871
J. Phys. Chem. 1993,97, 871-880
Weak Collision Effects in the Reaction C2H5 e C2H4 + H Y. Feng, J. T. Niiranen,+ A. Bencsura,$ V. D. Knyazev, and D. Gutman' Department of Chemistry, Catholic University of America, Washington, D.C. 20064
W. Tsang' Chemical Kinetics Division, National Institute of Standards and Technology. Gaithersburg, Maryland 20899 Received: June 1 1 , 1992; In Final Form: September 8, 1992
The unimolecular decomposition of C2Hs in helium has been investigated near the low-pressure limit ( T = 876-1094 K; P = 0.8-14.3 Torr). Rate constants (kl)have been determined as a function of temperature and pressure in the indicated ranges in time-resolved experiments. The reaction was isolated for quantitative study in a heated tubular reactor coupled to a photoionization mass spectrometer. Weak collision effects (fall-off behavior) were analyzed using a master equation analysis. Values of (&??)down for the exponential down energyloss probability were obtained for each experiment performed. The microcanonical rate constants, kl (E), needed to solve the master equation were obtained from a transition state model for the reaction which is described. The temperature dependence of these (&??)down determinations was apparent and fits the expression (&??)down = 0.255T'.0(*0.1)cm-I. It is shown that this expression (derived from experiments conducted between 876 and 1094 K) provides a reasonable representation of observed weak collision effects in helium down to 285 K. Values for (&??)down for C2H5 decomposition in other bath gases were obtained by reexamining published data on the fall-off of the C2H5 unimolecular rate constant in Nz,SF6, and C2H6. The experimental results and data simulation were used to obtain a parametrized expression for k l (T,M), the low-pressure limit rate constant for C2H5 decomposition in helium (200-1100 K); kl0 = 6.63 X 1 0 9 P . 9 9exp(-20,130 K/T) cm3 molecule-' s-l. Prior published experiments on both the forward and reverse reactions (C2Hs (M) e C2H4 H (M)) in the fall-off region were reevaluated and used in conjunction with an RRKM model of the transition state to obtain a new recommended expression for the high-pressure limit rate constant for the temperature range 200-1 100 K, kl" = 1.1 1 X 10'0T1.037exp(-l8,504/T) s-I. Parametrization of the density and temperature dependence of kl in helium according to the modified Hinshelwood expression introduced by Gilbert et al. is provided.
+
+ +
I. Introductioa The unimolecular decompositionof molecules and free radicals is an important class of elementary reactionsin high-temperature pyrolysis and combustion For quantitative modeling of the chemical, kinetics of these reactions, it is essential to understand the behavior of these elementary processes over wide ranges of temperature and pre~sure.~.5Typically, laboratory studies of unimolecular decompositions are conducted far from the conditions ( T and P) encountered in these high-temperature proccsses. Therefore, in constructing kinetic models, there is a continuing need to obtain unimolecular decomposition rate constants by extrapolations of kinetic behavior observed under much different conditions. Transitionstate theory has proven to be a very useful formalism for extrapolatingthe temperaturedependenceof the high-pressure limit thermal unimolecular rate constant.6-s At fixed temperatures, RRKM theory (e&, as modified for weak collision effects by Troe and co-workers9J0) or the direct use of the master equation7~8J~J* provides frameworks for reproducing pressure dependencies in terms of parameters such as collision efficiencies or averages of energy transfered per collision. Unfortunately, knowledge of the magnitudes of these collision parameters and their temperature dependenciesis still sparse.llJ2 For this reason, extrapolationsof fall-off behavior of unimolecularrate constants to the conditions of practical pyrolysis and combustion processes using any theory of unimolecular reactions continue to be problematic. ~~~~~~
Present address: Department of Physical Chemistry, Helsinki University, Meritullinkatu IC.SF-00170, Helsinki, Finland. 8 On leave from the Central Research Institute for Chemistry, Hungarian Academy of Sciences, P. 0. Box 17, H-1525 Budapest, Hungary. +
OO22-3654/ 58/2097-087 1$04.00/0
We have recently developed a new procedure to isolate and study the unimolecular decomposition reactions of polyatomic free r a d i c a l ~ . ~The ~ J ~experiments provide unimolecular rate constants both as a function of temperature and pressure which are used to obtain collision efficiency parameters for the reactions under study. To date we have investigated the kinetics of decomposition of the f0rmy1.l~ neopentyl,14 n-propylIs8 and isopropyl radi~a1s.I~~ Because of the size of the radicals and the pressure range which could be used in these studies (typically 2-20 Torr), all the experiments were conducted far from the low-pressure limit (where sensitivity to weak collision effects is greatest). HCO was an exception. However, HCO, with its paucity of bound vibrational states (only 15), is such an unusual speciesthat its energy-transferpropertiesare no guide to probable collision effects (and their temperature dependencies)involving larger molecules and radicals, ones with high densities of states. For the remaining reactions studied,l3-l5the closeness of reaction conditionsto the high-pressurelimit prevented the establishment of the temperature dependencies of the collision efficiency. In the current investigation, we have focused our attention on the unimolecular decomposition of the ethyl radical: (M)
+
C2H, C2H4 H (19-1) This reaction is virtually at the low-pressure limit under the conditions of our studies, 0.80-14.3 Torr and 876-1094 K. Unimolecularrate constants ( k l )were obtained over these ranges of pressure and temperature. In thecurrent investigation, knowledgeof kl"( 7')wascombind with the experimental unimolecular rate constants obtained near the low-pressure limit to obtain precise measures of ( u ) d o w n r Q 1993 American Chemical Society
872 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
averageenergy loss per collision, using a master equation analysis. Because of the excellent sensitivity of these experiments to weak collision effects and because of the accuracy and precision of the measured rate constants, it was also possible to obtain a direct measure of the temperature dependence of ( h E ) d o w n for this unimolecular reaction between 876 and 1094 K. To obtain as broad a picture of the temperature dependence of thecollisionefficiencyof reaction (1,-1) as possible (for helium, the bath gas used), recent prior determinations of k-I( T,M) of Pilling and co-workers,I6-18 Kurylo et al.,I9 and Michael et aL20 (who also used this bath gas) were analyzed in exactly the same manner as the rate constants obtained in the current investigation to obtain additional values of ( M ) d o w n at lower temperatures, in the range 285-825 K. Merging these determinations with our own provides a picture of the temperature dependence of this collision parameter which extends down to 285 K. Little knowledge exists of the temperature dependence of collision efficiencies in unimolecular reactions of combustion species at elevated temperatures.12J1J2 Tsang has analyzed the fall-off behavior of rate constants for the dissociation of small alkanes obtained from shock tube experiments (typically conducted at temperatures above 1500 K) as well as that of the reverse association reactions generally studied near ambient temperature.23 From differences in the collision efficiencies flc needed to account for the pressure dependencies using modified strong collision RRKM theory, gross trends were discerned. Tsang finds that ( h E ) d o w n (obtained from conversionsof 8, to ( u ) d o w n using the expression of Troe9)is considerably higher at combustion temperatures than at rmm temperature. For example, for CH4 in an argon bath gas, an increase from -100 cm-I at 298 K to -500 cm-1 near 2000 K is indicated by the analysis used. More direct measures of weak collision effects in thermal systems, such as studies of competitive multichannel thermal isomerization or decomposition reactions of small molecules at elevated temperatures report apparently contradictory findings regarding the temperature dependence of ( h E ) d o w n , the former studies report a decrease in this collision parameter with increasing temperature and the latter an increase with rising t e m p e r a t ~ r e . ~ ~ . ~ ~ More recently, there has been considerable interest in studying weak collision effects more directly, by monitoring collisional deactivation processes of polyatomic molecules that are highly vibrationally excited. Such studies provide still more fundamental information such as the dependence of ( AE) (average energy transfered at both up and down collisions) on total vibrational energy. Hippler and T r ~ e Quack , ~ ~ and T r ~ e Gilbert , ~ ~ and Smith,8 and Oref and TardyI2 have discussed specific examples and summarized general tendencies. Unfortunately, the conclusions that can be reached are of limited scope. They still do not provide a framework which can be used to determine collision efficiencies for molecules and free radicals whose unimolecular decompositions are important in high-temperature pyrolysis and oxidation processes. There is still no substitute for extracting this information from measured pressure and temperature dependencies of unimolecular rate constants. Thechemical kinetics of reaction (1,-1) was reviewed by Tsang and Hampson in 1986.4 Since then, new results have been reported.16-18 The current large body of both old and new information on the kinetics of reaction (1,-1) is critically reanalyzed here to obtain as accurate expressions for kl"(7') and k-l"(T) as possible. This exercise updates the rate constant expressions published in the earlier review.4 In the same review, Tsang and Hampson, using RRKM analyses, obtained bath-gas efficiencies for ethyl radical formation and decomposition for several bath gases from published rate constants (kland Ll). Using the current master equation model, these bath-gas efficiencies are redetermined and compared with that of helium.
Feng et al.
rl1
1 >
300 200
100
1
f
1 I
1
I
450
600
750
900
1050
T / K Figure 1. Plot of C2H5 first-order decay constants k'vs T for a series of experiments conducted at a fixed density, 9.55 X 10l6molecule cm-3, T, is the temperature above which unimolecular decomposition begins to be observable. The dashed line indicates extrapolated values of k' (equal to k2 below TJ to higher temperatures (15.5 SKI). The insert is the recorded C2H5 decay profile for the temperature of the filled plotted point; T = 899 K, decay constant k' = 207 f 5 s-I.
11. Experimental Section
Ethyl radicals were produced in a heatable tubular flow reactor at elevated temperature by pulsed laser photolysis, and their subsequent unimolecular decays were monitored in time-resolved experiments using photoionization mass spectrometry. Details of the experimental apparatus26 and procedures used have been described previouslyl3~14and so are only briefly reviewed here. Ethyl radicals were produced by the pulsed, 193-nm laser photolysis of 3-pentanone ( C ~ H ~ C O C ~ H2C2H5 S + CO). This radical precursor has been found to be a useful source of C2Hs for kinetic studies.27Initial conditions (3-pentanoneconcentration and laser intensity) were selected to provide low radical concentrations (less than 10" molecule cm-3) so that reactions between radical photolysisproducts had negligiblerates compared to that of the unimolecular decomposition under study. Pulsed unfocused 193-nm radiation (=4 Hz) from a Lambda Physics EMG 201 MSC excimer laser was directed along the axis of a heatable uncoated quartz tubular reactor (2.2-cm i.d.). Gas flow through the tube at =4 m s-l contained the radical precursor (0.003W.03%) and He carrier gas in large excess. The flowing gas was completely replaced between laser pulses. Gas was sampled through a hole (0.025-cm diameter) in the side of the reactor and formed into a beam by a conical skimmer before the gas entered the vacuum chamber containing the photoionization mass spectrometer. As the gas beam traversed the ion source, a portion was photoionized and mass selected. Ethyl radicals were ionized using the light from a chlorine resonance lamp (8.8-8.9 eV). Temporal ion signal profiles were recorded on a multichannel scaler from a short time before each laser pulse up to 20 ms following the pulse. Data from 1500 to 20 000 repetitions of the experiment were accumulated before the data were analyzed. The C2H5 ion signal profiles were fitted to an exponential function ([C~HS],= [ C ~ H S ] O ~ by - ~ "using ) a nonlinear leastsquares procedure. A sample decay profile and fitted decay function is shown in Figure 1. Experiments were performed to establish that the decay constants did not depend on the initial C2H5 concentration (provided that the concentration was kept low enough to ensure that radical-radical reactions had negligible rates in comparison to that of the unimolecular reaction) or the concentration of the radical precursor. The exponential decay constants depended only on temperature and density. At a fixed gas density, the C2H5 decay constants (k3 were observed to be essentiallyindependentof temperature below about
-
Weak Collision Effects in C2H5 s C2H4 + H
The Journal of Physical Chemistry, Vol. 97,No. 4, 1993 813 Unimolecular Rate Constants
500
0 [He]= 0.71E16
0 0
-
0 V
[He]= 1.33316 [He]= 4.21E16
v [He']=9,55E16 [He]=15,7E16
0
I
I
V
I
Y
-
A
*' 3 200 -
0
o 0.9
~~~~
1.10
1.2
1.15
670 K and then to increase rapidly with rising temperature after reaching a critical value Tc (see Figure 1). From experiments conducted at different total gas densities, it was found that Tc decreased when the density used in a set of experiments was increased. Based on these observations and additional tests, C2H5 was assumed to decay by two processes, reaction 1 and reaction 2, first-order heterogeneous loss:
-
1.1
1.0
1000 K / T
1000 K / T Figure 2. Plot of C2Hj unimolecular rate constants (kl vs lOOO/T) obtained from the measured decay constants (k') displayed in Figure 1. [He] = 9.55 X 10l6molecule cm-).
C,H,
v
Bimolecular Rate Constants
~
1.05
m
0 [He]= 1.33E16 V V
0
0.9
[He]= 4 . 2 l E 1 6 [ H e ] = 9.55E16 [He]= 15.7316
1.o
1000 K
1.1
/ T
1.2
F i p e 3 . Plot of first- andsecond-orderCzHs decompositionrate constants for different total gas densities (helium). (Units are atom ~ m - ~Near .) congruence of second-order rate constants indicates closeness of experimental conditions to the low-pressure limit. The solid line in the lower figure is the low-pressure limit unimolecular rate constant, k1° (see text).
heterogeneous loss (2) Thus k' = kl k2. Below about 670 K,kl was negligibly low, so k' = k2. Typically k2 was in the range 15-25 s-I. Sets of experiments were performed to determine kl as a by freezepumpthaw cycles prior to use. Helium gas was used function of temperature at five different fixed gas densities with as provided. Estimated uncertainty (1 u ) in the kl determinations helium as the carrier gas (0.71-15.7 X 10l6molecule ~ m - ~The ). is *lo% in the center of the temperature range, increasing to C2H5 exponential decay constant, k', was measured as a function f15% at the extremes. of temperature keeping the concentrations of all other gases constant. Calculations of kl from measurements of k' require III. Data Analysis subtraction of k2. While k2 (at the fixed total gas density of a In this sectionseveral aspectsof the unimolecular decomposition set of experiments)could be directlydetermined below Tc(because of C2H5 both as observed in this study and in prior investigations k' k2), it could not be measured above this temperature due are analyzed. First, the current body of kinetic data on k land to the additional loss of C2H5 by unimolecular decomposition. k-1 obtained near the high-pressure limit is reviewed and used to Values of k2 above Tcneeded for the data analysis were obtained obtain updated recommended expressions of the high-pressure by extrapolationassuming that k2 retains its observed temperature limit rateconstantskl"(T) and k-l"(T). Then,anRRKMvibrator independence below T,, up to highest temperature of the set of model for the transition state of reaction 1 is described which experiments (see Figure 1). accurately reproduces the high-pressure limit rate constants. The To minimize possible errors in the determinations of klcaused purpose of this transition state model is to provide microcanonical by the need to use an extrapolated temperature dependence of rate constants for the unimolecular decomposition of C2H5, k2 above T,, experiments to obtain kl were conducted at kl(E), used in the master equation calculations of kI(T,M) temperaturessufficiently high toassure that k'> 4k2. It was this described below as well as to extrapolate kl"(T) from 913 K to criterion that established the lowest temperature that could be the highest temperature of the current investigation, 1094 K. used to determine kl at each density. The highest temperature Finally, rate constants for reaction (1,-1) obtained in the fall-off that could be used at each density was determined by the fact region that have been reported by others for several bath gases that decay constants above 600 s-] could not be measured (He, N2, SF6, and C2H6) are reanalyzed with the same master accurately. The set of klvalues obtained at a density of 9.55 X equation model and used to provide a wider picture of weak 1016 molecule cm-3 from the k'determinations is shown on an collision effects in this unimolecular reaction. Arrhenius plot in Figure 2. A. High-Pressure Limit Rate Constants of Reaction 1. There The measured rate constants for reaction 1 are very close to have been numerous experimental investigations of reaction 1 the low-pressure limit under these experimental conditions. This which reported determinations of k l or k-l at or near the highis apparent in Figure 3 where both first- and second-order rate pressure limit, rate constants which provided values of kl" or constants for reaction 1 are shown on Arrhenius plots. There is kl-,either directly from experimentor from minor extrapolations. near congruence along a single line of second-order rate constants Studies conducted prior to 1986 have been reviewed by Tsang obtained over a 20-fold range of densities indicating an essentially and Hampson4 and used to provide recommended rate constant linear dependenceof the unimolecular rate constant on the helium expressions for both kl"(T) and k-l"(T). density over the range of conditions covered. The conditions of all experiments and the rate constants obtained are presented in The studies of reaction -1 have tended to be very direct in Table I. nature. Recent investigations have broadened the temperature The gases used were obtained from Aldrich ( C ~ H ~ C O C ~ H S , range of reported rate constants which now extends from 198 to 99+%) and Matheson (He, 99.995%). 3-Pentanonewas degassed 800 K.16-18.28929 This set of studies now provides a remarkably
+
-
Feng et al.
814 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
TABLE I: Conditions and Results of Experiments To Measure the Unimolecular Rate Constnot (kd for the Thermrl DecomwQition of Ethyl R8dicaL
0
master eq analysis
P
10-I6[M] lmolecule cm-3) ~
0.7 1 0.7 1 0.7 1 0.7 1 0.7 1 1.33 1.35 1.34 1.34 1.33 1.33 4.22 4.21 4.20 4.20 4.21 4.20 4.20 4.20 9.58' 9.56' 9.56' 9.55' 9.54' 9.54c 9.54' 9.53' 9.54' 15.7 15.7 15.7 15.7 15.8
10-'Z((C2Hs)2CO]kz
(K) 988 1030 1052 1073 1094 944 965 986 1008 1029 1051 90 1 923 933 943 953 963 974 984 877 888 899 910 921 931 941 952 957 876 89 1 907 917 927
(molecule cm-3) 2.14 2.14 2.14 2.14 2.14 4.63 4.70 4.67 4.67 4.63 4.63 4.46 4.45 4.44 4.44 4.45 4.44 4.44 4.44 4.23 4.22 4.22 4.21 4.21 4.21 4.21 4.20 4.21 5.09 5.09 5.09 5.07 5.10
(s-1) 25.0 25.0 25.0 25.0 25.0 19.4 19.4 19.4 19.4 19.4 19.4 18.5 18.5 18.5 18.5 18.5 18.5 18.5 18.5 15.5 15.5 15.5 15.5 15.5 15.5 15.5 15.5 15.5 18.8 18.8 18.8 18.8 18.8
kib (s-1) 77.5 132 189 272 326 68.0 97.7 142 190 260 344 92.7 136 161 197 245 284 338 392 128 146 191 232 277 327 396 485 536 177 234 319 389 449
(hE)down
(cm-1) 252 250 266 288 282 235 239 253 253 263 266 228 229 23 1 239 248 248 251 251 236 226 238 238 236 236 240 248 25 1 229 227 232 237 229
0 Temperatureuncertainty: f 3 K. kl= k'- kz. Experiment shown in Figures 1 and 2.
congruous picture of the values of kl"( T ) over this temperature range and, hence, is used here as the basis for new recommended expressions for both kl"(T) and k-I"(T). In Figure 4a, the determinations of kl"of Pilling and co-~orkers,~~J8 Sugawara et a1.,28 and Lee et al.29 are displayed. Lightfoot and Pilling,I7 who obtained values of Ll"from 285 to 604 K, combined their results with those of Sugawara et alaZ8(206-461 K) and Lee et al.Z9(198-320 K) to obtain a recommended expressionfor k-l"( 7') for the temperature range 198-604 K:
k-,"(T) = 4.39(f0.56)
X
lo-'' exp(-1087(*36)/T)
cm3 molecule-' s-' (I) Over this range, eq I provides k-1" values that are very close to those obtained from the expression,
k-,"(T) = 1.42 X 10-isT1.49exp(-499/T) cm3 molecule-' s-' (11) recommended in the earlier review by Tsang and Hampon (who suggest an uncertainty of +30%).4 In the current study, accurate values of kIm are needed at higher temperatures, up to 1 100 K, principally to obtain accurate values of klmup to this temperature (from Lim and the reaction thermochemistry). The high-temperaturevaluesof kl " are needed to test compliancewith experimental determinations (in the 6739 13 K range) and to interpret the experimental results obtained in the current investigation (conducted up to =1100 K). For extrapolating k-1" above 604 K, an RRKM transition state model (described in section IIIB) is preferred over extrapolating with
P
-10
2
-11
-
Sugawara e t al. (1981) Lightfoot and Pilling (1987) e t 01. ( 1 9 9 2 )
- Hanning-Lee
c)
c
1000K/T A
E C
-
Lin a n d Back (1966) Loucks and Loidler (1917)
- Simon r t
01. (1988)
- Trenwith (lg66) 0 - Pacay a n d Wimolasmna 0
- 5
0 ' 1.00
1.10
1.20 1.30 1000K/T
(1984)
I
1.40
1.50
Figure 4. Plot of reported values of kl" and k-1" vs lOOO/T. (a) Plot of k-," vs lOOO/T: Lee et al. (ref 29), Sugawara et al. (ref 28), Lightfoot and Pilling (ref 17), Hanning-Lee et al. (ref 18). (b) Plot of 'lk vs lOOO/T: Lin and Back (ref 32), Loucks and Laidler (ref 33), Simon et al. (ref 34), Trenwith (ref 35), Pacey and Wimalasena (ref 36). Lines through the data (solid line in a, dashed line in b) are from the RRKM model of reaction 1 described in the text, section IIIB.
the simple Arrhenius expression for kl"given by Lightfoot and Pilling," eq IV. The transition state model was developed to reproduce as well as possible all the values of k-'" considered by Lightfoot and Pilling plus the additional value at 800 K provided by Hanning-Lee et a1.I8 The model provides new recommended expressions for kl"based on data in the 198-800 K range and presumed still accurate up to 1100 K. When fit to a modified Arrhenius expression the result is as follows:
k-,"(T) = 1.795 X 10-12p454exp(-917/T) cm3 molecule-' s-' (111) The estimated accuracy of this expression is f2O% in the temperature range 198-1 100 K. The line through the data in Figure 4a represents the values of k I m ( T )obtained from the RRKM model. Agreement between RRKM rate constants and experiment is excellent. The k-im(7')values obtained from the transition state model of reaction 1, when used together with the now accurately known thermodynamic properties of C2Hs, yield the following new recommended expression for kl"(T) in the same temperature range:
k,"(T) = 1.11 X 10'0Ti.037exp(-l8504/T) s-' (IV) These parametrized expressions for kl'(T) and k-I"(T), eqs I11 and IV, provide values that are within 2% of the RRKM rate constants between 198 and 1100 K. The calculated thermodynamic properties of CzH5 used in the conversion of k I m ( T )to kl"(T) are given in Table 11. They include the use of the heat of formation of CzHs recently determined by Hanning-Lee et a1.18 and frequenciesand moments of inertia of this radical reported in the theoretical studies of Hase and Schlegel.3oJl Arrhenius expressions of klm(T)derived from five studies of reaction 1 conducted relatively near the high-pressurelimit (C2&
Weak Collision Effects in C2H5 s C2H4
+H
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 875
TABLE II: Thermodynamic Properties of C2Hs Used in Present Study' SO -(Go - Ho")/T Ho - Ho0 Go (J/mol K) (J/mol K) (J/mol K) (kJ/mol) 200 41.50 229.03 190.72 7.661 298.16 49.51 246.97 206.40 12.099 300 49.69 247.28 206.65 12.190 400 60.12 262.98 218.80 17.674 500 70.42 277.52 229.10 24.208 600 79.71 291.20 238.33 3 1.724 246.8 1 40.113 700 87.91 304.1 1 800 95.14 316.33 254.74 49.273 900 101.54 327.92 262.24 59.114 1000 107.17 338.91 269.36 69.555 1 100 112.1 1 349.36 276.16 80.525 1200 1 16.44 359.31 282.68 91.957 1300 120.23 368.78 288.94 103.795 1400 123.55 377.82 294.97 115.988 1500 126.45 386.44 300.78 128.49 1 frquencies: 3112,3033,2987,2920,2842, 1400(2), 1439, 1366, 1186, 1175, 1138,802,540cm-I product of moments of inertia 5: 1.252 X kg3 m6 moment of inertia of free rotor = 1.860 X loJ7 kg m2 symmetry no. of free rotor = 6
T (K)
0
AGr
(kJ/mol) 139.094 147.296 147.465 157.164 167.796 179.074 190.81 1 202.858 2 15.168 221.644 240.209 252.845 265.550 278.306 29 1.090
log KP -36.32 -25.80 -25.67 -20.52 -17.53 -15.59 -14.24 -13.24 -12.49 -1 1.89 -11.41 -1 1.00 -10.67 -10.38 -10.14
AHro 298 K taken from ref 18. Molecular structure and frequencies taken from ref 30.
TABLE III: Limiting High-Pressure Rate Expression for C2H5 Radical Decomposition A, factor @ - I )
2.7 x 1013 1.9 X 10'' 1.6 x 1013 8.9 X l o i 2 ki = 2.72 X 104 0
O
(kJ/moU 124.034 120.200 120.127 116.275 112.840 109.878 107.378 105.307 103.633 102.336 101.350 100.621 100.056 99.644 99.353
activation energy (W/mol) 159 171 159 159
T(K) 823-913 673-773 793-813 841-913 902
ref Lin and Back (32)' Loucks and Laidler (33)" Simon et al. (34) Trenwith (35) Pacey and Wimalasena (36)
Recalculatedusingk(2C2Hs+C~Hlo) = 1.8 X
cm3 molecule-1
s-I *4
was the bath gas in all cases) are given in Table 111, and values obtained are displayed in Figure 4b. Scatter in reported values is more significant than for the k-l"( 7') determinations displayed in Figure 4a, reflecting the less direct natureof these experimental investigations. All the expressions for &I"(?')in Table I11 are extrapolations of measured rate constants obtained relatively close to the highpressure limit. Still some uncertainty in kl"( 7') is introduced by theseextrapolations (estimated f30%). Comparison of theactual experimental rate constants obtained in the different studies indicates that the differences between kl" values are not due to extrapolation errors. The differencesalready exist in the measured rate constants. The most disparate results are the most recent, those reported by T r e n ~ i t h . ' Analysisof ~ theresultsof this study indicates that the reported values for &I" are only consistent with a heat of formation of C ~ Hwhich S is 6 kJ mol-' lower than that determined by Hanning-Lee et a1.I8 (whose heat of formation is supported by other recent ~tudies16,2~,~~.~*). This is a major disagreement. Accordingly, the results of Trenwith have been excluded from further consideration in this data analysis. The line through the data in Figure 4b represents &I" values from the RRKM model described in section IIIB below. The fit of calculation to experiment is as good as can be expected considering the disagreements between the experiment-derived expressions of klm(r ) (the results of Trenwith notwithstanding). The accuracy and the temperature dependence of klvalues from the RRKM model are clearly supported. For this reason we have accepted the model as one that not only provides accurate valuesof kl"( T )and kI"(T)in the 198-800 K temperature range of the investigations of reaction -1 which were used to develop the model but also is capable of predicting kl" and k-I" values up to 913 K (the upper limit of reported values of k l mdisplayed in Figure 4b) and beyond (at least up to the highest temperature of our investigation, 1094 K).
TABLE I V ArrheNus Parameters and Rate Co~trnCsfor C2Hs Decomposition
~~
200 300 400 500 600 700 800 900 1000 1100
12.93 13.09 13.21 13.31 13.40 13.47 13.53 13.59 13.63 13.68
155.52 156.27 157.11 157.99 158.87 159.75 160.67 161.54 162.38 163.22
-27.75 -14.17 -7.35 -3.23 -0.47 1.51 3.01 4.18 5.13 5.90
13.10 13.39 13.63 13.83 13.99 14.12 14.22 14.31 14.38 14.44
159.58 160.96 162.55 164.26 165.89 167.49 168.99 170.4 1 171.75 172.97
-28.81 -14.80 -7.73 -3.44 -0.55 1.54 3.11 4.35 5.35 6.17
Transition State Structure and Properties Used in Present Work frequencies: 500,700,800, 1O00, 1200, 1400, 1440, 1460, 1460, 1460, 3040,3040,3040,3040 cm-l reaction threshold = 154.78 kJ/mol Ia(C2H5) = Ia(C2Hs*) = 8.134 X l P 7 kg m2 (active) Ib(CzH5) = Ib(CzH5') = 37.848 X kg m2 (inactive) kg m2 (inactive) Ic(C2Hs) = Ic(C2Hs*) = 40.670 X symmetry no. = 1 Lennard-Jones parameters:'* e / k e = 46.95 K,u = 3.497 X 10-10 m
B. RRKM Model for Reaction 1. A transition state structure was developed for RRKM calculations (vibrator model, Le., the external moments of inertia of C2Hs are equal to those of the transition state for decomposition, I(CzH5) = I(CzH5')) which reproduces as closely as possible the kl" values between 198 and 800 K from the four studies considered which are displayed in Figure 4a. The transition state properties are given in Table IV. Also presented in Table IV are temperature-dependent Arrhenius parameters for kl" as well as log kl"values in the temperature range 200-1 100 K obtained from RRKM calculations and the transition state properties in this table. Agreement between calculated rate constants and experimental values is excellent as can be seen in Figures 4a and 4b. Also included in Table IV are k l m values and Arrhenius parameters derived from the transition state model reported by Hase and S ~ h l e g e l They . ~ ~ differ significantly from thosededuced from the present analysis. In their theoretical study, the transition state structure and frequencies (and hence Arrhenius A factors) were obtained from abinitio calculations,and the threshold energy (with the resulting activation energy) was used as an adjustable parameter to obtain the best agreement between calculated and measured rate constants. The Hase and Schlegel model does not fit the temperature dependence of the experimental values well,
et al.
876 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 L i g h t f o o t a n d Pilling (1987)
0.0
v
-
0
- [He]=15.7E16
0
[He]=1.30E18 [He]=4.20E18
--
-0.2
I
'E
260
1
BOOK
2
0
=' A ; 240
-0.a
V
-1.0 220
y/yl
Honning-Lee e t a i . (1992)
v I 1.6
1 I
1
1.8 2 . 0
I
1
I
I
2.2
2.4
2.6
2.8
I
log ( P / t o r r )
200 900
1000
1100
T/ K
Figure 5. Plot of values of ( m ) d o w n vs temperature derived from the values of kl in Table I. The line through the data is from the fit of the tempcraturedependenceof( m ) d o w n ( a P).Results wereobtainedfrom experiments conducted at different densitiesas indicated (units are atom cm-3).
a fact also noted by Davies and Pilling.39 The ab initio calculations predict a looser transition state than is indicated by experiment, forcingthe adoption of a threshold energy (or Arrhenius activation energy) which is too high to accurately reproduce rate constants except in the middle of the temperature range considered, 400lo00 K. These differences are apparent in the Arrhenius A and E factors in Table IV. C. Master Equation Determinations of kl( T'MI) and (AEh, Using CurrentResults. The master equation was used to reproduce fall-off from the high-pressure limit of each measured kl( T,[M]) listed in Table I. Only one parameter, ( h E ) d o w n , was adjusted to obtain agreement between a measured value of kl and the calculated value obtained using the master equation. (&%own is the constant in the "exponential down" probability distribution function used to characterize the collisional deactivation from an initial internal energy E to a final internal energy E' in the function:8,12*24
P(E',E) = A exp(-[E - E l / ( &??)down)
E'
A is a normalization constant. Used in this manner,
< E (v) (hE)down
provides the average energy lost in deactivating collisions. The UNIMOL suite of computer programs of Gilbert et al. was used in all the calculations.8~40The k(E) values for the calculations were obtained from the transition state model of reaction 1 described above. The UNIMOL program was modified slightly to compute collision frequencies, Z,II, from LennardJones parameters using the expression for the collision integral W 2 v 2 ) provided by Neufeld et al.41 These determinations of ( L \ E ) d o w n are included in Table I and plotted in Figure 5 vs T. D. Additional Murter Equation Determinations of kl( TJM]) and (AE)c,.fromResulLsofOtherStPdies. Usingprior reported rate constants (kl and k-l) for four bath gases (He, Nz,SF6, and C2H6) more traditional global fits to fall-off curves (sets of k l values vs P at a fixed temperature) were performed using the same master equation formalism. (If k-l values were reported, they were first converted to kl values for the data-fitting exercise using the indicated thermochemistry.) In each case, the best single value of ( h E ) d o w n was sought that minimized sum-ofsquares differences between calculated k values and reported values along a particular fall-off curve. Pilling and co-workers16-18have investigated the kinetics of reaction -1 both under irreversible conditions (at temperatures
Figure 6. Plot of k-,/k-l"values vs P (helium). Data plotted are those of Lightfoot and Pilling (lowest four temperatures, ref 17) and Hanning Lee et ai. (800 K, ref 18). Lines through the reported rate constantsare calculated values using the master equation based on global fits of the fall-off behavior at each temperature. Valuesof ( h E ) d o w n resulting from these global fits are in Table V.
below -600 K)I7and under conditions where the reaction relaxed to equilibrium (near 800 K).16,18The experiments provided k-l determinations from 285 to 604 KI7 and equilibrium constants near 800 K (as well as values of kl and k-1).16-18 The equilibrium constantsobtained by Hanning-Leeet alai6also yielded an accurate determination of the heat of formation of C2H5 which is used in thecurrent study in all the theoretical calculations. The kl values obtained from the experiments conducted near 800 K are less accurate than the k-l determinations obtained from the same experiments. Therefore, for the current comparisons,we obtained kl values near 800 K (for the ( h E ) d o w n calculations) only from thedeterminationsof k-l and theknown reaction thermochemistry. Kurylo et al.I9 as well as Michael et al." report fall-off of k-l in helium at 298 K. These results agree well with each other and were handled together to obtain a single determination of ( h E ) d o w n at this temperature. The optimum fits for all the studies considered together with the reported rate constants are displayed in Figures 6-10, (Calculated kl values were converted into k-l values before plotting where reaction -1 was studied). The value of (&!?)down obtained from each of these global fits is given in Table V.
IV. Discussion This study provides the first set of determinations of kl near the low-pressure limit and the first opportunity to obtain very precise values of ( h E ) d o w n for this reaction. The high accuracy and excellent precision of the experimental rate constants provide determinationsof ( h E ) d o w n which clearly indicatethe temperature dependence of this variable near 1000 K. In this section these observations are first discussed, and then the (&!?)do, values obtained arc merged with thoseobtained from theresultsof others obtained at lower temperatures to obtain a broader picture of the temperature dependenceof (&?.?)down in helium, one which extends from 285 to 1094 K. Finally, the values of ( u ) d o w n obtained from data fitting of the fall-offs of kl and k-Iin other bath gases are compared with those for helium to assess the role of bath-gas complexity on energy-transfer efficiency. A. The Temperature Dapendence of (AE)b,,, in Helium (Current Results). Because of the high precisios (*5%) and accuracy (ilO-l5%) of the measured unimolecular rate constants there is little scatter in the individual determinations of (&)down. The temperature dependence of this energy-transfer parameter is clearly apparent from the current determinations that were obtained between 876 and 1094 K which are displayed in Figure
Weak Collision Effects in C2Hs s C2H4 + H
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 811
0.2 Kurylo e t a l . (1970) Michael e t 01. (1973)
-
A
A8 A
0.0
3
7
n
5 \
2
'x
-o'2
v
0 -Ql
c
Y
,913K
4 ,
-
I
v
01 0
-
-0.4 0
L o u c k s a n d L a i d l e r (1967)
I
-0.6
I
I
-0.8 I 0.0
-1
2
I og ( P/ t o r r)
L 1.0
0.5
1.5
2.0
2.5
log(P/torr) Figure 7. Plot of k-llk-1" values vs P (helium). Data plotted are those of Kurylo et al. (ref 19) and Michael et al. (ref 20). The line through the reported rate constants is for calculated values obtained using the master equation based on a global fit of the fall-off behavior. Rate constants from both studies were fitted as one data set. The value of ( h E ) d o w n resulting from this global fit is in Table V.
Figure 9. Plot of reported values of kl vs P (C2H6). Data plotted are those of Lin and Back (solid circles, ref 32) and Loucks and Laidler (open circles, ref 33). Lines through the reported rate constants are calculated values obtained using the master equation based on global fits of the fall-off behavior. Values of ( h E ) d o w n resulting from these global fits are in Table V. 1901K I ~
0.5
/
0.4
807K 0.3
0.2
0.1
0.0
v
0
5
10
15
J
1
Simon e t al. (1988)
o/
P/tarr
Figure 8. Plot of reported values vs P (N2and SFs) at room temperature. Data plotted are those of Michael et al. (ref 20). Lines through the reported rate constants are calculated values obtained using the master equation based on global fits of the fall-off behavior. Values of (&!&own resulting from these global fits are in Table V.
5 . The values were fitted to the temperature-dependent function (hE)down A P . The line through the plotted points in Figure 5 represents the best fit: ( h E ) d o w n = 0.255T1.0(*0,1) cm-I. Determinations obtained from experiments conducted near the same temperature but at very different densities are in agreement as expected by the model. B. The Temperature Dependence of (AE)bwn in Helium (A Broader Look). Figure 11 provides a display of the temperature dependence of ( h E ) d o w n for reaction 1 in helium which extends from 285 to 1094 K. Values obtained in the current study are merged here with those obtained from the results of others (as described in section IIID and provided in Table V). A monotonic increase of ( h E ) d o w n with increasing temperature in this range is clearly apparent. The proportional dependence of ( h E ) d o w n on temperature, which is indicated by the results of our experiments conducted between 876 and 1094 K (indicated by thelinein Figure 11) provides a reasonablyquantitative indication of the lower temperature values particularly those below 700 K. Near 800 K, deviation of the values based on the experiments of Brouard et a1.I6 and Hanning-Lee et a1.18 from the indicated linear dependence on temperature is larger (40%) due to the apparently steeper dependence of ( h E ) d o w n on temperature displayed by the results of Pilling and co-workcrs.16-18 Taken
0
1
2
I 3
iog(P/torr)
Figure 10. Plot of reported values of kl vs P (CzH6). Data plotted are those of Simon et al. (open circles, ref 34) and Pacey and Wimalasena (solid circles, ref 35). Only one temperature used by Simon et al. is displayed because experiments were conducted over a vary narrow temperature range. Lines through the reported rate constants are calculated values obtained using the master equation based on global fits of the fall-off behavior. Values of ( h E ) d o w n resulting from these global fits are in Table V.
alone, these ( h E ) d o w n values suggest a higher power dependence on temperature, one proportional to Tl.S(*OJ). The agreement between the ( h E ) d o w n values near 900 K should be regarded as relatively good. Those near 800 K were derived from kldeterminations at pressures that are relatively near the high-pressure limit. The values of ( & ! ? ) d o w n above 876 K come from measurements of k l near the low-pressure limit (at pressures typically a factor of 100 times lower than were used by Pilling and co-workers near 800 K). Relatively small changes in some of the numbers used in the theoretical calculations (e.g., reaction thermochemistry, high-pressure limit rate constants at elevated temperatures, reaction threshold, etc.) could narrow apparent differencesin temperature dependenceand magnitude of ( h E ) d o w n indicated by the two sets of values. The parameter most likely to have the requisite error to provide a reconciliation of ( h E ) d o w n values is the high-pressure limit rate constant at elevated temperatures (*20-30%). The existing uncertainty in k-1" at 800 K is discussed by Hanning-Lee et a1.I8
878 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
TABLE V
Weak Collision Parameters for Reaction 1 Obtained from Fitting Fall-Off Data on Reaction (1,-1) pressure no. ME analysis refg (reaction)
Lightfoot and Pilling (-1) Kurylo et al. (-1) Michael et al. (-1) Kurylo and Michael* (-1) Lightfoot and Pilling (-1) Lightfoot and Pilling (-1) Lightfoot and Pilling (-1) Brouard et al. (-1) Brouard et al. (1) Brouard et al. (-1) Brouard et al. (1) Hanning-Lee et al. (-1) Hanning-Lee et al. (1) Brouard et al. (-1) Brouard et al. (1) current studyC(1) current studyC(1)
T range (K) (Torr) M = He 285 50-400 298 5-500 298 2.09-600 298 2.09-600 400 100-400 511 50-700 604 50-600 775 200 775 200 800 200 800 200 800 97-600 800 97-600 825 200 825 200 876 14.27 1094 0.804
of points
(E)down
4 10 11 21 4
1
68 105 98 94 89 136 184 3 24 210 312 183 280 278 313 229 229 282
5 5 3 3 2 2 9 9 3 3 1
(cm-’)
Michael et al. (-1)
M = N.7 298 0.991-14.71
5
120
Michael et al. (-1)
M = SFs 298 0.25-2.93
6
360
12 16 20 11 12
461 304 562 682 382 404 421 467 868 949 732 753
M = C2Hs Loucks and Laidler (1) Loucks and Laidler (1) Loucks and Laidler ( I ) Loucks and Laidler (1) Simon et al. (1) Simon et al. ( I ) Simon et al. (1) Simon et al. (1) Lin and Back (1) Lin and Back (1) Lin and Back (1) Pacey and Wimalasena (1)
673 707 743 773 793 801 807 813 823 873 913 901
5-631 4-646 3.9-631 3.4-562 3-300 3-300 1-300 1-300 66-603 40-501 35.5-209 10.1-250
10
12 11 6 7 6 15
Lightfoot and Pilling, ref 17;Kurylo et al., ref 19;Michael et al., ref 20;Brouard et al., ref 16;Hanning-Lee et al., ref 18;Loucks and Laidler, ref 33;Simon et al., ref 34;Lin and Back, ref 32;Pacey and Wimalasena, ref 36. Data from refs 19 and 20 analyzed together. Entries for only the lowest and highest temperatures of the current study.
*
-
L i g h t f o o t a n d Pllling (1987) H a n n i n g - L e e e t 01. ( 1 9 9 2 )
I
5 ‘c
E
h
0
300
-
250
-
200
-
I
00 00
V
150
50
0
0 0
0
200
400
600
800
1000
1200
T/K
Figure 11. Plot of values of ( h E ) d o v n vs temperature obtained from individual kl values in Table I (solid circles), average &-I values at a fixed pressure (open squares), and global fits of the fall-off of &-I values at a single temperature (shaded symbols). Brouard et al. (ref la), Lightfoot and Pilling (ref 17). Hanning-Lee et al. (ref 18), Kurylo et al. (ref 19), and Michael et al. (ref 20).
In the preceding paper,18 HanningLee et al. also report on master equation calculations using the exponential down probability function to fit the fall-off characteristics of k-1 values. Their values of ( h E ) d o w n at 285, 400, 51 1,604, and 800 K (47,
Feng et al. 57, 85, 118, and 201 cm-l) are 29 to 37% lower than the values we obtained at these same temperatures (68, 89, 136, 184, and 280 cm-I). In the two sets of master equation calculations (that of HanningLee et a1.I8 and one described here), a special effort was made to use thesame “data base”, i.e., virtually the same C2H5structure and frequencies, the same thermochemistry, and the same expressions to calculate collision frequencies. The principal difference in the two sets of calculations was the method used to obtain k ( E ) values needed in the master equation calculations. We used RRKM theory and HanningLeeet a1.I8used the inverse Laplace transformation of the high-pressure limit rate constant, k - i m ( T ) . In both of our studies, k ( E ) values were ultimately values displayed in derived from the same source, the set of k l m Figure 4a. In our case this information was used to obtain a transition state model for reaction 1 that reproduces these highpressure limit rateconstants. The model was then used tocalculate k ( E ) values using RRKM theory. In the study of Hanning-Lee et al.18 k ( E ) values were obtained more directly using inverse Laplace transformation of rate constant expressions, kIm( T), fitted to these same values of k-1m.18 The differences in ( h E ) d o w n values derived from the two sets of master equation calculations appear to be a consequence of the use of different assumptions regarding the “active” degrees of freedom of C2H5. In our calculations, all internal degrees of freedom and one external rotational degree of freedom were regarded as active when calculating densities of states ( n ( E ) ) , energy distribution functions (P(E)),and microcanonical rate constants (k(E)). The inverse Laplace transformation of the high-pressure limit rate constant returns the distribution function of k ( E ) n ( E ) . Hanning-Lee et al. used calculated complete rovibronic equilibrium distribution functions to obtain the n(E) values needed to obtain determinations of k ( E ) from k ( E ) n ( E ) . The same densities of states were used in the master equation. In the inverse Laplace transformation, this use of the complete rovibronic distribution function to obtain n(E) and then k ( E ) , both of which are needed in the master equation, appears to be equivalent (in the RRKM-based treatment) to assuming that all external rotors are active in C2H5. If we make this assumption, when calculating k ( E ) using microcanonical RRKM theory, we can reproduce the k(E) values obtained using the inverse Laplace method by Hanning-Lee et al. with considerable accuracy. Of course, as expected, if we use these k ( E ) values in the master equation (along with n ( E ) determinations continuing to assume that all rovibronic states are active), we obtain virtually the same ( h E ) d o w n values gotten by Hanning-Lee et al. At the five temperatures they consider, our values of ( h E ) d o w n obtained using these assumptions are 35, 51, 89, 123, and 210 cm-I. These comparisons of energy-transfer parameters derived from reproducing the pressure dependence of unimolecular rate constants illustrate how model-dependent conclusions are. This dependence hinges not only on the methodology used (e.g., a modified strong collision model or a master equation formalism) but also (in the case of master equation solutions) on the specific transition probability model preferred and the methodology and assumptions used (both explicit and implicit) to derive microcanonical rate constants. It is clear that highly quantitative comparisons between the results of different investigationsrequire not only accurate experimental results but also use of a common data reduction procedure. C. Reanalysis of Fall-Off Behavior of Reaction 1 for Other Bath Cases. The resultsof the global fitting of the fall-off behavior of k, and k-l for bath gases other than helium (Nz. SF6, and C2H6) are displayed in Figures 8-10. The gases N2 and SF6 were used by Michael et a1.20 in room temperature experiments to observe bath-gas effects in reaction -1. The fitted values of k-1 for these two bath gases are shown in Figure 8. While a bath-gas
Weak Collision Effects in C2H5 a C2H4 0
0 V
'*O0
V
400 200
0
1
I
-
+H
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 879
Lin and Back ( 1 9 6 6 ) Loucks and Laidler (1967) Pacey and Wimalasena (1984) Simon e t a!. (1988)
0 0
of C2H5, we found that the inclusion or exclusion of the energy in one external rotational degreeof freedom in the pool of available energy provides calculatedvalues of ( h E ) d w n that typically differ by 35%. Hence, the absolute accuracy of the ( u ) d w n values obtained in the current study is not expected to be high, perhaps f50%. However, the precision isexcellent sochangesin ( A E ) b with experimental parameters such as temperature and bath gas are worth noting. The temperature dependence of energy-transfermoments such as (h??)down are determined by many factors including substrate and bath-gas structure and complexity,interaction potential, mass effects, and degree of internal ~xcitation.8J~.2~.~~ This temperature dependence is itself expected 'to change with rising temperature due to several factors, e.g., the fact that molecular interactions become increasingly controlled by repulsive forces as temperature increases.43 For weak colliders, such as the one of interest here, studies of unimolecular processts in thermal systemshave provided a mixed pictureof the temperature dependence of ( L \ E ) d w n which is related to the kind of experiment from which this information is 0btained.I2,~~.~ For example in studies of two- and threechannel competitivethermal isomerizationsof deuterio- and tritiocyclopropanes Rabinovitch and co-workers have reported significant decreases in (&!&own with increasing temperatures for weak colliders. When this collision parameter is characterid by a power law (( b?down) a Ik) values of n between -2 and -3 are r e q ~ i r e d . ~ 5On , ~the ~ other hand, studies of the unimolecular dissociation of small molecules in shock tubes provide indications that this variable is essentially independent of temperature or increases slightly.11J2,24 Significantdecreasesin ( u ) d o w n with rising temperature have not been observed in "direct" experimentsthat permit measuring rates of energy transfer in sequences of collisions (e.g., by monitoring infrared fluorescenceor UV absorption of molecules excited with known amounts of internal ~ n e r g y ) . ~T ~h w .~~~~ latter studies usually provide measures of (AE),the average energy transferred per up and down collision. Reported temperature dependencies for (AE)parametrized in the same manner are in the range n = fl. The higher values of n (the positive values) are found for inefficient collision partners (such as the rare gases) and for smaller energized molecules (such as CS2 and S02). Our observed temperature dependence of ( A E ) d o w n for C2HsHe between 876 and 1094 K corresponds to one for (AE)0: P,U*O15. This result is very consistent with those of the direct experiments of energy transfer with weak colliders above loo0 K as summarized above. The relative bath gas collision efficiencies for C2H5 with different colliders is not unlike that found for other systems. Converting our determinations of ( h E ) d o w n at 298 K (the value for helium being taken from the linear fitting function) to values of ( AE)for helium, nitrogen, and SF6 yields the values -27,40, and -217 cm-1, respectively. These values are in the proportion 1:1.5:8. Comparable relative values of (AE)for the same bath gases have come from direct studies of energy transfer for other species including azulene,47-48toluene,U and ~ycloheptatriene.4~
1
600
700
800
900
1000
T / K
values VI Tfor C2H6 bath gas derived from the global fitting of the reported fall-off behavior of kl using the master equation model. Rate constants (kl)were obtained from the following studies: Lin and Back (ref 32), Loucks and Laidler (ref 33). Simon et al. (ref 34), Pacey and Wimalasena (ref 36). Alsoshown isaveragevalue derived for all studies considered in this temperature range of (M)Q,, (673-913 K), 582 cm-I. mguw 12. Plot of
(&?)down
effect on k-1 is obvious, the fall-off behavior displayed by the reported rate constants is not reproduced well by the theoretical model. The differences between calculated and reported rate constantscould well be associatedwith an experimentalproblem. There was a need to make large corrections to the apparent rate constants for reaction stoichiometry (secondary reactions), corrections which depended significantly on the pressure of the experiment. (This explanation could also account for similar poor agreement between the experimentaland calculated fall-off behavior in helium at ambient temperature as can be seen in Figure 7.) Several studies have reported determinations of kl vs P using C2H6 as the bath gas.32-36The results of four of these studies in the temperature range 673-913 K were fitted (see Figures 9 and 10). The fits tothereportedrateconstantsinC2Hsvaryinquality, but they can be regarded as generally good consideringthe indirect nature of the experiments. The relatively poor fit to the lowpressure results of Loucks and Laidler33 must be related to experimental errors. Their fall-off curves display either no curvature or, in one case (707 K),an inverted curvature from that expected on this kind of a fall-off plot. The dependenceof ( m ) d o w n on temperature for this bath gas was sought. The values obtained from the master equationglobal fits (all given in Table V) are displayed in Figure 12. Systematic differences in ( M ) d o w n derived from the four different studies considered are unfortunately of the magnitude of any likely change in ( U ) d o w n in this temperature range. So no indication of its temperature dependence is obtainable from these studies. Most values of ( U ) d o w n displayed in Figure 12 lie in the range 4 W 800 cm-I with an average value of 582 cm-l (representing temperatures near 800 K). The effect of bath-gas complexity on ( h E ) d o w n is very apparent both at room temperature and at 800 K. At room temperature, thevalues of ( M ) d o w n for He, N2, and SF6 are 94,120,360cm-I, respectively. A comparison can also be made at 800 K between the efficiency of helium and C2H6. The former is 280 cm-1 while the latter is 582 cm-I. D. ( A E L , , for CZHSmd Its Temperature Dependence: Comparisons. As demonstrated in section IVB,absolute values for collision parameters such as ( AE)hwn obtained from modeling fall-off of unimolecular rate constants from the high-pressure limit are highly model dependent, particularly with respect to how availability of energy is treated for the various degrees of freedom of the energized molecule. Using the simple transition state model discussed here for the unimolecular decomposition
V. ParameMzatiea of Rate C ~ ~ t r n t s We provide here a parameterization of kl in helium, one which will provide accurate rate constants throughout the range of temperatures (200-1 100 K)used in the current and prior studies of reactions 1 and -1 and through a much greater range of pressures than have been used (0.001-10 atm.). The modified Lindemann-Hinshelwd expression introduced by Gilbert et al. was used.lO
k, .
klolMl/kl'
+
1 k,o[M]/kl"- 'k,' The three temperature-dependent parameters needed in q VI
880 The Journal of Physical Chemistry, Vol. 97, No.4, 1993
are
Feng et al.
References .nd Notes
k," = 1.11 X 10'0T'.037exp(-l8,504/T) s-' (VII) kIo= 6.63 X 1 0 9 P 9 9exp(-20,130 K/T) cm3 molecule-' s-' (VIII)
Fa,, = 0.832 exp(-T/1203 K)
'.
(IX)
The limiting rate constant expressions were obtained from the RRKM model and master equation solutions, respectively, the latter assumingthe linear dependence of (At!?)&,, on temperature which fits the results of the current study, (At!?)down = 0.255f.O cm-1. A simulated data base of kl values (obtained using the master equation solutions) covering the pressure range 0.001-10 atm and the temperature range 20&1100 K was fitted to eq VI using the limiting expressions for kl" and k10 given above in order to derive the parameters of F,,,. The average deviation of the parametrizedvaluesof kl from themaster equation rateconstants over this large range of conditionsis 10%. The maximum deviation is 29% which occurs at the lowest temperatures considered.
VI. swnmiry The unimolecular decomposition of C ~ H in S helium has been studied near the low-pressure limit. Measurements of kl as a function of temperature were combined with knowledge of the high-pressure limit rate constant to characterize weak collision effects in this reaction. Master equation analyses of the data provided (&??)down values from 876 to 1094 K which display a proportional dependence on temperature, ( M ) h= 0.255F1.0(foJ) cm-I. A transition state model of reaction 1 was described which reproduces the high-pressure limit rate constant. The model was used to provide k(E) values for the master equation analyses. Additional master equation calculations performed to reproduce reported values of obtained using helium at lower temperatures indicate that this proportional dependence of ( u ) d o w n provides a reasonable representation of this weak collision constant at lower temperatures, down to 285 K. Bath-gas effects on the magnitude of (At!?)down in reaction 1 for N2,SF6, and C2Hs were reexamined and comparedwith those of helium. Finally, parametrized expressions are presented which are for use in obtaining klin helium over a wide range of pressures and temperatures.
AcLaorrkdgmeat. This research was supportedby the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, US.Department of Energy under Grant No. DE/FG05-89ER14015. The authors thank Dr. Irene R. Slagle for her assistanceand guidance. We also thank Professor M.J. Pilling, Dr. M. A. Hanning-Lee, and Dr. S.H.Robertson for many useful comments and discussions and for the use of their computer programs used toobtain k(E)values from kl"( T) by the inverse Laplace transform method.
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