J . Phys. Chem. 1987, 91, 3024-3030
3024
Unimdgcular Dlssoclation of Cyclohexene at Extremely High Temperatures: Behavior of the Energy-Transfer Collision Efficiency John H. Kiefer* and Jatin N. Shah Department of Chemical Engineering, University of Illinois at Chicago, Chicago, Illinois 60680 (Received: November 25, 1986; In Final Form: February 5, 1987)
The dissociation of cyclohexene has been observed in shock waves with the laser schlieren and pulsed laser flash absorption techniques over 1200-2000 K and 110-550 Torr, using 2% and 4% cyclohexene in krypton and 0.2% cyclohexene in argon. The inverse Diels-Alder molecular elimination to 1,3-butadiene and ethylene is clearly the dominant dissociation channel under all conditions. Rate constants derived for this reaction have a high precision, with rms deviations of only a few percent. Unimolecular falloff, although slight at the lowest temperatures, is clearly discernible for all temperatures. A simple RRKM calculation, using a nonspecific vibration model transition state, is fit to the measured rate constants, giving log k , = 15.57 - 65.7 (kcal/mo1)/2.303RTwith a barrier of 61.9 kcal/mol. These RRKM calculationswere performed with the energy-transfer collision efficiency derived from both fixed (pE)d, and ( pE)all.A good fit is achieved with either below 1500 K, but above this a fixed gives rates which are much too small, and this deficiency cannot be compensated by other changes, whereas a fixed (AE),IIprovides quite satisfactory agreement with the measurements for all conditions. This situation is similar to that indicated by other large molecule dissociations at extreme temperatures.
Introduction
TABLE I: Previous Rate Determinations for Cyclohexene
At least for moderate temperatures, below about 1300 K, dissociation of cyclohexene is evidently confined to the simple molecular process
Dissociation
0-6
+II
giving 1,3-butadiene and ethylene.' This reaction is thus the reverse of the familiar Diels-Alder cycloaddition. Recent studies of reaction 1 include a flow tube product analysis2 and several single-pulse shock tube investigations using both ~ o m p a r a t i v e ~and - ~ absolute6 rate techniques. Most recent is a shock tube study in which the reaction was followed with time-resolved UV absorption of the product b ~ t a d i e n e . ~Below 1200 K all these determinations are in fair agreement on both rate and Arrhenius parameters, as shown in Table I. It is only at higher temperatures that some anomalies begin to appear (see below). In this paper we report a rather extensive shock tube study of cyclohexene dissociation covering 1200-2000 K, using both the laser schlierena (LS) and pulsed laser flash absorption9 (PLFA) techniques. There are several reasons for this additional study. First, although reasonable behavior was reported therein for lower temperatures, in two instances6q7a severe "drop-off" of the unimolecular rate (a sudden and precipitous drop in apparent activation energy) occurred at higher temperatures. In the absolute rate, single-pulse measurements of Barnard and Parrott6 this drop-off begins at about 1200 K; by 1330 K the observed rate lies nearly an order of magnitude below an extrapolation of the (1) Tsang, W . In Shock Waves in Chemistry; Lifshitz, A,, Ed.; Marcel Dekker: New York, 1981; p 59. (2) Uchiyama, M.; Tomioka, T.; Amano, A. J. Phys. Chem. 1964, 68,
1878. ( 3 ) Tsang, W. J. Chem. Phys. 1965, 42, 1805. (4) Tsang, W. Int. J. Chem. Kinet. 1970, 2, 311. (5) Tsang,W. Int. J . Chem. Kine?. 1973, 5, 651. (6) Barnard, J. A,; Parrott, T. K. J. Chem. SOC.,Faraday Trans. 1 1976, 72, 2404. (7) Hidaka, Y.; Chimori, T.; Shiba, S.; Suga, M.Chem. Phys. Lett. 1984, 1 1 1 , 181. (8) Kiefer, J. H. In Shock Waves in Chemistry; Lifshitz, A,, Ed.; Marcel Dekker: New York, 1981; p 219. (9) Kiefcr, J. H.; Sitasz, J. S.; Manson, A. C.; Wei, H. C. In Proceedings of the 15th International Symposium on Shock Waves and Shock Tubes, Berkeley, CA, 1985; p 819.
0022-3654/87/2091-3024$01 .50/0
method
flow SP shock SP shock SP shock SP shock UV shock
temp range,
log A, s-]
Ea, kcal/mol
8 14-902 900-1050 950-1100 1000-1170 1009-1330 1120-1490
15.16 14.98 15.30 15.15 15.16 15.18
66.2 66.1 66.9 66.1 65.5 66.9
K
ref 2 3 4 5 6
I
low-temperature data. This may be just another example of the well-known difficulties with absolute rate, single-pulse rate determinations for large conversions of reactant,I0 but the similar drop-off above 1300 K in the time-resolved measurements of Hidaka et al.7 cannot be so dismissed. A change in mechanism at high temperatures seems unlikely, but the possibility should be investigated. This study is also motivated by the simplicity of the molecular dissociation. If reaction 1 is indeed the only significant dissociation channel even at very high temperature, the stability of the products then implies a negligible interference from secondary reactions. Eventually, the butadiene will dissociate to radicals at an appreciable rate," but the very early reaction, Le., that seen by the LS technique, will still arise from reaction 1 alone. With but one operative reaction whose LW is solidly known, the usual high precision of LS measurements should reflect a corresponding accuracy. Such accuracy can illuminate some aspects of the LS measurements and their analysis, in particular the omnipresent problem of time origin location.* Accurate rates at very high temperatures will also exhibit a well-defined unimolecular falloff, even for such a large molecule. It is just this combination of large molecule and high temperature for which the falloff is most sensitive to the behavior of the energy-transfer collision efficiency introduced into an RRKM m ~ d e l . ' ~ . ' ~ Experimental Section
The shock tube has been fully described.I4 All measurements were made on incident shock waves produced by spontaneous bursts of Mylar diaphragms with helium. Frozen (no reaction) (10) Skinner, G. B. In?. J . Chem. Kinet. 1977, 9 , 863. ( 1 1) Kiefer, J. H.; Wei, H. C.; Kern, R. D.; Wu, C. H. Int. J . Chem. Kinet. 1985, 17, 225. (12) Troe, J. Ber. Bunsenges. Phys. Chem. 1974, 78, 478. (13) Tardy, D. C.; Rabinovitch, B. S . J. Phys. Chem. 1985, 89, 2442. (14) Kiefer, J. H.; Manson, A. C. Rev. Sci. Insrrum. 1981, 52, 1392.
0 1987 American Chemical Society
Unimolecular Dissociation of Cyclohexene
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3025
TABLE II: Initial and Frozen Conditions for the LS Experiments shock PI, U, PZ? T2r no. mm/ps Torr K Torr 2 3 4 5 6 7 8 9 10 11 13 14 15 16 17 19 19 20 21 22 23 25 26 27 28 29 30 31 32 33 34
15.72 18.06 28.16 30.97 26.99 8.07 4.99 7.1 1 25.20 21.44 15.51 14.31 19.45 6.05 5.44 4.02 4.80 4.58 7.94 8.60 5.05 4.04 4.85 3.71 4.25 6.19 7.51 5.95 6.58 5.54 4.78
2.0% c-CnHln-Kr 0.963’0.940 0.844 0.817 0.840 0.888 1.020 0.928 0.831 0.897 0.968 0.976 0.929 0.918 0.961 1.043 1.027 1.042 0.894 0.858 0.970 1.045 0.964 1.084 1.034 0.894 0.857 0.936 0.906 0.953 1.004
394 430 538 554 510 172 141 165 467 464 394 369 454 138 136 119 138 135 171 170 128 120 122 119 123 133 148 141 146 136 131
1736 1667 1410 1342 1398 1525 1905 1635 1376 1549 1748 1771 1637 1607 1727 1973 1926 1970 1540 1445 1755 1981 1736 2105 1945 1540 1443 1657 1574 1706 1855
35 36 37 38 39 40 41 42 43 44 45 46 47 49 50 51 52 53 54 55 56 57 58 59 61 62 63 64 65 66 68 69 70 71 72 73 74 75
25.27 26.17 26.04 28.36 21.94 16.67 4.50 5.00 4.01 3.75 3.56 3.25 3.00 23.93 22.93 23.40 17.66 15.18 14.28 13.78 4.76 4.50 4.00 3.31 3.00 5.12 4.51 7.63 7.40 7.46 6.39 6.07 5.90 6.27 6.32 22.25 21.61 21.09
4.0% c-C6Hl0-Kr 0.820 475 0.824 496 483 0.815 520 0.811 159 0.865 0.924 400 125 0.992 0.946 126 119 1.027 1.021 110 112 1.053 113 1.108 1.133 109 0.832 462 463 0.850 457 0.837 390 0.888 0.948 383 369 0.957 367 0.970 117 0.935 0.984 123 117 1.015 107 1.067 1.157 114 137 0.975 129 1.004 152 0.844 148 0.846 151 0.852 0.869 135 138 0.901 133 0.898 0.882 136 137 0.881 440 0.842 444 0.858 0.885 430
1220 1227 1209 1200 1315 1449 1612 1501 1699 1686 1766 1913 1982 1244 1283 1255 1368 1506 1527 1559 1474 1592 1670 1803 2050 1571 1643 1272 1275 1288 1326 1397 1390 1354 1353 1267 1302 1296
conditions were calculated assuming complete relaxation of the cyclohexene with thermodynamic properties for this taken from the API tablesI5 and extended to higher temperatures by using
TABLE III: Initial and Frozen Conditions for the PLFA Experiments with 0.2%c-C&,,-Ar
u,
shock no.
Torr
mm/p
Torr
T2, K
3 4 5 6 7 8 9 12 13 14 15 17 18 19
30.13 29.08 27.14 24.74 21.05 19.22 12.70 23.57 24.98 21.99 20.21 22.12 19.54 17.87
1.060 1.061 1.070 1.058 1.146 1.165 1.079 1.126 1.098 1.143 1.043 1.113 1.189 1.195
406 393 372 332 333 314 177 359 362 345 394 343 333 308
1256 1257 1273 1252 1424 1463 1292 1383 1328 1417 1223 1400 1513 1526
p1I
p29
the frequencies of Net0 et a1.16 Initial and calculated frozen conditions for all the experiments are shown in Tables I1 and 111. The cyclohexene was Aldrich research grade. A G C analysis of the liquid showed less than 0.05% impurities. This liquid was vacuum degassed and distilled, and a middle vapor fraction retained. Mixtures of 2% and 4% cyclohexene in krypton and 0.2% cyclohexene in argon were prepared manometricalIy in a 50-L glass vessel and mixed with a Teflon-coated magnetic stirrer. The methods” and apparatus14 for the LS experiments have been described. Density gradients were calculated from measured angular deflections assuming a constant refractivity of the cyclohexene-krypton mixtures, with a molar refractivity R = 27.01 for cyclohexene.18 Complete conversion of cyclohexene to 1,3butadiene and ethylene would cause less than a 3% change in specific refractivity for even the 4% mixture. The PLFA technique and its apparatus have been presented el~ewhere;~ nonetheless, for easy reference, we offer a brief description here. The basic concept of the PLFA method is quite simple: we use the very short ( N 10 ns) pulse from an excimer laser (Lumonics TE-860) to take a “stopaction snapshot” of absorption in the shock flow. The excimer beam is first broadened to a diameter of 6 cm with an inverted telescope. After traversing the shock tube through 6 mm X 50 mm fused silica windows, the remaining radiation is focused, through several attenuators and a wavelength filter, onto a linear CCD (Fairchild 122) by a cylindrical lens. This CCD has 1728 silicon photoelements, 13 pm X 13 Nm, in a 2 2 5 ” length. The CCD is read serially at a 1-MHz rate, and the sequence of voltages generated by the charge amplifier, proportional to integrated intensity, are further amplified and recorded by a synchronized Lecroy 8210, 10-bit waveform digitizer. The excimer laser is fired by a piezoelectric pressure transducer located above the center of the fused silica windows. A blank recording of the laser pulse (lo)and of the signal in the presence of the shock wave (I) are both sent to a microcomputer (Intertech “Superbrain”) which forms and displays the ratio Z/Zo element by element. The result is an axial record of absorbance with a diffraction-limited resolution near 0.05 mm. With t = x / u , the corresponding time resolution will normally be better than 0.1 p s . To illustrate the PLFA method, as well as the basis for its application herein, we present a PLFA recording of 1,3-butadiene pyrolysis in Figure 1. An important feature is the shock-front signal, here located between 5 and 7 mm. This is essentially a shadowgram with some adjacent Fresnel diffraction. The axial extent of this signal undoubtedly restricts the actual resolution ~
~~
~~
~
(15) “Selected Values of Properties of Hydrocarbons and Related Compounds”; Research Project 44; American Petroleum Institute: Washington, DC, 1975. (16) Neto, N.; Lauro, C. Di.; Castellucci, E.; Califano, S. Spectrochim. Acta 1967, 23A, 1163. (17) AI-Alami, M. Z.; Kiefer, J. H. J. Phys. Chem. 1983, 87, 499. (18) Partington, J. R. An Advanced Treatise on Physical Chemistry; Longmans: London, 1953; Vol. 4.
3026
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987
Kiefer and Shah
.-
28
t(P4
t(W)
Figure 3. Laser schlieren density gradient profiles for experiments 53 (1506 K, 383 Torr) and 42 (1500 K, 126 Torr) of Table 11. The data and calculations are identified as in Figure 2.
k
\
" XII
10 10
20
30
40
tiP*s)
Figure 2. Laser schlieren density gradient profiles for experiments 51 (1255 K, 457 Torr) and 71 (1354 K, 136 Torr) of Table 11. The experimental gradients are (X), and the solid line shows the predicted gradients given by the mechanism presented in the text. The first few steeply falling points are caused by beam-shock front interaction.
t(P4
20
30
40
t(P4
Figure 4. Laser schlieren density gradient profiles for experiments 4 (1410 K, 538 Torr) and 13 (1748 K, 394 Torr) of Table 11. Data and calculations are identified as in Figure 2.
near the front to something greater than the estimate given above. How much so, however, is dependent on time origin location, a difficult problem which is considered later in this paper. Calculations The shock wave kinetics integration routine has been described.Iq Thermodynamic functions for cyclohexene were again those mentioned in the preceding Experimental Section. Butadiene properties were those of ref 11, and the properties of ethylene were taken from the J A N A F tables.20 The RRKM program used herein is a slight modification of one described previo~sly.'~Again, this program employs a direct state count, using the Ekyer-Swinehart algorithm2' with classical rotationz2for the transition state and the Whitten-Rabinovitch approximation for the molecular state density. Rotational and anharmonic corrections, and the energy-transfer collision efficiency, Pc, are all introduced through the formulas given by Troe;23,24this formulation produces Troe's expression for the low-pressure limit rate constant in that limit. The RRKM program has been slightly improved. Collision frequencies are now directly calculated from the approximate Lennard-Jones ex(19) Kiefer, J. H.; Mizerka, L. J.; Patel, M. R.; Wei, H. C. J. Phys. Chem. 1985,89, 2013.
(20) 'JANAF Thermochemical Tables "; Natl. Stand. ReJ Data Ser.
Figure 5. Laser schlieren density gradient profile for experiments 3 (1667 K, 430 Torr) and 8 (1904 K, 141 Torr) of Table 11. Data and calculations are identified as in Figure 2.
pression, eq 3.4 of ref 25. Previously, a rough "fit" of the hard-sphere frequency for the range of considered temperatures was used. The calculation of & now includes the normalization correction of Gilbert et a1.,26using their eq 4.7 Pc
= 1
- (1 - ( P , o ) l / 2 ) exp(
(US., Natl. Bur. Stand.) 1971, No. 37. (21) Stein, S . E.; Rabinovitch, B. S. J. Chem. Phys. 1973, 58, 2438. (22) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: New York, 1972. (23) Trw, J. J. Chem. Phys. 1977, 66, 4745. (24) Troe, J . J . Phys. Chem. 1979, 83, 114.
(25) Troe, J. J . Chem. Phys. 1977, 66, 4759.
--)
Eo - E FEkT
dE]]-'
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3027
Unimolecular Dissociation of Cyclohexene
TABLE IV: Kinetic Mechanism for Cvclohexene Pvrolvsis _ _ _ _ _ _ _ ~
reaction no.
reaction"
log A,b cgs
1 2 3 4 5 6
C6H10 -* C4H6 + C2H4 C4H6 -* C2H3 + C2H3 C2H3+ M -* C2H2+ H + M H C4H6 C4H4 + H + H2' CAHA -* CAHi + H H-+-C4H6-+-H2 + C2H2
12.37 39.08 14.00 11.30 14.54
0.00 -7.20 0.00 0.00 0.00
66.6 50.6 14.5 60.0 14.5
13.78
0.00
14.5
14.30 14.30 14.79 13.00 13.00 13.30
0.00 0.00 0.00 0.00 0.00 0.00
14.5 40.0 14.5 0.0 14.5 14.5
10.40
0.00
0.0
14.48 11.70
0.00 0.00
14.5 0.0
+
+
+
d
C2H3
7
C2H3 + C4H6 -* C2H4 + C2H3
8 9 10 11 12 13
4"
14
I
4
15 16
I
10.0
I
+ C2H2
H + C4H3 --+ C4H2 + Hz C4H3 H + C4H2 H + C2H4 C2H3 + H2 H + C2H3 C2H2 + H2 C2H3 + C4H4 -* C2H4 + C4H3 C2H3 + C4H6 -* C4H4 + C2H4 + H C2H3 + C4H6 --* C6H6 + H2 + H H + C4H4 -* C4H3 + H2 H + C4H6 -P C2H4 + C2H3 +
4
+
E, kcal/mol
n d
d
@Thereverse of each reaction is included through detailed balance. bThe rate constants have the form A7" exp(-E/RT). 'This and other reactions are composites of more than one elementary step. dPressure dependent; see text.
c
r s, I
L
I
I
I
1
5
10
15
20
T(K)
y o
1qoo
1400
15
1300
I
DIS1.W)
Figure 6. PLFA records of butadiene formation in the dissociation of
cyclohexene, the butadiene absorption recorded at 220 nm. The dots represent measured concentrations, and the solid lines are calculated profiles (see text). The three experiments shown are, in ascending order, numbers 5 (1273 K, 372 Torr), 7 (1424 K, 333 Torr), and 19 (1526 K, 308 Torr) of Table 111.
-
where Eo is the reaction barrier,f(E) the Boltzmann fraction in E E + dE, FE = S&flE) dE/flEo)kT, and
In the calculation of FE,eq 8 of ref 24 was still used, but variation of the a(E) in the Whitten-Rabinovitch state density was included through a simple iteration scheme. Results Example LS semilog gradient profiles are shown in Figures 2-5, and PLFA absorption records in Figure 6. In all cases the semilog gradient profiles are concave upward, the expected behavior for a simple endothermic dissociation with no significant contribution from secondary reactions. Bond fission to radicals usually generates a chain reaction producing an acceleration of the endothermic rate with a consequent upward convexity of the semilog profile^.'^ The qualitative appearance of these profiles thus suggests cyclohexene dissociation is at least dominated by reaction 1, and we begin our analysis assuming this is the only reaction of the cyclohexene. To deal with any high-temperature decomposition of the butadiene product, we include those reactions important in the early stages of butadiene decomposition." The resulting mechanism is listed in Table IV. The results of modeling the LS profiles with the above minimal mechanism are exhibited in Figures 2-6. As is quite evident, this simple mechanism provides a completely satisfactory description (26) Gilbert, R. G.; Luther, K.;Troe, J. Ber. Bunsenges. Phys. Chem. 1983, 87, 169.
1
$0
sf0
(I/l)X$
710
8!0
Figure 7. Rate constants for reaction 1 as determined by extrapolation/interpolation of measured and computed gradient profiles. Each point is taken from a separate L s experiment: ( 0 ) 2% C-C6H,,,-Kr, 369-538 Torr; (M) 4%, 367-520 Torr; (0)2%, 120-172 Torr; (0) 4%, 110-152 Torr. The solid lines illustrate least-squares fits to the data,
whose expressions are given in the text. of these experiments. And these examples are typical; the mechanism describes all the experiments this well. Decomposition of the butadiene product is significant only for frozen temperatures over 1900 K and has an evident effect only in Figure 5, where the gradient beyond about 2 ps in the second example is dominated by this decomposition. To produce the fit shown here, one adjustment was made which should be mentioned. The experimental gradients were all shifted 0.2 ps to shorter time; Le., the "time origin" was delayed by 0.2 ps from that suggested by theory2' and previous pra~tice.",'~This shift, which has a negligible effect on the derived rates, was indicated by the initial analysis, which consistently generated gradient profiles that paralleled the experiments on the underside. (27) Kiefer, J. H.; Al-Alami, M. 2.;Hajduk, J.-C. Appl. Opt. 1981, 20, 221.
3028
Kiefer and Shah
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987
The location of the time origin in LS experiments is an old problem;E the small delay introduced here is discussed further below. Initial rates and rate constants for (1) were derived by a combination of extrapolation and interpolation of computed profiles. The rate constants thus derived are shown in Figure 7. Here the lines illustrate separate least-squares fits of the usual modified Arrhenius expression, A P exp(-E,/RT), to the high- and lowpressure data, giving high pressure (360-550 Torr): log k (s-I) = 96.37 - 23.6 log T - 11 1.2/6 low pressure (107-170 Torr): log k (s-’) = 102.26 - 25.3 log T - 115.5/0 10
where 0 = 2.303RT with R in kcal/(mol K). These rate constants show a very high precision, with rms deviations of but a few percent. The above rate coefficient expressions were used to generate the “final” profiles shown in Figures 2-6. The rate coefficients of Figure 7 are completely determined by the zero-time gradients (see ref 8) which may be quite closely estimated, except perhaps at the highest temperatures, by “eyeball” extrapolation. Here the modeling merely provides a more rational basis for extrapolation which seldom alters such a prior visual estimate by more than 20%. These rate constants are then very nearly independent of the kinetic modeling and, indeed, of the detailed shape of the gradient profiles. Yet with rate constants for (1) thus established, all the kinetic and thermodynamic parameters of this mechanism are set, and the entire time dependence of the gradient is determined. The excellent agreement of calculated and experimental gradients, over the length of the observations, and for all conditions, is then quite decisive evidence for the simple mechanism adopted. Dissociation of cyclohexene must indeed by strongly dominated by reaction 1 over the entire 1200-2000 K range of these experiments. The force of the above contention is quantitatively demonstrated by the sensitivities shown in Figure 8. Here the consequences of 30% alterations in the rate of (1) are illustrated. The changes in just rate demonstrate that the measured gradients indeed show a high sensitivity to the rate of this reaction. But we have also shown the result of compensating a 30% drop in rate with a corresponding increase in AH for (1). Together these changes maintain the product rAIY and thus the initial density gradient; this calculation then properly starts at the point indicated by extrapolation of the experimental profile. The results now show one consequence of some contribution by an additional dissociation path for cyclohexene having a larger AH, as would be the case for bond fission. Possible fission reactions of cyclohexene might involve ring opening or C-H scission, either of which, through further dissociation of the large formed radical, should carry an effective heat of reaction in excess of 100 kcal/mol. A 30% increase in overall AH, clearly unacceptable in light of Figure 8, would then reflect no more than a 10% contribution from any bond-fission path. Actually, even a smaller contribution from any such dissociation to radicals is undoubtedly excluded since this would also initiate chain acceleration with a still further rise in late gradient. Toward further confirmation of the above simple mechanism, we have observed the variation of the 1,3-butadiene concentration using the PLFA technique. Three example PLFA recordings of butadiene concentration with distance behind the shock, x = ut, are shown in Figure 6. Also shown are the computed profiles of butadiene concentration we obtained using the previously derived mechanism and rate constants, using absorption coefficients obtained from some PLFA experiments on butadiene pyroly~is.~ These computed profiles have been positioned on the abscissa, Le., along the shock tube axis, for best agreement. Note that this positioning now locates a space/time origin for the reaction, and at least for the two higher temperature experiments, this optimum origin is quite late in the shock-front signal, well into the region of positive signal. Since both LS and PLFA recordings of shock-front passage are essentially shadowgrams, and thus quite
80
30
40
t(ILS)
Figure 8. Sensitivity calculations for experiments 3 (1667 K, 430 Torr) and 51 (1255 K, 457 Torr). The data and ”optimum-fit”calculations are identified as in Figure 2. The short dashed lines illustrate the effect of reducing the rate constant of reaction 1 by 30% from its optimum value. The long-short dashed line shows the result of attempting to compensate this reduction with a 30% increase in heat of reaction.
similar, we may regard this as some additional evidence for a “late” location of the time origin as suggested by LS measurements. The agreement of computed and observed butadiene profiles in Figure 6 also serves as further confirmation of the assumed kinetic mechanism.
Discussion Below about 1900 K, the laser schlieren and pulsed laser flash absorption measurements presented here can be modeled to experimental accuracy with just one reaction, the retro Diels-Alder molecular elimination
-
C - C ~ H I O C4H6 + C2H4
(1)
At higher initial temperatures some decomposition of the 1,3butadiene product is evident, but no additional reactions of the cyclohexene are indicated. In fact, no more than a few percent contribution by scission to radicals is possible given the excellent agreement obtained. Of course, this is just as might have been anticipated for cyclohexene. The lowest energy path for bond scission would be H-atom separation with a barrier near 95 kcal/mol and a high-pressure A factor of roughly 1015s-1.28 Even at 2000 K this would leave k , < 4 X lo4 s-’, almost 2 orders of magnitude less than the measured rate of (1) at low pressures. The simplicity of the molecular elimination, with its unambiguous product distribution and heat of reaction, allows a correspondingly unambiguous determination of its rate constant. As seen in Figure 6, the precision of this determination is high, and its accuracy should then be commensurate. Such accuracy affords a rather stringent test of the LS method and of the assumptions made in its analysis. For one thing, the higher temperature ( T2 > 1400 K) data consistently indicate a time origin for reaction delayed about 0.2 I.LS from prior estimates,27and the PLFA records are suggestive of a similarly late time origin location. An error of such small size in the location derived from optical calc~lations~~ is no surprise, given the serious uncertainties in the parametrization of these calculations. However, it is also possible the need for this delay indicates a slight induction time for the unimolecular process.29 The actual origin of this delay can perhaps be elicited by investigating other similarly uncomplicated reactions. The low-temperature end of the derived rate constants for reaction 1, shown in their entirety in Figure 7, is compared to the earlier shock tube data in Figure 9. Clearly, the high-temperature “drop-off‘ anomaly seen by Barnard and Parrott,6 and at higher temperatures by Hidaka et al.? is not present here. The LS rates essentially indicate a continuation of the steeper temperature variation found at lower temperatures by these authors, but they (28) Benson, S . W. Thermochemical Kinetics; W h y : New York, 1968. (29) Dove, J. E.;Troe, .I. Chem. Phys. 1978, 35, 1 .
Unimolecular Dissociation of Cyclohexene
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3029 0
1 (mX1a47h
SL
Figure 9. Comparison of LS rate constants for reaction 1 (cf. Figure 7), below 1450 K,with some previous measurements of Table I.
still do not fully agree with a direct extrapolation of these data. In all cases the lowest temperature LS rates are higher than such an extrapolation would predict. The nearest data are that of Barnard and Parrott, but even for this an extrapolation to 1200 K is 25% below the LS rates. This discrepancy still seems none too serious, since the lowest temperature LS data show a 20% scatter and that of Barnard and Parrott is even greater. It is somewhat distressing to find a factor of 3 disagreement with the one other dynamic measurement, namely, the kinetic UV absorption study of Hidaka et al.' However, they apparently did not correct for temperature drop during dissociation, a drop which is readily calculated to reach 60 K in some of their 1.O% cyclohexene experiments. This much temperature change can cause nearly a factor of 2 change in the rate constant at 1300 K. This might also explain the anomalously low rate constants they report above 1300 K. In hopes of extracting further information from the LS rate constants for reaction 1, we have constructed a simple, nonspecific RRKM model of this reaction and have adjusted its parameters to produce a best fit to the measured rates. The results of two such calculations are shown in Figure 10. Both assume the same vibration model transition state (TS) whose parameters are presented, along with those of the molecule, in Table V. Here the frequencies assigned to the TS are just those of the molecule, except for a few of the lower frequencies which were reduced to give a sufficient high-pressure A factor. This model is then "nonspecific" in that it could be just as applicable, or inapplicable, to any dissociation channel. That only the A factor is sensitive to TS frequency assignment is well-known22and is confirmed here. Several quite different TS frequency assignments were tried; none of these produced any significant change in the temperature dependence of the derived k,. The k , resulting from the above model TS is adequately described by log k,
(S1)= 15.57
- 65.7/8
8!0
Figure 10. Comparison of two RRKM model calculations with the rate data for reaction 1 (cf. Figure 7). The solid lines illustrate an optimum fit using a constant (AE), = -73 cm-l, the upper line for 350 Torr and the lower line for 150 Torr. The dashed lines shqw the results of a fixed (AE)down = 550 cm-I, for the same pressures. This value of (AE)down was chosen for good agreement below 1500 K. TABLE V Cvclohexene RRKM Vibrational Freauencies (cm-'Y -
molecule 175 280 393 452 495 640 720 789 822 878 905 917 966 1009 1038 1068 1084 1138 1124 1222 1227
transition state 1241 1264 1321 1338 1343 1350 1435 1438 1447 1456 1653 2840 2860 2865 2882 2929 2940 2940 2960 3022 3065
85 180 300 300 350 400 7 20 789 822 878 905 917 966 1009 1038 1068 1084 1138 1124 1300
b
600 1300 1321 1338 1343 1350 1435 1438 1447 1456 1653 2840 2860 2865 2882 2929 2940 2940 2960 3022 3065
"LJ collision parameters (c-C6HIO-Kr)u = 4.852 8, and c / k = 248 K. Symmetry number of molecule and transition state is 2. One rotation is active in both the molecule and transition state, where it is taken to be unchanged. The product of the moments of inertia of the two inactive rotations is assumed to double in the TS. bImaginary.
is discernible here only because of the high precision of measurement. One consequence is a k , which still lies well above the measurements even for quite low temperatures. For example, the calculations predict a 22% difference between k , and the rate for 0.5 atm at lo00 K. The usual procedure of equating rate constants measured for large molecules above 1000 K with k , may introduce significant error in some cases. The TS and resulting k , are common to both calculations of Figure 10; these two differ only in the treatment of the energytransfer collision efficiency @., As indicated, one calculation derives @, from a constant ( AE)afland the other from a constant (AE),,,. Although there is no good theoretical reason for constancy of either, it has been suggesting that (AE)allis at most weakly temperature dependent,30 and these two choices seem to cover a I
over 1200-1 800 K, with a barrier Eo = 61.9 kcal/mol. The small excess of activation energy (over the barrier) is characteristic of a vibration model TS and may or may not be a real feature of the reaction. It could be an artifact which is here compensated by the somewhat "low" barrier. An interesting feature, nicely shown by both the measured rates and the RRKM calculations, is the persistence of a slight pressure dependence even at the lowest included temperatures. This is undoubtedly a common phenomenon in large molecules, which
(30) Luu, S . H.; Troe, J. Ber. Bunsenges. Phys. Chem. 1974, 78, 166.
3030 The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 sufficient range of 3/, behavior. In both cases the values of /3, were obtained from eq 1and 11, given earlier. For constant (&)down, corresponding ( AE)aIIvalues were first obtained from Troe’s reIationI2
Equation 111 has been criticized by Tardy and RabinovitchI3 mainly because it does not allow for a sign change in ( AE),’,at high temperatures. There is no question but that such a sign change will occur, but it seems quite unlikely to be of any importance in the present instance. The sign change in may be adequately located by considering the ratio of the probability for up transitions to that for down transitions across a small energy gap 6EI3 Pup/Pdown
-
[ p ( E + 6 E ) / p ( E ) I exp(-6E/kT)
Following Tardy and Rabinovitch, we may obtain a notion of this ratio’s behavior by use of classical state densities, p ( E ) . Then, for small 6E, we have Pup/Pdown 1 [ ( S - 1 ) / E - l/kT]GE + ...
-
+
where s might be the usual “effective” number of classical oscillators.22 This ratio crosses unity, and so ( AE)aIIchanges sign, at E = (s - 1)kT. Below this “critical” energy ( AE)aIIis positive, and above it, negative. Note that eq 111 provides only negative values. If we retain the classical approximation, then unimolecular reaction is dominated by a small range of energies near the maximum of the integrand in Kassel’s integral.22 In the highpressure limit this maximum is at E = Eo (s - l)kT, with Eo the barrier, and this is far above the critical energy for sign change. Although falloff will reduce the energy of this maximum, even for the low-pressure limit it never goes below the critical energy. Given the moderate falloff observed, it seems most unlikely the dissociation of cyclohexene, under present conditions, can involve molecular energies for which (AE)allis positive. We note that improved state densities, e.g., the semiclassical (Marcus-Rice) approximation,22 or a rough Whitten-RabinovitchZ2 formula, do not substantially alter the difference between the energy of maximum rate and the critical energy. When used with eq I and 11, eq 111 is part of a coherent picture of energy transfer in unimolecular reactions, and this usage seems fully justified here. = -73 cm-l As shown in Figure 10, the selection of ( produces a reasonably close fit to the measurements at all temperatures. The separation of high- and low-pressure rates is slightly exaggerated, but this is not far outside experimental = 550 cm-I genuncertainty. The selection of a fixed (AE)down erates rate constants nearly identical with those of the constant (AE),”calculation below 1500 K, but at higher temperature they now drop far below both this calculation and the measurements. This deviation is so sudden and severe it cannot possibly be compensated by alteration in any other parameters of the RRKM
+
Kiefer and Shah calculation. Within the context of Troe’s formulation of energy-transfer collision e f f i c i e n ~ y , ~the ~ - very ~ ~ high temperature dissociation rates for cyclohexene are quite consistent with a but not at all with constant ( AE)down. constant ( As we noted above, it is unlikely the treatment of ( AE)a,lgiven by Tardy and Rabinovitch13 is applicable to cyclohexene dissociation under present conditions. Nonetheless, it is worth observing and ( that the hyperbolic tangent relation between ( AE),’] they suggest also gives a very sharp drop in ( and thus &, for a fixed ( but now this occurs at a much lower temperature. Here the calculated (AE),IIcrosses zero below 1600 K and is positive at higher temperatures. Before attempting to extract any further meaning from the above results, we should mention some corroboration. Previous LS investigations of large molecule dissociation at high temperature have shown very similar behavior. For the dissociation of propane,” 1,3-butadiene,I1 and benzene,I9 a good fit to the unimolecular falloff at all temperatures was again obtained with will again result in fixed ( AE)aI1.For these a constant ( AE)down rates which are too low at the highest included temperatures. Of course, a fixed ( hE)all implies ( AE)down is increasing with temperature, whether one relates these through (111) or the hyperbolic tangent equation offered by Tardy and Rabin~vitch.’~ Whereas, some few previous s t ~ d i e s ~have l - ~ actually ~ suggested implying a slight negative temperature dependence for ( AE)down, and a very strong negative temperature dependence of ( of /3c.31 This is of course quite unacceptable here. These experiments were performed at much lower temperatures; perhaps this trend reverses at high temperature. A more likely possibility is an increase of ( h E ) d o w n with molecular energy which overcompensates any drop with temperature. There is some evidence for such an increase,34and dissociation of these large molecules is sampling very high molecular energies at the highest temperatures of the LS experiments. Regardless of the origin of the energy-transfer collision efficiencies here indicated, it is clear that Troe’s formulation of this p r ~ b l e m , using ~ ~ - ~a ~constant ( AE)all,provides a very satisfactory empirical description of the unimolecular falloff in these large molecules up to very high temperatures, E o / R T < 20. This conclusion also seems to have some generality: the reactions cited here include C-C scission in propane and butadiene, C-H scission in benzene, and a concerted molecular elimination in cyclohexene. Acknowledgment. The authors thank Dr. A . L. Wagner for helpful discussions of RRKM theory. This research was supported by the U S . Department of Energy under Grant No. DE AC0278ER2,4159. (31) Klein, I. E.; Rabinovitch, B. S.; Jung. K. H. J. Chem. Phys. 1977,87, 3833. (32) Krongauz, V. V.; Berg, M. E.; Rabinovitch, B. S. Chem. Phys. 1980, 47, 9. (33) Brown, T. C.; Taylor, J. A,; King, K. D.; Gilbert, R. G. J . Phys. Chem. 1983,87, 5214. (34) Barker, J. R. J . Phys. Chem. 1984.88, 11.
