J. Phys. Chem. 1986,90,445-450
Observation of Equilibration in the System H the Heat of Formation of C2H,
445
+ CPH, +C2H5. The Determination of
Mark Brouard, Phillip D. Lightfoot, and Michael J. Pilling* Physical Chemistry Laboratory, Oxford, OX1 3QZ, U.K. (Received: July 22, 1985)
The approach to equilibrium in the system H + C2H4== CzH5 has been observed directly by excimer laser flash photolysis/resonance fluorescence over the temperature range 775-825 K and at a total pressure of 200 torr (helium). Forward and backward rate constants have been determined and the equilibrium constant has been calculated. Combination with calculated entropies gives a value of the heat of formation of the ethyl radical, AHfe298(C2H5), of 28.36 f 0.40 kcal mol-].
Introduction Thermodynamic parameters for free radicals are of importance in chemical kinetics because they permit calculation of the rate parameters for a reverse reaction from measured values for the forward reaction. Many reactions are much more easily measured in one direction, a particularly extreme case being that of radical recombination/dissociation reactions where the recombination reaction may be measured directly, and to high precision, over a very wide range of temperature.14 Dissociation, on the other hand, is difficult to measure in i ~ o l a t i o n . The ~ ~ ~high precision, now attainable in pulsed or discharge flow measurements, of rate constants for radical reactions places demands, which are rarely met, on the quality of the thermodynamic parameters, if this precision is to be matched in the thermodynamically estimated reverse rate constants. Thus an uncertainty of 1 kcal mol-’ in AH is reflected in an uncertainty of over 50% in the estimate of the rate constant at 1000 K. A corollary of this argument is that, if precise data for both forward and backward rate constants can be determined, then the thermodynamic parameters may be defined to much higher precision than has generally proved possible ’ ~ rate to date. While this approach has often been e m p l ~ y e d , the data available have generally been obtained in different laboratories, using different techniques, and significant, and often unassessable, errors result. The ethyl radical exemplifies the ambiguities that frequently result from this approach. The generally recommended value for AHfez9g(C2H5) of 25.9 f 1.O kcal mol-’, which is based originally on the halogenation studies,1° has frequently been questioned and there is increasing support for a value of around 28 kcal mo1-1.399311-13 Recently Cao and Back9 reviewed the general evidence for an upward revision of and made a detailed analysis of the available rate data for the reaction system:
H
+ C2H6 + H2 + CZHS
(1) Macpherson, M. T.; Pilling, M. J.; Smith, M. J. C. J . Phys. Chem. 1985,89, 2268.
(2) Tulloch, J. M.; Macpherson, M. T.; Morgan, C. A,; Pilling, M.J. J. Phys. Chem. 1982, 86, 3812. (3) Parkes, D.A.; Quinn, C. P. J. Chem. SOC.,Faraday Trans. 1 , 1976, 72, 1935. (4) Adachi, H.;Basco, N.; James, D. G. L. I n f . J. Chem. Kinef. 1979, 11, 995. (5) Lin, M. C.; Back, M. H. Can. J. Chem. 1966, 44, 505. (6) Kanan, K.;Purnell, J. H.; Sepehrad, A. Int. J . Chem. Kinet. 1983,15, 845. (7) Rossi, M. J.; Golden, D. M. I n f . J . Chem. Kinet. 1983, IS, 1283. (8) Baghal-Vayjooee, M. H.; Colussi, A. J.; Benson, S. W. Inf. J. Chem. Kinef. 1979, 11, 147. (9) Cao, J. R.; Back, M. H. In?. J. Chem. Kinef. 1984,16, 961. (10)McMillen, D.F.;Golden, D. M. Annu. Reu. Phys. Chem. 1982, 33, 493. (1 1) Castelhaus, A. L.; Marriot, P. R.; Griller, D. J. Am. Chem. Soc. 1981, 103, 4262. (12) Pacey, P. D.; Wimalasena, J. H. J. Phys. Chem. 1980, 84, 2221. (13) Tsang, W. Int. J . Chem. Kinet. 1980, 10, 821.
0022-3654/86/2090-0445$01.50/0
Despite the considerable scatter in the data, the authors found overwhelming support for the higher heat of formation but were unable to recommend a precise figure. In this paper, we describe measurements, using excimer laser flash photolysis/resonance fluorescence, on the reaction system: H CzH4 + C2H5 (1, -1)
+
Equilibration is observed directly and analysis of the data enables rate constants for both the forward and backward reactions to be evaluated. These are then combined with spectroscopic estito determine A P l and the heat of formation of mates of ASel are C2H5. The advantage of this approach is that kl and measured under idential conditions by a direct, time-resolved technique. The concurrent determination of the rate constants is particularly important because of their strong pressure dependence in the experimental temperature range. Detailed attention is paid to the assessment of the uncertainty in the final value derived for AHfe29s(C2H,);Cao and Back9 have emphasized that much of the ambiguity surrounding the heat of formation of C2HSstems from the use of unrealistic error limits. Experimental Section The apparatus is described in detail e1~ewhere.l~Modifications of the cell housing were required in order to attain the elevated temperatures required in this work. The cell was heated by eight 200-W Watlow Firerod cartridge heaters and the temperature of the flowing reactant gases was measured by two Thermocoax chromel-alumel thermocouples placed above and below the photolysis region. The thermocouples also acted as sensors for the temperature-controlled power supply for the cartridge heaters. The temperature of the gases was stable during experimental measurements and the variations across the cell ( f 2 K) were well within the quoted accuracy of the thermocouples ( f 4 K at the temperatures studied). ArF excimer pulse energies of 15-150 mJ were employed to photolyze ethylene as the source of H atoms at a repetition rate of 3 Hz. The pulse energy incident on the cell was varied by placing quartz flats in front of the entrance window and was measured on a Gentec joulemeter. End product analysis experiments were performed by transferring the contents of a quartz reaction cell to a Varian Vista 401 gas chr~matograph’~ with a 2-m n-octanelPorasi1 C column, with flame ionization detection.
Results and Discussion Photolysis System. At ambient temperatures, the absorption cross section of ethylene at 193 nm is too small to produce significant photolysis.I6 The enhanced population of the torsional (14) Brouard, M.; Macpherson, M. T.; Pilling, M. J.; Tulloch, J. M.; Williamson, A. P. Chem. Phys. Lett. 1985, 113, 413. (15) Baggott, J. E.;Brouard, M.; Coles, M. A.; Davis, A,; Macpherson, M. T.;Pilling, M. J., to be submitted for publication. (16) McDiarmid, R.; Charney, E. J. Chem. Phys. 1967, 47, 1517.
0 1986 American Chemical Society
446
The Journal of Physical Chemistry, Vo1. 90, No. 3, 1986
(v4) and C-C stretching ( v 2 )vibrations at elevated temperatures ]:ads toea significant red shift in the long wavelength tail of the AIBI,-XIA, absorption ~pectrurn,’~ and H atoms could be readily detected by Lyman cy resonance fluorescence, following 193-nm photolysis, with [HI,=, k 5 X 1O’O cm-3 at the ethylene concentrations appropriate to our kinetic measurements ( [C2H4] 2.5 X 1014~ m - corresponding ~) to [H],=o/[C2H4]ratios in the range 0.1-1%. A comparison of the fluorescence signal obtained from photolysis of N 2 0 / H e / H 2 mixtures, which we have previously used and characterized as our source of H, showed that, even at the maximum pulse energies and ethylene concentrations (6.45 X 1014~ m - employed ~) in our fluorescence experiments, the initial H atom concentration was less than 4 X 10l2~ m - which ~ , is close to the limit of signal linearity for typical reversed resonance lamp conditions.18 The bulk of our measurements were conducted at significantly lower atom concentrations, thus ensuring signal linearity. In addition to acetylene, the major end products of the photolysis of 1 torr of C2H4 in 200 torr of He at 750 K were n-butane, ethane, but- 1-ene, and butadiene in typical ratios 1.0:0.35:0.1:0.1. The presence of butadiene and but-1 -ene demonstrates the formation of vinyl in the initial photolysis, with an overall scheme:
-
C2H4 C2H4 C2H4 H
P1
P2
P3
R1
3
h
4.0
c 9 $
Y
3 0.0 c
.-ul v,
-4.0 -2.5 -10.0,
I
I
2.5
0.0 I
I
I
I
5.0
I
I
7.5 Timelms I
I
v)
C2H5
+ -
+
+ CzH4
-5.0
C4H10
C2H6 R4
R5
I-C4H8
C4H6
+H
+
0.0
-1.0
C2H4
-
At the ethylene concentrations employed in the end product experiments, H radical reactions (e.g. H C2H5 2CH3) are unimportant. In addition to the C2 and C4 products specified in the scheme, small amounts of unsaturated C6 hydrocarbons were also found with a total yield comparable to that of l-C4H8 C4H6. They presumably arise from reactions of C2H3 with C2H4. Thus, in addition to H, vinyl radicals are produced in the initial photolysis, with [H]t=o/[C2H3]r=o 2 5 . Vinyl radicals have previously been implicated in the photolysis of C2H4 in this wavelength range. Borrell et a l l 9 demonstrated that C2H3 is formed in the steadystate photolysis of ethylene at 184.9 nm, with a pressure independent (>50 torr) quantum yield of 0.2. In flash photolysis experiments covering the 150-190-nm range, Back and Griffiths20 could find no evidence for vinyl radical production at low pressures (- 1 torr), while minor products, derived from C2H3, could be detected at high pressures. There is evidence, therefore, not only for the production of C2H3b u t also for its formation in a n initially vibrationally excited state. The threshold for reaction P2 lies at 103 kcal mol-’, while that for P3 lies at 145.9 kcal mol-’ 21,22 only slightly below the energy of 193-nm photons (148.1 kcal mol-’), although the later will be supplemented by the thermal energy of C2H4. Any translationally excited hydrogen atoms formed in P2 or P3 will be very rapidly
+
c
-
C2H5 + C2H3 CzH,
a
$,
.-
+ H2 C2H3 + H
riz
2C2H5
I
C2H2 + 2H
+ C2Hd R3
-2 8.0
C2H2
+
2C2H5
Brouard et al.
+
(17) Meyer, A. J.; Mulliken, R. S. J . Chem. Phys. 1969, 50, 1026. (18) Kurylo, M. J.; Peterson, N. C.; Braun, W. J . Chem. Phys. 1970, 53, 2776. (19) Borrell, P.; Cervenka, A.; Turner, J. W. J . Chem. SOC.E 1971,2293. (20) Back, R. A.; Griffiths, D. W. L. J . Chem. Phys. 1967, 46, 4839. (21) JANAF Thermochemical Tables, 2nd ed. Natl. Stand. Ref. Dura Ser., Natl. Bur. Stand. 1971, No. 37. (22) Ayranci, G.; Back, M. H. Int. J . Chem. Kine?. 1981, 23, 897.
2.0
1.0
3.0
Timelms
Figure 1. (a) Time dependence of the Lyman a resonance fluorescence signal at 825 K, following photolysis of C2H, ([C2H,] = 1.90 X I O l 4 in 200 torr of He. A plot of the residuals resulting from an analysis according to eq 1 is shown below the decay trace. Laser energy = 127 mJ/pulse. (b) Time dependence of the Lyman a resonance fluorescence signal at 298 K in 100 torr of He. [C2H4] = 1.55 X l O I 5 cm-3, [ N 2 0 ] = 1.0 X lOI5 ~ m - [H2] ~ , = 2 X 1OI6 ~ m - ~Laser . energy = 58 mJ/pulse. The residuals refer to a first-order analysis.
moderated and it is also likely that any vibrationally excited vinyl radicals which do not dissociate on a short time scale will also be thermalized, under the pressure conditions employed, well before the time scales applicable to reactions 1, -1 are approached. Experiments with vibrationally excited CH3 have demonstrated relaxation times of 5 1 0 ps a t a total pressure of 10 torr.23 It is necessary, therefore, only to consider potential contributions to the H atom signal from the reactions of thermalized vinyl radicals. Under the conditions employed in the resonance fluorescence experiments the major complication introduced by the presence of C2H3 is a small contribution to the H atom decay by H C2H3 and C2H5 C2H3recombination. As will be demonstrated, these contributions may be eliminated by extrapolation to zero pulse energy. The rate constant for thermal decomposition of C2H3may be estimated from the limiting high-pressure rate constant for the reverse reaction
+
+
H
+ C2H2
R-7
C2H3
*
obtained by Payne and Stief (k7= (9.2 2.6) X exp~ major dif1-(2410 140)/1.987T) cm3 molecule-’ s - ~ ) . * The ficulty in such a procedure is the uncertainty in AHfe298(CZH3), Ayranci and Backz2giving a value of 63.4 kcal mol-’ and Macmillen and Goldenlo one of 70.4 kcal mol-’. These two disparate
*
(23) Smith, M. J. C . , D.Phi1. Thesis, Oxford University, 1985. (24) Payne, W. A,; Stief, L. J. J . Chem. Phys. 1976, 64, 1150.
The Journal of Physical Chemistry, Vol. 90, No. 3, 1986 447
Heat of Formation of CzH5 estimates, combined with Payne and Stief s expression for k-7m, give values for k7"' at 800 K of 1.2 and 87 S-I, respectively. Allowance for substantial falloff effects (greater than a factor of 5 at 200 torr) demonstrates that decomposition of the comparatively small amounts of vinyl produced in these experiments should have a negligible effect. Time-Resolved Measurements. Time-resolved resonance fluorescence experiments were performed at temperatures in the range 775-825 K, and at a total pressure of 200 torr of helium, so as to reduce the first-order rate constant for diffusional loss of H from the monitoring zone ( k d )to values in the range 70-100 s-I, over the temperature range studied. At each temperature, kd was measured in separate experiments, using N 2 0 / H 2 / H e (1/250/4000) mixtures and 193-nm photolysis to generate the H atoms. Diffusive loss was sensibly exponential on the experimental time scale and kd was determined to a precision of around f 1 0 % ( f 2 standard deviations). Figure l a shows a typical resonance fluorescence decay curve, following the photolysis of 16.3 mtorr of CzH4 in 200 torr of H e at 825 K. The form of the approach to equilibrium, with a superimposed first-order diffusional loss, may be appreciated by comparison of Figure l a with Figure lb, which shows the decay in the presence of C2H4, following photolysis of an N,O/H,/He mixture at 285 K, which is exponential with no significant contribution from the back reaction. Experiments were conducted with an order of magnitude range of laser energies (15-1 50 mJ) and ethylene concentrations were varied by at least a factor of two. Provided the kinetic scheme may be limited to the reactions:
H
+ C2H4 7 CzHS 1
kd
H diffusive loss Then the time dependence of the H atom concentration is given by (1) [HI/[Hlo = { A exp(A+t) + B e x ~ ( h t ) ) where [HIo is the concentration at t = 0, and A* = - {(kl'
+ k-l + kd) + [(k,' + k-1 k,J2 - 4k-1kd]'/~]/2 A = (k1' + kd + h)/(A- A+) B = (kif + kd + A+)/(A+ - A_) kif = ki[CzH41
The decay curves were analyzed according to eq I, using a nonlinear least-squares program, based on the Marquardt alg ~ r i t h mwith , ~ ~ three variable parameters, kl, k-,, and the zero time H atom signal (a [HI,). The technique has been described previously in the context of decay measurements.2 It does not rely explicitly on the determination of A , B, A+, and A_; instead known parameters, such as kd,are kept fixed in the fitting process and the x2 surface is searched as a function only of the three unknown parameters. Nevertheless, the four constants contained in eq I describe the shape of the curve and the success of the fitting procedure depends on their relative sizes, B determining the magnitude of the rapidly decaying component and A that of the more slowly decaying one. At the higher temperatures, k l f and k-I contribute significantly to all four parameters and both may be determined to good precision. At lower temperatures, where k l f is much greater than k-I and kd, h tends to -kif, A / B to k-l/kl',and A+ t o -k-]kd/kl. Thus, even when k-] is comparatively small, it can still be determined to good precision, provided it is comparable to, or greater than kd, because it is essentially contained in eq 1 as a ratio to the well-determined parameter k l a t no stage is it calculated from the difference of two large numbers, as might be anticipated from a casual interpretation of eq I. For each set of temperature/concentration conditions, at least (25) Bevington, P. R. "Data Reduction and Error Analysis for the Physical Sciences"; McGraw-Hill: New York, 1969.
I lo' 10
I
I
I
I
I
'250
50 70 Laser intensitylarbitrary units I
30
Figure 2. Plot of the apparent experimental rate constants, kfS' and k4:$, vs. the relative laser pulse energy. The error bars represent f 2 a and the straight lines are the reciprocal variance weighted linear least-squares fits. T = 825 K, [C2H4] = 3.83 X 1014 ~ m - ~ .
five separate experiments were performed. Under all the conditions studied, the residuals were randomly scattered around zero (cf. Figure l a ) but both and kl (to a lesser extent) decreased as the flash energy was increased (Figure 2) presumably because of an increasing contribution to the H atom decay from atom/ radical and radical/radical reactions. In the following section we examine, using simulated decay curves, how the true values of k l and k-l may be determined. Simulations of Decay Curves. The experiments described in the last section returned values of kl' and k-, which depend on the laser energy. The purpose of the present section is to discuss the simulation of decay curves in which allowance is made for atom/radical and radical/radical reactions. These simulated curves, generated from a complete and realistic set of chemical reactions, are then analyzed according to the simple scheme (reactions 1, -1, and d) employed in the previous section and a comparison is made between the rate constants returned from the simple analysis and the equivalent input values in the full numerical simulation. Close agreement between the two sets of parameters demonstrates the validity of the simple analysis and the unimportance of the atom/radical reactions-we should expect such a situation to pertain at low laser energies. In particular, we shall attempt to construct an empirical extrapolation which will allow us to employ rate constants returned under conditions where the radical/radical reactions are significant (cf. Figure 2), to obtain estimates of k , and k-, exclusive of contributions from such reactions. Throughout these simulations, it is necessary to employ absolute atom concentrations, which approximate to the experimental values. The latter were estimated by comparison with experiments on N 2 0 / H 2 / H e mixtures, which were in turn calibrated against experiments on acetone photolysis, in which both H and CH3 were monitored (the latter in absolute terms via its absorption), while the [CH,],/[H], ratio was determined by end product a n a l y s i ~ . ' ~It~ should ~~ be emphasized that it is not necessary to determine the absolute value of [HI, accurately, since the purpose of the simulations is to construct extrapolations to [HI, = 0 from a concentration range which is reasonably representative of experiment. Simulated hydrogen atom decay curves were generated by using the numerical integration code FACSIMILE." The kinetic scheme employed included reactions 1 and -1, diffusional loss of H, and (26) Kirwan, S. P.; Lightfoot, P. D.; Pilling, M. J., to be submitted for publication. (27) Chance, E. M.; Curtis, A. R.; Jones, I. P.; Kirby, C. R. FACSIMILE: A Computer Program for Flow and Chemistry Simulation and General Initial Value Problems, H.M.S.O., London, 1977, No. C13.
448
Brouard et al.
The Journal of Physical Chemistry, Vol. 90, No. 3, 1986
the radical, atom/radical reactions:
:-
-
I
R3
+
8
-
C2H6 + CZH,
k3 = 2 X lo-'' cm3 molecule-] H
/
C4H10
2C2H5
R8
+ C2H5
k8 = 1 x
@-
-200
SKI
2CH3
cm3 molecule-I s-I
-150
R9
+ CzH5 C3Hg CH4 + CzH4
CH3
-+
k9 = 7.5
X
CH3
lo-" cm3 molecule-] s-*
RIO
+H
CH4
k , , = 1 x IO-]' cm3 molecule-'
s-l l 3
4
0
8 [HI l ~ r n - ~
R11
CH3 + CH3 __+ C2H6
k , , = 5 x lo-" cm3 molecule-' s-I The effect of C2H3radicals is discussed explicitly below. Simulations were performed on the same time scale (4 or 10 ms), with the same number of data points (700), and with the same diffusional rate constant ( k d )as used in the analysis of the experimental data. In order to obtain realistic decay curves, it was necessary to estimate kl and kl and this was achieved by linear extrapolation of the experimental plots of apparent rate constants, kt;:', k!;:;, vs. laser intensity, I , to I = 0, for each temperature and each ethylene concentration. This procedure will be justified a posteriori. [HI, was varied from 5 X lolo to 1.6 X lo'* cm-3 to reproduce, approximately, the range covered experimentally as the laser intensity was varied. The simulated H atom profiles were then transferred to the microcomputer and analyzed according to eq 1, using the program employed for analysis of the experiPlots mental data, to give apparent rate constants, kft; and of these rate constants vs. [HI, (and, therefore, vs. I) for a given set of experimental conditions (ethylene concentration, time scale, and temperature) were essentially linear (Figure 3) and a linear least-squares extrapolation to [HI, = 0 returned values for k , and k-l which were, within the uncertainty caused by the scatter of the data points, equal to the values employed in the simulations, demonstrating the validity of a linear extrapolation of the experimental data to obtain k l and k-,. The effect of vinyl radicals on the H atom decay was examined by incorporating the reactions
+ CzH3
H
k I 2= 1 x C2H5
RIZ
CZH,
cm3 molecule-' s-l
+ C2H3
-
kl3 = 5 x
CH3
-
R13
l-C4H8
2C2H4 cm3 molecule-' s-l
-
+ C2H3
R14
C3H6
kl4 = 7.5 x lo-" cm3 molecule-' s-' 2C2H3
R15
C4H6
kl5 = 2 x IO-'' cm3 molecule-'
SKI
into the scheme employed previously, and analyzing as before. Figure 3 shows a plot of the apparent rate constants obtained with [C,H,], = [HI,. The dependence of k$ on I is steeper than before but is still linear and extrapolates back to give a good
Figure 3. Plot of the apparent simulated rate constants, kil: and k?& vs. the initial hydrogen atom concentration, [HI,: (0)k , , (A) IC-', [C,H,lo = 0, k16 = 0; ( 0 )ki, (A)k-1, [ C ~ H ~=.[HI,, IO ki6 = 0; (m) k i , (X) k..,, [C2H3],= [HI,, k , , = k,/10. For each simulation, T = 800 K, [C,H,] = 5.2 X 1014 cm-3,k , = 3 X lo-', cm3 molecule-' s-', and k-, = 170 s-'.
estimate of k-]. (Note that the [C2H3],/[H], ratio pertaining in the experiments was C0.2.) Finally, simulations and fits were performed with a kinetic scheme incorporating abstraction from C2H4: H
+ CzH4
R16
H2
+ C2H3
with k I 6 = O . l k , . The resulting estimates, IC$' and Pita, are shown in Figure 3. In this case, the extrapolation to I = 0 gives a poor estimate of both k , and k-, (Le. they differ significantly from the values of k l and k-, employed in the simulations). We conclude that the effects of radical, atom/radical reactions on the H atom decay may be eliminated, and good estimates of k , and k-, made, by linear extrapolation of k&P' and k!i$ to I = 0, provided abstraction from ethylene is not significant. Contributions from reaction 16 should be discernible, however, from an apparent dependence on [C,H4] of the estimated rate constants. It should be stressed that the contribution from radical/radical reactions is a comparatively minor one. The maximum effect occurs at high temperatures (cf. Figure 2) where absorption by C2H4 is strongest and where, as a consequence, the maximum atom concentrations are generated for a given ethylene concentration. The largest effects observed, at 825 K, corresponded to a 30% decrease in k-, and a 15% decrease in k,' over the range of laser energies studied. The experimental rate constants depended only very weakly on I at 775 K. Determination of Rate Constants and Thermodynamic Parameters. Table I shows estimates of k l and k-' obtained by a weighted linear least-squares extrapolation to I = 0. Each estimate refers to a specific time scale, ethylene concentration, and temperature. Each value of kt;!' and k4;$ used in the extrapolations was weighted according to its reciprocal variance, as determined in the analysis program. The uncertainties quoted in Table I refer to the 95.45% uncertainty limits, determined from the larger of the external and internal standard deviations and including a student's t . At least five measurements were made for each set of conditions. The estimates of k l and k-' show no systematic dependence on [C2H4],demonstrating that reaction 16 is unimportant under our conditions. The values for k l were compared with estimates obtained by extrapolation of direct measurements of kl obtained at lower temperatures (196-600 K).28-30 Significant corrections for falloff were necessary, but these were
The Journal of Physical Chemistry, Vol. 90, No. 3, 1986 449
Heat of Formation of C2H5
TABLE I: Experimental Data for Reaction 1, -1 and the Heat of Formation of C2H5 T/K 775 775 775 800 800 825 825 825
10’2kl/cmg molecule-’ S-I 2.74 f 1.38 3.64 f 0.18 3.31 f 0.28 2.97 f 0.18 2.98 f 0.24 2.93 f 0.20 2.59 f 0.26 2.86 f 0.23
1O4[C,H4]/crn” 1.90 2.50 5.00 2.60 5.20 1.90 3.83 6.45
k-,/s-l 103.4 f 61.2 91.6 f 5.3 92.1 f 3.4 151.5 f 21.4 171.7 f 30.5 381.4 f 23.4 342.3 f 42.1 362.0 f 31.9
In (Kp/atm) 12.43 f 0.78 12.83 f 0.08 12.73 f 0.09 12.09 f 0.15 11.97 f 0.19 11.12 f 0.09 11.11 f 0.16 11.15 f 0.12
AHf,e298(C2HS)/ kcal mol-’ 28.81 f 1.20 28.19 f 0.12 28.35 f 0.14 28.16 f 0.24 28.35 f 0.30 28.56 f 0.15 28.57 f 0.26 28.51 f 0.20
(k-I /s-l)ca,cdo 18 24 22 42 42 86 76 84
“Calculated from the experimental value of kl, assuming AHfe29s(C2HS)=26 kcal mol-’.
determined e~perimentally.~’However, Pacansky and S ~ h r a e d e r ~ ~ have calculated the frequencies and demonstrated good agreement in those cases where experimental values are available; their vibrational frequencies were, therefore, adopted in calculating Se298(CzH5). (See Appendix.) The standard enthalpy change for reaction 1 , -1 was calculated from Ap298
=T
I
hSe298
- R In Kp(Tj +
1
T
298
ACp d In T -
where ASeZg8= -20.57 cal mol-’ K-I. ACp was calculated from spectroscopic parameters at 50 K intervals over the temperature range 300-1000 K and fitted to a polynomial. A linear fit was found to give adequate precision: ACp = [-3.59
I
I
20
I
I
I
I
I
I
40 60 80 Intensitylarbitrary units
Figure 4. Plot of (kl/k-l)FPt vs. relative laser pulse energy. The error bars represent f 2 u and the straight line is the reciprocal variance weighted linear least-squares fit. T = 825 K, [C2H4]= 3.83 X l O I 4 ~ m - ~ . effected by using parameters from fits in the falloff region over the temperature range 285-600 K.30 Good agreement was obtained; e.g. at 800 K, the extrapolated value for kl is 2.76 X cm3 molecule-’ SKI,thus validating the techniques employed in the present investigation. Figure 4 shows a plot of the ratio of the rate constants, k ~ ~ ~ t /vs. k flash ! ~ y energy ~ and demonstrates the weak dependence of the apparent equilibrium constant on I . Table I includes estimates of the equilibrium constant, Kp: Kp = k l / ( k - l R T ) The tabulated uncertainty in Kp was determined from the sum of the variances of k l and k-,, i.e. the covariance was assumed to be zero. This is almost certainly not correct, but the method of analysis did not permit an estimate of the covariance to be made. However, as we shall demonstrate below, the main uncertainty in AHfe298(C2H5) derives primarily from other sources, so that the neglect of the covariance is of no consequence. The temperature range investigated was very limited because, for T 5 750 K , the reverse reaction was too slow (k-’ 5 50 s-l), while, for T 5 850 K , the plots of the apparent rate constant vs. I became nonlinear. As a consequence, A I P for reaction 1, -1 cannot be determined graphically. However, the reactants are all well characterized spectroscopically, so that their entropies may be calculated to high precision. The worst-characterized species is C2H5,where 1 0 of the 15 vibrational frequencies have been (28) Lee, J. H.; Michael, J. V.;Payne, W. A,; Stief, L. J. J . Chem. Phys. 1978, 68, 1817. ( 2 9 ) Sugawara, K.; Okazaki, K.; Sato, S. Bull. Chem. SOC.Jpn. 1981.54, 2812. (30) Lightfoot, P. D.; Pilling, M. J., to be submitted for publication.
+ 2.01(T/1000 K ) ] cal mol-’ K-’
“$298(CZH5) was calculated from A P 2 9 8 , by using AHHfe298(H) = 52.07 kcal mol-’ 33 and AHfe298(C2H,) = 12.54 kcal mol-’,2’ for each set of experimental conditions. The quoted confidence limits were calculated only from the uncertainties in k , and k-’, by using the method of propagation of errors. A weighted mean of the values listed in Table I gives AHfe298(C2H5)= (28.36 f 0.08) kcal mol-’. Table I also contains values for k-, calculated from the forward rate constant, k l , assuming ~ i f f ~ ~ ~ =~ ~ ( C ~ H ~ 26 kcal mol-’. The discrepancy between the calculated and measured values of k-’ greatly exceeds experimental uncertainty and clearly illustrates the incompatibility of the present results and the low value of AHfe298(CzH5) recommended by McMillen and Golden.lo The estimate of the uncertainty in AHfe298(C2H5) contains no contribution from potential systematic errors: (i) The experimental system was designed primarily for the measurement of fast atom reactions, which have comparatively weak temperature dependences. As a consequence temperature measurement is not very precise and the uncertainty in the experimental temperature was f 4 K. Since Kp depends strongly on temperature this comparatively small uncertainty in T i s reflected in a large uncertainty in AHfe298(C2H5), which may be calculated from the experimental dependence of In Kp on 1 / T , giving a contribution to the fractional error ( U / A H ~ ~ ~ ~ ~ ( C , H , ) ) of 0.0074. (ii) Potential errors in the measurement of the concentration of C2H4arise from uncertainties in the total pressure, the temperature, and the flow rates, which we estimate result in percentage errors of -l%, OS%, and 2%, respectively, giving an overall contribution to the fractional error in AHfe298(C2H5) of 0.0013. (iii) The first-order rate constant for diffusional loss, kdrwas determined in separate experiments to a precision of better than *lo%. Errors in kd primarily affect k-l and have the largest effect when the latter is small, i.e. at 775 K. The incorporation in the analysis program of a 10% increase in kd from its measured value of 70 s-’ at 775 K returned ratios of kl/k-’ which were changed by f52.6%,thus making a maximum contribution to the fractional (31) Pacansky, J.; Dupuis, M. J. Am. Chem. SOC.1982, 104, 415. (32) Pacansky, J.; Schraeder, B. J. Chem. Phys. 1983, 78, 1033. (33) Wolniewicz, L. J. Chem. Phys. 1983, 78, 6173.
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The Journal of Physical Chemistry, Vol. 90, No. 3, 1986
error in the heat of formation of 0.0018. (iv) It is difficult to assess the uncertainty in me298. The entropies of H and C2H4 may be calculated with a precision far exceeding that of the present measurements and CzH5 represents the major potential source of error. The model employed for C2H5, and the sensitivity of the measured heat of formation to specific frequencies, is discussed in the Appendix. (v) Finally, we must consider the effects of reaction 16 which cannot be eliminated by extrapolation to I = 0 and which could lead to an underestimation of k-, and of diffHfe298(czHS). Reaction 16 has not been well characterized experimentally and estimates of k,6, which have been made either indirectly or at elevated to 5 x cm3 molecule-' temperatures, range from 2 x at 800 K.3e39 Reaction 16 would increase the first-order loss of H, Le. it would, in effect, increase kd linearly with CzH4. If such a contribution were pronounced (i.e. if kI6[C2H4]were greater than -40 s-l) then the residuals would demonstrate an increased decay at long times. In addition, no systematic dependence of K p on C2H4 was found at 825 K, where the most accurate and systematic experiments were conducted, suggesting that any contribution from (R16) is negligible. Combining the internal and external estimates of variance gives Afffe298(C2H5) = 28.36 f 0.40 kcal mol-', where the quoted uncertainty refers to f 2 a but is exclusive of any error in the entropy and heat capacity of C2HS. This value is in good agreement with those of Cao and Back,g Pacey and Wimalasena,I2 and TsangI3 but is significantly higher than the value recently recommended by McMillen and Golden.lo Finally, we note that, with improved temperature measurement, the technique is capable of very precise estimates of heats of formation for free radicals. Acknowledgment. W e are grateful to Dr. C. F. Melius for giving us details of his calculations on C2H5 prior to publication.
Brouard et al. We thank the SERC for an equipment grant and research studentships to M.B. and P.D.L. Appendix The entropies of C2H4 and C2Hs were calculated from spectroscopic data. For CzH4, the data of Duncan et al.4034'were employed. The following spectroscopic constants were used for C2H5: (a) internal 1.12 amu A2 rotational constants:42 (b) external 24.5, 22.8, 4.90 amu A2 vibrational frequencied2 31 12,3033, 2987,2920,2842, 1462*, 1440, 1427*, 1366, 1175, 1138,948*, 713*, 540 The starred vibrational frequencies were calculated32 from C N D O and I N D O procedures with force constants adjusted to fit the appropriate CH3 and CH2 group frequencies in C2H6and C2H4. Pacansky and CoufaP have demonstrated that free rotation occurs around the C-C bond. Recently, Melius and Wagner44performed detailed ab initio calculations (HF/6-3 1G) on C2Hs and obtained significantly different frequencies. The most important changes concern the lowest two frequency modes, where they found, after the conventional 10% reduction of the calculated values, frequencies of 414 and 778 cm-I. Use of their frequency set increases difffe298(C2H5) by 0.39 kcal mol-'. It should, however, be noted that the 540-cm-I band, which corresponds to the pyramidal motion of the -CH2 group, appears very strongly in the infrared spectra of matrix-isolated ethyl radicals and shows changes on deuteration which agree well with values calculated by Pacansky and S ~ h r a e d e r It . ~seems ~ unlikely that this frequency is as small as 414 cm-'. The greatest uncertainty, therefore, concerns the 7 13-cm-' band. A change in frequency of 65 cm-' for this band of -0.1 kcal mol-'. leads to change in Registry No. H , 12385-13-6; CzH4, 74-85-1; CzH,, 2025-56-1.
(34) Baldwin, R. R.; Simmons, R. F.; Walker, R. W. Trans. Faraday SOC., 1966.62, 2486. (35) Just, Th.; Roth, P.; Damm, R. In?. Symp. Combust. 1976, 16, 961. (36) Skinner, G. 9.; Sweet, R. G.; Davis, S.K. J . Phys. Chem. 1971, 75, 1.
(37) Benson, S. W.; Haugen, G. R. J . Phys. Chem. 1967, 71, 1735. (38) Peeters, J.; Mahren, G. Combust. Inst. European Symp. 1973, 53. (39) Lamplskii, Yu. P. Kine?. Catal. 1974, 15, 938.
(40) Duncan, J. L. Mol. Phys. 1974, 28, 1177. (41) Duncan, J. L.; McKean, D. C.; Mallinson, P. D. J . Mol. Spectrosc. 1973, 45, 221. (42) Hase, W. L.; Schlegel, H. B. J . Phys. Chem. 1982, 86, 3901. (43) Pacansky, J.; Coufal, H. J . Chem. Phys. 1980, 72, 5285. (44) Melius, C. F., private communication.
