Reaction of nitrogen dioxide with hydrocarbons and its influence on spontaneous ignition. A computational study

Abstract

Estimates are made, by using BHandHLYP/6-311G** density functional molecular orbital theory, of the activation energies and frequency factors for the reaction of NO₂ with methane, ethane, propane, isobutane, and benzene. For the aliphatic hydrocarbons, over the temperature range 600–1100 K, the rate of formation of a new isomer of nitrous acid, HNO₂, is very similar to that for the formation of the common isomer, HONO. This complicates our description of the acceleration of spontaneous ignition of diesel fuels by organic nitrates. These rate data are used in a reduced kinetic model to examine the effect of NO₂ upon the spontaneous ignition of some linear- and branched-chain aliphatic hydrocarbons. It is concluded that, under typical diesel engine operating conditions, the spontaneous ignition of linear-chain paraffins is accelerated by the presence of NO₂, but may be retarded for heavily branched-chain isomers. An Appendix discusses the relative importance of tunnelling in hydrogen-transfer reactions.

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Introduction

Organic nitrates, principally 2-ethylhexyl nitrate, are commonly used as accelerants to improve the ignition quality of hydrocarbon fuels in diesel engines. The basic mechanism by which these work was, until now, thought to be quite simple: viz.^1,2

The OH radicals are responsible for initiating the destruction of the hydrocarbon, and the NO is converted back to NO₂ to begin another cycle. Because reactions (2)–(4) constitute a chain, the the alkoxyl radical formed in reaction (1) makes only a minor contribution, principally through dissociation to form formaldehyde and a smaller (Alk₋₁) radical.^2–4 Hence, at this exploratory stage, we need only concentrate on the reactions of NO₂ itself; this simplification is supported by the experimental observation that NO₂ or an equivalent amount of organic nitrate are about equally effective in promoting ignition in a real engine.¹

The simplicity of the reaction scheme is compromised by recent theoretical predictions,⁵ described more extensively below, that in the reaction of NO₂ with aliphatic hydrocarbons the formation of a new isomer isonitrous acid, HNO₂, can be competitive with the formation of ordinary nitrous acid, HONO. In contrast with HONO, HNO₂ cannot dissociate to give OH radicals, nor can it isomerise to HONO at the temperatures of interest, but it still provides a vehicle for ignition acceleration through the reaction

which is important because O₂ is present in large excess before ignition occurs.

Fuels from different sources respond differently to the addition of 2-ethylhexyl nitrate,^6,7 but the reasons for this remain only speculation at the present time. Understanding is hampered by the lack of kinetic data on the hydrogen abstraction reactions between NO₂ and hydrocarbons, of which there seem to be only two examples, namely methane and propane.

In this paper, we report the results of molecular orbital structure calculations on the reactions of NO₂ with methane (for calibration), at the primary, secondary and tertiary positions, respectively, in ethane, propane and isobutane, and (for future use) with benzene. Rate constants for these reactions are calculated by standard transition state theory, and these are then used in a reduced kinetic model⁸ to simulate the effect of NO₂ on the spontaneous ignition of four isomeric dodecanes and two isomeric hexadecanes.

Molecular orbital calculations

Ab initio calculations were performed by using the GAUSSIAN 98 molecular orbital program package,⁹ at the BHandHLYP/6-311G** density functional level (DFT) of approximation. The computational procedures were identical with those described previously;¹⁰ however, the justification given for their use needs, perhaps, some minor amplification.

DFT methods are known to be problematic for the simple hydrogen-transfer reaction¹¹ H + H₂ → H₂ + H and for addition of H atoms to radicals and to radical cations.¹² Nonetheless, they have gained acceptance as a tool for computation of a variety of H-abstraction reactions, as exemplified by several recent surveys^13–15 of DFT and ab initio calculations for reactions of the form X − H + Y → X + H − Y, where DFT methods were shown to provide generally reasonable results for transition state properties. In particular the study of Susnow et al. reported that DFT in combination with transition state theory is capable of providing accurate predictions of pre-exponential factors as well as the curvature of the Arrhenius plots.¹⁵ Our choice of the BHandHLYP/6-311G** DFT method for calculation of self-abstraction in alkyl peroxyl radicals and cyclisation of alkyl hydroperoxyl radicals was originally based¹⁰ on the recommendation of Truong and coworkers,¹⁶ performance comparisons by Durant,¹⁷ and reasonable agreement with high-level calculations where known.¹⁰ In addition, we have observed that the BHandHLYP/6-311G** method transition state properties for the reaction¹⁸ H₂ + NO₂ → H + cis-HONO are in better agreement with QCISD/6-311G** results reported by Lin and coworkers than their (U)MP2 results;¹⁹ they attributed a spurious MP2 frequency for the transition state to spin-contamination, to which DFT is known to be less susceptible.²⁰ The BHandHLYP/6-311G** method also provides minimum energy paths for hydrogen abstraction reactions with a series of fluoromethanes (CH_nF_4−n, n = 1–4) that closely mimic those obtained from a higher level MP4 method.²¹ Hence, our contention that the DFT-BHandHLYP method provides a viable alternative to the more costly ab initio methods.

The principal results are listed in Table 1, confirming the observation of Yamaguchi et al.⁵ that the unknown isomer HNO₂ is formed in preference to the trans-HONO ground state. We also confirm¹⁸ their result that the isomerisation barrier between HNO₂ and trans-HONO is in excess of 50 kcal mol⁻¹. However, they overlooked the possible formation of a third product, cis-HONO, having the lowest activation energy of all. Hence, rather than it being necessary to reformulate completely our ideas on how nitrate ignition improvers function, the existence of this new pathway only adds a little complication.

Table 1 Calculated BHandHLYP/6-311G** energies (ΔE) and enthalpies (ΔH)^a of reaction and activation (in kcal mol⁻¹) for the reactions of NO₂ with selected hydrocarbons at 0 K

trans-HONO				cis-HONO				HNO2
Product RH	ΔE	ΔE‡	ΔH	ΔH‡	ΔE	ΔE‡	ΔH	ΔH‡	ΔE	ΔE‡	ΔH	ΔH‡
a Zero point energies calculated by using unscaled frequencies. b CCSD(T)/6-311G(2d,p)//DFT calculation. c CCSD(T)/6-311 + + G(2d,p)//DFT calculation. d Secondary C–H bond. e Tertiary C–H bond.
CH4	27.0	39.4	25.0	36.2	26.3	33.7	24.3	31.0	35.1	36.3	34.2	34.3
28.8^b	41.3^b	28.2^b	35.1^b	37.3^b	38.3^b
27.5^c	40.7^c	28.0^c	35.5^c	35.3^c	36.8^c
C2H6	23.0	35.4	20.8	31.9	22.3	29.7	20.1	26.5	31.2	31.8	30.0	29.3
C3H8^d	19.8	32.4	17.6	28.8	19.1	26.6	16.9	23.2	27.9	28.3	26.8	25.6
C4H10^e	17.2	30.4	15.3	26.7	16.5	24.2	14.6	20.8	25.3	25.4	24.5	22.6
C6H6	34.5	41.8	33.8	37.6	33.8	36.8	33.1	33.1	42.6	39.9	43.0	37.5

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Confidence in the results of Table 1 is justified by the close coincidence between the DFT predictions for the reaction with methane and those of two higher-level sets of calculations, also listed there. Moreover, the agreement between experiment and theory for the ΔH values of these reactions lies between + 1.2 and + 1.6 kcal mol⁻¹ for four of them, and is + 3.2 kcal mol⁻¹ for the propane case. Somewhat disappointing, however, is the fact that several high-level approximations place cis-HONO below trans-HONO, for example: CCSD(T)/6-311G(2d,p) by 220 cm⁻¹, CCSD(T)/6-311 + + G(2df,2p) by 150 cm⁻¹, whereas it should be 180 cm⁻¹ above;²² our BHandHLYP/6-311G** DFT calculation puts it 250 cm⁻¹ below, but the CCSD(T)/6-311 + + G(2d,p) approximation in Table 1 actually puts it near the correct position, about 195 cm⁻¹ above.

Throughout Table 1, the activation energies follow the somewhat unexpected pattern of being lowest for the formation of cis-HONO and highest for the formation of trans-HONO, with that for the production of HNO₂ being in between. The lower energy path [italic v]ia the cis-configuration arises because the transition state comprises a 5-membered ring, with its attendant stabilisation; similar effects have been noted elsewhere.²³ The formation of the trans-tautomer is by far the slowest reaction, but the other two reactions are of comparable rates because the transition state for the formation of cis-HONO is much tighter than that for the formation of HNO₂. The transition state structures are very similar for all five hydrocarbons in each reaction type, and only one set of transition configurations, for methane, is shown in Fig. 1; additional structural parameters are listed in Table A1 of Appendix 1. The three diagrams illustrate the more constrained nature of that for the cis-reaction product, where it is evident that the cis-conformation is already nascent at the saddle point. Notice also the very small values for the lowest frequency ω₁ in the transition state for the formation of HNO₂, found in Table A3.

Fig. 1 Structures of the three possible transition states for the reaction of NO₂ with CH₄; distances are in Å and angles in degrees. For NO₂ itself, the bond lengths are 1.173 Å and the bond angle is 135.1°.

Finally, we remark in passing that the computational study of the reactions of NO₂ is much more intricate than might be assumed: not only are there multiple pathways, but when reactions with, for example, alcohols are examined, potential minima are sometimes to be found between reactants and the transition state, and between the transition state and the products.²⁴

Rate constants

Rate constants were calculated by standard transition state theory methods²⁵ for a series of temperatures 600⩽T/K⩽1100, and then fitted to a strict Arrhenius equation by least squares, with results as listed in Table 2. It is seen that the derived activation energies are consistently some 1–2 kcal mol⁻¹ higher than the 0 K values shown in Table 1.

Table 2 Activation energies E (in kcal mol⁻¹) and frequency factors A (in units of mol⁻¹ cm⁻³ s⁻¹ per H atom) for the reactions of NO₂ with RH in the range 600–1100 K

trans-HONO		cis-HONO		HNO2
Product RH	E	A	E	A	E	A
a Secondary C–H bond. b Tertiary C–H bond.
CH4	40.7	2.1 × 10^14	35.2	8.7 × 10^13	38.4	9.5 × 10^14
C2H6	36.7	5.7 × 10^13	31.1	2.2 × 10^13	33.8	1.6 × 10^14
C3H8^a	33.8	1.4 × 10^13	28.1	5.8 × 10^12	30.3	3.0 × 10^13
C4H10^b	31.9	2.1 × 10^13	25.8	9.3 × 10^12	27.6	2.8 × 10^14
C6H6	43.0	4.3 × 10^14	38.2	7.4 × 10^13	42.2	2.5 × 10^14

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For the aliphatic hydrocarbons, the rates of formation of cis-HONO and of HNO₂ are similar throughout the temperature range of interest, but for benzene, the formation of cis-HONO is faster than the other two below about 1400 K.

There is a severe lack of experimental kinetic data for these reactions: Slack and Grillo²⁶ give an activation energy for the reaction with methane as 30.0 kcal mol⁻¹, originally with an A-factor of 7.0 × 10¹¹ but later revised to 1.2 × 10¹³ mol⁻¹ cm⁻³ s⁻¹; and for propane, Titarchuk et al.²⁷ give an activa tion energy of 22.6 kcal mol⁻¹, with an A-factor of 2.4 × 10¹¹ mol⁻¹ cm⁻³ s⁻¹.

It has been our experience with this DFT method that although the reaction energies are in good agreement with experiment, the fitted Arrhenius activation energies are often too high.¹⁰ In fact, inclusion of NO₂ in the modelling calculations with these activation parameters gives rise to an insignificant acceleration in the ignition rate. Since the computed frequency factors should be fairly reliable, we standardised the computed activation energies by subtracting 5 kcal mol⁻¹ from each of them. Thus, for the two cases where experimental measurements exist, the selected (experimental) values for the activation energies are: CH₄, 30.2 (30.0) kcal mol⁻¹; C₃H₈, 23.1 (22.6) kcal mol⁻¹. Note, the implication that the initial pathway in these experiments leads to cis-HONO and not to the slightly more stable tautomer,²² but no distinction between the two forms of HONO was made in the modelling.

Consequently, for the modelling calculations, we have used the A-factors listed in Table 2, but have reduced the activation energies for attack of NO₂ at the primary, secondary, and tertiary C–H sites to 26.1, 23.1, and 20.8 kcal mol⁻¹, respectively, for the formation of HONO, and to 28.8, 25.3, and 22.6 kcal mol⁻¹, respectively, for the formation of HNO₂.

Modelling calculations

(a) Refinement of the model

The original reduced kinetic model of Griffiths et al.⁸ was stated to be slightly defective in that it treated all (1,nt) self-abstraction reactions RO₂ → Q̇;OOH as having the same rate constant. Since we wish to examine a series of isomeric dodecanes containing tertiary C–H bonds in different environments, this could now be important. We have therefore extended the model slightly by treating the (1,4t) and (1,5t) reactions separately.

Previous calculations had shown that the second of these two reactions should have both a lower activation energy and a lower A-factor.^10a We adopted an arbitrary procedure, starting with the estimated Arrhenius parameters for these two classes of reaction,^10a and iterating until the ignition delay for heptamethylnonane–air mixtures was the same as we had found previously by using the original model.²⁸ Consistency was then checked by calculating the ignition delays for the normal C₈, C₁₀, C₁₂ and C₁₄ paraffins and comparing the values with those for artificial mixtures of n-hexadecane and heptamethylnonane having the same Cetane numbers. Whereas before, the latter values always exceeded the former (see Fig. 2 of ref. 28) by between 0.02 and 0.06 ms, now the corresponding differences do not exceed 0.01 ms, as shown in Table 3. The rate constant values used were 5 × 10¹² exp(−17000/T) s⁻¹ for the (1,4t) reaction and 2.5 × 10¹² exp(−16000/T) s⁻¹ for the (1,5t) reaction, per H atom; these two values are not unique but, as shown, they provide consistent results.

Table 3 Comparison of computed ignition delays (in ms) for n-paraffins with those for artificial mixtures of n-hexadecane and heptamethylnonane having the same CN, with and without inclusion of (1,5t) self-abstraction reactions

Without (1,5t)			With (1,5t)
RH	CN	Molecule	Mixture	Difference	Molecule	Mixture	Difference
C8H18	64	2.11	2.13	0.02	2.05	2.05	0.00
C10H22	77	2.00	2.06	0.06	1.98	1.99	0.01
C12H26	88	1.97	2.00	0.03	1.95	1.96	0.01
C14H30	96	1.94	1.96	0.02	1.92	1.93	0.01

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(b) The inclusion of NO₂ reactions

Before the modelling can proceed, it is necessary to establish the fate of HNO₂ in the system. Several possibilities were explored, but the only viable one was found to be:¹⁸

For completeness, the analogous reaction between O₂ and HONO was also considered. Here too, there are two possible transition states with two quite different rate expressions:¹⁸

Of these two, the latter is always the faster, but its inclusion in the reaction set did not affect any of the computed ignition delays.

Another new reaction which must be considered is that of NO₂ with formaldehyde, to form HNO₂, since the parallel reaction to form HONO is a crucial part of the modelling.¹ However, unlike the hydrocarbon cases, the reaction with formaldehyde to form HNO₂ has an activation energy about 5 kcal mol⁻¹ higher than that to form HONO,^5b,18 and was therefore excluded.

In our previous work,²⁸ we had examined the ignition behaviour of four isomeric dodecanes, n-dodecane, 3-ethyldecane, 4,5-diethyloctane, and 2,2,4,6,6-pentamethylheptane, since ignition quality measurements by Petrov et al. were available for these hydrocarbons.²⁹ We now also include calculations on n-hexadecane and heptamethylnonane, which are used to define fixed points on the Cetane number scale. The effect of NO₂ on the spontaneous ignition of stoichiometric mixtures in air was estimated by use of the reduced kinetic model of Griffiths et al.,⁸ augmented by a minimal set of reactions involving nitrogen-containing species. Rate constants for these reactions were the same as we had used previously in the modelling of the original Bone and Gardner³⁰ ignition measurements on CH₄–O₂–NO₂ mixtures,¹ as follows:

were taken from Burcat;³¹ those for

from Slack and Grillo;^26b and

from Lin et al.³² Where appropriate, the reverse reactions were also included.

Likewise, the compression–ignition simulation model remained unchanged from the previous one: air, initially at 358 K and 1 atm, was compressed to 1/12th of its volume in a time of 22 ms; fuel and the NO₂ (when present) were injected homogeneously 2 ms before the end of the compression stroke, at about 745 K and 10 atm. The results of this modelling are given in Table 4, where ignition delay is defined as the time taken to reach 1000 K on a trajectory leading to complete combustion.

Table 4 Effect of adding 0.1% of NO₂ on the ignition delay (in ms) for spontaneous ignition of four isomeric dodecanes and two hexadecanes, with fuel injection occurring at 745 K, 2 ms before the end of the compression

Mechanism(s) included
Hydrocarbon	CN	[NO2] = zero	HONO	HNO2	both
a Cetene number, approximately commensurate with Cetane number.^29
n-Dodecane	82^a	1.95	1.89	1.92	1.89
3-Ethyldecane	55^a	1.99	1.95	1.96	1.95
4,5-Diethyloctane	23^a	2.14	2.00	2.02	2.00
2,2,4,6,6-Pentamethylheptane	10^a	2.77	2.46	2.53	2.51
n-Hexadecane	100	1.92	1.85	1.87	1.85
Heptamethylnonane	15	2.60	2.96	—	—

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It is clear throughout Table 4 that the HONO mechanism is somewhat more efficient than the HNO₂ mechanism with the chosen rate parameters. The only significant difference appears when there is multiple methyl group substitution and a paucity of secondary C–H bonds, whence there is also a slight interference between the two mechanisms. Moreover, it is only for such cases that the inclusion of the (1,5t) reactions makes any significant difference. For 4,5-diethyloctane, their inclusion only changes the computed ignition delays by 0.02–0.03 ms in the presence of NO₂, but for 2,2,4,6,6-pentamethylheptane, NO₂ has no effect when only (1,4t) reactions are considered, but causes a significant acceleration when the (1,5t) reactions are added to the reaction set.

In the special case of heptamethylnonane, one has to raise the pre-compression temperature from 358 to 373 K for ignition to occur in the presence of 0.1% of NO₂ when the (1,5t) reactions are missing (corresponding to an increase of injection temperature from 745 to 776 K). On the other hand, when these reactions are included, the HONO mechanism is retarded somewhat but the HNO₂ mechanism still extinguishes the combustion.

Given the complexity of the reaction set, it is often not easy to assign changes in ignition behaviour definitively to distinct sub-sets of reactions within the mechanism. However, some trends are fairly obvious. Among the dodecanes, the HONO mechanism is essentially as shown in reactions (1)–(4), with the OH concentration elevated by a factor of 3–4 in the period before ignition. Acceleration of ignition by the second mechanism can only occur through the reaction of O₂ with HNO₂ to give NO₂ and HO₂, and the pre-ignition concentrations of HO₂, H₂O₂, and of OH are all elevated by factors of 2–5 or more. However, the two mechanisms do not reinforce each other because rather over half of the NO₂ is trapped as HNO₂, so that the rates of formation of HONO by reactions (3), (16) and (17) are lowered, with a corresponding reduction in the rate of formation of OH radicals.‡ Of course, the magnitude of this effect is sensitive to the choice of the rate parameters for reaction (6) between O₂ and HNO₂, for which we have used the virgin computed result, without any manipulation.

The reason for the retardation in heptamethylnonane with the HONO mechanism is less clear. Comparison of the population distributions between n-hexadecane and heptamethylnonane reveals that the presence of NO₂ increases the OH radical concentration in the former case, but reduces it in the latter. This traces back to a much greater difference in activation energy between abstraction at the primary sites to give R₁ radicals and at the tertiary sites to give R₃ radicals for NO₂ than there is for O₂, thereby altering the balance between the peroxyl radical concentrations in favour of R₃O₂. In the Griffiths model,⁸ this radical isomerises less readily into the chain-generating Q̇;OOH radical than does R₁O₂, and the resulting loss of OH radicals is not compensated by the additional formation [italic v]ia HONO dissociation.

Extinction of the heptamethylnonane combustion by the inclusion of the HNO₂ mechanism stems from a similar cause. The activation energy for its formation by reaction at the tertiary site is much less than it is from the primary site. Not only does this accentuate the R₃O₂/R₁O₂ disparity, but it increases the fraction of the NO₂ trapped as HNO₂, thus reducing the rates of formation of HONO by reaction with RH and with other minor constituents such as CHO and CH₂O.

The possibility that NO₂ may actually retard the spontaneous ignition of heptamethylnonane can only be considered as tentative, depending as it does on the theoretical prediction that there is a greater difference in activation energy between primary and tertiary C–H bonds for attack by NO₂ than the model assumes⁸ that there is for O₂ . Despite recent advances in computational methods, the estimation of activation energies still remains a difficult challenge. For this reason too, refinements to transition state theory such as allowance for restricted rotation or tunnelling were ignored: both types of correction are minuscule relative to that due to the mismatch between the computed and the observed activation energies, noted above.§

Comments

The general perception in the petroleum industry is that the better the diesel fuel, i.e. the higher the Cetane number (CN), the more strongly it responds to the addition of nitrate; such a correlation can be seen in a study of 13 different fuels by Nandi and Jacobs.⁷ In practice, it is usual to express the effect as the change in CN for a given concentration of nitrate, but since the relationship between CN and ignition delay is non-linear, quantitative comparison with the results shown in Table 4 is difficult. However, for the poorest fuel, CN = 13, that they studied,⁷ there was no improvement in ignition delay upon addition of 2-ethylhexyl nitrate, which is qualitatively consistent with the behaviour that we have found for heptamethylnonane. It is obvious that one can construct suitable mixtures with other hydrocarbons that will give a null response to NO₂.

An attempt to do just this led to a rather unexpected result: NO₂ retarded the ignition of n-hexadecane–heptamethylnonane mixtures until the mol fractions were almost equal, with CN = 55. Contrast this with the behaviour of 3-ethyldecane in Table 4, for which CN is raised from ∽55 to the mid-80s by the addition of 0.1% of NO₂. Thus, not only may the absence of secondary hydrogen atoms in a hydrocarbon result in its ignition being retarded by nitrates, but also it can lead to different responses from mixtures of hydrocarbons of the same CN. Here again examples of such behaviour are known: in Nandi and Jacobs' study,⁷ three fuels of CN = 44 (samples 6, 7 and 8) were improved to CN = 50, 46, and 47 respectively by the addition of 0.1% of 2-ethylhexyl nitrate.

Comparison of the results in Table 4 with those of Nandi and Jacobs⁷ suggests that the calculated reductions in ignition delay are somewhat too large. A mole fraction of 0.1% of NO₂ in dodecane corresponds approximately to the inclusion of 0.1 wt.% of 2-ethylhexyl nitrate in the fuel. This would raise the CN of a relatively non-aromatic fuel from ∽55 to ∽65, whereas in our calculation, the CN of 3-ethyldecane is raised from ∽55 to ∽88 (the same delay as for n-dodecane itself).

It would, of course, be possible to manipulate the model so as to achieve weaker increments in CN, but that would not improve our understanding of the phenomenon. To that end, the measurement of the rates of reaction of NO₂ with a series of hydrocarbons (with a definitive discrimination between the relative rates of formation of HONO and of HNO₂) and the establishment of the fate(s) of HNO₂ in combustion systems are fundamental. In addition, the determination of the CN improvement for a selection of pure hydrocarbons by 2-ethylhexyl nitrate in a test engine (or in a static bomb) are prerequisites. Aromatic hydrocarbons must be included in these studies since most fuels contain considerable amounts of aromatic components,⁷ and the aromatic centres themselves, as our calculations suggest, are rather inert to NO₂ attack.

This could then be followed by a more extensive modelling calculation in which the reactions of NO₂ with important reaction intermediates, such as aldehydes, would be included. But it has to be remembered that CN improvement may not be a function only of the hydrocarbon content of the fuel, but that trace hetero-impurities can have a pivotal effect, as we have found recently when di-tert-butyl peroxide is used as an ignition improver.³⁴

This Appendix contains three tables of auxiliary data from the DFT calculations on reactions used in the modelling calculations. Further details can be obtained by electronic mail from 〈wtchan@yorku.ca〉; also, all 15 sets of transition state co-ordinates are available in XMOL-xyz form as Electronic Supplementary Information.† These data are provided so as to enable others either to verify or to reproduce the results of these molecular orbital calculations.

Table A1 Bond distances (in Å) and bond angles (in degrees) for R–H–ONO and R–H–NO₂ transition state structures in reactions of NO₂ with selected hydrocarbons

trans-HONO			cis-HONO				HNO2
Product RH	C–H	H–O	∠CHO	C–H	H–O	C–O	∠CHO	C–H	H–N	∠CHN
a Secondary C–H bond. b Tertiary C–H bond.
CH4	1.252	1.252	176.5	1.345	1.191	3.042	177.7	1.471	1.182	180.0
C2H6	1.229	1.296	174.7	1.317	1.226	3.064	177.0	1.437	1.206	175.1
C3H8^a	1.212	1.339	172.9	1.298	1.258	3.117	175.0	1.411	1.225	177.7
C4H10^b	1.196	1.379	175.6	1.283	1.284	3.165	172.5	1.390	1.242	177.7
C6H6	1.272	1.208	178.4	1.372	1.157	3.142	172.3	1.548	1.132	179.9

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Table A2 Total BHandHLYP/6-311G** energy (in E_h), point group and electronic state designation for individual reactants and products in reactions of NO₂ with selected hydrocarbons

Reactant	State	E	ZPE	Product	State	E	ZPE
CH4	T d	−40.4973	0.0459	CH3	D 3h ^2A2^″	−39.8241	0.0304
CH3CH3	D 3d	−79.7927	0.0767	CH3CH2	C s ^2A′	−79.1257	0.0608
(CH3)2CH2	C 2v	−119.0901	0.1063	(CH3)2CH	C s ^2A′	−118.4283	0.0905
(CH3)3CH	C 3v	−158.3885	0.1351	(CH3)3C	C 3v ^2A1	−157.7309	0.1199
C6H6	D 6h	−232.1563	0.1037	C6H5	C 2v ^2A1	−231.4711	0.0902
NO2	C 2v ^2A1	−205.0119	0.0094	trans-HONO	C s	−205.6422	0.0218
				cis-HONO	C s	−205.6433	0.0218
				HNO2	C 2v	−205.6292	0.0234

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Table A3 Total BHandHLYP/6-311G** energy (in E_h), imaginary and 1st harmonic frequencies (ω_i and ω₁, in cm⁻¹) for transition state structures in reactions of NO₂ with selected hydrocarbons; (unless otherwise stated, all species are C_s and ²A′)

trans-HONO				cis-HONO				HNO2
Product RH	E	ZPE	ω i	ω 1	E	ZPE	ω i	ω 1	E	ZPE	ω i	ω 1
a State designation C1, ^2A. b State designation C2v , ^2A1.
CH4	−245.4465	0.0503	1974i	39	−245.4554	0.0510	2010i	73	−245.4513	0.0521	1710i	6
CH3CH3	−284.7482^a	0.0806	1722i	37	−284.7574^a	0.0812	1943i	54	−284.7540^a	0.0822	1801i	9
(CH3)2CH2	−324.0503	0.1100	1438i	36	−324.0596	0.1105	1852i	50	−324.0570	0.1114	1820i	18
(CH3)3CH	−363.3520	0.1388	1120i	36	−363.3619	0.1392	1737i	49	−363.3599	0.1400	1815i	4
C6H6	−437.1016	0.1065	2051i	11	−437.1096	0.1073	1919i	25	−437.1046^b	0.1093	820i	13

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Allowance for tunnelling is often made in these types of calculation by the use of an implied partition function Q_T in the rate constant expression.^14,15 This is estimated (in the limit of small barrier width and height) by using Wigner's approximation³⁵

Since the temperature range over which we integrate the set of kinetic equations is from 745 and 1000 K, we display in Table A4 the estimated corrections for this method at 750 and at 1000 K. The correction is seen to lie generally between 30 and 60% for the reactions of interest in this study.

Table A4 Estimated Wigner tunnelling corrections Q_T at 750–1000 K for reactions of NO₂ with selected hydrocarbons; the higher value of each pair is for the lower temperature

RH	trans-HONO	cis-HONO	HNO2
a Secondary C–H bond. b Tertiary C–H bond.
CH4	1.62–1.35	1.65–1.37	1.47–1.27
C2H6	1.47–1.27	1.61–1.35	1.52–1.30
C3H8^a	1.33–1.19	1.56–1.32	1.53–1.31
C4H10^b	1.20–1.12	1.49–1.28	1.56–1.32
C6H6	1.67–1.38	1.59–1.33	1.11–1.06

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However Wigner's formula tends to overestimate tunnelling effect³⁶ and it is desirable to use more sophisticated methods such as the zero-curvature tunnelling (ZCT) or the small-curvature tunnelling (SCT) approximations.³⁷ Such methods, however, entail use of variational transition-state theory and are beyond the scope of this paper. As an example of the degree of overestimation, we cite the hydrogen transfer reaction between OH and fluoroethane

where the value of the imaginary frequency is ω_i = 1920i cm⁻¹ and the barrier height is 1.95 kcal mol⁻¹.³⁸ At room temperature, Wigner's method yields a value of Q_T = 4.58, instead of a value of 2.04 obtained from the SCT method;³⁹ at 1000 K, the effect was reported to be negligible while Wigner's method gives a value of 1.32.

We present here an alternative method for judging the relative importance of tunnelling in our series of reactions. Since most molecular orbital computational methods characterise the transition state by an imaginary harmonic oscillator frequency, ω_i, a useful procedure is to match the curvature of this hypothetical harmonic oscillator to that of a symmetric Eckart potential. Defining these two potential functions as

and

the condition for equal curvature at the saddle point is⁴⁰

where μ is the reduced mass of the tunnelling particle, i.e. μ∽1 u, and ω_i is in cm⁻¹.

The transmission coefficient for a symmetric Eckart barrier at energy E is⁴⁰

where

Extension to unsymmetric barriers (if needed) is logically straightforward, but cumbersome.

If we take E₀ = 30 kcal mol⁻¹ and ω_i = 1800i cm⁻¹, then the characteristic Eckart length L = 1.468 Å, and

Transmission coefficients for this example are listed in Table A5.

Clearly, for barriers of this kind, it is not necessary to go more than 1 kcal mol⁻¹ below the saddle point. The contribu tion to the rate is then found by integrating over the fraction of collisions lying in the 1 kcal mol⁻¹ band below the barrier, i.e.

Table A5 Transmission coefficients κ_T(E) for tunnelling below the barrier at energy E (in kcal mol⁻¹) with E₀ = 30 kcal mol⁻¹ and ω_i = 1800i cm⁻¹

E	απ	exp(−απ)	κT(E)
28.0	7.72	4.44 × 10^−4	4.44 × 10^−4
29.0	2.86	2.10 × 10^−2	2.06 × 10^−2
29.5	1.93	0.145	0.127
29.6	1.54	0.214	0.176
29.7	1.16	0.314	0.239
29.8	0.77	0.462	0.316
29.9	0.39	0.680	0.405

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Considering the two temperatures that span our range of integration: at 750 K, the number of collisions with energy along the line of centres lying between E = 29 and 30 kcal mol⁻¹ is 95% of the number between E = 30 kcal mol⁻¹ and ∞; for 1000 K, it is 54%. The first point to note is that the correction therefore reduces the activation energy slightly, but not by enough to eliminate the discrepancies already discussed above. Taking a mean value of κ_T(E) over this range of about 0.2, then the tunnelling correction is about 20% at 750 K, 10% at 1000 K—perhaps only a third of those shown in Table A4 for the formation of cis-HONO or of HNO₂; these are trivial in comparison with the uncertainties presently existing in the computational methods.

Acknowledgements

Work supported by the Natural Sciences and Engineering Research Council of Canada and, in part, by Esso Petroleum Canada.

References

1. P. Q. E. Clothier, B. D. Aguda, A. Moise and H. O. Pritchard, Chem. Soc. Re[italic v]., 1993, 22, 101.
2. M. A. R. Al-Rubaie, J. F. Griffiths and C. G. W. Sheppard, Some Obser[italic v]ations on the Effecti[italic v]eness of Additi[italic v]es for Reducing the Ignition Delay Period of Diesel Fuels, SAE Technical Paper Series, 1991, 912333..
3. H. O. Pritchard, Combust. Flame, 1989, 75, 415.
4. P. Q. E. Clothier, H. O. Pritchard and M.-A. Poirier, Combust. Flame, 1993, 95, 427.
5. (a) Y. Yamaguchi, Y. Teng, S. Shimomura, K. Tabata and E. Suzuki, J. Phys. Chem. A, 1999, 103, 8272; (b) K. Tabata, Y. Teng, Y. Yamaguchi, H. Sakurai and E. Suzuki, J. Phys. Chem. A, 2000, 104, 2648.
6. R. Galiasso and E. Rodriguez, A Low Emission Diesel Fuel, SAE Technical Paper Series, 1993, 932731..
7. M. K. Nandi and D. C. Jacobs, Cetane Response of di-tertiary-Butyl Peroxide in Different Diesel Fuels, SAE Technical Paper Series, 1995, 952368..
8. J. F. Griffiths, K. J. Hughes, M. Schreiber and C. Poppe, Combust. Flame, 1994, 99, 533.
9. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. Cioslowski, J. V. Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle and J. A. Pople, GAUSSIAN 98, Revision A.6, Gaussian, Inc., Pittsburgh, PA, 1998..
10. (a) W.-T. Chan, I. P. Hamilton and H. O. Pritchard, J. Chem. Soc., Faraday Trans., 1998, 94, 2303; (b) W.-T. Chan, H. O. Pritchard and I. P. Hamilton, Phys. Chem. Chem. Phys., 1999, 1, 3715.
11. B. G. Johnson, C. A. Gonzales, P. M. W. Gill and J. A. Pople, Chem. Phys. Lett., 1994, 221, 100.
12. M. T. Nguyen, S. Creve and L. G. Vanquickenborne, J. Phys. Chem., 1996, 100, 18422.
13. H. Basch and S. Hoz, J. Phys. Chem., 1997, 101, 4416.
14. S. Skokov and R. A. Wheeler, Chem. Phys. Lett., 1997, 271, 251.
15. R. G. Susnow, A. M. Dean and W. H. Green, Chem. Phys. Lett., 1999, 312, 262.
16. Q. Zhang, R. Bell and T. N. Truong, J. Phys. Chem., 1995, 99, 592.
17. J. L. Durant, Chem. Phys. Lett., 1996, 256, 595.
18. W.-T. Chan and H. O. Pritchard, to be published..
19. C.-C. Hsu, M. C. Lin, A. M. Mebel and C. F. Melius, J. Phys. Chem., 1997, 101, 60.
20. J. Baker, A. Scheiner and J. Andzelm, Chem. Phys. Lett., 1993, 216, 380.
21. D. K. Maity, W. T. Duncan and T. N. Truong, J. Phys. Chem., 1999, 103, 2152.
22. G. Herzberg, Molecular Spectra and Molecular Structure, III, Electronic Spectra and Electronic Structure of Polyatomic Molecules, Van Nostrand, New York, 1966, p. 614..
23. J. M. Bofill, S. Olivella, A. Solé and J. M. Anglada, J. Am. Chem. Soc., 1999, 121, 1337.
24. D. Shen and H. O. Pritchard, unpublished results..
25. S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York, 1941, pp. 191–195..
26. (a) M. W. Slack and A. R. Grillo, Ele[italic v]enth International Shock Tube Symposium, University of Washington Press, Seattle, WA, 1977, p. 408; (b) M. W. Slack and A. R. Grillo, Combust. Flame, 1981, 40, 155.
27. T. A. Titarchuk, A. P. Ballod and V. Ya. Shtern, Kinet. Katal. (Engl. Transl.), 1976, 17, 932.
28. S. M. Heck, H. O. Pritchard and J. F. Griffiths, J. Chem. Soc., Faraday Trans., 1998, 94, 1725.
29. A. D. Puckett and B. H. Caudle, Ignition Qualities of Hydrocarbons in the Diesel-fuel Boiling Range, United States Department of the Interior, Bureau of Mines, 1948, I.C. 7474..
30. W. A. Bone and J. B. Gardner, Proc. R. Soc. London, Ser. A, 1936, 154, 297.
31. A. Burcat, Combust. Flame, 1977, 28, 319.
32. C.-Y. Lin, H.-T. Wang, M. C. Lin and C. F. Melius, Int. J. Chem. Kinet., 1990, 22, 455.
33. K. A. Holbrook, M. J. Pilling and S. H. Robertson, Unimolecular Reactions, Wiley, Chichester, 1996, pp. 108–110..
34. P. Q. E. Clothier, S. M. Heck and H. O. Pritchard, Combust. Flame, 2000, 121, 689.
35. J. I. Steinfeld, J. S. Francisco and W. L. Hase, Chemical Kinetics and Dynamics, Prentice Hall, Englewood Cliffs, NJ, 1989, p. 320..
36. I. Shavitt, J. Chem. Phys., 1959, 31, 1359.
37. D. G. Truhlar, B. C. Garrett and S. J. Klippenstein, in Theory of Chemical Reaction Dynamics, ed. M. Baer, CRC Press, Boca Raton, FL, 1985, vol 4..
38. S. Sekusak, H. Gusten and A. Sabljic, J. Phys. Chem., 1996, 100, 6212; S. Sekusak, H. Gusten and A. Sabljic, J. Phys. Chem., 1997, 101, 967.
39. S. Sekusak, K. R. Liedl, B. Rode and A. Sabljic, J. Phys. Chem., 1997, 101, 4245.
40. D. Rapp, Quantum Mechanics, Holt, Rinehart and Winston, New York, 1971, pp. 168–170..

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Footnotes

† Electronic Supplementary Information available. See http://www.rsc.org/suppdata/cp/b0/b006088g/

‡ Incidentally, this sequestering of the NO₂ further helps to justify the exclusion of the reaction between NO₂ and formaldehyde to form HNO₂.

§ For the reactions to form HNO₂, where the problem is most acute (see Table A3), the difference between restricted rotor and simple harmonic partition functions amounts to less than 20% for the tabulated cases; for larger R, the effect declines because of the appearance of an effective reduced mass in the restricted rotor formulae.³³ The neglect of tunnelling is addressed in Appendix 2.

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