soc. 1934, 56, 284.

ARTICLE

(10) Kossiakoff, A.; Rice, F. O. Thermal Decomposition of Hydrocarbons, Resonance Stabilization and Isomerization of Free Radicals. J. Am. Chem. Soc. 1943, 65, 590. (11) Sundaram, K. M.; Froment, G. F. Modeling of Thermal Cracking Kinetics. 3. Radical Mechanisms for the Pyrolysis of Simple Paraffins, Olefins, and Their Mixtures. Ind. Eng. Chem. Fundam. 1978, 17, 174. (12) Kissin, Y. V. Free-Radical Reactions of High Molecular Weight Isoalkanes. Ind. Eng. Chem. Res. 1987, 26, 1633. (13) Yu, J.; Eser, S. Thermal Decomposition of C10-C14 Normal Alkanes in Near-Critical and Supercritical Regions: Product Distributions and Reaction Mechanisms. Ind. Eng. Chem. Res. 1997, 36, 574. (14) Chang, J.; Fan, L.; Fujimoto, K. Enhancement Effect of Free Radical Initiator on Hydro-Thermal Cracking of Heavy Oil and Model Compound. Energy Fuels 1999, 13, 1107. (15) Chang, J.; Tsubaki, N.; Fujimoto, K. Elemental Sulfur as an Effective Promoter for the Catalytic Hydrocracking of Arabian Vacuum Residue. Fuel 2001, 80, 1639. (16) Chang, J.; Fu, Y.; Shibata, Y.; Yoshimoto, M.; Fujimoto, K.; Tsubaki, N. Promotional Effect of Oxidation Pretreatment on HydroThermal Cracking of Canadian Oil Sand Bitumen. Fuel 2005, 84, 1661. (17) Yu, J.; Eser, S. Supercritical-Phase Thermal Decomposition of Binary Mixtures of Jet Fuel Model Compounds. Fuel 2000, 79, 759. (18) Edwards, T.; DeWitt, M. J.; Shafer, L.; Brooks, D.; Huang, H.; Bagley, S. P.; O~ na, J. O.; Wornat, M. J. Fuel Composition Influence on Deposition in Endothermic Fuels. Presented at the AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference, Canberra, Australia, November 2006; Paper AIAA 2006-7973. (19) Wang, Z.; Lin, R.; Fang, W.; Li, G.; Guo, Y.; Qin, Z. Triethylamine as an Initiator for Cracking of Heptane. Energy 2006, 31, 2773. (20) Wang, Z.; Guo, Y.; Lin, R. Effect of Triethylamine on the Cracking of Heptane under a Supercritical Condition and the Kinetic Study on the Cracking of Heptane. Energy Convers. Manage. 2008, 49, 2095. (21) Wang, Z.; Lin, R.; Guo, Y.; Li, G.; Fang, W.; Qin, Z. Tributylamine as an Initiator for Cracking of Heptane. Energy Convers. Manage. 2008, 49, 1584. (22) Yanovskiy, L. S.; Sapgir, G. B.; Strokin, V. N.; Ivanov, V. F. Endothermic Fuels: Some Aspects of Fuel Decomposition and Combustion at Air Flows. Presented at the ISABE Conference, Florence, Italy, September 1999; Paper 99-7067. (23) Liu, G.; Han, Y.; Guo, W.; Zhang, X.; Wang, L.; Mi, Z. Supercritical Initiative Cracking of Endothermic Fuel Model Compound n-Dodecane with 1-Nitropropane. Presented at the AIAA/ASME/SAE/ ASEE Joint Propulsion Conference Exhibit, Hartford, CT, July 2008; Paper AIAA 2008-5127. (24) Liu, G.; Han, Y.; Wang, L.; Zhang, X.; Mi, Z. Supercritical Thermal Cracking of n-Dodecane in Presence of Several Initiative Additives: Products Distribution and Kinetics. Energy Fuels 2008, 22, 3960. (25) Liu, G.; Han, Y.; Wang, L.; Zhang, X.; Mi, Z. Solid Deposits from Thermal Stressing of n-Dodecane and Chinese RP-3 Jet Fuel in the Presence of Several Initiators. Energy Fuels 2009, 23, 356. (26) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery, J. A., Jr.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, revision A.02; Gaussian, Inc.: Wallingford, CT, USA, 2009. 9061

dx.doi.org/10.1021/ie200515g |Ind. Eng. Chem. Res. 2011, 50, 9054–9062

Industrial & Engineering Chemistry Research (27) Becke, A. D. A New Mixing of Hartree-Fock and Local DensityFunctional Theories. J. Chem. Phys. 1993, 98, 1372. (28) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785. (29) Perdew, J. P.; Wang, Y. Accurate and Simple Analytic Representation of the Electron-Gas Correlation Energy. Phys. Rev. B 1992, 45, 13244. (30) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, Molecules, Solids, and Surfaces: Applications of the Generalized Gradient Approximation for Exchange and Correlation. Phys. Rev. B 1992, 46, 6671. (31) Head-Gordon, M.; Pople, J. A.; Frisch, M. J. MP2 Energy Evaluation by Direct Methods. Chem. Phys. Lett. 1988, 153, 503. (32) Frisch, M. J.; Head-Gordon, M.; Pople, J. A. A Direct MP2 Gradient Method. Chem. Phys. Lett. 1990, 166, 275. (33) Frisch, M. J.; Head-Gordon, M.; Pople, J. A. Semi-Direct Algorithms for the MP2 Energy and Gradient. Chem. Phys. Lett. 1990, 166, 281. (34) Petersson, G. A.; Bennett, A.; Tensfeldt, T. G.; Al-Laham, M. A.; Shirley, W. A.; Mantzaris, J. A Complete Basis Set Model Chemistry. I. The Total Energies of Closed-Shell Atoms and Hydrides of the First-Row Elements. J. Chem. Phys. 1988, 89, 2193. (35) Foresman, J. B.; Frisch, A. Exploring Chemistry with Electronic Structure Methods, 2nd ed.; Gaussian, Inc.: Pittsburgh, PA, USA, 1996. (36) Hratchian, H. P.; Schlegel, H. B. Accurate Reaction Paths Using a Hessian Based Predictor-Corrector Integrator. J. Chem. Phys. 2004, 120, 9918. (37) Hratchian, H. P.; Schlegel, H. B. Finding minima, transition states, and following reaction pathways on ab initio potential energy surfaces. In Theory and Applications of Computational Chemistry: The First Forty Years; Dykstra, C. E., Frenking, G., Kim, K. S., Scuseria, G. E., Eds.; Elsevier: Amsterdam, 2005; p 195. (38) Hratchian, H. P.; Schlegel, H. B. Using Hessian Updating to Increase the Efficiency of a Hessian Based Predictor-Corrector Reaction Path Following Method. J. Chem. Theory Comput. 2005, 1, 61. (39) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 Theory for Molecular Energies of First- and Second-Row Compounds. J. Chem. Phys. 1991, 94, 7221. (40) Ochterski, J. W.; Petersson, G. A.; Montgomery, J. A., Jr. A Complete Basis Set Model Chemistry. V. Extensions to Six or More Heavy Atoms. J. Chem. Phys. 1996, 104, 2598. (41) Montgomery, J. A., Jr.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. A Complete Basis Set Model Chemistry. VII. Use of the Minimum Population Localization Method. J. Chem. Phys. 2000, 112, 6532. (42) Montgomery, J. A., Jr.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. A Complete Basis Set Model Chemistry. VI. Use of Density Functional Geometries and Frequencies. J. Chem. Phys. 1999, 110, 2822. (43) McClurg, R. B.; Flagan, R. C.; Goddard, W. A., III. The Hindered Rotor Density-of-States Interpolation Function. J. Chem. Phys. 1997, 106, 6675. (44) Ayala, P. Y.; Schlegel, H. B. Identification and Treatment of Internal Rotation in Normal Mode Vibrational Analysis. J. Chem. Phys. 1998, 108, 2314. (45) McClurg, R. B. Comment on: “The Hindered Rotor Densityof-States Interpolation Function [J. Chem. Phys. 106, 6675 (1997)] and “The Hindered Rotor Density-of-States” [J. Chem. Phys. 108, 2314 (1998)]. J. Chem. Phys. 1999, 111, 7163. (46) Eyring, H. The Activated Complex and the Absolute Rate of Chemical Reactions. Chem. Rev. 1935, 17, 65. (47) Laldler, K. J.; King, M. C. The Development of Transition-State Theory. J. Phys. Chem. 1983, 87, 2657. (48) Truhlar, D. G.; Issacson, A. D.; Garrett, B. C. Theory of Chemical Reaction Dynamics; CRC Press: Boca Raton, FL, USA, 1985. (49) Mcquarrie, D. A.; Simon, J. D. Physical Chemistry: A Molecular Approach; University Science Books: Sausalito, CA, USA, 1997. (50) Truhlar, D. G.; Garrett, B. C. Variational Transition-State Theory. Acc. Chem. Res. 1980, 13, 440.

ARTICLE

(51) Garrett, B. C.; Truhlar, D. G.; Grev, R. S.; Magnuson, A. W. Improved Treatment of Threshold Contributions in Variational Transition-State Theory. J. Phys. Chem. 1980, 84, 1730. (52) Wigner, E. Calculation of the Rate of Elementary Association Reactions. J. Chem. Phys. 1937, 5, 720. (53) Szaleniec, M.; Borowski, T.; Sch€uhle, K.; Witko, M.; Heider, J. Ab Initio Modeling of Ethylbenzene Dehydrogenase Reaction Mechanism. J. Am. Chem. Soc. 2010, 132, 6014. (54) Willems, P. A.; Froment, G. F. Kinetic Modeling of the Thermal Cracking of Hydrocarbons. 1. Calculation of Frequency Factors. Ind. Eng. Chem. Res. 1988, 27, 1959. (55) Luo, Y. R. Comprehensive Handbook of Chemical Bond Energies; CRC Press: Boca Raton, FL, USA, 2007. (56) Frey, H. M.; Walsh, R. The Thermal Unimolecular Reactions of Hydrocarbons. Chem. Rev. 1969, 69, 103. (57) Nazin, G. M.; Manelis, G. B. Thermal Decomposition of Aliphatic Nitro-Compounds. Russ. Chem. Rev. 1994, 63, 313. (58) Serrano, A. J.; Loro~ no, M.; Cordova, T.; Chuchani, G. Theoretical Calculations of the Thermal Decomposition Kinetics of Several tertNitroalkanes in the Gas Phase. THEOCHEM 2008, 859, 69. (59) Khrapkovskii, G. M.; Shamov, A. G.; Nikolaeva, E. V.; Chachkov, D. V. Mechanisms of the Gas-Phase Decomposition of C-Nitro Compounds Inferred from Quantum Chemical Calculations. Russ. Chem. Rev. 2009, 78, 903. (60) Wang, Q.; Ng, D.; Mannan, M. S. Study on the Reaction Mechanism and Kinetics of the Thermal Decomposition of Nitroethane. Ind. Eng. Chem. Res. 2009, 48, 8745. (61) Shamsutdinov, A. F.; Shamsutdinov, T. F.; Chachkov, D. V.; Shamov, A. G.; Khrapkovskii, G. M. The Influence of Molecular Structure on the Change of the Arrhenius Factor of Gas-Phase Elimination of Nitric Acid from Nitroalkanes. Int. J. Quantum Chem. 2007, 107, 2343. (62) Yamauchi, N.; Miyoshi, A.; Kosaka, K.; Koshi, M.; Matsui, H. Thermal Decomposition and Isomerization Processes of Alkyl Radicals. J. Phys. Chem. A 1999, 103, 2723. (63) Tsang, W.; Bedanov, V.; Zachariah, M. R. Master Equation Analysis of Thermal Activation Reactions: Energy-Transfer Constraints on Falloff Behavior in the Decomposition of Reactive Intermediates with Low Thresholds. J. Phys. Chem. 1996, 100, 4011. (64) Viskolcz, B.; Lendvay, G.; K€ortvelyesi, T.; Seres, L. Intramolecular H Atom Transfer Reactions in Alkyl Radicals and the Ring Strain Energy in the Transition Structure. J. Am. Chem. Soc. 1996, 118, 3006. (65) Hammond, G. S. A Correlation of Reaction Rates. J. Am. Chem. Soc. 1955, 77, 334. (66) Xiao, Y.; Longo, J. M.; Hieshima, G. B.; Hill, R. J. Understanding the Kinetics and Mechanisms of Hydrocarbon Thermal Cracking: An Ab Initio Approach. Ind. Eng. Chem. Res. 1997, 36, 4033. (67) Willems, P. A.; Froment, G. F. Kinetic Modeling of the Thermal Cracking of Hydrocarbons. 2. Calculation of Activation Energies. Ind. Eng. Chem. Res. 1988, 27, 1966. (68) Vandeputte, A. G.; Sabbe, M. K.; Reyniers, M. F.; Speybroeck, V. V.; Waroquier, M.; Marin, G. B. Theoretical Study of the Thermodynamics and Kinetics of Hydrogen Abstractions from Hydrocarbons. J. Phys. Chem. A 2007, 111, 11771. (69) Slack, M. W.; Grillo, A. R. Shock Tube Investigation of Methane-Oxygen Ignition Sensitized by NO2. Combust. Flame 1981, 40, 155. (70) Slutskii, V. G. Effect of Nitrogenous Promoters on the LowTemperature Ignition Lag of Methane Mixtures. Combust., Explos. Shock Waves (Engl. Transl.) 1985, 21, 138. (71) Ballod, A. P.; Shtern, V. Y. The Gas-Phase Nitration of Alkanes. Russ. Chem. Rev. 1976, 45, 721. (72) Chan, W.; Heck, S. M.; Pritchard, H. O. Reaction of Nitrogen Dioxide with Hydrocarbons and Its Influence on Spontaneous Ignition. A Computational Study. Phys. Chem. Chem. Phys. 2001, 3, 56.

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