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J . Phys. Chem. 1990, 94, 1432-1439

1432

Detailed Kinetic Modeling of Autocatalysis in Methane Pyrolysis A. M. Dean Corporate Research, Exxon Research and Engineering Company, Annandale, New Jersey 08801 (Received: May 10, 1989; In Final Form: August 3, 1989)

A detailed kinetic model is developed to explain the observed acceleration in rates of product formation during the pyrolysis of methane at 0.58 atm and 1038 K. This model, obtained by a systematic simplification of a much larger one, consists of 44 reactions involving 25 species. The model includes a number of reactions involving chemically activated complexes formed by radical addition and recombination reactions, with rates for unimolecular reaction and bimolecular stabilization of these energized complexes evaluated with a QRRK formalism. A combination of reaction rate analysis as well as sensitivity analysis is used to show that these chemically activated reactions lead to surprisingly rapid production of cyclopentadiene at very low extents of conversion ( ~ 0 . 1 % and ) that dissociation of cyclopentadiene accounts for the acceleration in rate. Cyclopentadiene is produced by the addition of allyl radicals to acetylene; the chemically activated linear adduct undergoes cyclization prior to collisional stabilization. An optimization procedure was used to obtain an accurate description of the observed kinetics without compromising the theoretical plausibility of the rate constants.

Introduction Although methane pyrolysis has been extensively studied,I some aspects of the kinetics remain puzzling. In particular, Back and Back' have discussed a sharp increase in the rate of formation of ethane at very low levels of conversion (~0.1%). They performed an exhaustive search for possible experimental artifacts that might produce this acceleration and concluded that it was probably a real effect, not attributable to reactions on surfaces, etc. They speculated that it might be due to homogeneous reactions of the reaction products but concluded that the mechanism of this acceleration in rate was in considerable doubt. Roscoe and Thompson2 addressed this autocatalysis in methane pyrolysis by proposing a detailed model that appeared to satisfactorily describe the acceleration in rate. This mechanism was consistent with the spirit of Back and Back's discussion and appeared to present a nice solution to the autocatalysis issue. However, a closer inspection of this mechanism, as described below, indicated that it appeared to have been constructed in such a way that forward and reverse rate constants for a specific elementary reaction were not necessarily related by the appropriate equilibrium constant. When the reverse rate constants were obtained explicitly from the forward values with microscopic reversibility, it was observed that the model results of Roscoe and Thompson were dramatically affected and virtually all of the autocatalysis disappeared. This reaction system is particularly important since it has the prospect of providing an explanation for the process of rapid molecular weight growth. It is well-known that methane pyrolysis can produce copious quantities of tar and coke. In fact, such production has been the primary reason for not using methane as a feedstock in thermal cracking processes.' This propensity to form very high molecular weight species is directly attributable to the stability of methane. At temperatures sufficiently high that it is thermodynamically possible to produce significant yields of hydrocarbons such as acetylene and benzene from methane, the most favored thermodynamic products are graphite and hydrogen. It has proved exceedingly difficult to kinetically trap significant quantities of such products without concomitant formation of unwanted tar and coke. Thus, the data of Back and Back that illustrate this tendency toward an accelerating rate of production of heavier hydrocarbons under conditions where a careful product analysis has been done offer a nice opportunity to explore details of the kinetics of the growth process. The unusually rapid rate of growth at high temperatures is also exhibited in flames as formation of soot. Although there has been ( I ) Back, M. H.; Back, R. A. In Pyrolysis: Theory and Industrial Practice; Albright, L. F., Crynes, B. L., Corcoran, W.H., Eds.; Academic

Press: New York, 1983; pp 1-24 and references cited therein. (2) Roscoe, J. M.;Thompson, M. J. Inr. J . Chem. Kinet. 967-990.

1985, 17,

0022-3654/90/2094- 1432$02.50/0

progress recently in characterization of this rapid rate of soot formation, attempts to accurately model this growth process have been only partially successful. For example, the extensive study by Frenklach et al.3 required rate constants for radical addition reactions that are several orders of magnitude larger than usually observed for these types of reactions. There appear to be three possible pathways to rapid molecular weight growth: (1) cycloaddition reactions between olefins and dienes (Diels-Alder); (2) ion-molecule reactions with their very large rate constant^;^ and (3) radical addition to unsaturates. A major difficulty with (1) is that the equilibrium constants for this type of reaction suggest that, at the high temperatures where growth is observed to occur, the cyclic species would tend to dissociate to the linear (smaller) fragments. A major difficulty with an ionic mechanism is that it appears unlikely that the concentration of ions in a pyrolytic environment (where chemionization reactions involving O2could not occur) could be sufficiently high to have any impact. By default, our attention (like that of Frenklach et a].) has focused upon radical addition reactions. Previously, we demonstratedS that new reaction channels could open up at higher temperatures in chemically activated reactions (such as radical addition to unsaturates). Methane pyrolysis appeared to offer an opportunity to see if these new channels could account for the observed autocatalysis.

Results and Discussion Roscoe-Thompson Mechanism. The mechanism presented by Roscoe and Thompson2 was used to calculate concentration-time profiles for the conditions (neat CH4, T = 1038 K, P = 0.58 atm) reported by Back and Back.l These calculations utilized the CHEMKIN package6 with the LSODE' stiff equation solver. Two sets of calculations were done. In the first, following Roscoe and Thompson, the system was treated as a set of irreversible reactions, with use of their rate constants. Most of the reverse reactions were also entered as irreversible reactions with rate constants explicitly assigned. In the second set of calculations, the system was treated reversibly, with the forward rate constants so designated by Roscoe and Thompson; here, all reverse reactions were automatically considered, with rate constants obtained via com(3) Frenklach, M.; Clary, D.W.; Gardiner, W. C., Jr.; Stein, S. Twentyfirst Symposium (International) on Combustion; The Combustion Institute:

Pittsburgh, PA, 1986; pp 1067-1076. (4) Calcote, H. F. Combust. Flame 1981, 42, 215-242. (5) Dean, A. M. J . Phys. Chem. 1985,89, 4600-4608. (6) Kee, R. J.; Miller, J. A.; Jefferson, T. H. CHEMKIN: A GeneralPurpose, Problem-Independent, Transportable, Fortran Chemical Kinetics Code Package. Report SAND80-8003; Sandia National Laboratory: Livermore, CA; 1980.(7) Hindmarsh, A. C. In Scientific Computing, Stepleman, R. S., et al., Eds.; North-Holland: Amsterdam, 1983; pp 55-64.

0 1990 American Chemical Society

The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1433

Kinetic Modeling of Autocatalysis in Methane Pyrolysis n

larger than that observed for methyl.) Another result of chemical activation at high temperatures is the possibility of providing the energy required for larger radicals to cyclize:I0

(22

Lp

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TIME (SEC) Figure 1. Comparison of predicted and observed production of ethane from methane pyrolysis at 1038 K and 0.58 atm. Key: (A)observed, from ref 8 (0.57 atm); (+) observed, from ref 1; (-D-) predicted using Roscoe-Thompson irreversible mechanism; (-0-) predicted using Roscoe-Thompson reversible mechanism.

puted equilibrium constants. The results of these calculations are compared to two sets of observations for ethane production in Figure 1. One set of the observations is taken from a figure in ref 1 , and the other is taken from a table in an earlier paper by Chen et ai.* Although the irreversible calculation does a good job of replicating the observed increase in ethane production near 1500 s, it is immediately obvious that virtually all of this acceleration is removed when the system is treated reversibly. A comparison of rate constants reveals that the most likely cause for the discrepancy is that the value used by Roscoe and Thompson for the reaction H

+ C2H4

-+

C2H5

is over a factor of 100 larger than that obtained via microscopic reversibility with their value for ethyl dissociation. (Much of the problem here seems to lie with the choice of rate constant for dissociation of ethyl; the value used in the mechanism is a factor of 20 lower than that recommended by TsangS9) Use of the mechanism employing microscopic reversibility forces the rate constant for addition of hydrogen atoms to ethylene to be unrealistically low. Since the Roscoe-Thompson sensitivity analysis shows this addition to be important for ethane production at later times and that it has a positive sensitivity coefficient, i.e., an increase in rate constant will increase the ethane yield, it is not surprising that this lower value led to a substantially decreased rate of ethane production, thus minimizing the acceleration. Attempts to improve the fit by adjusting some of the forward rate constants, consistent with theoretical guidelines, were not successful. It appears that the mechanism proposed by Roscoe and Thompson is not adequate to explain the observed acceleration in rate for ethane production when proper account is taken of microscopic reversibility. Generation of Energized-Complex Mechanism. Another possible mechanism to explain the autocatalysis is the onset of new reaction channels in radical addition reacti0ns.j For example, high temperatures significantly increase the probability that the adduct formed when C H 3 adds to C2H4 will have enough energy to break a C-H bond: CH3

+ C2H4 * C3H7'

C3H6

+H

Such a reaction opens up two possibilities for more rapid growth. The first is prpduction of a stable molecule of higher molecular weight. The second, perhaps more important, aspect is that these reactions convert relatively unreactive methyl radicals into much more reactive hydrogen atoms. (For example, both abstraction and addition rate constants for H atoms are orders of magnitude (8) Chen, C. J.; Back, M. H.; Back, R. A. Can. J . Chem. 1976, 54, 3 175-3 184. (9) Tsang, W. J . Phys. Chem. Ref. Data 1986, 15, 1087-1279; Ibid. 1987, 16, 471-508.

Detailed analysis5 indicated that the temperature and pressure conditions under which rapid growth has been observed to occur were such that these types of chemical activation were indeed possible and that they could be responsible for rapid growth. To apply this approach to the case of methane pyrolysis, a large mechanism was assembled to attempt to describe the chemistry in this system. This mechanism included our estimates for the rate constants for both radical addition and recombination rate constants that were obtained with use of the methods described in ref 5. The approach used was to estimate the unimolecular reaction rates of the energized adduct formed in the reaction with a quantum version of RRK theory. Previous studies5*" have demonstrated the validity and utility of this method. One improvement in the estimation method was that the preexponential factors and barriers for reactions of the energized complexes were obtained via a specific calculation of the appropriate equilibrium constant as opposed to the generic estimates used earlier. Rate constants for radical isomerizations were assigned on the basis of thermochemical kinetic arguments; A factors accounted for the loss of rotors in the transition state, and barriers were estimated on the basis of ring strain, enthalpy change, and intrinsic activation energy for the hydrogen-transfer process. For example, the 1,3 H shift CH,CH=C'H

* C'H2CHXH2

was described as having A = 5 X 10l2s-' and E, = 36 kcal/mol. The high-pressure limiting rate constants for the recombination and addition rate constants were taken from literature sources such as Tsang9 and the C R C tables.I2 Apparent rate constants for the various reaction channels were computed as functions of both pressure and temperature. These values obtained a t a pressure of 0.58 atm were then fitted to an expression in modified Arrhenius form over the widest temperature range within the interval 300-2500 K such that this expression yielded rate constants within 10% of the QRRK-computed values over the range. Since these modified Arrhenius parameters are fits to apparent rate constants and reflect the competition between various unimolecular dissociation channels and collisional deactivation, no physical or chemical significance should be attributed to the value of any specific parameter; these are best interpreted as simply a convenient way to describe the complex temperature dependence. Moreover, it is important to note that these QRRK estimates are applicable only at 0.58 atm; the extent of collisional deactivation scales with pressure and thus the apparent rate constants must be considered to be functions of pressure as well as temperature. These estimates were combined with our best estimates for hydrogen-transfer and disproportionation rate constants, relying heavily on the compilations of T ~ a n g .Abstraction ~ reactions that formed resonantly stabilized radicals were assigned rate constants based upon the CRC tabulations.12 Preexponential factors were scaled to reflect the number of abstractable hydrogens, and activation energies were adjusted to reflect abstraction of primary, secondary, or tertiary hydrogens. Similarly, activation energies were lowered for abstraction reactions involving even weaker C-H bonds (such as in cyclopentadiene) with use of Evans-Polanyi relationships. Other critical inputs to the model are the updated estimates of the enthalpies and entropies of alkyl radicals; these were based on Tsang.13 (These values were used to compute the equilibrium constants needed for the analysis of the Roscoe(IO) Westmoreland, P. R.; Dean, A. M.; Howard, J. B.; Longwell, J. P. J . Phys. Chem., in press. ( 1 1 ) Westmoreland, P. R.; Howard, J. B.; Longwell, J. P.; Dean, A. M. AIChE J . 1986, 32, 1971-1979. (1 2) Handbook of Bimolecular and Termolecular Gas Reactions; Kerr, J. A,, Ed.; CRC Press: Boca Raton, FL, 1981; Vol. I and 11. (13) Tsang, W. J . Am. Chem. Soc. 1985, 107, 2872-2880.

1434 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990

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TIME (SEC) Figure 2. Comparison of predicted and observed production of ethane observed, from methane pyrolysis at 1038 K and 0.58 atm. Key: (0) from ref 8 (0.57 atm); (A)observed, from ref 1; (-m)predicted using Table I mechanism. Thompson mechanism as well.) The final mechanism consisted of 438 reactions involving 122 species. This mechanism was systematically condensed via successive applications of sensitivity ana1y~is.I~For the conditions of the Chen et al. experiments, it was possible to obtain essentially the same results with a much smaller mechanism involving 44 reactions among 25 species. Optimization and Comparison to Observations. Although this smaller mechanism gave definite indications of autocatalysis, the magnitude was somewhat less than observed in the experiments. Since the sensitivity analysis indicated that a few reactions were particularly important for all of the observed species and since it is clear that there remains significant uncertainty as to the accurate assignment of rate constants, a constrained optimization procedureI5 was employed to try to improve the model description while requiring the rate constants to remain theoretically plausible. It was observed that small changes in only three rate constants markedly improved the fit. The first of these changes involved the initiation reaction: CH,

C H 3 + H, k = 2.2

X

Dean

IO7 s-I

500.0

1000.0

1500.0

2000.0

2500.0

30000

TIME (SEC) Figure 3. Comparison of predicted and observed production of ethylene observed, from methane pyrolysis at 1038 K and 0.58 atm. Key: (0) from ref 8 (0.57 atm); (A)observed, from ref 1; (-D-)predicted using Table I mechanism.

I I

I

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TIME (SEC) Figure 4. Comparison of predicted and observed production of acetylene from methane pyrolysis at 1038 K and 0.58 atm. Key: (0)observed, from ref 8 (0.57 atm); (A)observed, from ref 1; (--O-) predicted using Table I mechanism.

This value is only 40% higher than that recommended by TsangI' (suggested uncertainty of 50% for k,, presumably more when in falloff region). The value used for C2H6

+ CH3

C2HS+ CH,,

k = 5.7 X lo9 cm3 mol-]

s-l

is approximately a factor of 2 lower than that recommended by Tsang. However, there are few data points in this particular temperature range, and the value used here is still consistent with upward curvature on Arrhenius plots frequently observed with hydrogen abstraction reactions; simple extrapolation of lower temperature datal2 would suggest a value near 2 X lo9. The last rate constant adjusted was for methyl abstraction from cyclopentadiene:

+ CH3

= 0+

i

/

5000

CH4, k = 2.2 x 10'' cm3 mol

-' s-'

This rate constant is only 25% higher than one would estimate on the basis of C R C data on abstraction of species that form resonantly stabilized radicals when one accounts for the unusually weak bond in cyclopentadiene. Thus, it appears that the adjustments made in the mechanism are completely consistent with known uncertainties and do not represent excessive parametrization in an attempt to achieve a better fit. (14) Lutz, A. E.: Kee, R. J.; Miller, J. A. SENKIN: A Fortran Program for Predicting Homogeneous Gas Phase Chemical Kinetics with Sensitivity Analysis; Report SAND87-8248; Sandia National Laboratory: Livermore,

CA, 1988. ( I 5) Byrne, G.D.; Dean, A. M.; McCroskey, P. S. In Proceedings ofthe Second International Symposium on Computational Chemistry on Cray Supercomputers; Cray Research, Inc.: 1988; 209-21 9.

10000

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3( 10.0

TIME (SEC) Fipre 5. Comparison of predicted and observed production of propylene from methane pyrolysis at 1038 K and 0.58 atm. Key: (0) observed, from ref 8 (0.57 atm); (A)observed, from ref 1; (-0-) predicted using Table I mechanism.

This mechanism is listed in Table I, and the results for ethane production are compared to the observations in Figure 2. Note that the mechanism accurately predicts the autocatdlysis. Figures 3-5 illustrate the comparisons between predictions and observations for C2H4,C2H2,and C3H6. The fit to ethylene is excellent, but the model overpredicts acetylene and slightly underpredicts propylene. Since both acetylene and propylene are present at much lower concentrations, it is possible that some of this discrepancy might be related to analytical difficulties. (Note the scatter in the C2H2data in Figure 4.) However, it is certainly possible that there remain some mechanistic problems. Attempts to improve

The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1435

Kinetic Modeling of Autocatalysis in Methane Pyrolysis

n 0 = OVERALL REACTION A = CY13PD=CY13PDS+H

+ = CH3+CY13PD=CH4+CY13PD5

X = H+CYl3PD=HZ+CY13PD5 0 = CYPENE4.=CY13PD+H V = CCC +CY13PD5.=OCC+CY13PD tX = C’CC +CZH2=CY13PD+H

0

-r

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TIME (SEC) Figure 6. Comparison of predicted and observed product ion of ethane from methane pyrolysis at 1038 K and 0.58 atm. Key: (A)observed, from ref 8 (0.57 atm); (+) observed, from ref 1; (-0-) predicted using Table I mechanism; (-O-) predicted using Table I mechanism without reactions involving species with more than four carbon atoms. the fit to these species were not successful without resorting to extensive manipulation of rate constants. In the author’s opinion, such manipulation was unjustified and was not pursued seriously. Nevertheless, the overall success of this mechanism in describing most of the observations suggests that this mechanism indeed captures the essence of the observed autocatalysis. The results of a sensitivity analysis for this mechanism are shown in Table I1 as an ordered list of the ten most important reactions for the major products observed by Chen et al. (In the case of methane, only seven reactions are shown because all others had normalized sensitivity coefficients of less that 1 X 10” at all times.) An interesting feature of the sensitivity analysis is that the same group of reactions are seen to be important for all of the major products. Note that only 14 reactions are listed in Table 11. If one extends the optimization procedure to include more of these reactions, the fit to the data is not observed to improve significantly. More importantly, adding additional reactions generally had little effect upon the ”optimized” values of the three rate constants described above. This provides additional evidence that the values used to obtain the fits in Figures 2-5 have physical significance. Of course, as in any modeling exercise, the rate constants obtained should not be considered to be as reliable as those directly measured in simpler systems designed to yield such rate data. Reasons for Autocatalysis. We have seen above that the predicted profiles were sensitive to reactions of cyclopentadiene. This was quite surprising since the overall extent of reaction in this system is very low (0.09%conversion at 1600 s) and one might not have expected the reactions to produce such a large molecule, much less to have it play a dominant role in the system. This sensitivity is illustrated in Table 11, where a number of reactions involving cyclic C5 species are listed. This conclusion is reinforced by removing the reactions involving the larger species (any reaction that involved a species with more than four carbons) from the mechanism; the autocatalysis is completely suppressed. These results are shown in Figure 6. It is noteworthy that this behavior is quite similar to that observed with the Roscoe-Thompson mechanism when that mechanism incorporated microscopic reversibility. It illustrates that the main problem with that mechanism is simply that it did not consider enough of the molecular weight growth chemistry. The most important reactions involved in the production and decay of cyclopentadiene are shown in Figure 7. Note that the major path for production is the abstraction of a hydrogen from methane by cyclopentadienyl (reaction 25 running backwards). Although this reverse reaction is very endothermic with a correspondingly small rate constant, the large concentration of methane shifts this reaction toward this direction. Once formed, the cyclopentadiene undergoes unimolecular dissociation. This dissociation is really the critical step for the acceleration in rate; not only does this regenerate the cyclopentadienyl radical, but it

1040 0

520 0

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m

2000 0

2600 0

TIME (SEC) Figure 7. Reactions of cyclopentadiene with the largest net rates of

production and destruction are plotted as a function of time, based on the mechanism in Table I at 0.58 atm and 1038 K. Note that the reaction CH3 + CY 13PD = CH4 + CY 13PD5’ is running backwards to produce cyclopentadiene.

-1

1040 0

520 0

1560 0

2060 0

2600 0

TIME (SEC)

Figure 8. Reactions of hydrogen atoms with the largest net rates of production and destruction as a function of time, based on the mechanism in Table I at 0.58 atm and 1058 K. Note that two reactions, ethyl dissociation and cyclopentadiene dissociation, are the main sources of hydrogen.

also produces hydrogen atoms that we expect to be particularly reactive. Note that these rates become important just as the acceleration in rate is observed for ethane. These two reactions serve as an efficient route for radical production: ’

Q=Q+H Net:

CH4

=

CH3

+

H

Figure 8 shows the most important reactions for production of hydrogen atoms; note that dissociation of cyclopentadiene is as important as B scission of ethyl radicals. Thus, the essential role of cyclopentadiene in acceleration of methane pyrolysis becomes clear: this molecule has the sufficiently weak C-H bond required to allow onset of a new chain-branching reaction. Role of Energized Complexes in the Acceleration. Given the importance of cyclopentadiene, attention was focused on the routes for its production. A potential energy diagram is shown in Figure 9. We want to ascertain the relative importance of “equilibrated” vs “energized-complex” mechanisms in this reaction sequence. In the conventional approach (here termed equilibrated to denote the dominance of deactivating collisions), the cyclopentadiene production would be considered to be the result of three distinct reactions: (1) formation of a linear adduct from the addition of allyl to acetylene in which the excess energy has been removed in deactivating collisions; (2) isomerization of this adduct to the cyclic species, where the energy required for the isomerization

1436 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 TABLE I: Methane Pyrolysis Mechanism" no. reactionb 1 CH4 = CH, + H (1038 K, 0.58 atm) C2H6 = 2CH3 (1038 K, 0.58 atm) 2 3 CHg + CH3 = C2H5 + H

4

5 6 7 8

9 IO I1

I2 13 14 15 16 17 18

19 20

21 22

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

C=CC = C=CC' + H (900-2100 K) C=CC = C2H3 + CH3 (900-2100 K) CECC = CZCC' + H (600-2100 K) CY I3PD = CY 13PD5' + H (600-2100 K) CH3 + CY13PD5' = CHD (900-1200 K) CH, + CY 13PD5' = CYC6H7 + H (900-1200 K) CHD = CYC6H7+ H (600-2100 K) 2CY13PD5' = NAPH + H2 (900-2100 K) C2HS= C2H4 + H (1038 K, 0.58 atm) C2H3 = C2H2 + H (600-2500 K) CYPENE4' = C=CCC=C' (900-2100 K) CYPENE4' = CY13PD + H (900-2100 K) CYC6H7 = C6H6 + H (900-2100 K) CH4 + H = CH, + H2 C2H6 + H = C2H5 + H2 CY13PD + H = H2 + CYl3PD5' C2H6 + CH, = C2H5 + CH4 C2H4 + CH, = CIH, + CH4 C=CC + CH, = C=CC' + CH4 C z C C + CH, = CcCC' + CH4 C=C=C + CHj = C=CC' + CH4 CY 13PD + CH, = CH4 + CY 13PD5' CHD + CH3 = CYC6H7 + CH4 H + C=CC = CCC' (600-2500 K) H + C=C=C = C=CC' (600-2 100 K) H + C E C C = CC=C' (600-2500 K) H + C=CC = C=CC' (600-2100 K) CH, + C2H4= CCC' (900-2100 K) CH3 + C2H4 = C=CC + H (600-2500 K) CH, + C2H2 = CC=C' (600-2500 K) CH, + C2H2 = C=CC + H (600-2100 K) CH, + C2H2 = C=CC' (900-2100 K) CH3 + C=CC = C=CCC + H (600-2500 K) CH, + C=C=C = C2H3 + C2H4 (600-2500 K) C=CC' + C2H2 = C=CCC=C' (300-2500 K) C=CC' + C2H2= CYPENE4' (600-2500 K) C=CC' + C2H2 = CY 13PD + H (600-2500 K) CH, + C2H5 = CH4 + C2H4 (900-1700 K) CH, + C=CC' = CHI + C=C=C C=CC' + C=CC' = c=cc + c=c=c C=CC' + CY 13PD5' = C=C=C + CY 13PD

Dean

Ah

2.21E-07 5.71E-03 1.80E+I 2 5.888+47 1.478+56 8.79E+29 8.13E+24 1.01E+67 2.44E+41 4.35E+22 4.30E+36 8.94E+04 1.93E+28 1.52E+58 1.028+58 2.70E+57 2.20E+04 5.40E+02 4.00E13 2.70E-01 4.20E+11 3.80E+11 3.80E+11 3.80E+I 1 3.11E+ll 5.00E+II 1.37E+42 7.89E+23 4.22E+35 2.41 E+29 3.658+48 1.98E+22 7.28D+34 6.83D+16 1,10D+45 3.37E+24 3.46E+33 8.388+30 3.42E+52 2.95E+32 5.50E+11 1.OOE+ 12 1.OOE+ 12 1.OOE+ I2

n 0 0 0

-9.483 -1 1.426 -4.41 1 -2.981 -1 5.663 -7.989 -2.230 -6.268

E, 0 0

IO 400 107 303 117905 98 213 78 682 35 064 39 259 79 806 45671

0

-4.783 -1 3.062 -1 3.066 -13.102 3.0 3.5 0

4.0

0 51 123

60616 60 155 48 985 8 750 5 200 3 000 8 280

0 0 0 0 0 0

1 1 100 9 000 9 000 9 000 5 500

-8.823 -3.365 -7.023 -4.780 -10.896 -2.879 -6.921 -1.442 -9.449 -3.554 -6.058 -6.242 -12.194 -5.829

15 979

0 0 0 0

6 300 8 099 12441 15505 27 860 25 442 18 338 21 184 35412 29 636 33915 12824 27 978 25 733 0 0 0 0

refs see text 9 9 QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK 9 QRRK QRRK QRRK QRRK 9 9 estC see text 12 12 estd estd see text estC QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK QRRK 9 est/ est/ estf

kA1038 K)h kA1038 K)h 2.21 E-07 2.5 1E+13 5.71E-03 8.768+12 I.l6E+10 4.32E+13 3.74E-04 6.26E+13 7.54E-04 1.77E+13 9.10E-05 6.83E+13 2.24E-01 5.96E+13 2.39E+l2 1.79E+02 1.06E+09 6.13E+ 13 9.94E+ 13 1.28E-01 4.35E-05 1.29E+08 5.39E+12 8.94E+04 6.82E+Il 1.24E+03 2.99E+08 1.O5E+06 6.24E+ 12 8.56E+05 3.51E+ 12 3.97E+07 1.70E+ 10 3.54E+Il 1.568+12 3.788+09 9.34E+ 12 1.05E+06 5.66E+09 2.84E+08 1.93E+09 1.228+09 4.848+09 7.1 5E+06 4.848+09 3.20E+07 4.848+09 8.86E+06 2.15E+10 5.04E+04 2.368+30 1.61E+05 1.45E+ 12 2.53E+05 1,10E+12 2.72E+00 4.92E+03 6.65E+11 5.01E+ll 4.478+00 6.78E+09 3.628+06 1.80E+08 5.51E+I 1 1.34E+l0 1.448+05 1.06E+08 1.54E+ll 1.22E+09 1.58E+01 3.70E+07 1. I2E+12 1.348+08 1.04E+07 2.49E+09 1.03E+07 7.30E+09 1.06E+05 2.95E+09 3.13E+ll 5.50E+11 2.92E-01 1.OOE+ 12 3.57E+03 2.41E+06 1.OOE+ 12 1.53E+09 1.00E+12

" k = A T " exp(-EJRT), units = s-' (first order) or cm3 mol-' s-' (second order). bAbbreviations: C=CC' = allyl; CY13PD = 1,3-cyclopentadiene; CY 13PD5' = cyclopentadienyl;CHD = 1,3-cyclohexadiene; CYPENE4' = 4-cyclopentenyl; CYC,H7 = 1,3-hexadienen-5-y1;NAPH = naphthalene. CBasedon H + C=CCC H2 + C=CC'C (ref 12) with lower E, to account for weaker C-H bond (Evans-Polanyi plot). dBased on CH, + C=CC CH4 + C=CC'. CBasedon CH, + C=CCC (ref 12) with A X 2 (twice the number of abstractable H) and lower E, to account for weaker C-H bond (Evans-Polanyi plot). /Assumed faster than CH, + C2H5since involving resonantly stabilized radicals (Allara, D. L.; Shaw, R. J . Phys. Chem. Ref Data 1980, 9, 523-559). fQRRK signifies that the rate constant listed is a modified Arrhenius fit to the calculated chemically activated apparent rate constants at 0.58 a t m over the specified temperature range. hThese values follow the pattern 2.21E-07 = 2.21 X I 0-7.

-

-

is obtained via collisional activation; and (3) collisional activation of the cyclic species so that it could undergo p scission to cyclopentadiene and hydrogen atoms. The energized-complex approach does not assume that the initially formed linear adduct will always be collisionally quenched; it allows the unimolecular cyclization and/or p scission to compete with collisional stabilization. Note that the two approaches will converge in the limit of high pressure where all adducts must be collisionally deactivated prior to any unimolecular reaction. So the question is really whether, under the conditions of these experiments, the pressure is low enough to make the time required for the bimolecular energy-transfer processes implicit in the conventional approach to be longer than that required for unimolecular reactions of the energized adducts. Two separate analyses indicate that the energized-complex route plays an important role under the conditions of the Chen et al. experiments. The first of these is based upon the computed rate constants for production of the various C, species by the addition of allyl to acetylene. The Q R R K treatment incorporated in the mechanism shown in Table I has three ways to make cyclopentadiene:

Equilibrated

Energized Complex Followed by Equilibrated (ECE1)

0'--Q

+H

(ECE2)

Energized Complex

The rate of production of cyclopentadiene was computed for the first two pathways by using a steady-state analysis in conjunction with the appropriate rate constants in Table I. These rates were compared to that of the direct path ECl. It was found that the

The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1437

Kinetic Modeling of Autocatalysis in Methane Pyrolysis

5;N

* C=CC'

+

C,H,

C=CCC=C"

0. Q +

+ H

d l

100 3

L

90

-s

8 :

80

Lf

7

C=CCC=C'

P LU

70 9

50

Figure 9. Potential energy diagram for the addition of allyl radical to acetylene, illustrating the possible pathways for reactions of the intermediates.

equilibrated route E1-E3 only accounted for approximately 10% of the total rate, while the "mixed" route ECE1-ECE2 accounted for approximately 60% of the total rate. An inspection of Figure 9 offers an explanation of these results. Note that the isomerization barrier is below the entrance barrier and that there is a deep well for the cyclic intermediate. This will result in a rapid unimolecular reaction for this channel, sufficiently rapid to compete favorably with deactivation. However, there remains a sizable barrier for p scission of the cyclic radical. This, coupled with the relatively low A factor for the 6-scission reaction (2.4 X 1O1j s-I), results in significant deactivation to the cyclic species. This is converted to the final products in a subsequent step. Thus, the direct formation of cyclopentadiene as well as its cyclic precursor via unimolecular reaction prior to stabilization was found to be important in this system. Another way to test the importance of these routes is to modify the mechanism in Table I to reflect only the conventional routes to cyclopentadiene. Thus, reactions 39 and 40 were removed, and the rate constant for reaction 38 was increased (to 8 X lo9 cm3 mol-' s-I) to reflect its rate if no other channels were available for the initially formed linear adduct other than stabilization or redissociation to the reactants. Figure 10 compares the production of cyclopentadiene with this modification to the original one. Note that substantially more cyclopentadiene is formed when one allows the chemically activated adducts to react. This change in production of cyclopentadiene is also manifested in increased ethane yield; Figure 1 1 clearly illustrates the role of the energized adducts in increasing the magnitude of the observed acceleration. In general, one finds that temperature is a much more dominant variable than pressure in determining the importance of energized-adduct chemistry. The unimolecular rate constants scale approximately exponentially in temperature while the stabilization scales linearly with pressure. Thus, the observation that the energized-complex reactions play a role in methane pyrolysis at 1038 K suggests that they are to be expected to play an even more important role at higher temperatures, even if the pressure is raised. It appears that the autocatalysis is a manifestation of the extremely rapid production of higher molecular weight species and that this production is augmented via reactions of chemically activated adducts. As we found in our earlier studies, these adducts have enough internal energy at these temperatures to react unimolecularly before collisional stabilization can occur; this significantly increases the overall rate of production of heavier species because no time is required for collisional deactivation and subsequent activation to get to these same products. An interesting feature of the reaction mechanism is the fact that, although addition of allyl to acetylene is seen to be critical, the analogous reaction, addition of allyl to ethylene, is not needed, even though ethylene is present in much larger concentrations than acetylene. Analysis of these two systems illustrates some important ideas about cyclization of radicals. The major difference in these

10000

5000

60

15000

20000

25000

3000 0 30000

TIME (SEC) Figure 10. Comparison of the production of cyclopentadiene via a conventional vs an energized-complex treatment. Key: (-O-) predicted using Table I mechanism (includes energized-complex routes); (-0-) predicted using Table I mechanism where direct routes to cyclic products from allyl addition to acetylene were omitted.

P

I

T

500.0

1000.0

1500.0

2000.0

2500.0

3000.0

TIME (SEC) Figure 11. Comparison of the production of ethane via a conventional vs an energized-complex treatment. Key: (-O-) predicted using Table I mechanism (includes energized-complex routes); (-e) predicted using Table I mechanism where direct routes to cyclic products from allyl addition to acetylene were omitted.

C=Cc' + C = C

C=CCCC*'

0" -

$

+

H

ioi C.CC.C'

?

130

g

.

i Y

-

110

-

A

+

c c I

+?

c.cc.ccsc~' p

I

I !

0"-0+ H

i

9i C=CC=CC.C'

90 -

70 -

50

-

Figure 12. Comparison of the potential energy diagrams for (a) addition

of allyl radical to ethylene and (b) addition of butadienyl radical to acetylene. The differences in these diagrams account for the markedly different behavior of these two addition reactions. cyclizations is the energetics. The potential energy diagram for allyl addition to ethylene is shown in Figure 12a, and can be compared to that for addition to acetylene in Figure 9. Note the first difference is the depth of the initial well; it is deeper for

1438 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990

Dean

TABLE 11. Sensitivity Information' no.

reactionb

I

CH4 = CH3 + H H + C=CC = CCC' CH, + CY13PD = CH4 + CY13PD5' C2H6 = 2CH3 C2H3 = C2H2 + H CY 13PD = CY 13PDS' + H C=CC' + C2H2 = CYPENE4'

26 24 2

13 7

38 1 19

24 26 13 7 38 31 11

39 1

24 19 26 2 7 13 38 31 39 1

13 26 19 24 2 7 31 28 34 1

26 24 2

19 31 13 7

38 5 1

26 24 2

13 7 38 31 19

39

60 I

2400

CH, -0.002

-0.008 -0.002 -0.002 0.002 -0.002 -0.001

-0.001

CHI = CH, + H C2H6 + CH3 = C2H5 + CH4 CH3 + CY13PD = CH, + CY13PD5' H + C=CC = CCC' C2H3 = C2H2 + H CY13PD = CY13PD5' + H C=CC' + C2H2 CYPENE4' CHp + C2H4 = C=CC + H 2CY13PD5' NAPH + H2 C=CC' + C2H2 CY13PD + H

C2H6

CHI = CH3 + H CH, + CY13PD = CH4 + CY13PD5' C2H6 + CH3 = C2H5 + CH4 H + C=CC = CCC' C2H6 = 2CH3 CY 13PD = CY 13PDS' + H C2H3 = C2H2 + H C=CC' + C2H2 = CYPENE4' CH3 + C2H4 = C=CC + H C=CC' + C2H2 = CY 13PD + H

C2H4

0.701 -0.592 0.006 0.001

0.003

0.797 -0.677 0.025 0.069 0.040 0.032 0.026 0.030

1.49 -0.605 0.245 0.321 0.249 0.210 0.177 0.140 -0.035 0.097

1.70 -0.592 0.632

1S O

0.134 0.252 0.07 1 -0.131 0.129 0.060 0.108 0.031 0.059

2.27 0.560 0.300 0.406 -0.384 0.381 0.330 0.308 0.177 0.169

0.080 -0.074 0.107

1.61 0.559 0.286 0.129 0.102 -0.268 0.092 0.125 -0.114 0.100

1.86 0.682 0.439 0.173 0.407 -0.398 0.259 0.191 -0.138 0.106

1.48 0.572 0.012 -0.202 0.164 0.249 0.024 0.013 0.006 -0.140

I .75 0.659 0.137 -0.295 0.140 0.287 0.137 0.115 0.08 1 -0.137

2.23 0.886 0.540 -0.498 0.210 0.386 0.340 0.317 0.229 -0.095

1.18 0.076 0.01 1 -0.068 0.043 0.015 0.013 0.033

1.69 0.272 0.141 -0.216 0.198 0.138 0.118 0.119 0.124 0.065

2.52 0.639 0.598 -0.521 0.491 0.398 0.328 0.278 0.188 0.180

0.014 1.17

-0.029

1.08 0.009 0.253 -0.065 -0.028 0.012 -0.049

-0.020

-0.028

0.439 -0.045 -0.078

0.0 10 0.005

C2H2

CH4 = CH3 + H C2H3 = C2H2 + H H + C 4 C = CCC' C2H6 + CH3 C2H5 + CH4 CH3 + CY13PD = CH4 + CYl3PD5' C2H6 = 2CH3 CY 13PD = CY 13PD5' + H CHI + C2H4 = C 4 C + H H + C=CC = CC=C' CHg + C2H2 = C=CC' CHI = CHg + H H + C=CC = CCC' CHI + CY 13PD = CHa + CY 13PD5' C2!& = ~ C H , C2H6 + CH, = C2H5 + CH4 CHI + C,Hd = C=CC + H C2H3= d2H2+ H CY13PD = CY13PD5'+ €I C=CC' + C2H2 = CYPENE4' C=CC = CzH3 + CHj

reaction times. s 1204 1839

1.69 0.678 0.115

0.420 -0.324

1.52 0.561 0.183 0.177 0.009 -0.227 0.01 1

0.050

-0.037 0.086

c=cc

CH4 = CH, + H H + C=CC = CCC' CH3 + CY 13PD CH4 + CY l3PD5' C2Hn = 2CH3 C;H; = C2H; + H CY13PD = CY13PD5' + H C=CC' + C2H2 = CYPENE4' CH3 + C2H4 = C=CC + H C2H6 + CHj = C2H5 + CH4 C=CC' + C2H2 = CY13PD + H

1.63 0.607

-0.289 0.410 0.264 0.006 -0.087 H2

1.1 1

0.021 -0.05 1 0.010

0.009 0.172

0.1 17 0.007

0.500

0.390 0.357 0.276 0.2 18 -0.212 0.151

'Each entry is the normalized sensitivity coefficient, A i / X i (&Yi/&4,),. (Blank entries mean that the absolute value was less than 0.0005.) bSee Table I for a list of abbreviations. addition to acetylene by about 8 kcal/mol. Furthermore, the difference in energy between the linear and cyclic intermediates is greater in the acetylene case by about 9 kcal/mol. Most of this difference can be attributed to the fact that C=CCC=C' is a relatively high-energy vinyllic radical whereas C==CCCC' is not.

These differences in energetics continue with respect to formation of final products; addition to acetylene is overall exothermic while addition to ethylene is slightly endothermic. Each of these differences contribute to enhancement in rate for formation of cyclic products for addition to acetylene. The shallower well for ethylene

The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1439

Kinetic Modeling of Autocatalysis in Methane Pyrolysis addition will lead to more stabilization of the linear adduct since the barrier to cyclic species is comparable to the entrance channel; for acetylene it is 11 kcal lower. Similarly, the difference in energy between linear and cyclic intermediates means that the equilibrium constant for the cyclization involving C=CCC=C' is much more favorable, by a factor of 60 at 1038 K, than for the cyclization of C=CCCC'. Thus, we expect cyclization, whether it occurs via stabilized or energized adducts, to be much more favored for the case of C=CCC=C'; under the present conditions, this difference is enough to make contributions from allyl addition to ethylene unimportant. In general, the shallower wells in the ethylene system, coupled with the greater entropy of the reactants with respect to the intermediates, favor redissociation of the adducts back to reactants, thus making this an inefficient channel for molecular weight growth. These observations can be generalized to other addition reactions. The following factors will increase the probability that direct production of cyclic species can occur via an energized-complex mechanism: (1) a deep well for the linear adduct-this will tend to make the barrier to cyclization lower than the entrance barrier, thus increasing its unimolecular rate; (2) an increase in the exothermicity of the cyclization reaction-this is needed to offset the entropy loss upon cyclization; and (3) a final cyclic product with high stability-this will result in low-energy exit channels relative to the entrance and will lead to faster unimolecular rates. One such system where these factors are especially significant is the formation of benzene via the following sequence: C=CC=C*

+ C2H2eC=CC=CC=C*'

Z

0" 0 ===

0

+ ti*

The potential energy diagram is shown in Figure 12b. Here, the linear adduct is seen to have a much deeper well than that resulting from the allylic addition shown in Figure 12a. There are two reasons for this: (1) addition to acetylene is generally more exothermic than to ethylene; (2) the adding radical here is vinyllic, as contrasted to the resonantly stabilized allyl; thus there is no loss of resonance upon addition as is the case with allyllic species. The cyclization is also much more favored in the benzene system since the combination of an unstable vinyllic linear radical and a very stable cyclic radical results in an exothermicity of 43 kcal/mol for the cyclization. Furthermore, note that the final exit channel for benzene production, due to its unusual stability as an aromatic molecule, is much lower than the entrance. As a result, this path to benzene can be very important. In fact, at 1 atm and 1200 K, the rate of production of benzene via the direct reaction of the energized complex accounts for over 90% of the total reaction of the initially formed linear adduct. Here, the overall exothermicity is sufficient to compensate for the loss in entropy upon cyclization. Such compensation is particularly important at high temperatures where the TPS term plays a larger role. Concluding Remarks. This description of methane pyrolysis illustrates that even this seemingly simple system, where one works

at extremely small extents of reaction, is in fact extremely complicated. There appear to be three reasons for the success of the present mechanism: (1) It was possible to use a quick but accurate method to estimate apparent rate constants for addition and recombinations reactions, thus making assembly of the large initial mechanism feasible. (2) By starting with a very large mechanism and then rigorously pruning via sensitivity analysis, it was possible to identify important reactions of large species such as cyclopentadiene that were not considered by earlier workers, who simply never considered that the mechanism could be so complicated. (3) The mechanism included the possibility of unimolecular reactions of energized adducts arising from radical addition reactions prior to their collisional stabilization; these reactions substantially increased the rate of production of cyclopentadiene, which then serves as a new chain-branching center. It appears that chemically activated reactions like those identified in this analysis might be expected to play a major role in a variety of systems. Given the importance of these reactions in this system at 1038 K, one expects an even greater role at higher temperatures where the unimolecular rates of the initially formed adducts will be even faster. One way to estimate the potential impact is to observe the shift in falloff curves for radical recombination reactions with temperature. This behavior directly tracks the competition between unimolecular reaction and bimolecular stabilization, and we see dramatic effects here. For example, increasing the temperature from 900 to 1350 K for methyl radical recombination shifts the falloff curve by a factor of 50 toward higher pressures-an increase from 1350 to 2200 K shifts it another factor of 60.16 Thus, it is to be expected that one will need to account for the chemically activated reaction channels in some circumstances even at pressures of many atmospheres. It is also important to note that these effects are not limited to small molecules. Although one associates a slower unimolecular rate with an increasing number of vibrational modes since there is less likelihood of localizing the critical energy in the reaction coordinate, this is only to be expected at a fixed energy. In chemically activated systems, the energy in the complex is composed of both the energy released when the new bond is formed and the internal energy of the reactants. As the reactants become larger, their internal energy necessarily increases, and thus, the energy in the complex will increase. As a result, one finds that the possibility of unimolecular chemistry is not particularly sensitive to molecular complexity. All of these factors suggest that it is important to explicitly consider the possible role of these types of reactions in virtually any gas-phase system at high temperatures.

Acknowledgment. I have benefited from many stimulating discussions on various aspects of molecular growth with J. W. Bozzelli, P. R. Westmoreland, and R. L. Woodin. Registry No. CH,, 74-82-8; C2Hz,74-84-0 cyclopentadiene, 542-92-7. (16) Warnatz, J. In Combustion Chemistry; Gardiner, W. C., Jr., Ed.; Springer-Verlag: Berlin, 1984; pp 197-360.