J . Phys. Chem. 1987, 91, 3373-3379 the previously reported value of (0.6 f 0.7) X lo4 M-' s-'. However, the latter value obtained under pseudo-first-order conditions is subject to larger errors compared to 4-CB because the peak potential (E,) of 4-BB is much closer to the Eo' of the catalyst. As mentioned in the Introduction, in the pseudo-firstorder method the catalytic current is separated from background current for direct reduction of substrate by nonlinear regression. Values of k , are then computed from their proportionality to ( I c / I d ) 2 , where 1, is the limiting catalytic current, and I d is the peak current of the catalyst alone. Analysis of errors2 in the pseudo-first-order method predict that, under conditions of significant overlap of catalytic and background I-E curves, a decrease in ,??' - Epof 40 mV leads to about a fourfold increase in relative standard error in k , . E, for 4-BB is 42 mV more positive than for 4-CB,I7 and for this reason the pseudo-first-order-derived k , for 4-BB has significantly worse precision than that of 4-CB. In fact, k , of 4-BB/phenanthridine approaches the upper limit that can be estimated by the pseudo-first-order method. As an additional check on the 4-BB system, kl was estimated vs. log from a working curveg of (F/RT)(E*' [RTC*,k,/Fv], where E p j 2is the half-peak potential of the catalytic I-E curve. This method gave 7 X IO4 M-' s-' , i ngood agreement with the second-order simulation/regression (Table IV). Further evidence of the validity of that value is that kl values of a series of catalysts reacting with 4-BB were generally about 10-fold greater than for 4-CB.4 The present value also better follows the trend4 of increasing kl with a decrease in Eo' - E,. Thus, the second-order simulation/regression method appears to extend the upper limit of k l determinations by at least an order of magnitude over the pseudo-first-order method. Poorer precision for 4-BB than for 4-CB suggests a decrease in precision as k , increases, especially in cases where the substrate is reduced at potentials only slightly more negative than the catalyst. Precision should be optimized for such systems with higher k l values by using smaller concentrations of substrate. In such cases, confirmation of the rds by examining the influence of C** on fitted kinetic parameters is highly recommended. Recent results in our laboratoryz5showed that, for systems in which the substrate is reduced at potentials much more negative (25) Connors, T. F.; Arena, J. V.; Rusling, J. F., unpublished results.
3373
than E" of the catalyst, the upper limit of k , is much greater than the value found for the phenanthridine/4-BB system. Specifically, the regression/simulation was applied to reduction of 1,2-dibromobutane by electrogenerated vitamin Blh [Co(I) form]. For this system, direct reduction of substrate occurs about 0.8 V more negative than the fl of the catalyst. Moreover, no substrate anion radical exists, eq 3 was formally considered a very fast step, and the reaction of Co(1) and substrate is the rds. Analysis of voltammetric data at scan rates between 10 and 25.6 V s-I gave25a rate constant of (6.1 f 2.2) X IO6 M-' s-l. This application required minor changes in the simulation to include unequal diffusion coefficients of substrate and catalyst, planar diffusion, and smaller values of the heterogeneous rate constant for electron transfer. Since the ratio (d) of diffusion coefficients of substrate:catalyst was correlated with kl in the regression analysis, d was fixed at an experimentally estimated value. Successful estimation of k , in this case can be used with the dimensionless kinetic parameter X = ( R T / f l P , ( k i / ~ )which , governs relative catalytic ~ u r r e n tto , ~obtain an upper limit of estimable k l . For the 1,2-dibromobutane/vitaminB12rsystem, the largest X used for regression was 15.4. Thus, assuming good data can be obtained at u = 50 V s-' and C*, = 0.3 mM for X = 15.4 yields an estimated upper limit on kl of lo8 M-' s-l, subject to the limitationof fixed d . In summary, the method to elucidate kinetic behavior of homogeneous electrocatalytic reactions is general and applicable to any catalytic system following the assumptions, offering to provide more precise and accurate values of rate constants. This is also important in the separate context of estimating thermodynamic parameters from free energy relationships for compounds which undergo slow electron transfer at electrode^.^,^ In addition, when mixed kinetic control can be attained, determination of both k , and k2 from the same experiment is possible. Assumptions of the simulation models are easily modified to apply to slightly different systems.
Acknowledgment. This work was supported by US. PHS Grant No. BO3154 awarded by the National Institute of Environmental Health Sciences. Registry No. 4-CB, 205 1-62-9;4-BB, 92-66-0; phenanthridine, 22987-8; phenanthridine anion radical, 345 18-98-4.
Temperature and Pressure Dependence of the Rate Constant for the Addition of H to
Phillip D. Lightfoot and Michael J. Pilling* Physical Chemistry Laboratory, Oxford, OX1 3Q2,U.K. (Received: October 10, 1986; In Final Form: February 9, 1987)
-
The rate constant for the reaction H + C2H4 C2HShas been measured, as a function of temperature and pressure, over the ranges 285 Q T/K d 604 and 50 Q p/Torr d 600, in a helium diluent, by laser flash photolysis/resonance fluorescence. The limiting high- and low-pressure rate constants, k,"(T) and klO(T ) , were obtained from nonlinear least-squares fits to the data using the Troe factorization procedure, with a transition-state model proposed by Hase and Schlegel. The values for kl"(T) were combined with previous data, obtained at lower temperatures, to give an overall Arrhenius expression: k,"(7') = (4.39 f 0.56) X lo-" exp(-(1087 A 36)/7') cm3 molecule-' s-l, where the uncertainties represent 95% confidence limits, as returned directly from the data analysis. The falloff parameters obtained from the fits to the experimental data were employed to provide estimates of k l ( P , T ) over the temperature range 600-1200 K.
Introduction The addition of hydrogen atoms to ethylene H + C,H, --.* C2H5
C2H5 CzH4 + H (R2) are of central importance in the high-temperature pyrolysis of alkanes and, in particular, of ethane, especially at the high conversions which pertain in industrial crackers. (R2) is also important in combustion. ( R l ) has been widely studied at low +
(~1)
and the reverse decomposition 0022-3654/S7/2091-3373$01.50/0
0 1987 American Chemical Society
3374 The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 temperatures by both flow and static techniques,'-29and there have been two extensive determinations of the high-pressure limiting rate constant, k,", using flash photolysis/resonance f l ~ o r e s c e n c e ~ ~ and pulse radiolysis/resonance absorption,28which together cover the temperature range 193-461 K. The reaction has also been the subject of theoretical investigation, especially by Hase and c o - w ~ r k e r s .There ~ ~ ~ ~is general agreement that there is a small activation barrier to (RI) and that, in consequence, the reaction occurs on a type I potential surface, with a transition state whose geometry does not depend significantly on excess energy within the thermal range. Hase and S ~ h l e g e l developed )~ a model for this transition state which agrees satisfactorily with both k,"( 7') and k2"( 7'). Although reaction R1 was the first to be studied by flash photolysis/resonance fluorescence4 and several room temperature studies of the effect of pressure on the rate constant have been p ~ b l i s h e d , ~ * ~ , ~the J ~falloff J ~ J * behavior ,~~ at high temperatures is not well established. This omission has severe consequences for the modelling of high-temperature cracking since, at typical temperatures, the reaction is well into the falloff regime. The aim of the present investigation is to extend the temperature range and study the reaction in the falloff region at elevated temperatures. The resulting experimental data, when coupled with the transition-state model of Hase and S ~ h l e g e lprovide , ~ ~ a good test
(1) Yang, K. J . Am. Chem. Soc. 1962,84, 719. (2) Strunin, V. P.; Dcdonov, A. F.; Lavrovskaya, G.K.; Talrose, V. T. Kinet. Caral. 1966, 7, 610. (3) Michael, J. V.; Weston, Jr. R. E. J . Chem. Phys. 1966, 45, 3632. (4) Braun, W.; Lenzi, M. Discuss. Faraday SOC.1967, 44, 252. (5) Brown, J. M.; Thrush, B. A. Trans. Faraday SOC.1967, 63, 630. (6) Michael, J. V.; Osborne, D. T. Chem. Phys. Lett. 1969, 3 , 4 0 2 . (7) Wooley, G. R.; Cvetanovic, R. J. J . Chem. Phys. 1969, 50, 4697. (8) Cvetanovic, R. J.; Doyle, L. C. J . Chem. Phys. 1969, 50, 4705. (9) Knox, J. H.; Dalgleish, D. G. Znt. J. Chem. Kinet. 1969, 1 , 69. (IO) Westenberg, A. A,; de Haas, N. J . Chem. Phys. 1969, 50, 707. (11) Eyre, J. A,; Hikada, T.; Dorfman, L. M. J . Chem. Phys. 1970, 53, 1281. (12) Kurylo, M. J.; Peterson, N. C.; Braun, W. J . Chem. Phys. 1970, 53, 2776. (13) Halstead, M. P.; Leathard, D. A,; Marshall, R. M.; Purnell, J. H. Proc. R. Soc. London 1970, A316, 575. (14) Barker, J. R.; Keil, D. G.; Michael, J. V.; Osborne, D. T. J . Chem. Phys. 1970, 52, 2079. (15) Hikada, T.; Eyre, J. A.; Dorfman, L. M. J . Chem. Phys. 1971, 54, 3422. (16) Penzhorn, R. D.; Darwent, B. de B. J . Chem. Phys. 1971,55, 1508. (17) Teng, L.;Jones, W. E. J . Chem. Soc., Faraday Trans. 1 1972, 68, 1267. (18) Michael, J. V.; Osborne, D. T.; Suess, G. N. J . Chem. Phys. 1973, 58, 2800. (19) Mihelcic, D.; Schubert, V.; Hofler, F.; Potzinger, P. Ber. Bunsen-Ges. Phys. Chem. 1975, 79, 1230. (20) Cowfer, J. A.; Michael, J. V. J . Chem. Phys. 1975, 62, 3504. (21) Pratt, G.; Veltman, I. J. Chem. Soc., Faraday Trans. I 1976, 72, 1733. (22) Gordon, E. B.; Ivanov, B. J.; Perminov, A. P.; Balalaev, V. E. Chem. Phys. 1978, 35, 79. (23) Lee,J. H.; Michael, J. V.; Payne, W. A,; Stief, L. J. J . Chem. Phys. 1978, 68, 1817. (24) Isikawa, Y . ;Yamabe, M.; Ncda, A.; Sato, S.;Bull. Chem. Soc. Jpn. 1978, 51, 2488. (25) Oka, K.; Cvetanovic, R. J. Can. J. Chem. 1981, 57, 777. (26) E M ,R.; Potzinger, P.; Reimann, B.; Camilleri, P. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 407. (27) Sugawara, K.; Okazaki, K.; Sato, S. Bull. Chem. SOC.Jpn. 1984, 54, 358. (28) Sugawara, K.; Okazaki, K.; Sato, S. Bull. Chem. SOC.Jpn. 1981,54, 2872. (29) Sugawara, K.; Okazaki, K.; Sato, S . Chem. Phys. Lett. 1981, 78,259. (30) Hase, W. L.; Mrowka, G.; Brudzynski, R. J.; Sloane, C. S. J . Chem. Phys. 1978, 69, 3548. (31) Hase, W. L.; Wolf, R. J.; Sloane, C. S . J . Chem. Phys. 1979. 71, 2911. (32) Hase, W. L.; Ludlow, D. M.; Wolf, R. J.; Schlick, T. J . Phys. Chem. 1981. 85. 958.
(33) Hase, W. L.; Wolf, R. J.; Sloane, C. S . J. Chem. Phys. 1982, 76, 2771. (34) Hase, W. L.;Schlegel, H. B. J . Phys. Chem. 1982, 86, 3901. (35) Hase, W. L.; Buckowski, D. G.; Swamy, K. N. J. Phys. Chem. 1983, 87, 2754. (36) Swamy, K. N ; Hase, W. L. J. Phys. Chem. 1983, 87, 4715
Lightfoot and Pilling 210 - 0 3 c
I
1
-1 0
I
I
I
00
10
20
30 tlms
Figure 1. Hydrogen atom decay trace at 285 K, 100 Torr of He, laser [N,O] = pulse energy = 58 mJ, [C,H,] = 1.55 X lOI5 molecule 1.0 X l O I 5 molecule ~ m - [H,] ~ , = 2.0 X 10l6molecule cm-). In the final fit, the points have been weighted by a quadratic fit to the moduli of the residuals obtained from a previous unweighted fit.
of Troe's factorization model for a type I p ~ t e n t i a l . ~ ' By . ~ ~fitting the data to this model, parameters are determined which permit extrapolation to temperatures and pressures which pertain in industrial crackers (- 1100 K, 2 atm). Extrapolation to 800 K shows excellent agreement with the limited data obtained in a recent study of equilibration in the H + C2H4/C2H5system,39 and further extrapolation, when coupled with thermodynamic parameters, shows satisfactory agreement with experimental measurements of k2 at temperatures in the range 673-913 K.
Experimental Section The apparatus is described in detail e l ~ e w h e r e . ~Hydrogen ~.~ atoms were produced by excimer laser flash photolysis of N 2 0 at 193 nm in the presence of H2 and with He as buffer gas. The time dependence of the H atom concentration was monitored by resonance fluorescence, following excitation by a microwave powered Lyman a discharge lamp. The temperature and laser power were controlled and measured as before. Total pulse energies in the range 15-1 10 mJ were employed, and the total area of the laser beam at the reaction zone was 10.5 cm2, The data, obtained by averaging, typically, over 500 shots, were analyzed by using a nonlinear least-squares fitting routine based on the Marquardt algorithm4' with two variable parameters, the apparent first-order decay constant, kapp,and the fluorescence intensity at zero time, Io. Figure 1 shows a typical decay trace at 285 K. The scatter of the residuals about zero demonstrates the decreasing noise level with decreasing signal strength and illustrates the importance of weighting the data. Poisson statistics are inapplicable and so a more heuristic approach was adopted. Initially, the data were fitted to an exponential by using an unweighted procedure and a quadratic was then fitted to the moduli of the resulting residuals. The coefficients of this fit were then employed to calculate the variance for each data point and the reciprocal variance was then used as a weighting factor in a subsequent fit of the data to an exponential decay. In practice, comparatively small changes were observed in the fitting parameters, so that further iteration w a s n o t necessary; an important benefit of this procedure is that it provided a good estimate of the sample variance and, hence, of the parameter uncertainties. At the higher temperatures, the increase in the rate constant enabled experiments to be performed at lower ethylene concentrations. An enhancement of the signal to noise ratio resulted, (37) Troe, J. Ber Bunsen-Ges. Phys. Chem. 1983, 87, 161. (38) Gilbert, R.; Luther, K.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 169. (39) Brouard, M.; Lightfoot, P. D.; Pilling, M. J. J . Phys. Chem. 1986, 90,445. (40) Brouard, M.; Macpherson, M. T.; Pilling, M. J.; Tulloch, J. M.; Williamson, A. P. Chem. Phys. Lett. 1985, 113, 413. (41) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969.
The Journal of Physical Chemistry, Vol. 91, No. 12, 1987 3375
Rate Constant for the Addition of H to C2H4 since C2H4 absorbs strongly at 121.6 nm. Figure 1 illustrates, therefore, the poorest signal/noise ratio studied.
Results and Discussion 1 . Characterization of the Photolysis System. A problem associated with the precise measurements of atom/molecule reactions is interference from atom/radical reactions. A small, but significant, contribution to the decay of the atom can arise from such processes even when a sensitive technique such as resonance fluorescence is employed. The interfering radicals arise in two ways: (i) as a coproduct, with the atom of interest, in the initial photolysis; (ii) as a product of the atom/molecule reactions. The second radical source cannot be eliminated from the reaction system but the first can, by careful choice of the atom precursor. In this section we discuss the use of N z O photolysis at 193.3 nm in the presence of H 2 and demonstrate that, at least at higher temperatures, it acts as a clean H atom source. N 2 0 was employed as the photolysis source at 193.3 nm N20
-
N2
+ O'D
(R3)
H2 was added to convert O'D to H O'D
+ H2
-
OH
+H
(R4)
At the higher temperatures, H2 also converts OH to H OH
+ H2
-+ H
H20
whilst, a t lower temperatures, OH reacts with C2H4 OH
+ C2H4
products
(R6)
The A'A-X'Z+ transition of N 2 0 peaks at 180.6 nm and has cm2.4z an absorption cross section at 193.3 nm of 1.1 X Photolysis in this region has been extensively studied by Cvetanovic et a1.43-45and by Preston and Barr,& who demonstrated that the quantum yield (4) for O'D production exceeds 0.95 whilst 4(N4S) < 0.02 and +(03P) < 0.03. N o evidence was found for 0's production. Direct photolytic production of 0 3 P is therefore insignificant, but deactivation of O'D to 0 3 P would present a complication, since 0 3 P C2H4 has a rate constant similar to k , and has a channel leading to H.47,48 The reaction between O'D and H2is rapid, with a temperature-independent rate constant cm3 molecule-' s - ' ; ~ at ~ least 95% of the reaction of 1.25 X proceeds via (R4) at room t e m p e r a t ~ r e . ~The ~ importance of physical quenching of O'D by H2 at higher temperatures was checked by examining the time dependence of the H atom resonance fluorescence signal in N20/Hz/He mixtures at temperatures up to 800 K. The hydrogen pressure was varied and, under the conditions employed, 03P would have been discerned by a short-time buildup in the H atom signal following the reaction
+
0 3 P + H2
-
H
+ OH
(R7)
N o such buildup was observed, and the H atom decay, under all conditions, was compatible with diffusive showing that 03P production by quenching of O'D by H2was also insignificant at higher temperatures. Quenching by the buffer gas was obviated by employing helium.52 (42) Okabe, H. Photochemistry of Small Molecules; Wiley: New York, 1978. (43) Yamazaki, H.; Cvetanovic, R. J. J . Chem. Phys. 1964, 41, 3703. (44) Preston, K. F.; Cvetanovic, R. J. J . Chem. Phys. 1966, 45, 2888. (45) Paraskevopoulos, G.; Cvetanovic, R. J. J. Am. Chem. SOC.1969, 91, 7572. (46) Preston, K. F.; Barr, R. F. J . Chem. Phys. 1971, 54, 3347. (47) Browarzik, R.; Stull, F. J . Phys. Chem. 1984, 88, 6004. (48) Hunziker, W. E.; Kneppe, H.; Wendt, H. R. J. Photochem. 1981.17, 377. (49) Schofield, K. J . Photochem. 1978, 9, 55. (50) Wine, P. H.; Ravishankara, A. R. Chem. Phys. 1982, 69, 365. (51) Brouard, M. D.Phi1. Thesis, Oxford University, 1986.
The final potential complications concerning O'D are its reactions with N 2 0 and C2H4. O'D reacts rapidly with NzO, with cm3 molecule-' s-153 a total rate constant of 1.2 x O'D
+N20
-
N2
+ O2
2N0
(R8a) (R8b)
and a branching ratio, ksb/k8,of 1.6.54 Reaction of O'D via (R8) was kept below 5% of the total by employing [H2]/[N20] ratios of at least 20:l. DeMoreS5showed that O'D reacts rapidly with CzH4 at 87 K by both addition and insertion mechanisms O'D
+ CzH4
-
products
(R9)
Employing his rate constant ratio (k9/k4 = 2.6) suggests that, at the highest ethylene concentrations employed at 285 K, up to 25% of the O'D would react with C2H4. However, plots of the experimental rate constant, k l , vs. [CzH4]were found to be linear over the range [C2H4]/[H2]= 0.015 (where (R9) is unimportant) up to [C2H4]/[Hz]= 0.1, demonstrating that any reaction of O'D with C2H4 does not significantly affect the analysis of reaction. At higher temperatures, the ratio of the ethylene and hydrogen concentrations was such that (R9) was insignificant. Ideally, sufficient H2should be employed to ensure that (R5) is complete at times short compared with the experimental time scale of 1 5 0 0 ps, thus requiring k5[H2] 5 2 X lo4 s-I. Baulch et al.54give k5 = 4.19 X 10-'9p,44exp(-1281/T) cm3 molecule-' s-I, so that, a t 300 K, hydrogen pressures of -100 Torr are required. Such high pressures are unacceptable in a study of the pressure dependence of kl and would also lead to a significant decrease in sensitivity because of H2quenching of the Lyman a resonance fluorescence. In the experiments at 285 K, therefore, the hydrogen pressure was kept low (
