1593
USEOF TUBULAR FLOW REACTORS FOR KINETIC STUDIES OVER EXTENDED PRESSURE RANGES
The Use of Tubular Flow Reactors for Kinetic Studies over Extended Pressure Rangeslalb by Robert V. Poirier and Robert W. Carr, Jr.* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota (Received January 22, 1971)
66466
Publication costs assisted by the U. S. Atomic Energy Commission
The continuity equations for first- and second-order chemical reactions, coupled with radial diffusion, were solved numerically for isothermal, incompressible, viscous-flow tubular reactors. Radial concentration profiles were averaged by integrating over reactor geometries closely approximating the experimental arrangements for some techniques commonly used in tubular flow reactor experiments. The applications of the results to gas-phase systems were explored, and a comparison with experiments on hydroxyl radical kinetics was made. The calculated and experimental results are in accord with one another. A simple method is described in which plots of average concentration vs. axial distance can be used to extend the useful pressure range of tubular flow reactors for fundamental rate studies from the low-pressure regime where rapid radial diffusion rates allow use of a uniform residence time assumption to higher pressures where parabolic concentration profiles exist.
Introduction When using steady-flow tubular reactors for kinetic studies, one encounters the problem of relating axial distance along the reactor with reaction time. I n the case of gas phase reactions, the experimental conditions are most frequently such that laminar flow occurs. The resulting parabolic velocity profile gives rise to a distribution of residence times for the reactive species in which those traveling near the wall remain in the reactor longer and experience larger extents of reaction than those traveling nearer the center of the reactor. This difficulty may be circumvented by operating at sufficiently low pressure and concentration that diffusional dispersion becomes extremely rapid. One can then assume, to a close approximation, that the radial concentration profile is flat. The concentration of reactive species is therefore independent of radial distance, and reaction time is equal to axial distance divided by average velocity. I n apparatus of the type and dimensions commonly used for kinetic studies it is usually a good assumption a t pressures of approximately 1Torr or less. The necessity of operating tubular reactors under conditions where the radial concentration profile is uniform has imposed severe limitations upon a widely used and otherwise versatile technique. With diffusion rates great enough to maintain a flat profile, the rapid transport to the reactor walls may allow heterogeneous reactions to occur at rapid rates and thus complicate the kinetic analysis. This difficulty will, of course, become more severe as operating pressure decreases. I n addition to heterogeneous effects, limitations imposed by detector sensitivity and increasing importance of viscous pressure drop impose restrictions upon operation a t the lower pressures. Experiments are thus
limited to a narrow range of pressures which handicaps the kinetic investigation. For example, the formation of chemically activated intermediates may go undetected, or the possible importance of a termolecular combination reaction may not be adequately tested experimentally. I n the operation of a tubular flow reactor a t higher pressures it is necessary to know the concentrations of reacting species as a function of both radial and axial distance in order to extract accurate values of reaction rate coefficients from the data. The required information can be obtained by solution of the continuity equation for diffusive and convective flow in the presence of chemical reaction. Bosmorthza obtained approximate solutions for a viscous-flow tubular reactor and discussed the effects of reactor size upon extent of reaction compared to the plug-flow situation. He also discussed conditions where the effects of either axial or radial diffusion can be neglected. Walkerzbobtained asymptotic solutions to the continuity equation for concurrent first-order homogeneous and heterogeneous reactions in Poiseuille flow with both radial and axial diffusion, The results were used as a guide for estimating the error introduced through using a one-dimensional model. Numerical integration techniques have been used to treat the case of first-order3 and second-order4 homogeneous reactions in order to determine the effects of a (1) (a) Work supported by U.S. Atomic Energy Commission, Contract No. AT(11-1) 2026. This is AEC document No. COO-2026-5; (b) paper presented in part a t the 160th National meeting of the American Chemical Society, Chicago, Ill., Sept 1970. (2) (a) R. C. L. Bosworth, Phil. Mag., 39, 847 (1948); (b) R. E . Walker, Phys. Fluids, 4, 1211 (1961). (3) F. A. Cleland and R. H. Wilhelm, A.I.Ch.E. J., 2 , 489 (1956). (4) J. P. Vignes and P. J. Trambouze, Chem. Eng. Sci., 17, 73 (1962). The Journal of Physical Chemistry, Vol. 76, No.10,1971
1594
ROBERT V. POIRIER AND ROBERT W. CARR,JR.
nonuniform concentration profile upon extent of reaction (ie.,reactor productivity). Krongelb and Strandberg6 also used numerical methods to examine mass transport with second-order homogeneous and firstorder heterogeneous reactions, and discussed conditions for minimizing the wall reaction. The effects of termolecular combination reactions and convective flow on measurements of atom removal by catalytic surfaces in “side-arm diffusion” apparatus have been examined by Wise and Ablowa6 The results were also applied to evaluation of homogeneous atom combination kinetics. The problem of diffusion and chemical reaction also arises in investigations of ion-molecule reactions in flowing afterglow apparatus, and approximate treatments in which the highly efficient removal of ions at the reactor surface are taken into account have recently been published.? I n spite of the published work on solutions of the continuity equation in tubular flow reactors, to our knowledge there is not yet a general scheme for obtaining kinetic parameters from experiments where a parabolic concentration profile exists. The earlier approximate treatments2*tb and the later precise numerical results3v4were concerned almost solely with the qualitative and quantitative effects of a nonuniform residence time distribution upon the extent of a chemical reaction, which is the important chemical engineering question. The other investigation^^-^ have treated special cases having the factor of a catalytic surface in common. The objective of this research was to solve the continuity equation for some simple kinetic cases and to present the results in such a form as to be useful to kineticists for extracting reasonably accurate values of rate constants for elementary reactions from data obtained with a tubular reactor over a wide pressure range. Accordingly, numerical solutions for first- and secondorder kinetics were obtained. The theoretical model was verified by comparison with experimental results for some hydroxyl radical reactions.
tary chemical reactions. The only cases of importance will be n = 1 and n = 2 since termolecular combination reactions, where the third body is an inert heat bath molecule, will follow a pseudo-second-order rate law. Bimolecular reactions between different chemical species can be made to follow pseudo-first-order kinetics by adding one reactant to large excess. The case of second-order reaction between different species present in similar concentrations was not treated in this investigation. Assuming that axial diffusion is negligible in comparison to bulk flow in the axial direction, the dimensionless form of the continuity equation is
with 8 = 1 at X = 0, b8/bX = 0 at u = 0, and - ab8/du = p8 at u = 1; where u = r/R, 8 = c/co, and n = kinetic order of the chemical reaction. For n = 1, X = kx/vo, a = D/kR2, and p = k,/kR, while for n = 2, X = kCox/vo,a = D/lccoR2,and p = k,/lccoR. Equation 2 was solved for 8 as a function of u and X in an isothermal system by the method of finite differences for n = 1 and n = 2 on the University of Minnesota CDC 6600 digital computer. The case of homogeneous firstor second-order reaction with concurrent first-order wall reaction was handled by letting the parameter p in the boundary condition - a b8/bu = p8 have positive values. For homogeneous reaction only, p was set equal to zero. For first-order kinetics, the treatment followed Cleland and Wilhelm.3 The treatment of the second-order case is outlined in the Appendix. I n each case the size of the increments was decreased until no significant gain in accuracy resulted. Concentrations were averaged over radial direction by three different methods =
1’
8 du
(3)
8udu
(4)
1
cz = Z i
Theory The conditions under which many flow experiments are done may be modeled by laminar, incompressible flow. Also, a system composed of a diluent plus several components at low concentration can be treated as a binary system composed of the diluent and the component of interest.8 Under these conditions the continuity equation for a binary system of constant density with chemical reaction is
+ z2-k kC”
(1)
with the boundary and initial conditions c = co at x = 0, &/at = 0 at r = 0, and -Dac/& = k,C at r = R. The form Of the rate law used in eq ” kc‘’ aJ’ises since this work is Solely concerned with elemenThe Journal of Physical Chemistry, Vol. 76,No. 10,1971
1
c3 = 4 l [l - uz]8udu
(5)
Equation 3 is the average along a principal diameter of the reactor, and is the average measured by the apparatus used for the experiments described below. Equation 4 averages over a cross section, such as might be (5) 8 . Krongelb and M. W. P. Strandberg, J . Chem. Phys., 31, 1196 (1959). (6) H . Wise and C. M. Ablow, ibid., 35, 10 (1961). (7) R. W. Huggins and J. H. Cahn, 1.A p p l . Phys., 38, 180 (1967); E. E. Ferguson, F. C. Fehsenfeld, and A. L. Schmeltekopf, Advan. A.M ~ ZPhys., . 5, 1 (1969) ; R. c. Bolden, R. s. Hemsworth, M . J. Shaw, and N. D. Twiddy, J. Phgs. B, 3,46 (1970); A. L.Farragher, Trans. Faraday Soc., 66, 1411 (1970). (8) R. B. Bird, W. E. Stewart, and E. N. Lightfoot, “Transport Phenomena,” Wiley, New York, N. y., 1960, 570.
USE OF TUBULAR FLOW REACTORS FOR KINETICSTUDIES OVER EXTENDED PRESSURE RANGES
1595
approximated by an esr detection system, and ( 5 ) is the L I cup mixing'' average. The concentration that would be measured by mass spectrometric detection with a pinhole leak on the reactor axis would be obtained by calculating e us. h at u = 0, and no averaging procedure need be done.
Experimental Section The apparatus was similar to that used by Kaufman and Del G r e ~ o . The ~ reactor was a quartz tube 3 cm in diameter by 150 cm long. Reactants diluted by helium could achieve flow velocities of up to 6 X lo3 cm sec-l with a Kinney KDH 130 high-capacity vacuum pump. At the upstream end a 2-cm diameter glass ring formed from 3-mm 0.d. Pyrex tubing and pierced with 30 small holes appropriately spaced around the inner and outer circumference served as a dispersing inlet for NOz. The effectiveness of mixing was tested by observing chemiluminescence from the reaction of atomic oxygen with nitric oxide.1° The emission was of uniform intensity at the inlet, and no striations were observable downstream. Also, an experiment was done in which helium was added to the nitrogen dioxide before the inlet ring. This increased the velocity of the gas emerging from the ring, thus increasing the effectiveness of mixing. KO change in the hydroxyl decay characteristics were observed, which was taken as evidence in favor of good mixing without added helium. Hydroxyl radical concentrations were determined by the line absorption technique. The light source consisted of 0.6% by volume of water vapor in argon at 18 Torr flowing in a 13-mm 0.d. quartz tube through a Fehsenfeld, Evenson, and Broidall designed microwave cavity. The cavity was powered by a Raytheon PGM-10 generator operating at 10% of power, and was tuned such that a hydroxyl rotational temperature of approximately 500°K was maintained. A narrow beam of light from the source passed through the reactor three times within a 3-mm axial distance before entering the 0.5-m grating spectrometer which isolated either the P12 or the Q14line, which were interchangeably used as absorption sources with no difference in results. The IP28 photomultiplier output was displayed on a 1-mV strip chart recorder. The light source and monochromator traversed the axial reactor direction on their lathe-bed carriage mounting. Flow rates were measured with Brooks rotameters which were calibrated, with the exception of NO2,with a "wet test meter." The NOa rotameter was calibrated by a weighing technique. Nitrogen dioxide was supplied from a Pyrex reservoir of liquid maintained a t 0". All other gases were taken directly from cylinders. The 99.9% helium and argon were from Air Reduction Co., 99.970 hydrogen from National Cylinder Gas, while 99.9% carbon monoxide, 99.5% nitrogen dioxide, and CP ethylene were from Matheson. Atomic hydrogen was generated by passing molecular
x Figure 1. Solutions of the continuity equation for a first-order reaction. Average concentration = C1, p = 0.
A
Figure 2. Solutions of the continuity equation for a first-order reaction. Average concentration = C2, p = 0.
hydrogen containing 3% water vapor through a watercooled Fehsenfeld, Evenson, and Broidall microwave cavity coupled to a PGM-10 generator. The cavity was on a quartz side arm located 10 cm upstream from the nitrogen dioxide inlet. The yield of atomic hydrogen was measured by a tandem platinum calorimetric probe technique. l 2 Complete dissociation of hydrogen was obtained for the flow rates most commonly used in this work. The temperature in the reaction zone was measured by a glass-enclosed copper-constantan thermocouple which was inserted through a close fitting rubber sleeve at the downstream end of the reactor. Under typical operating conditions the temperature variation in the reaction zone varied as much as 20-30°K in the axial (9) F. P. Del Greco and F. Kaufman, Discuss. Faraday SOC.,33, 128 (1962). (10) P. Harteck, R. R. Reeves, and G. Mannella, J. Chem. Phys., 29, 1333 (1958). (11) F. C.Fehsenfeld, K. M. Evenson, and H. P. Broida, Nat. Bur. Stand. Rept., No.8701 (1964). (12) R . V. Poirier, Ph.D. Dissertation, University of Minnesota, Minneapolis, Minn., 1970. The Journal of Physical Chemistry, Vol. 76,No. 10, 1971
ROBERTV. POIRIER AND ROBERT W. CARR,JR.
1596
4.01
I
I
I
I
1.0
2.0
3 .O
40
x
I 5.0
Figure 5. Solutions of the continuity equation for a second-order reaction. Average concentration = Cll p = 0.
x Figure 3. Solutions of the continuity equation for a first-order reaction. Average concentration = Ca, p = 0.
I
ob
I.o
I
I
I
30
4.0
5.0
I
2 ,o
x
Figure 6. Solutions of the continuity equation for a second-order reaction. Average concentration = Ci,p = 0.
x Figure 4. Solutions of the continuity equation for a first-order reaction. C?..Ois concentration evaluated along the reactor axis, p = 0.
direction. The reactor walls were coated with G.E. “Drifilm,” a mixture of silanes, to retard wall reactions. For the range of flow conditions used in these experiments, the Reynolds numbers varied from Re = 0.1 to 1.0. The maximum reactor entrance length (calculated from Le = 0.035 d Re, ref 8 p 47) was 0.2 cm.
Results Computer solutions of the continuity equation are displayed in Figures 1-9. For the first-order homogeneous reaction, Figures 1-3 are the solutions for the different average concentrations given by eq 3-5, respectively, and Figure 4 is the computer result for concentration along the reactor axis. Figures 5-8 are the corresponding solutions for a second-order homogeneous reaction, while Figure 9 displays the theoretical results for with a second-order homogeneous reaction and a first-order wall reaction. I n each of these plots the ordinate was deliberately chosen as the functional form of concentration (In c and l/c) commonly plotted vs. The Journal of Physical Chemistry, Vol. 76, No. 10,lD7l
ob
I
I
I
I
IO
20
3.0
40
x
I
so
Figure 7. Solutions of the continuity equation for a second-order reaction. Average concentration = C2, p = 0.
time for first- or second-order reactions. The abscissa is a dimensionless distance which may be converted to time if the concentration profile is assumed to be uniform. The figures are therefore in a conventional form used by kineticists. The concentration profile is flat only when a = m , and for this condition the slopes of Figures 1-8 are related to the true value of the reaction the plots for both firstrate constant. For a! < and second-order reactions show slight curvature near the origin, and a nearly linear region of lower slope than &a!=
a.
The concentration of hydroxyl is related to optical density by Beer’s law whenlo/I < i.5.1s 1
[OH] = H-log
Io I
(13) A. G . C. Mitchell and M.W. Zemansky, “Resonance Radiation and Excited Atoms,” Cambridge and New York, 1961.
USE OF TUBULAR FLOW REACTORS FOR KINETICSTUDIES OVER EXTENDED PRESSURE RANGES
1597
+=-ob
I
I
I
1.0
20
3.0
x
I
I
4.0
5.0
Figure 8. Solutions of the continuity equation for a second-order reaction. Cr-0 is concentration evaluated along the reactor axis, p = 0.
2.0 I
I
I
40
6.0
lrnsecl
Figure 10. Second-order hydroxyl decay plot. Least-squares slope = 8.2 msec-1. Experimental conditions: W N O ~ = 0.0180 m mol sec-1, W H ~= 0.0675 m mol sec-l, w t o t a l = 1.76 m mol sec-1: (OH), = 8.7 X 10-8 Torr; ij = 61.1 m sec-1; T = 336°K.
L
Pressure Dependence o f pnd Order Plots
a:O.2
slope E 6.5rnsedl 20
a.0.04
ob
I
I
I
I
I
10
20
30
40
50
x
slope=S.Ornsed' 1
1.0
0
1
,
,
/
1
2.0
,
,
,
,
1
1
,
,
3 .O
I,rnsec
Figure 9. Solutions of the continuity equation for a second-order reaction and wall reaction. Average concentratioii Ci, B = 0.07.
Figure 11. Pressure dependence of second-order hydroxyl decay plots; W N O ~= 0.0231 m mol sec-', W H ~= 0.068 m mol sec-l.
and a plot of log os. time should be linear if hydroxyl decays by a second-order rate law. 'Figure 10 shows such a plot obtained under typical experimental conditions and at lo^ pressure," e . g . , 1.05 Torr. The value of the extinction coefficient, H, for the P12 line was empirically determined. Its value was 4.55 f 0.25 X los cc mol-l at 340°K, which is in satisfactory agreement with the value 4.3 X lo8 cc mol-l which was calculated from the P12 line oscillator strength measured by Golden, Del Greco, and Kaufman.14 The linearity of Figure 10 and other similar plots obtained during this investigation supports the use of hydroxy! decay data to test the theoretical model. Accordingly, experiments were done over the pressure range 1-15 Torr. Work at higher pressures was not possible owing to the pump capacity limitation. Representative data are given in Figure 11. I n this series of experiments, all flow rates except the helium diluent were kept constant, and all other experimental conditions were the same.
The reproducibility of the slopes of the second-order plots was tested by doing a number of experiments at the same initial conditions. The conditions chosen were those of low [OH], where, owing to the low light absorbance, the greatest error is to be expected. The slopes were reproduced to within 10%. At higher [OH], such as in the experiments of Figure 11, the reproducibility was considerably better.
Discussion The computer results clearly show that experimental , plotted in data taken at pressures where a < ~ 3 and the conventional way, may appear to yield satisfactorily linear first- or second-order plots for homogeneous reactions. The expected curvature in the vicinity of the origin may be obscured by random experimental error, or by effects caused by inefficient mixing of the reactanh Thus the uncritical use of the relationship (14) D. M. Golden, F. P. Del Greco, and F. Kaufman, J . Chem. Phys., 39, 3034 (1963). The Journal of Physical Chemistry, Vol. 76, N o . 10,1971
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ROBERT V. POIRIER AND ROBERT W. CARR,JR.
t = z/fj to calculate reaction time in flow regimes where it is no longer warranted leads to rate constants which are smaller than the true value. The deviation may be as much as a factor of 2 in unfavorable cases. In practice, however, the calculation of reaction time by use of the uniform concentration profile approximation is probably satisfactory for 0.5 ,< a < a , or for even smaller values of a if accuracy is not necessary. It is apparent from Figures 1-8 that to obtain accurate values of rate constants in tubular reactors one must obtain data with experimental conditions such that a is large, or operate at other conditions and estimate the deviation of the apparent rate constant from the true value with the aid of theoretical results such as presented in Figures 1-9. The latter case may be necessary if it is not experimentally feasible to do the experiments with suitably large values of a. For example, in the second-order homogeneous case, if wall reactions are not negligible the behavior illustrated in Figure 9 is predicted, where the wall reaction is most pronounced at large values of a. I n the first-order homogeneous case with competing first-order wall reaction it is apparent that the first-order plots will remain linear, but will yield a larger spread of slopes between a = 0 and a = ~0 than those predicted by the first-order homogeneous case alone. An experimental procedure would be to operate at conditions such that the effect of diffusion is small (a small) and estimate a correction factor to the apparent rate constant based on Figures 1-8, where the wall reaction is not included. To estimate such a correction one must have values for D and IC. If an experimental value for D is not available, it can be estimated by the methods described in the literature. l6 The experimentally obtained apparent rate constant can be used as an estimate for the required rate constant when initially calculating a. At the extremes of a large and a small in the case of no wall reaction, and a small when wall reactions are important, the slopes of the theoretical curves are relatively insensitive to a and hence to estimated values of D and IC. Thus, only a rough estimate of D is required if the experiments can be conducted under such conditions. The calculation of a and X can then be done and an estimate made of the deviation of the apparent rate constant from the true rate constant. The corrected value can then be used to obtain an improved correction factor if this becomes necessary. The theoretical model was tested with the bimolecular disproportionation of hydroxyl radicals. Also, a limited test was achieved in the reaction of hydroxyl with ethylene, which followed a pseudo-first-order rate law under the experimental conditions used. According to previous investigation^,^*^^-*^ the atomic hydrogen-nitrogen dioxide reaction system can be described by reactions a-d
H
+ NO2
kzt
=
OH
+ NO
The Journal of Physical Chemistry, Vol. 76, N o . IO, 1971
(4
+ OH 0 + OH 0 + NO2
OH
kb
=
H2O
ko
=
H
kd
=
0 2
+0
(b)
+ + KO
(4
0 2
(d)
Since reaction a is very fast,17and since the experiments reported here were done with the initial conditions [H]o/[NOZ]O= 10, nitrogen dioxide is nearly completely consumed (98%) in the first centimeter of the approximately 20-cm reaction zone. Since IC, ‘v l o k b , a steady-state assumption on atomic oxygen has been made in the past, leading to the following rate law for the downstream region where the rates of reactions a and d are negligible.
(7) The validity of eq 7 was tested by solving the four simultaneous rate equations for reactions a-d on the CDC 6600 digital computer using the Runge-Kutta method. Using the cited values for the rate constants, and an initial excess of atomic hydrogen over njtrogen dioxide, the results of Figure 12 were obtained. Also plotted on Figure 12 is the solution of eq 7 assuming the same k b as used in the computer simulation. It is apparent that eq 7 is an adequate representation of the rate of disappearance of hydroxyl. A wall removal of hydroxyl was considered, and it was concluded that while a wall reaction may be occur-
I
0
I
I
1.0
20
30
I , mr
Figure 12. Computer simulation of hydroxyl decay for reactions a-d. Parameters chosen: k, = 5 X lo-” cc molecule-’ sec-1, k b = 2.5 x 10-12 cc molecule-‘ sec-l, IC, = cc 1.7 X lo-” cc molecule-1 sec-1, kd = 5.3 X molecule-’ sec-l; 0 , C o , ~ 0=2 6 X 1013 molecules CC-’, CU,H = 6 X 1014molecules cc-l: A, C o , ~ o=Z 3 X l O I 4 molecules cc-1, CO,H = 4.0 x 1014 molecules cc-1. (15) J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Liquids,” Wiley, New York, N. Y . , 1954. (16) F . Kaufrnan and F . P. Del Greco, Sump. (Int.) Combust. [Proc.], Qth, 659 (1963). (17) L. F. Phillips and H. I. Schiff, J. Chem. Phys., 37, 1233 (1962). (18) G. Dixon-Lewis, W. E. Wilson, and A. A . Westenberg, ibid., 44, 2877 (1966).
(19) A . A. Westenberg and PI’. De Haas, ibid., 43, 1550 (1965). (20) J. E. Breen and G. P. Glass, ibid., 52, 1082 (1970).
USE OF TUBULAR FLOW REACTORS FOR
IhNETIC
STUDIES OVER EXTENDED PRESSURE RANGES
ring it is not necessary for an adequate explanation of the qualitative experimental observations. During the early stages of experimentation slight curvature of the inverse hydroxyl concentration us. time curves, obtained with a large excess of H, was noticed. Consequently, the walls of the reactor were coated with "Drifilm," a solution of silanes. Xo appreciable changes in the curves were apparent. It was concluded that some curvature should be expected as indicated by an exact solution of the proposed mechanism of reactions a-d, Figure 12. Computer simulation revealed that the curvature is greatly increased when there is only a slight excess of atomic hydrogen, and this was experimentally verified. The value of k b obtained by setting the slope of cc moleFigure 10 equal to 3JCbI.x was 2.04 X cc cule-' sec-'. This is between the values 2.65 X molecule-' sec-' reported by Dixon-Lewis, Wilson, and Westenberg,18 and 1.4 X cc molecule-' sec-' reported by Kaufman.21 Although it was not possible to ascertain whether wall reactions involving hydroxyl occurred in these experiments, a correction for a possible first-order wall removal of hydroxyl, having a rate constant 16," = 80 sec-', obtained from independent experiments,22reduced k b to 0.68 X 10-l2 cc molecule-' sec-'. This is close to the value 0.84 X 10-l2 cc molecule-' sec-l reported by Breen and Glass,20who are the only others to have made such a correction. Furthermore, values of k b obtained from plots similar to Figure 10 showed a previously unreported dependence upon nitrogen dioxide flow rate; k b decreased with increasing nitrogen dioxide flow rate, all other conditions being held constant. The range of rate constants obtained, assuming eq 7, was from 4.65 X cc molecule-' sec-l to 0.91 X cc molecule-' sec-' vhich encompasses the range of previously reported values. Rate constants at the lower end of the range were obtained under conditions close to those used by Kaufman, et a1.,16 while those at the upper end of the range were obtained at conditions approaching those of Westenberg, et al.'81'g Equations a-d are incapable of accounting for this effect even if removal of hydroxyl at the wall is included. Regardless of the correct value of k b , the apparent linearity of Figure 10 as well as other similar plots supports the use of hydroxyl decay data as a test of the theoretical model for a second-order reaction. Experimental data were obtained at constant flow rates of nitrogen dioxide and atomic hydrogen but at varying pressures, Figure 11. The slopes of these plots decrease with increasing pressure as js predicted by theory, except at 15 Torr, where a slight increase over the slope at 10 Torr was observed. The binary diffusion coefficient for hydroxyl in helium was calculated15 to be D = 830/P cm2 sec-', from which the parameter 01 was calculated to be 0.5 at 1.05 Torr, 0.2 at 2.65 Torr, and 0.05 at 10.0 Torr. The slope of the experimental
1599
curves decreased by a factor of 1.45 over the 10 Torr pressure range involved, while inspection of Figure 5 indicates that a decrease in slope, due to the change in the radial concentration profile, of a factor of about 1.2 would be expected for the range of 01 covered. At 10 Torr, the apparent second-order rate constant is 1.40 X 10-l2 cc molecule-' sec-'. Correction by the theoretical factor of 1.2 yields k b = 1.68 X 10-l2 cc molecule-' sec-', compared with the experimental value, cc molecule-' sec-l, obtained at 1.05 Torr. 2.0 X The approximately 20y0 discrepancy may be due in part to experimental error, and in part to the effects of wall removal of hydroxyl which makes a greater contribution to the rate of hydroxyl decay at 1 Torr than at 15 Torr. The increase in the apparent value of k b observed at 15 Torr may be due to the occurrence of
have reported k, = 0.85 X Caldwell and cm6 molecule-2 sec-I. Using this value along with k b = 2.0 X 1O-lZ cm3 molecule-' sec-', R,/Rb = 1.3 X at 1 Torr and 0.19 at 15 Torr, where R stands for differential reaction rate. Thus at 15 Torr the rate of termolecular hydroxyl combination becomes an important additional path for hydroxyl decay, whereas at 1 Torr it is negligible. I n view of the preceding considerations a precise, quantitative test of the second-order case was not attempted. The general agreement between theory and experiment , however, was considered satisfactory. A pseudo-first-order reaction, the addition of hydroxyl t o ethylene in excess ethylene, was briefly studied over the narrower pressure range 1-5 Torr. The apparent first-order rate constant changed by approximately the amount predicted in Figure 1 over the range of the parameter 01 spanned by the experiments. The maximum effect of increasing pressure predicted by the results of this work is to cause the apparent rate constants to be about 50% smaller than their true value. The most deviation is predicted for axial sampling, i.e., mass spectrometry, while other experimental methods predict lesser deviations. We feel that the approximations made in solving eq 1 are not severe, since experiments can usually be designed to approximate isothermal and isobaric conditions. If these experimental conditions are attained, then the analysis presented here can be reliably used to estimate rate constants of elementary chemical reactions where data have been taken at "high" pressures in tubular flow reactors. (21) F. Kaufman, Ann. Geophys., 20, 106 (1964). (22) A. Pastrana, unpublished results, University of Minnesota. (23) J. Caldwell and R. A. Back, Trans. Faraday Soc., 61, 1939 (1965).
The Journal of Physical Chemistry, Vol. 76, N o . 10, 1971
1600
ROBERT V. POIRIERAND ROBERTW. CARR,JR.
Symbols C
concentration inittial concentration diffusion coefficient reactor diameter transmitted intensity of light from source incident intensity of light from source homogeneous rate constant heterogeneous rate constant pressure radius of reactor radial distance temperature time axial velocity at r = 0 flow rate axial distance
co D d
I Io k kw
P R r
T t
vo W
x
P
D/kCoR2 k,/kCoR
H
extinction coefficient
x
kCox/Vo
a
The predictive equation AX
I=
Appendix In difference form, eq 2 for n = 2 becomes
+
Cj+lK+'
Cj-lK+l
- 2CfK+1+ Cj+lK
a
c
is used to predict a value for CfK+l = PJK+l, Writing the equations in linear form
I+
+ C*-lK - 2CjK
~(Au)~ Cj+lK+l
-
C,-1K+1
+
Cj+lK
- Cj.2
4ujAu
1-
[ cfK+l
2
+
CfK12
or
COR -
CIK
+ CoK+l - CIK+' =
CNK
+
CN-lK+l
-
CNK+l
2Au
=
+
= -CoK
+
rBo AI 0
L i
at u = 1
Thus, one has N 1 simultaneous nonlinear partial differential equations. Using a method similar to that used by Bruce, Peaceman, Rachford, and Rice,24the equations are linearized
- ClK+1
=:
ANCN-lK+l
I
I=
@CNK+'
CoKfl
AjCj-IK+'
0
Writing the equations in matrix form
at
2Au
-
+ DoCIK+' = W " + BjCjK+l + DjCj+i"+' WjK + BNCNK+' = WNK
BoCoK+'
where the subscript refers to the increment in the radial direction and the superscript to the increment in the axial direction. The boundary conditions become
D~ BI A1
o
o
...I
Di B2
0
. ' * I [ C , K + l ] = [W,"]
De
...
J
Applying the appropriate operations to this equation, it becomes
C1K
(24) G. H. Bruce, D. W. Peaceman, H. H. Rachford, and J. D. Rioe, Trans. A I M E , 198, 79 (1953). The Journal of Physical Chemktry, Vola76,No. 10, 1071
1601
ADDITION REACTIONS OF H AND D ATOMS
From this equation the concentration is easily calculated.
where fo =
DO -'
Bo
fN
=
CjK+' = g,
0
WO Bo
go = -
- f,C,+P+'
The procedure is to obtain a value of PjK+lfrom the predictive equation, calculate g,, j j , and then CjK+I, compare these values with the predicted values, and if they are not within the required error repeat the procedure using the calculated CjKtl values as the predictive values. Repeat this procedure until the required accuracy is attained. A computer program has been written to solve these equations for first- and second-order reactions. l 2
Mass Spectrometric Studies of Rate Constants for Addition Reactions of Hydrogen and of Deuterium Atoms with Olefins in a Discharge-Flow System at 3 0 0 ° K by E. E. Daby, H. Niki,*and B. Weinstock Scient& Research S t a f , F w d Motor Company, Dearborn, Michigan 48131
(Reeeiaed December .29, 1970)
Publication costs assisted by the Ford Motor Company
Absolute rate constants for the addition reaction of H and of D atoms with a number of olefinic hydrocarbons have been determined directly at 25" using mass spectrometric detection in a fast discharge-flow system. The rate constants obtained range from 0.7 to 9 x 10'1* cm8 molecule-' sec-', and their ratios agree well with relative values reported by CvetanoviE and coworkers. The relevant reaction scheme is H
+
+ (o1efin)I
kf(f) kf
R1*; Rl* k,_ Rz (olefin)rI;R1* -% R1. Under the experimentalconditionsused,the observed bimolecular rate constants yield krcs, directly in most cases. The systematic variations of rate constant with molecular structure of the olefins are discussed.
Introduction When atomic hydrogen reacts with olefinic hydrocarbons, chemically activated alkyl radicals are formed. The subsequent unimolecular dissociation of these adducts has been the subject of extensive experimental and theoretical studies.' The absolute rate constants for the initial addition reactions are known to a much lesser degree2 with the exception of ethylene. Values of the absolute rate constant for the addition of atomic hydrogen to ethylene have been determined by a variety of experimental techniques, but show an order of magnitude spread and a critical choice of the most reliable value is difficult to make.3 Several measurements of relative rates for olefins also lack consistency, with the exception of three independent photochemical studies by CvetanoviC: and coworker^.^-^ In Table I, mean
values of the relative addition rates obtained by Cvetanovi6's group are compared with those of other systematic studies.',* In a previous note from this laboratory, mass spec(1) B. 8. Rabinovitch and D. W. Setser, Advan. Photochem., 3, 1 (1964). (2) For reviews mior t o 1964: R . J. CvetanoviE, ibid., 1, 115 (1963); B. A. Thrush, Progr. React. Kinet., 3, 65 (1965). (3) See, for example, Table I of ref 6. (4) K . R. Jennings and R. J. Cvetanovi6, J . C'hem.*Phys.,35, 1233 (1961). (5) G . R. Woolley and R. J. CvetanoviO, ibid., SO, 4697 (1969). (6) R . J. Cvetanovii! and L. C. Doyle. ibid., 50, 4705 (1969). (7) (a) P. E. M . Allen, H. W. Melville, and J. C. Robb, Proc. Roy. Soc., Ser. A , 218, 311 (1953); (b) J. N. Bradley, H. W. Melville, and J. C. Robb, ibid., 236, 318 (1956). (8) K. Yang, J. Amer. Chem. Soc,, 84, 719, 3795 (1962).
The Journal of Physical Chemistry, Vol. 76, No. 10, 1971
