Thermal Decomposition of Ethanol: 1 - Chemistry - Emory University

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Thermal Decomposition of Ethanol: 1 - Chemistry - Emory University

eaking C-H, O-H and the forming H-H bond lengths are 2.193, 1.003 and 1.651 Å,respectively, which are significantly different from our values. The reason for the differencecould be attributed to the larger basis set with the d and p polarization functions included in ourcalculation. The dissociation barrier height at the G2M level is 86.1 kcal/mol, which is muchmore reasonable than the value, 112.1 kcal/mol, obtained at the CI(QC)/4-31G level. 13CH 2 CHOH + H 2 . In this channel, two H atoms, each from the CH 3 and CH 2 groups inCH 3 H 2 OH, form a HHCC four-member-ring transition state TS3, resulting in H 2 elimination toproduce H 2 and CH 2 CHOH. The cleaving C-H bonds corresponding to CH 3 and CH 2 groups inTS3 are 0.343 and 0.643 Å longer than those in C 2 H 5 OH; the C-C bond length, 1.443 Å, isbetween 1.517 Å in C 2 H 5 OH and 1.328 Å in the product CH 2 CHOH and the C-O bond length1.353 Å is close to the value 1.367 Å in CH 2 CHOH. This H 2 -elimination process has a largebarrier, 106.3 kcal/mol, which is again much lower than 168.5 kcal/mol reported in ref. 13.Comparing the two H 2 -elimination processes in channels (3) and (4), we see that TS3 is about 20kcal/mol higher than TS2. The difference may be ascribed to the fact that a strong stereorepulsionforce from the CH 3 group exists in TS3 during the H 2 -forming process and a largerenergy is released from the forming C=O bond (comparing with the C=C bond) in TS2.CH 4 + CH 2 O. This path involves the abstraction of the H atom from the OH group byCH 3 producing CH 4 and CH 2 O via TS4. In this process, with the C-C bond lengthening, theHOCH 2 moiety in CH 3 CH 2 OH undergoes a large rotation leading to the HOCC dihedral anglechange from 180.0° in CH 3 CH 2 OH to -15.0° in TS4. TS4 has a loose structure, compared withCH 3 CH 2 OH; the cleaving C-C bond lengthens by 1.179 Å, but the breaking O-H bond onlylengthens by 0.075 Å. The forming C-H bond is 1.910 Å. The result of an IRC 20 calculationshows that TS4 connects CH 4 + CH 2 O and CH 3 CH 2 OH. The barrier for CH 4 elimination fromCH 3 CH 2 OH is 99.7 kcal/mol.CH 4 + CHOH. As shown in Fig. 1, one of the two H atoms of the CH 2 group inCH 3 CH 2 OH can migrate toward the C atom in the CH 3 group to produce CH 4 + CHOH via athree-center transition state, TS5. Surprisingly, the barrier of this channel is 14 kcal/mol lowerthan that of the analogous channel mentioned above. The stronger O-H bond (c.f. with thesecondary C-H bond) and the large rotation of the HOCH 2 moiety required in TS4 may beresponsible for its higher barrier.CH 3 CH + H 2 O. In channel 1, C 2 H 5 OH eliminates H 2 O via a four-member-ringtransition state, in which one of the H atoms in the elimination process comes from the CH 3group. In this channel the H 2 O-elimination process involves the OH group and one of the Hatoms in the CH 2 group via a three-member-ring transition state. The transition state, TS6, isshown in Fig. 1. The breaking C-H and C-O bonds in TS6 are 0.351 and 0.532 Å longer thanthose in C 2 H 5 OH and the forming O-H bond, 1.065 Å, is close to that of 0.97 Å in H 2 O. Thedissociation barrier height for this process is 82.9 kcal/mol. Comparing with the analogouschannel (1), this process is more endothermic because of the higher heat of formation of s-CH 3 CH.In principle, CH 3 CH may be formed in the ground electronic triplet state by surfacecrossing.Because of the required spin change, the small energy difference between the singletand triplet states, 3.0 kcal/mol (see Fig. 2), and the tight 3-centered transition state, contributionfrom this product channel giving t-CH 3 CH + H 2 O, is expected to be small.Theoretically, several investigators 27-31 have reported the singlet-triplet splitting ofCH 3 CH. The singlet-triplet energy difference, ∆E ST = 3. 0 kcal/mol predicted at the G2M levelof theory, is much closer to the value, 3 ± 2.0 kcal/mol predicted by Khodabandeh and Carter 275


at the GVB-CI/TZ2p and MCSCF/TZ2P levels. Our value is also in reasonable agreement withthose, 5.6 30 and 5.2 31 kcal/mol, obtained by correlated electron pair approximation (CEPA)theory with a near-TZ2p-quality basis set and at the HF*SD(CI)/TZ2pf level, respectively.However, it is lower than those of 10.7 28 and 10.0 kcal/mol 29 predicted at the MP4//UHF/6-31G(d) and HF*SD-CI/DZp levels, respectively. The larger difference may simply be due to thesmall basis sets used in their work. The reliability of the singlet-triplet splitting energy obtainedby the G2M method can be confirmed by comparison of the experimental and calculated resultsfor the CH 2 radical. For CH 2 , the ∆E ST value calculated by the G2M method, 9.6 kcal/mol, 32 isclose to the experimental value, 9.0 kcal/mol 33 and the most recent value, 9.9 kcal/mol 34calculated at the [6e, 12o]MRMP(SD)/cc-pVTZ level. Our G2M results for CH 2 and CH 3 CHindicate that substitution of one of the H atoms in CH 2 by a CH 3 group reduces the singlet-tripletsplitting energy by 6.6 kcal/mol, the predicted reduction is close to 6.0 kcal/mol reported byKhodabandeh and Carter. 27 Interestingly, if the two H atoms in CH 2 are replaced by two CH 3groups, the singlet-triplet energy separation of dimethylcarbene was found to be 5.3 kcal/mol 35 infavor of the singlet state.CH 3 CH 2 + OH. Similar to channel (2), the reaction occurs without a well-definedtransition state. The CH 3 CH 2 + OH association potential function was computed variationally tocover the range of C-O from 1.427 to 4.0 Å; the computed potential energies could be fitted tothe Morse function with b = 2.06 Å -1 . The predicted dissociation energy, 94.8 kcal/mol, is 3kcal/mol higher than the value, 91.8 kcal/mol, estimated by Marinov. 32. Rate constant calculationsThe rate constants for the unimolecular decomposition reactions (1) – (8) have beencomputed with the Variflex code of Klippenstein et al. 24 The Lennard-Jones parameters forCH 3 CH 2 OH and Ar employed in the calculation, σ = 4.317 and 3.465 Å and ε/κ = 450.2 and113.5 K, respectively, were taken from ref. 36. Energy transfer per downward collision,, was assumed to be 400 cm -1 . The energies given in Fig. 2 and the moments of inertiaand vibrational frequencies presented in Table 1 were used.The predicted results for the temperature range 700 – 2500 K at 1 atm (Ar) and the highpressurelimit are presented graphically in Fig. 3. In Fig. 4, we compare the predicted highpressurerate constants for k 1 and k 2 with literlature values as well as with the result of ourpreliminary analysis of experimental data obtained by shock-tube pyrolysis/CO-laser absorptionspectrometry and static-cell pyrolysis/FTIR spectrometry. 11 The individual rate constantexpressions predicted at various pressures, including the values at the high–pressure limit, aresummarized in Table 2.The results summarized in Fig. 3 and Table 2 clearly show that product yields for thevarious channels are strongly influenced by pressure and temperature of the system. Below 10atm, channel (1) producing C 2 H 4 + H 2 O is dominant in the whole temperature range of 700 –2500 K. At the high-pressure limit over 1500 K, however, the formation of CH 3 + CH 2 OH bychannel (2) becomes dominant (see Fig. 3(a)). Notably, the predicted values for these twochannels, as shown in Fig. 4, agree closely with the earlier reported data. In addition, at hightemperature and high pressure, the formation of CH 3 CH 2 + OH products also becomescompetitive with the two key channels. It should therefore be included in the modeling ofethanol combustion reaction. On the other hand, the H 2 -elimination reactions producingCH 3 CHO and CH 2 CHOH are not competitive because of the high barriers involved (see Fig. 3).6


3. Comparison with the O ( 1 D) + C 2 H 6 Data of Lee et al.As mentioned in the introduction, Lee and co-workers 12 studied the fragmentation of achemically activated ethanol in a cross-molecular beam experiment. They concluded that at theenergy available from the O ( 1 D) + C 2 H 6 reaction, 140.7 kcal/mol above C 2 H 5 OH, the majorproducts detected were CH 3 + CH 2 OH and C 2 H 5 + OH, accounting for about 70% and 25% ofthe total yield, respectively. However, for the latter product pair, a major fraction of the yieldderived from the direct abstraction reaction, according to their angular distribution measurement.Most interestingly, the H 2 O-elimination process producing C 2 H 4 was found to be negligible andthe H- and H 2 -elimination reactions were also reported to be of minor significance, accountingfor only 3% and 2% of the total yield, respectively.It is worthwhile to compare this important result with our predicted relative specificrate constants, k Ei , computed at the energy generated by the O ( 1 D) + C 2 H 6 reaction. Our resultsshow that at the available energy above C 2 H 5 OH (140.7 kcal/mol), the branching ratios formingCH 3 + CH 2 OH (channel 2) and C 2 H 5 + OH (channel 8) amount to 61.4% and 16.4%,respectively, which are in qualitative agreement with 70% and 25% reported by Lee and coworkers.12 The higher value for C 2 H 5 + OH measured in experiment may be attributed to thecontribution from the direct abstraction reaction. 12 However, the branching ratio for H 2 O-elimination producing C 2 H 4 + H 2 O (channel 1) was predicted to be 19%, which was notobserved in the experiment but could be attributed to the low detectivity of C 2 H 4 and H 2 O. 38 Thepredicted H 2 -elimination channels (3 and 4) only account for 0.2%, compared with theexperimental value of 2%. 12 In addition, CH 4 -elimination channels (5 and 6) and channel 7amount to 0.4 and 2.6% of the total rate, respectively, which were also not observed in theexperiment. 12IV. ConclusionThe kinetics and mechanisms for the thermal decomposition of CH 3 CH 2 OH have beeninvestigated by high-level molecular orbital (G2M) and variational RRKM calculations over awide range of reaction conditions. At pressures below 10 atm, the decomposition of CH 3 CH 2 OHoccurs primarily by the dehydration reaction producing C 2 H 4 + H 2 O. At the high-pressure limit,and over 1500 K, however, the production of CH 3 + CH 2 OH becomes dominant and thedecomposition reaction is controlled by chain processes. H 2 -molecular elimination processeswere found to be unimportant through out the temperature range investigated.At the internal energy corresponding to the O( 1 D) + C 2 H 6 reaction, the predicted k Ei ratiosfor CH 3 + CH 2 OH, C 2 H 5 + OH and H 2 + CH 3 CHO/C 2 H 3 OH agree qualitatively with the result ofa recent cross-molecular beam study by Lee and co-workers. 12AcknowledgementsThis work was supported in part to JP by the Basic Energy Sciences, Department ofEnergy under grant No. DE-FG05-91-ER14192 and in part to RSZ and MCL by the Office ofNaval Research under grant No. N00014-J-89-1494.References:1. W. Tsang, Int. J. Chem. Kinet. 1976, 8, 173.2. T. K. Choudhury, M. C. Lin, C.-Y. Lin and W. A. Sanders, Combust. Sci. Technl. 1990, 71,219.3. N. M. Marinov, Int. J. Chem. Kinet. 1999, 31, 183.7


4. C. F. Cullis, E. J. Newitt, Proc Royal Soc London 1956, 237 A, 530; and 1956, 242S, 516.5. G. R. Freeman, Proc Royal Soc London 1960, 245A, 75.6. J. A. Barnard, H. W. D. Hughes, Trans Faraday Soc 1960, 56, 55.7. K. M. Bansal, G. R. Freeman, J. Am. Chem. Soc 1968, 90, 7190.8. J. Brown, C. F. H. Tipper, Proc Royal Soc London 1969, 312A, 399.9. A. A. Borisov, V. M. Zamanskii, A. A. Konnov, V. V. Lisyanskii, S. A. Rusakov, G. I. S.Skachkov, Russ. Chem. Phys 1991, 8, 121; ibid. 1992, 9, 2527.10. J. Li, A. Kazakov, F. L. Dryer, Int. J. Chem. Kinet 33: 859, 2001.11. J. Park, R. Chen, J. Chen, M. C. Lin, "Experimental and Computational Studies of theUnimolecular Decomposition of Ethanol," 2001 Eastern States Section Fall TechnicalMeeting, Hilton Head Island, S.C., Dec. 4-7, 2001.12. J. Shu, J. J. Lin, Y. T. Lee, and X. Yang, J. Chem. Phys. 2001, 115, 849.13. T. Yamabe, M. Koizumi, K. Yamashita, and A. Tachibana, J. Am. Chem. Soc. 1984, 106,2255.14. N. I. Butkovskaya, Y. Zhao, and D. W. Setser, J. Phys. Chem. 1994, 98, 10779.15. A. D. Becke, J. Chem. Phys. 1993, 98, 5648.16. A. D. Becke, J. Chem. Phys. 1992, 96, 2155.17. A. D. Becke, J. Chem. Phys. 1992, 97, 9173.18. Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. 1988, B37, 785.19. W. Hehre, L. Radom, P. V. R. Schleyer, and J. A. Pople, Ab initio Molecular OrbitalTheory (Wiley, New York, 1986).20. C. Gonzalez, H. B. Schlegel, J. Phys. Chem.1989, 90, 2154.21. Mebel, A. M.; Morokuma, K.; Lin, M. C. J. Chem. Phys. 1995, 103, 7414.22. M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M . W. Gill, B. G. Johnson, M. A. Robb, J. R.Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A.Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L.Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J.Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez and J. A. Pople, GAUSSIAN 98,REVISION A.1; Gaussian, Inc., Pittsburgh PA, 1998.23. MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J.Knowles, with contributions from J. Almlöf, R. D. Amos, A. Berning, D. L. Cooper, M. J.O. Deegan, A. J. Dobbyn, F. Eckert, S. T. Elbert, C. Hampel, R. Lindh, A. W. Lloyd, W.Meyer, A. Nicklass, K. Peterson, R. Pitzer, A. J. Stone, P. R. Taylor, M. E. Mura, P. Pulay,M. Schütz, H. Stoll and T. Thorsteinsson.24. S. J. Klippenstein, A. F. Wagner, R. C. Dunbar, D. M. Wardlaw, and S. H.Robertson, VARIFLEX: VERSION 1.00, 1999.25. R. G. Gilbert, S. C. Smith, Theory of Unimolecular and Recombination Reactions,Blackwell Scientific, Carlton, Australia, 1990.26. K. A. Holbrook, M. J. Pilling, and S. H. Robertson, Unimolecular Reactions, Wiley, 1996.27. S. Khodabandeh and E. A. Carter, J. Phys. Chem. 1993, 97, 4360.28. B. T. Luke, J. A. Pople, M. B. Krog-Jesperson, Y. Apeloig, M. Karni, J. Chandrasekhar, P.R. Schleyer, J. Am. Chem. Soc. 1986, 108, 270.29. T. K. Ha, M. T. Nguyen, L. G. Vanquickenborne, Chem. Phys. Lett. 1982, 92, 459.30. H. J. Kohler, H. Lischka, J. Am. Chem. Soc. 1982, 104, 5884.31. M. M. Gallo, H. F. Schaefer III, J. Phys. Chem. 1992, 96, 1515.8


Table 1. Moments of Inertia (I A , I B , I C ) and Vibrational Frequencies of the Species Involvedin the C 2 H 5 OH Decomposition Reaction Computed at the B3LYP/6-311G(d, p) level.Molecule I A , I B , I C (a. u.) Frequencies (cm -1 )CH 3 CH 2 OH 51.53, 192.67,221.58252, 288, 417, 827, 902, 1035, 1107, 1180,1278, 1302, 1406,1461, 1481, 1504, 1531, 2965, 2989, 3034, 3101, 3106, 3839TS1(C 2 H 4 +H 2 O)67.12, 197.09,235.28-2008, 347, 419, 544, 603, 742, 830, 838, 1032, 1125, 1227,1234, 1414, 1461, 1521, 1612, 3118, 3129, 3196, 3225, 3741TS2(CH 3 CHO+H 2 )51.19, 195.61,215.36-2208, 252, 438, 616, 816, 892, 966, 1039, 1153, 1254, 1368,1387, 1412, 1467, 1483, 1898, 2175, 2926, 3028, 3098, 3137TS3(CH 2 CHOH+H 2 )50.05, 193.89,227.57-1795, 217, 479, 548, 606, 922, 980, 1000, 1066, 1180, 1225,1350, 1397, 1427, 1516, 1585, 1815, 2962, 3180, 3220, 3776TS4(CH 4 +CH 2 O)67.50, 278.58,329.84-1010, 97, 227, 300, 401, 535, 818, 898, 1011, 1117, 1279,1389, 1418, 1451, 1550, 2562, 2988, 3086, 3129, 3131, 3303TS5(CH 4 +CHOH)55.75, 277.87,307.65-1142, 85, 236, 428, 627, 694, 772, 1001, 1134, 1164, 1211,1365, 1437, 1445, 1486, 2216, 3010, 3050, 3163, 3185, 3695TS6(CH 3 CH+H 2 O)57.34, 264.12,298.42-558, 209, 230, 406, 438, 696, 815, 939, 1019, 1120, 1271,1363, 1430, 1501, 1562, 2334, 2965, 3014, 3027, 3094, 3810CH 2 CHOH 28.08, 173.08,201.16262, 483, 717, 845, 957, 978, 1145, 1292, 1349, 1438, 1733,3133, 3160, 3252, 3869CH 2 OH 9.37, 60.43, 69.10 439, 593, 1061, 1207, 1367, 1488, 3111, 3255, 3841CH 3 6.30, 6.30, 12.60 505, 1403, 1403, 3104, 3283, 3283H 2 O 2.25, 4.13, 6.38 1638, 3811, 3909H 2C 2 H 4 12.25, 59.67, 71.92 835, 973, 973, 1066, 1239, 1380, 1472, 1692, 3122, 3137,3193, 3221CH 3 CHO 31.55, 178.11,198.53160, 508, 777, 883, 1126, 1135, 1375, 1426, 1460, 1471,1825, 2856, 3022, 3077, 3136CH 3 CH (singlet) 15.02, 69.11, 73.39 478, 614, 960, 1119, 1266, 1304, 1350, 1513, 2830, 2902,2987, 3079CH 3 CH (triplet) 13.02, 73.51, 75.21 188, 761, 994, 1071, 1101, 1389, 1453, 1454, 2943, 2978,3045, 3206CH 3 CH 2 17.40, 79.40, 85.68 113, 476, 814, 980, 1063, 1192, 1401, 1465, 1483,1483,2942, 3033, 3076, 3138, 3239CH 4 11.42, 11.42, 11.42 1341, 1342, 1342, 1560, 1560, 3026, 3131, 3131, 3132CH 2 O 6.34, 46.23, 52.57 1202, 1270, 1539, 1826, 2869, 2918CHOH 6.46, 49.52, 55.97 1022, 1210, 1336, 1478, 2720, 352010


Table 2. Equations for Individual Rate Constants in Units of s -1 Predicted for Different Pressures1×10 -8 atm 1 atm 10 atm infinite pressurek 1 2.19×10 50 T -13.32 exp(-34310/T) 2.22×10 38 T -7.56 exp(-38450/T) 7.32×10 23 T -3.14 exp(-35452/T) 6.99×10 13 exp(-34190/T)k 2 2.90×10 46 T -14.73 exp(-48520/T) 4.46×10 66 T -15.18 exp(-53930/T) 1.12×10 50 T -9.95 exp(-51162/T) 3.68×10 26 T -2.95 exp(-45640/T)k 3 1.82×10 41 T -13.08 exp(-33790/T) 1.18×10 49 T -11.30 exp(-48350/T) 3.32×10 27 T -4.57 exp(-44427/T) 2.47×10 13 exp(-43370/T)k 4 1.44×10 34 T -14.88 exp(-37300/T) 1.28×10 69 T -18.07 exp(-61530/T) 6.22×10 31 T -6.24 exp(-53416/T) 5.62×10 13 exp(-54190/T)k 5 2.23×10 21 T -10.36 exp(-48100/T) 2.26×10 79 T -20.29 exp(-63800/T) 4.47×10 38 T -7.54 exp(-53978/T) 1.76×10 15 exp(-51720/T)k 6 2.15×10 20 T -8.14 exp(-34940/T) 2.62×10 61 T -14.32 exp(-52860/T) 2.76×10 41 T -8.03 exp(-49438/T) 6.10×10 14 exp(-43670/T)k 7 1.91×10 44 T -14.58 exp(-46000/T) 1.20×10 61 T -14.24 exp(-52540/T) 7.64×10 41 T -8.19 exp(-49277/T) 4.04×10 14 exp(-43200/T)k 8 5.55×10 42 T -14.80 exp(-52500/T) 6.32×10 71 T -16.98 exp(-58290/T) 1.54×10 54 T -11.19 exp(-56009/T) 3.62×10 27 T -3.15 exp(-50350/T)11


Figures captionsFig.1. The optimized geometries of the reactants, transition states and products computed at theB3LYP/6-311(d, p) level.Fig.2. Schematic energy diagram for the dissociation of CH 3 CH 2 OH computed at theG2M(RCC2) level.Fig.3. Predicted rate constants for the dissociation reactions (1) - (8) at infinite pressure (a) and 1atm (b). The numbers in the figure correspond to each reaction channel given in theintroduction.Fig. 4. Comparison of the predicted and experimental high-pressure limit decomposition rateconstants for (a) CH 3 CH 2 OH Ø H 2 O + C 2 H 4 , (b) CH 3 CH 2 OH Ø CH 3 + CH 2 OH.In (a), solid line, predicted high pressure rate constant; open circle, shock-tube data (ref.11); solid circle, static-cell data (ref. 11); 1, ref. 14; 2, ref. 6. In (b), solid line, predictedhigh pressure rate constant; open circle, shock tube experimental data (ref. 11); 1, ref. 37;2, ref. 1; 3, ref. 14; 4, ref. 312


1.254 OOO 1.380112.51.3261.35357.3 1.0021.4431.433 C C1.860C C74.4 1.511 1.469C1.420C1.4361.7431.006TS1(C1) TS2(C1) TS3(C1)1.036O0.9661.910 97.91.2911.099O2.048 103.640.54C 2.260 104.9 OCCC C C 29.6 1.0652.69620.71.45156.561.3821.2221.483TS4(C1) TS5(C1) TS6(C1)1.093C107.71.517 C1.1000.961O1.4271.0951.1051.083C C C1.4871.092 1.474C1.112C 2 H 5 OH (Cs) CH 3 CH 2 (Cs) CH 3 CH (Cs)122.5CC1.3281.081O 0.9601.3671.0881.085119.0 122.2109.1C0.9621.367 OC1.2001.0811.109OO1.090124.8110.51.204C C1.5061.095 1.11CH 2 CHOH (Cs) CH 2 OH (Cs) CH 2 O (C2v) CH 3 CHO (Cs)103.9Fig. 1121.8 1.085C C1.327C 2 H 2 (D2h)120.0C1.080CH 3 (D3h)0.962OH 2 O (C2v)1.125 107.0 116.0 0.977C O1.307CHOH (Cs)CCH 4 (Td)1.09113


TS3 (106.3)CH 3 CH + H 2 O(S. 80.3)CH 3 CH + H 2 O(T. 77.3)CH 3 CH 2 + OH (94.8)CH 3 + CH 2 OH (87.5)TS6vdw (82.9)(81.3)TS4 (99.7)TS2 (86.0)TS1(66.6)TS5 (84.3)CH 4 + CHOH(65.8)C 2 H 5 OH (0.0)CH 2 CHOH + H 2 (26.9)CH 3 CHO + H 2 (14.7)CH 4 + CH 2 O (9.0)C 2 H 4 + H 2 O (6.5)Fig. 214


2010(a)038ln(k; s -1 )-10-20-30-40-5012675410087(b)ln(k; s -1 )-10-20-303126-4054-500.4 0.6 0.8 1.0 1.2 1.4 1.61000/TFig. 315


2015(a)105ln (k 1)0-512-10-1515(b)10ln (k 2)50-5-10-151-200.4 0.6 0.8 1.0 1.2 1.421000/T34Fig. 416

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