J. Phys. Chem. 1988, 92, 6924-6929
6924
Isomerization of Cyclopropene to Allene and Propyne at Elevated Temperatures. Expettmental, ab Initio, and Model Calculatlons M. Karni, I. Oref,* Department of Chemistry, Technion, Israel Institute of Technology, Haifa 32000, Israel
S . Barzilai-Gilboa, and A. Lifshitz* Department of Physical Chemistry, The Hebrew University, Jerusalem 91 904, Israel (Received: January 22, 1988; In Final Form: April 8, 1988)
Results of an experimental and theoretical investigation of the isomerization of cyclopropene to propyne and allene allene
k A cyclopropene A propyne kl
k4
are reported. Shock tube experimentswere carried out over the temperature range 800-1200 K, and ratios of [propyne]/[allene] at different temperatures and pressures were measured. Ab initio calculationswere done, from which energy levels and vibrational frequencies of two transition states were evaluated. RRKM and absolute rate theory calculations based on these parameters produced the following high-pressure rate coeffcients (at 800 K): kl = loL3."exp(45.1 X 103/RT) PI, kz = 1013.85exp(-43.7 X 103/RT) s-l, k3 = lOI4.O3exp(-70.8 X 103/RT) s-l, and k4 = 1014.18 exp(40.4 X 103/RT) s-', where R is expressed in units of cal/(K mol). RRKM calculated rate coefficients were used to compute ratios of [propyne]/[allene] at different temperatures and pressures and to compare them with the experimental results. A very good agreement between experiment and theory is obtained.
Introduction The gas-phase isomerizations among the three C3H4isomers allene
A cyclopropene k4
k2
kl
propyne
is of ongoing interest for various reasons. It is a small system, it can be investigated experimentally in the absence of free-radical reactions, and it is amenable to theoretical interpretation. However, although a considerable volume of experimental results and theoretical calculations have been accumulated on this system, a complete picture is still missing. It has been suggested1 (with some experimental evidence2) that cyclopropene might be an intermediate in the direct allene F? propyne isomerization, but theoretical support for such a hypothesis is still lacking. It would therefore be interesting to first compute energy levels of both reactants and transition states and to calculate their vibrational frequencies and then to calculate rate coefficients for all the reactions involved and to compare them with the available experimental results. Whereas the allene $ propyne (ALL, PR) reaction has been investigated by several groups of investigators both experimentally and theoretically, there are only two previous experimental studies on cyclopropene (CP) isomerization and at a very limited temperature range. In the early study of Srinivasan3 the isomerization of cyclopropene to propyne was studied in a conventional oven heating over the temperature range 472-502 K at a total pressure of -60 Torr. The author obtained a first-order rate coefficient of kz = 1012.'3exp(-35.2 X 103/RT) s-l, where R (in this and all the following expressions) is given in units of cal/(K mol). He indicated that unless the reactant was highly diluted with an inert gas, an "unclean" reaction that was greatly influenced by surface reactions did occur. Bailey and Walsh' reported a very thorough investigation of cyclopropene isomerization over the temperature range 466-5 16 K and a wide pressure range. At 70 Torr the reaction was close to the first-order limit, and the following Arrhenius equation was obtained: k2 = lOI3.O9exp(-37.3 X 103/RT)s-l. On the basis of a measured [PR]/[ALL] ratio at one temperature and assuming that the Arrhenius A , factors for (1) Bailey, I. M.; Walsh, R. J . Chem. SOC.,Faraday Trans. I 1978, 74, 1146. (2) Hopf, H.; Priebe, H.; Walsh, R. J . Am. Chem. SOC.1980,102, 1210. (3) Srinivasan, R. J . Am. Chem. SOC.1969, 91, 6250.
0022-3654/88/2092-6924$01.50/0
-
kz and k4 are identical, they obtained the following rate coefficient for the cyclopropene allene isomerization path: k4 = 1013.25 exp(-43.4 X 103/RT) SI. In addition, by using the equilibrium constant for allene s cyclopropene, they obtained an estimate for k3 = 1013.05exp(-63.7 X 103/RT) s-l. The direct isomerization of allene to propyne was studied by several groups of investigators using flow techniques* and shock tubesk7 covering together a wide temperature and pressure range. Lifshitz, Frenklach, and Burcat4 studied the reaction over the temperature range 1030-1220 K employing the single-pulse shock tube technique and reported for the allene propyne direction k = lOI3.I7exp(40.4 X 103/RT)s-I with a slight pressure effect corresponding to a power dependence of 0.2. Bradley and West5 reported for the same rate coefficient k = 10'4.48exp(-92.7 X 103/RT) s-l in the temperature range 1440-1700 K and at pressures ranging between 3.9 and 5.3 atm. The very high value of the activation energy was shown later to be erroneous. The isomerization of propyne to allene was studied by Hidaka, Chimori, and Suga6 behind reflected shocks in the temperature range 1170-1440 K. They obtained k = exp(45.7 X 103/RT) s-l. Saito et aL7 reported k = 10'4.34exp(-68.1 X 103/RT) s-' in the temperature range 1300-2100 K. Honjou, Pacansky, and Y o ~ h i m i n ehave ~ . ~ done SCF a b initio calculations on the C3H4surface, but they did not report the vibrational frequencies of the transition structures. They proposed the following minimum-energy path for the system
-
ALL
TS1
CP
TS3
PR
I21
131
00
68.4
21 9
111
63 4
-07
where energy values relative to allene are given in kcal/mol. Saito et al.7 have done S C F calculations and evaluated the configuration and frequencies of the highest energy transition state (4) Lifshitz, A.; Frenklach, M.; Burcat, A. J. Phys. Chem. 1975, 79, 148. (5) Bradley, J. N.; West, K. 0. J . Chem. SOC.,Faraday Trans. 1 1975, 71, 967. (6) Hidaka, Y.; Chimori, T.; Suga, M. Chem. Phys. Lett. 1985,119,435. (7) Kakumoto, T.; Ushirogouchi, T.; Saito, K.; Imamura, A. J . Phys. Chem. 1987,91, 183. (8) Honjou, N.; Pacansky, J.; Yoshimine, M. J . Am. Chem. SOC.1984, 106, 5361. (9) Honjou, N.; Pacansky, J.; Yoshimine, M. J . Am. Chem. SOC.1985, 107, 5332.
0 1988 American Chemical Society
Isomerization of Cyclopropene to Allene and Propyne
The Journal of Physical Chemistry, Vol. 92, No. 24, 1988 6925
in the cyclopropene to allene reaction. However, the complete configuration of the reaction coordinate for the isomerization of cyclopropene to propyne and aliene was not reported. The present work reports on various facets of cyclopropene isomerization: an experimental study of the cyclopropene isomerization in a shock tube, an a b initio calculation of the reaction coordinate, RRKM and absolute rate theory calculations of rate coefficients, and comparison of the calculations with the experimental results. Thus, for the first time a comprehensive treatment of the cyclopropene isomerization is obtained and its relation to the allene propyne reaction is discussed.
Experimental Section Apparatus. The isomerization of cyclopropene was studied behind reflected shocks in a pressurized driver, 2-in.-i.d. singlepulse shock tube. The driven section made of “double tough” Pyrex tubing was 3.5 m long and was divided in its middle by a 2411.-i.d. ball valve. The driver had a variable length up to a maximum of 2.7 m and could be varied in small steps to obtain the best cooling conditions. A 36-L dump tank was connected to the driven section at 45O angle toward the driver, near the diaphragm holder, to prevent reflection of transmitted waves and to reduce the final pressure in the tube to facilitate the extraction of gas samples. The driven section was separated from the driver by Mylar polyester film of various thicknesses, depending upon the desired shock strength. A more detailed description of the single-pulse shock tube and the mode of its operation have already been described in an earlier publication.’0 After the tube was pumped down to approximately 2 X Torr, the reaction mixture was introduced into the section between the ball valve and the end plate and pure argon was introduced into the section between the diaphragm and the valve, including the dump tank. After the shock was fired, gas samples were collected from the tube through an outlet in the driven section (near the end plate) in 150-cm3 glass bulbs and were analyzed on a Packard 800 series gas chromatograph using a flame ionization detector. Reflected shock parameters were calculated from the measured incident shock velocities by using the three conservation equations and the ideal gas equation of state. The thermodynamic properties of cyclopropene were taken from the article by Bailey and Walsh’ and were extrapolated to temperatures higher than 1000 K by using the Wilhoit polynomials.” The incident shock velocities were measured with two miniature, high-frequency pressure transducers (Vibrometer Model 6QP500) placed 244 mm apart near the end plate of the driven section. The signals generated by the shock wave passing over the transducers were fed through a home-built piezo amplifier to a Nicolet Model 3091 digital oscilloscope. Time intervals between the two signals shown on the oscilloscope were obtained digitally with an accuracy of 2 ps (out of about 450), corresponding to approximately 10 K at high temperatures and 5 K at the lower temperature range of the study. A third transducer (PCB Model 113A26), placed at the center of the end plate, provided measurements of the reaction dwell times (approximately 2 ms) with an accuracy of -5%. Cooling rates were approximately 5 X lo5 K/s. Materials and Analysis. Cyclopropene was synthesized by using the procedure of Closs and Krantz,’* by reacting allyl chloride with sodium azide. The material obtained with this procedure consisted of a 1:l mixture of cyclopropene and ammonia. The latter could not be removed from the mixture by low-temperature vacuum distillation because of the almost identical boiling points of the two compounds. The ammonia was therefore removed by passing the gaseous mixture slowly into an aqueous solution of dilute sulfuric acid. The cyclopropene was dried by passing it through a coil at -30 “C and was then collected in a container immersed in liquid nitrogen. To prevent polymerization of the (10) Lifshitz, A.; Moran, A.; Bidani, S . Inr. J. Chem. Kinet. 1987,19,61. (1 1) Wilhoit, R.C. Thermodynamic Research Center CurrentData News; Texas ACM University: College Station, TX, 1975; Vol. 3, No.2. (12) Closs, A.; Krantz, T. J . Org. Chem. 1966, 31, 638.
8
IO
12
14 IOOOO/T
16
18
20
:2
(8)
Figure 1. The ratio [PR]/[ALL] vs 1/T. Total gas concentrations (experimental points): (V),(1-1.2) X lV5mol/cm3; (0),(2-2.5) X mol/cm3; (A), (4-5.0) x mol/cm3; (o),(8-10) x mol/cm3; (-),
calculated,present work; (---), extrapolated using Walsh’s values
for rate coefficients; (+), a point taken from Walsh’s data.
cyclopropene passing through the acid solution, its pH was kept around 3 by titrating with sulfuric acid at a rate just enough to react with the ammonia. The cyclopropene which was purified in this manner did not contain any ammonia. A gas chromatographic analysis showed small quantities of both allene and propyne. The gas was kept at liquid nitrogen temperature and was used as a source for all the experiments performed. The argon and the helium driver gas used in the experiments were obtained from the Matheson Gas Co. and were listed as 99.9995 and 99.999% pure, respectively. These materials were used without further purification. Reaction mixtures containing 1% of the purified cyclopropene in argon were prepared and stored at atmospheric pressure in 12-L glass bulbs. Both the bulbs and the vacuum manifold were pumped down to better than 1 X Torr before the preparation of the mixtures. Cyclopropene, allene, and propyne were analyzed a t 0 OC on a 3-m column of 10% &fl-oxydipropionitrile bonded to Porasil C. Since allene was eluted from the column very close to cyclopropene with an incomplete separation, we could determine its concentration with reasonable accuracy only when it approached that of the remaining cyclopropene. (Srinivasan3 in his study reported propyne as the only product of the isomerization probably because of the same reason.) This analytical drawback imposed some limitation on the extent of information that could be deduced from the experiments. Attempts to improve the separation by decreasing the column temperature, increasing its length, or varying the percentage of the bonded &3’-oxydipropionitrile were not very successful. Also, switching to different columns did not yield better results.
Experimental Results To determine the extent of decomposition of cyclopropene and ratios of [PR]/ [ALL] at different temperatures, some 25 tests were run with mixtures containing 1% cyclopropene in argon, covering the temperature range 780-1 180 K. Total densities behind the reflected shocks ranged between 2 X IO” and 1 X lo4 mol/cm3. The results of these experiments are presented in Figure 1 where ratios [PR]/[ALL] are plotted in a semilog plot against the reciprocal temperature. As has already been mentioned, very high ratios of [PR]/[ALL] could not be determined with reasonable accuracy but only roughly estimated because of the incomplete separation of allene from cyclopropene. This behavior occurs a t low temperatures where the extent of cyclopropene isomerization is moderate. On the other hand, at higher temperatures where these ratios decrease and the
Karni et al.
6926 The Journal of Physical Chemistry, Vol. 92, No. 24, 1988
-
TABLE I: Ab Initio Total" and RelativebEnereies on the C& Surface isomer 6-3 lG*"" AEb MP2/6-31GS"*' ~~
allene ProPYne cyclopropene TS 1 TS3
-1 15.86110
-1 -1 -1 -1
15.86432 15.82305 15.74780 15.73954
0.0 -2.02 23.88 71.09 76.28
I
-
-116.23310 -116.23951 -1 16.20393 -1 16.11758 -116.13568
AEb 0.0 -4.02 18.30 72.49 61.13
MP3/6-31G*'gc -1 16.25764 -1 16.25882 -116.22512 -116.14223 -116.15277
AEb 0.0 -0.74 20.40 72.42 65.81
ZPEd
U b . c
37.30 37.71 37.97 33.75 35.08
0.0 -0.38 21.0 69.3 63.9
"In hartrees. bEnergies relative to allene in kcal/mol. cGeometries optimized with the 6-31G* basis set. dZero-pint vibrational energy (kcal/mol) calculated with the 6-31G* basis set. e A t MP3/6-31G* including a correction factor of 0.87 of the calculated ZPE.
allene can be more accurately analyzed, most of the cyclopropene has already decomposed and the determination of rate parameters under such conditions is very questionable. The only meaningful quantity that could be obtained from these experiments is the ratio [PR]/[ALL]. This is therefore the quantity on which the comparison between the experiment and the calculation is based. Calculational Results and Discussion Introductory Remarks. To compute [PR]/[ALL] ratios at different temperatures and pressures and to compare them with the experimental values, a detailed description of the potential energy surface around the transition-state configurations and the reaction coordinate are needed. The resulting energy levels and vibrational frequencies are prerequisite for absolute rate theory and RRKM calculations. The rate coefficients that are calculated by using the above information can then be used to describe the dynamics of the system. Our approach will therefore be as follows. First, the results of ab initio calculations will be described. Threshold energies and transition-state configurations as well as frequencies will be given. Then, an analytical expression for the time-dependent concentrations of allene, propyne, and cyclopropene will be developed. Following that, absolute rate theory rate coefficients will be calculated. Next, the dependence of the ratio [PR]/[ALL] on the pressure will be checked by using RRKM expressions, and finally, a comparison between experimental and calculated results will be presented. Ab Initio Calculations. As has been mentioned before, partial ab initio SCF calculations on the C3H4surface were done in the past. Honjou, Pacansky, and Yoshimine8g9have studied the C3H4 surface using 4-31G and double-f plus polarization function basis set including CI. However, they did not report the vibrational frequencies of the transition structures. Saito et al.7 have calculated the configuration and frequencies of the highest energy transition state using 3-21G and 4-31G basis sets and symmetry-adapted cluster expansions. We have used the Honjou et al.899configuration and have run a complete calculation optimizing the two maxima on the reaction coordinate of the cyclopropene isomerization. Following the notation of Honjou et al.8.9 the highest maximum in the cyclopropene to propyne reaction coordinate is denoted by TS3 and the one in the cyclopropene to allene reaction coordinate by TS1. The energies and equilibrium geometries of allene, cyclopropene, propyne, and the transition states TS1 and TS3 were calculated by standard ab initio SCF-MO methods using 82 series of programs. The transition states TS1 and TS3 were located by using Baker's quasi-Newton-like algorithm15 at the 6-31G*16 level of theory. Electron correlation was estimated by using Mraller-Plesset perturbation theory" up to the third level (denoted MP3/6-31G*//6-31G* for a single-point MP3/6-31G* calculation a t the 6-31G* optimized geometry). Zero-point vibrational energies (ZPE) and vibrational levels were calculated (13) GAUSSIAN 82 ('Release A"): Binkley, J. S.;Whiteside, R. A.; Raghavachari, K.; Seeger, R.; DeFrees, D. J.; Schlegel, H. B.; Frish, M. J.; Kahn, L. R.; Pople, J. A. 'The Carnegie-Mellon Quantum Chemistry Archive", Carnegie-Mellon University, Pittsburgh, 1982. (14) whiteside, R. A.; Frisch, M. J.; Pople, J. A. "The Carnegie-Mellon Quantum Chemistry Archive", CarnegieMellon University, Pittsburgh, 1983. (15) Baker, J. J. Comput. Chem. 1986, 7, 385. (16) Hariharan, P. C.; Pople, J. A. Theor. Chim. Acta 1973, 28, 213. (17) Maller, C.; Plesset, M. S.Phys. Reu. 1934, 46, 1618. Pople, J. A.; Binkley, J. S.;Seeger, R. Inr. J. Quantum Chem., Quantum Chem. Symp. 1916, 10, 1.
Propyne Figure 2. Ab initio calculated energetics of the reaction coordinate for the isomerization of cyclopropene to propyne and allene. All numbers are in kcal/mol. TABLE II: Comparison between Experimental, Calculated, and Scaled Frequencies of cyciopropene (in cm-') %bsda
Vcald
k d e db
3152 2909 1653 1483 1105 905 996 815 3116 1043 1011 769 2995 1088 569
3510 3221 1885 1671 1264 1020 1109 979 3460 1275 1167 885 3278 1199 697
3054 2802 1640 1454 1100 887 965 852 3010 1057 101s 770 2852 1043 606
"Reference 23.
b~sceld
1OOA\v/v,w' -3.1 -3.7 -0.8 -2.0 -0.4 -2.0 -3.1 4.5 -3.4 1.3 0.4 0 -4.8 -4.1 6.5
= 0 . 8 7 ~ '~A ~v = ~ vow . - vmId.
TABLE 111: Frequencies of the Five Species Involved in the Isomerization Reactions allenl 3015, 1443, 1073, 865, 3007, 1957, 1398, 3086 (2), 999 (2), 841 (2), 355 (2) propynd 3334, 2918, 2142, 1382,931, 3008 (2), 1452 (2), 1053 (2), 633 (2), 328 (2) cyclopropeneb 3152, 2909, 1653, 1483, 1105, 905, 996, 815, 3116, 1043, 1011, 769, 2995, 1088, 569 3004, 2925, 2814, 2105, 1744, 1384, 1166, 1028, 974, TSl' 927, 794, 784, 473, 414 TS3' 3046,2932, 2854, 2355, 1556, 1420, 1149, 1021 (2), 957, 937, 862, 644, 592
"Observed, ref 22. bObserved, ref 23. cCalculated scaled values.
by using the 6-31G* basis set. The results of the calculations are shown in Figure 2. Table I gives the results of the energy calculations. The calculated frequencies of propyne, cyclopropene, and allene were scaled down by a factor of 0.87 to give the best agreement with the observed ones.18 This is demonstrated in Table I1 where the observed, calculated, and scaled vibrational frequencies of cyclopropene are shown together with the percent (18) Pople, J. A.; Schlegel, H. B.; Krishnan, R.; Defrees, D. J.; Binkley, J. S.;Frisch, M. J.; Whiteside, R. A.; Hout, R. F.; Hehre, W. J. Int. J . Quantum Chem., Quantum Chem. Symp. 1981, 15, 269.
Isomerization of Cyclopropene to Allene and Propyne TS 1 -
The Journal of Physical Chemistry, Vol. 92, No. 24, 1988 6927
TABLE IV: Parameters Used in the Calculations of the Rate Coefficients reaction PR -.CP CP -.PR ALL CP CP TS 3 -
I
1;
2 4 2
k3 4
Egb,e 1.17 1.05 1.04 0.934
66.5 42.5 69.5 48.6
[PRI, = [PRlm - ( b W 3 - m1)/(28m1)1 exp(-mlt) + (bk2(k3 - m2)/(28m2)l exp(-m2t)
Figure 3. Optimized configurationof the transition states for the isom-
erization of cyclopropene to propyne (TS3) and to allene (TSl). LH4CiC2C3 = -148.42', LHsClC&3 = 32.06', LH6C3CzCI 141.25', and LH7C3C2CI= -127.76'. deviation of the scaled from the observed values. As can be seen, the agreement is good. Similar results (not shown here) were obtained for allene and propyne. Table I11 gives the frequencies of propyne, cyclopropene, allene, and the transition states TS 1 and TS3. The frequencies of the stable molecules given in the table are the observed ones, and those for the transition states are the calculated (scaled) ones. The configuration of the transition states is given in Figure 3. These are formed by the appropriate 1,Zhydrogen shift.*s9 When TSl is formed, one H atom moves from the Cz to the C3 carbon and the C 1 4 3 bond is broken, forming allene. Similarly, when TS3 is formed, one H atom moves from C3 to C1 to form the methyl group and the C 1 4 3 bond breaks to yield propenylidene which isomerizes to propyne later on. There are therefore two distinct transition states on the reaction coordinate from allene to propyne. A transition state for the direct isomerization of allene to propyne where H atom bridges C1 and C3 of the allene (1,3-hydrogen shift) is about 20 kcal/mol above7+ TSl. The energetics of the reaction coordinate agrees with that of Honjou et aL9 even though the calculational procedure is different. The frequencies of TS1 and its configuration are in reasonable agreement with those of Saito et ala7 Model Calculations. The kinetic behavior of a complex system such as the cyclopropene isomerization depends very much on the conditions in which a particular experiment is done. A comparison between results at 500 K to those obtained at 2000 K might be erroneous for a variety of reasons. Is the system in the first-order region under a particular set of temperature and pressure conditions? Is the system in a steady-state condition? Do backreactions take place and, if so, under what experimental conditions? To answer the above questions, the following discussion is directed. For the kinetic scheme k3
k2
3
condition of [PR], = [ALLIo = 0 and [CP], = b there is, to the best of our knowledge, no analytical solution published, and one is provided below:
,--m79
ALL
u/u+"
'Path degeneracy. bObtained from the ab initio calculations. cValues (in kcal/mol) changed by an average value of 0.2 kcal/mol to obtain the best fit to the experimental data.
X
I
--
rate constant kl
allene s cyclopropene h
A propyne ki
the following differential equations describe the formation and decay of the various species that are involved in the isomerization reaction. d[PR]/dt = -kl[PR]
+ k2[CP]
d[CP]/dt = kl[PR] - (k2 + k4)[CP] d[ALL]/dt = -k3[ALL]
+ kS[ALL]
+ k,[CP]
An exact solution of the above set of equations for the initial conditions of [ALL], = b and [PRl0 = [CP], = 0 (where [ALL] and [PR] are interchangeable because of the symmetry of the system) is known.19 However, for the present case with initial (19) Lowry, T. M.; John, W. T. J. Chem. Soc. 1910,97, 2634.
[ALL], = [ALL], - (bk4(kl - ml)/(2Dml)l exp(-mlt) + (bk4(kl - m2)/(28m2)1 exP(-mZt) [CPI, = [CPI- - W 1 - ml)(ks - m1)/(28ml)l x exp(-mlt) + (b(k1 - m2)(k3 - m2)/(28m2)1 exp(-mzt) where m1 = u - 8, m2 = u + 0, u = 0.5Rki, 6 = klk3 + klk4 + k2k3, 8 = (a2 - 6)'12, [PR], = bk2k3/6, [ALL], = bklk4/6, and [CP], = bklk2/6. The concentrations of cyclopropene, allene, and propyne as a function of time can be calculated from the integral equations presented above provided the rate coefficients kl, k2, k3, and k4 are available and their dependence on the temperature is known. We next divert our attention to the calculations of the rate coefficients. Calculations of the Rate Coefficients. The rate coefficients defined in the previous section were calculated by an expression that is an extension of RRKM theoryZo and by the statistical mechanical formulation of absolute rate theory.21 The pressure-dependent RRKM expression is
where B(E) and k(E) are the Boltzmann distribution function and the RRKM energy-dependent unimolecular rate coefficient, respectively. w is the collision frequency, and the prime indicates the reverse reaction or one of the two parallel reactions. The calculated and observed frequencies of the various species are given in Table 111, and the parameters used in the calculations are given in Table IV. The calculated values of the A factors and activation energies are given in Table V. Direct counts of densities and number of states were used for computing k(E). The results of sample calculations are shown in Figure 4 where k2/kz,, is given as a function of the pressure. The pressures at which the experiments were done are marked on the lines. While the lowtemperature, high-pressure points are close to the high-pressure limit, the high-temperature, low-pressure points are well in the falloff region. This is of concern where evaluation of the individual rates of reaction is important; however, in the present work the ratio [PR]/[ALL] is the experimental variable. In such a case, the degree of falloff is of a minor concern. This is shown in Table VI where the ratio kz/k4, which is an indicator for the ratio [PR]/[ALL], is given at various temperatures and pressures. In the pressure range studied, IO3-lO4 Torr, the ratio varies by a fraction of percent at the lowest temperature to about 4% at the highest temperature, well below the experimental error. For that reason the high-pressure values of the rate coefficients were used. (20) Dewar, M. J. S.;Gardiner, W. C.; Frenklach, M.;Oref, I. J. Am. Chem. SOC.1987, 109, 4456. Tardy, D.C.; Rabinovitch, B. S.;Larson, C. W. J. Chem. Phys. 1966.45, 1163. (21) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; WileyInterscience: New York, 1972. (22) Shimanouchi, T. Narl. Stand. Re/. Data Ser. (US.,Natl. Bur. stand3 1972, N O . 39. (23) Yam, T. Y.; Eggers, D. F. J. Phys. Chem. 1979,83, 502.
Karni et al.
6928 The Journal of Physical Chemistry, Vol. 92, No. 24, 1988
"In units of s-I.
13.73 13.78 13.82 13.85 13.87 13.90 13.92 13.93 13.96 13.98 13.99 14.02
64.76 64.86 64.98 65.10 65.23 65.35 65.47 65.58 65.79 65.97 66.12 66.26
13.34 13.38 13.42 13.46 13.49 13.52 13.54 13.56 13.60 13.63 13.65 13.66
500 600 700 800 900 1000 1100 1200 1400 1600 1800 2000
13.81 13.88 13.94 13.99 14.03 14.07 14.10 14.12 15.15 14.18 14.21 14.23 14.25
43.22 43.37 43.49 43.60 43.71 43.81 43.91 44.00 44.08 44.24 44.37 44.48 44.77
49.61 49.84 50.03 50.20 50.35 50.49 50.63 50.75 50.86 5 1.07 51.24 51.39 51.32
13.89 14.00 14.08 14.13 14.18 14.22 14.25 14.27 14.29 14.33 14.35 14.37 14.39
70.18 70.34 70.48 70.63 70.77 70.91 71.05 71.17 7 1.29 71.51 71.69 7 1.85 7 1.99
units of kcal/mol. TABLE VI: The Ratio k J k , as a Function of Temperature and
Pressure temperature, K p, Torr lo2
IO3 lo4 lo5 lo6
800 40.22 38.88 38.47 38.41 38.29
700 72.47 70.04 69.56 69.41 69.41
900 25.67 24.56 24.21 24.08 24.11
loo0 17.86 17.03 16.67 16.54 16.57
1100 13.28 12.60 12.28 12.23 12.13
1200 10.40 9.84 9.56 9.46 9.44
1400 K 0.3v
0.0 2
,
,
,
,
3
4
5
6
- PR
log P (torr)
Figure 4. Ratio of the rate coefficient for isomerization of cyclopropene to propyne (k2)to its value at infinite pressure (k2J vs pressure (full lines). The full circles indicate pressures at which the experiments were done.
- 1 f
XJ
0.4
I
rAL
4 0.2
1000 K
1.2
s"0.0-12L LOG TIME (KC)
Figure 6. Concentrationsof cyclopropene (CP), propyne (PR), and allene (ALL) and the ratio [PR]/[ALL] vs time at T = 1400 K.
0.21 ,)A, 0.o -8
,
,
,fALL
1
6 a P I-
-6
-4
-2
0
2
4
6
0.8 ,
0.6-
LOG TIME (sod
Figure 5. Concentrations of cyclopropene (CP), propyne (PR), and allene (ALL) and the ratio [PR]/[ALL] vs time at T = 1000 K.
The results of the various model calculations are presented in Figures 5-7. In each figure which corresponds to a given temperature the concentrations of cyclopropene, propyne, and allene and the ratio [PR]/[AL] are plotted against the time in a log-log plot. There are four distinct time regions at each temperature. The first is the rise of the concentrations of propyne and allene and the decline in the concentration of cyclopropene. The second is a pseudo steady state in which the rate coefficients kz and k4 dominate owing to their higher values which result from the lower
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0.0
-12
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LOG TIME (sac)
Figure 7. Concentrations of cyclopropene(CP), propyne (PR), and allene (ALL) and the ratio [PR]/[ALL] vs time at T = 1800 K.
Isomerization of Cyclopropene to Allene and Propyne I .9c
TABLE VII: Calculated [PRl/[ALL] Ratios at Pseudo Steady State (SS). at Eauilibrium. and at 2 ms T, K pseudo SS equilibrium 2 ms
700 800 900 1000 1100 1200 1400 1600 1800 2000
55.3 30.3 18.9 13.0 9.59 7.42 4.97 3.68 2.90 2.41
3.48 3.18 2.96 2.80 2.67 2.57 2.42 2.32 2.24 2.17
55.3 30.3 18.9 13.0 9.55 7.09 2.61 2.32 2.24 2.17
activation energies for the isomerization of cyclopropene. As time progresses, back-reactions bontrolled by kl and k3 start to play a role, and there is an increase in the concentration of allene with time and a corresponding decrease in the concentration of propyne. At still longer times the fourth region is obtained in which an equilibrium between propyne and allene is achieved. The temporal behavior of the rate of cyclopropene isomerization to allene and propyne is summarized in Figure 8 in which the logarithm of the ratio [PR]/[ALL] is plotted vs the logarithm of the time. It is clear that if care is not taken to analyze the data according to the experimental conditions, erroneous conclusions can be obtained. In the present shock tube experiments the reaction time was 2 ms. Examination of the figures implies that over the temperature range in which the experiments were done the system is in the pseudo steady state. However, above 1200 K at 2 ms the system is in the third region and above -1500 K the system is at equilibrium. This information is summarized in Table VI1 where the ratio [PR]/[ALL] is given at the pseudo steady state, at equilibrium, and a t 2 ms which is the reaction dwell time in the shock tube. As can be seen, at the lower temperature range [PR]/[ALL] is the pseudo-steady-state value while at the hightemperature range it is the equilibrium value. In the range 1200-1500 K it is neither of the two. The experimental and calculated results are presented in Figure 1. The points are experimental data under various pressure conditions. The full line represents calculations where the rate coefficients were obtained from our ab initio energies and frequencies as described before. The best agreement with the experimental results was obtained when the value of the threshold energy that was used in the model and RRKM calculations was larger and the value of Eo.4 was smaller by 0.2 kcal/mol than the calculated values. This is well within the range of the reliability of our ab initio calculations from which these values were taken. The broken line represents the extrapolation of Walsh's values' to higher temperatures with the assumption4 of Kq = 2.3. The apparent agreement between the two lines is not surprising since the data are given in a temperature range in which the reaction
The Journal of Physical Chemistry, Vol. 92, No. 24, 1988 6929 is in the first and second regions and the ratio [PR]/[ALL] is determined by the ratio of k2/k4. That is to say, the difference AJZ, = Ea4- Ea2determines the temperature dependence of the isomerization. The value of E, obtained by our calculations is 6.7 kcal/mol compared with Walsh's' 6.03 kcal/mol, close enough for all practical purposes. At temperatures above 1400 K we expect the agreement to be not as good. Our results of E,, = 71.5 kcal/mol (temperature dependent) are close to Saito's results' for the direct isomerization of allene to propyne, namely, E, = 68.1 kcal/mol. On the same figure, one experimental point at 495 K which is taken from Walsh's data' is shown. It is the only point for which the ratio [PR]/[ALL] is reported. The agreement is embarrassingly good. The calculated ratio is 383 whereas Walsh's value is 370. The theoretical temperature dependence of the ratio [PR]/ [ALL] in the pseudo-steady-state region can be approximated by the expression [PR]/[ALL] = exp(6000/RT) which emanates from the dominating values of k2 and k4. For the equilibrium case [PR]/[ALL] = exp(lOOO/RT). That is to say, the variation of [PR]/[ALL] with temperature is much more pronounced in the pseudo-steady-state case than at equilibrium. The results of the model calculations explain very well the dynamics of cyclopropene isomerization. We would like now to check whether the model supports Walsh's suggestion' that the allene to propyne isomerization occurs via cyclopropene as an intermediate. To probe this point, Lowrey and John's e~pression'~ for the case where [ALL], = b and [CP], = [PR], = 0 was used to evaluate ratios of [PR]/[ALL] as a function of the time. The rate coefficients used in the expression were the same ones that were used in the calculations of the cyclopropene isomerization. The computed ratios [PR]/[ALL] were substituted in the equation for the direct allene s propyne rate coefficient^:^ k = -EQ In (1 - [PR],/[ALLIo/EQl/t, where E Q = Kq/(Kq+ 1). The value of k was calculated (in the range 900-1200 K) to be k = 1014.86 exp(-72.9 X 103/RT) s-'. The agreement between this expression and the experimental and the calculated results reported in the introduction is good. Saito et al. give k = 1015.08exp(-92.4 X 103/RT)s-l for the direct propyne allene isomerization rate coefficient and k = 1014.97exp(-92.8 X 103/RT) s-' for the reverse direction-a large deviation from the experimental values. It shows that the isomerization of allene takes place via cyclopropene with an activation energy of -20 kcal/mol less than that required for the direct allene + propyne isomerization via 1,3-hydrogen shift.
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Conclusion A comprehensive study of the isomerization reaction of cyclopropene to propyne and allene is presented. Experimental ratios of propyne to allene are given a t different temperatures and pressures. Calculations using an analytical expression presented here agree very well with the experimental ratios. The rate coefficients used in the model were calculated by RRKM and absolute rate theory. The reaction coordinate parameters and frequencies needed for such calculations were obtained by SCFMO ab initio calculations. The ratio [PR]/[ALL] was found to be only slightly dependent on the pressure even though the individual rate coefficients may not be precisely at their highpressure values. At moderate temperatures under the present experimental conditions, the forward rate coefficients dominate and the system is at a pseudo steady state. At higher temperatures the systems goes through a transition region and finally settles at the equilibrium ratio. Our results support Walsh's suggestion' that direct allene-propyne isomerization goes through a cyclopropene route. Acknowledgment. This work was supported by a grant from the US.-Israel Binational Science Foundation. The authors thank Professor W. C. Gardiner for many stimulating discussions. Registry No. CP, 2781-85-3; PR, 74-99-7; ALL, 463-49-0.
