9359
J. Phys. Chem. 1991, 95,9359-9366 100 I
a
dc discharge gave an NF(b) concentration of (2-3) X 10" molecules ~ m - as ~ ,well as highly vibrationally excited NF(b) molecules. Even higher concentrations of NF(b) can be expected with optimization of the discharge design, In addition to a more complete study of vibrational relaxation, the discharge source would permit investigation of energy-pooling reactions between NF(b) and other excited molecules to give higher energy species. The vibrational constants of the upper and lower states of N F have been refined from the band position measurements of the NF(b,u'll l--LX,u") transitions. The refined spectroscopic constants fit the band origins of the v ' l 2 transitions? as well as all the band positions from the full range of u'and ~"levels. It is noteworthy that the togx anharmonicity , terms are adequate to describe the vibrational energies of the NF(X) and (b) states up tov'oru''= 11. Since the stretch frequency of NF2, 1076 cm-l, is within 50 cm-I of that for NF(b), the vibrational relaxation mechansim for collisions of NF2 with NF(b,u') is probably V-Vexchange. The relaxation rate constant for u' = 1 by NFz, 3.5 X IO-" cm3 s-I, is large, and for u ' = 5 it has increased to 12 X IO-'' cm3 s-'. The mechanism for the vibrational relaxation by CH, is not so evident. Although the endoergic energy defect is 200 cm-I for V-Vrelaxation with CH,, the large rate constants imply a V-V mechanism rather than V + RT. Since the energy defect increases with u', this may explain why the rate constants do not increase much with vibrational level. A faster rate of vibrational relaxation of NF(b) by H e relative to Ar is expected for a V-T relaxation mechanism. Nevertheless, the relaxation rate constants for even the high v levels by Ar seem remarkably small. Additional investigation with different relaxation gases studied as a function temperature could give more insight into the vibrational relaxation mechanism@) for NF(b). The direct detection of NF(a) as a product in the quenching of NF(b) by CH4 and H 2 0 confirms the E-Vquenching mechanism that previously had been indirectly inferred.2 The agreement between the calculated and experimentally measured NF(a) concentrations from quenching of NF(b) by CH4and HzO seems to identify NF(a) formation as the exclusive product channel for the presently accepted 6-s radiative lifetime of NF(a).I4J5
-
0
1
2
14
3
4
5
[H20] 1 10 moiec c m 3
Figure 8. Decay of NF(b) (a) and the formation of NF(a) (b) as a function of [H,O].The [NF,] was 1.2 X IOi2 molecules ~ m - and ~ , the reaction time was 0.015 s. The arrow shows the complete conversion of NF(b) to NF(a) without the radiative decay of NF(b), the removal of NF(a) by wall quenching, or quenching of NF(a) by H20.
0.7s. Thus, the
7,
values from different studies are consistent.
Discussion and Conclusions The formation of NF(b) and SiC12(i33Bl)'6by passing NF2 or SiCI.,, respectively, through a simple dc discharge shows that in a few fortuitous instances, de discharges can generate high concentrations of metastable molecular states in a flow reactor. This (16)
Du,K.; Chen, X.;Setser, D. W. Chem. Phys. Left. 1991, 18, 344.
Acknowledgment. This work was supported by the US.Air Force Office of Scientific Research (Grant 88-0279).
Profiles of Chemlcal Waves in the Ferroin-Catalyzed Beiousov-Zhabotinskii Reaction Eugenia Mori, Igor Schreiber, and John Ross* Department of Chemistry, Stanford University, Stanford, California 94305 (Received: May 24, 1991)
We report measurements and numerical calculations of profiles of chemical waves in the ferroin-catalyzedBelousov-Zhabotinskii (BZ) reaction at various concentrations of reactants and study waves travelling in an excitatory as well as an oscillatory medium. The fronts of the waves in the excitatory medium are generally broader and somewhat bent, while the fronts of waves in the oscillatory mediums are narrower and straighter. In both cases, the measured concentration of ferriin, which defines the structure of the wave increases with increasing initial concentration of sulfuric acid and sodium bromate and decreases with increasing initial concentration of malonic acid. We model the waves by modifying the existing extended Oregonator model, and the concentrationsof ferriin of the numerically calculated wave profiles agree better with measurements than those of previous models.
1. Introduction
Chemical waves, which are travelling variations of concentration of chemical species or other state variables, may occur in systems far from equilibrium. Many experimental and theoretical studies have been reported on this subject.14 In this article we investigate ( I ) Vjdal, C.; Pacault. A. In Euolution of Order and Chaos; Haken, H., Ed.; Springer-Verlag: Berlin, 1982; p 74.
0022-3654191 12095-9359$02.50/0
the structure of chemical waves in the Belousov-Zhabotinskii (BZ) reaction catalyzed by ferroin ( 1 , IO-phenanthroline ferrous sulfate complex). The overall BZ reaction is a metal-catalyzed bromination of malonic acid occurring in an acidic solution: (2) ROSS,J.; Miiller, S.C.; Vidal, C. Science 1988, 240, 460. (3) Tyfon, J. J.; Fife, P. C. J . Chem. Phys. 1980, 73, 2224. (4) Muller, S. C.; Plesser, T.; Hess, B. fhysica D 1987, 2 4 0 , 87.
0 1991 American Chemical Society
9360 The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 2Br03-
+ 3MA + 2H+
-
2BrMA
+ 3C0, + 4 H 2 0
where MA and BrMA are shorthand for malonic acid and bromomalonic acid, respectively. Throughout the process, the metal may be in its original oxidation state or it may be oxidized one valence number higher. If the system is well mixed, some of the species involved may show sustained temporal oscillations in their concentrations and if the mixture is spread in a thin layer and not stirred, one can perturb the system to induce spatial variances in the concentrations of some of the species. These spatial variations in concentration may propogate through space and are then called chemical waves. If the solution sustaining waves has temporal oscillations of concentrations when stirred, the bulk solution is called oscillatory. If the solution has no oscillations in concentrations when stirred but on perturbation undergoes large variations in concentrations of some of the species, the bulk solution is called excitatory. Chemical waves can propagate in either an oscillatory or excitatory bulk solution. In the ferroin-catalyzed BZ system, chemical waves of ferriin (1 ,IO-phenanthroline ferric complex, the oxidized form of ferroin) are apparent to the eye as ferroin is red and ferriin is blue in solution. A wave appears as a ring of blue expanding through the red background. In some of the prior work, Wood and RossS made quantitative spectroscopic measurements on chemical waves of ferriin propagating through an excitatory solution of the ferroin-catalyzed BZ reaction, and Pagola and Vida16 observed waves travelling in an oscillating solution of the ferroin-catalyzed BZ reaction. NagyUngvarai et ale7studied experimentally and modeled chemical waves in the cerium-catalyzed BZ reaction. We record the wave profile, Le., the concentration of ferriin in solution as a function of space, at a number of concentrations of the initial reactants sulfuric acid, sodium bromate, malonic acid, and ferroin, and to better spatial resolution than in our previous work. The blue waves of ferriin propagating in the red ferroin solution are imaged by shining blue light on the solution in a glass Petri dish, and we record spatially resolved transmission of the light. The bulk medium is either excitatory or oscillatory. We observe the qualitative structure of the wave, the width of the wave front, the ferriin concentrations comprising the wave, and the wave velocity. We also model the waves numerically by modifying the extended Oregonator model used by NTH.I4 We compare our experiments with our model and with previous experiments done on both ferriin and cerium waves. In the waves studied, two qualitatively different wave profiles are observed: a steep, narrow wave front and a bent front that has both a steep and a less steep component, similar to observations in the cerium-catalyzed BZ reaction.' The width of the wave front is defined here as the distance required for the wave to attain full height against the red background. Widths of the steep and narrowest fronts are approximately 90 pm. Within the relatively narrow concentration ranges explored, no obvious trends are noted in the width of the wave front to within the experimental scatter of the measured widths, except the width of a wave travelling through an excitatory medium tends to increase with increasing initial bromate concentration. 11. Experimental Details
The apparatus is a modified version of that used in ref 5 . The 488-nm line of an argon ion laser is expanded to a beam width ( 5 ) Wood,P. M.; Ross, J . J . Chem. Phys. 1985, 82, 1924. (6) Pagola, A.; Vidal, C. J . Phys. Chem. 1987, 91, 501. (7) Nagy-Ungvarai, Z.; Mliller, S.C.; Tyson, J. J.; Hess, B.J . Phys. Chem. 1989, 93, 2760. (8) Ruesser, E. J.; Field, R. J . Am. Chem. Soc. 1979, 101, 1063. (9) Pagola, A.; Ross, J.; Vidal, C. J . Phys. Chem. 1988, 92, 163. (IO) W e t , J. M.; Ross. J.; Vidal, C. J . Chem. Phys. 1987, 86, 4418. ( I I ) Field, R. J.; K M s , E.; Noyes, R . M. J . Am. Chem. SOC.1972, 94, 8649. (12) Rovinsky. A. B.; Zhabotinskii, A. M. J . Phys. Chem. 1984.88, 386. (13) Dwyer, H. A. Physica D 1986, 20, 142. (14) Nagy-Ungvarai, Z.; Tyson, J. J.; Hess, B.J . Phys. Chem. 1989, 93, 707.
Mori et al. TABLE I: Standard Reactant Concentrations (M)Used for Both tbe Excitatorv and Oscillatorv Series std reactant concn ser A (excitatory) ser B (oscillatory) 0.500 [HzSO~I 0.333 0.310 [BrO3-1 0.199 0.500 [MA1 0.0430 [ferroin] 0.00294 0.00400 [BrMA] 0.0662 0.000 ~~
~
~
of 19 mm and directed with normal incidence into a Petri dish that contains the reaction solution. Part of the beam is transmitted through the dish, expanded again to magnify the image of the wave, and focused on a Reticon diode array that has 1024 photodiodes with a 25-pm center-to-center spacing. The magnification of the image is performed with a microscope objective consisting of two lenses with focal lengths 1 cm and 3.5 cm separated by 3 cm. The objective is positioned 2 mm beneath the Petri dish, and the detector is positioned 20.5 cm beyond the last objective lens, the distance at which an image on the Petri dish focuses on the detector. The magnification of the wave image is 8.9X, the field of view is 2.85 mm, and the spatial resolution after averaging every two points together is 5.6 pm. For comparison, the resolution of the waves imaged in the cerium-catalyzed BZ system' is 19.5 pmlpixel. The Reticon RL-1024G detector is controlled by a Reticon 77 R C lOOB motherboard which is in turn connected to an IBM PC/XT. One scan takes 5.2 ms, and scans were taken every second. The following chemicals are used without further purification: Fisher sulfuric acid, sodium bromate, sodium bromide, ferroin, and Aldrich malonic acid. Prepared solutions are passed through a 0.2-pm filter, and the amount poured into the Petri dish of 1 k m diameter forms a layer 0.88 mm deep. Waves are initiated by a 5-ms, 2-V pulse through Ag and Pt wires with diameter 0.25 ".I5 The temperature is 22.8 f 0.5 "C. The transmission of light is measured as function of space and time, and the concentration of ferriin a t that point in space and time is then determined by solving the Beer-Lambert law log ( I / I o ) = -d(erd[Fe2+]
+ t,,[Fe3+])
with the constraint [Fez+]
+ [Fe3+] = [FeZ+Io
for [Fe3+]. In the above equations, I is the intensity of light recorded by a single diode, Io is the intensity of the background reading for that diode, d is the depth of solution, and td and tm are the extinction coefficients for ferroin and ferriin. It is understood that Fez+ and Fe3+are shorthand for ferroin and ferriin, respectively. The values of the extinction coefficients used are erd = 10090 M-' cm-l and cox = 200 M-' cm-Is5 For each set of concentrations, about six waves are generated and measured. Typically only three or four of the waves have a regular and even profile, and only those waves are analyzed. For each wave, the asymptotic width of the front is determined, and the three or four asymptotic widths corresponding to the waves of a given set of initial concentrations are averaged to give the final determined width for that set of initial concentrations. The initial concentrations for the standard mixture of the excitatory solution are 0.333 M sulfuric acid, 0.199 M sodium bromate, 0.043 M malonic acid, 0.0662 M bromomalonic acid, and 0.00294 M ferroin, while the initial concentrations of the standard mixture for the oscillating solution are 0.50 M sulfuric acid, 0.31 M sodium bromate, 0.50 M malonic acid, 0.0040 M ferroin, and no bromomalonic acid. The initial concentrations of the reactants are calculated by assuming completion of the following reaction: Br03- + 2Br-
+ 3H++ 3MA
-
3BrMA
+ 3H20
where malonic acid and bromomalonic acid are abbreviated by ( 1 5) Showalter, K.; Noyes, R. M.; Turner, H. J . Am. Chem. SOC.1979, 101. 1463.
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9361
Ferroin-Catalyzed Belousov-Zhabotinskii Reaction
Series-A
Series43
1
1
1.o
0.0
2.0
3.0
dlalallcc ( a l l l l )
0.0
1.0
2.0
3.0
distance (mm)
distancc (mm)
1 0.0
distaricc
(iiiiii)
1.0
dis1:incc
2.0
3.0
(niin)
Figure 1. Experimentallyobtained wave profiles. The standard initial concentrations used in profile a of each series are listed in Table I. In profiles b-d the reactant concentration, which differs from the standard concentration, is listed, and all other concentrations are as in the standard for that series. In wave c of series A, two fronts travelling in opposite directions are shown.
MA and BrMA, respectively. These sets of standard concentrations are the same as those used in refs 5 and 6 and are displayed in Table 1 as well. 111. Experimental Results Two sets of experiments are performed. In series A the bulk medium is excitatory, and in series B the bulk solution is oscillatory. A standard set of initial concentrations, given in section 11, is chosen for both cases. Wave measurements are taken on the standard solution and then on a solution that has the same concentrations as the standard mixture but with the initial concentration of one reactant altered. Hence, the wave profile is monitored in its dependence on the concentration of one chemical species for both an excitatory and oscillatory medium, while the other concentrations remain a t their standard values. Figure 1 displays obtained profiles for each series. With the exception of profile c in series A, all profiles display one wave front
travelling to the left. Profile c of series A displays two fronts of the ring, both travelling outward. The first column shows profiles from series A, where the bulk medium is excitatory, and the second column shows series B, where the bulk medium is oscillatory. The corresponding standard concentrations were used in obtaining the top profile a of each series, the bromate concentration was changed in obtaining the profile b of each series, the sulfuric acid concentration was changed in obtaining profile c of each series, and the malonic acid concentration was changed in obtaining profile d of each series. Two qualitatively different types of wave fronts are observed: a straight front and a bent front. The straight front, line segment xy in Figure 1, series B, profile a, appears as a steep, linear front approximately 90 pm wide. The bent front, seen in Figure 1, series A, profile a, consists of two distinct segments: a first segment extending from point x to y is linear and steep, and a second segment extending from pointy to z is less steep and either linear
9362 The Journal of Physical Chemistry. Vol. 95, NO. 23, 1991
Mori et al,
TABLE 11: Comdlrtion of the Widths of Chemical Waves Measured for Various Reactant Concentrations‘ reactant upper bound of 1st width, pm 2nd width, pm reactant varied concn, M 1st width, pm Series A: Quiescent Medium, [bromomalonic acid],, = 0.0662 M sulfuric acid 0.333 116 f 27 155 f 31 311 f 27 sodium bromate
malonic acid ferroin
sulfuric acid sodium bromate malonic acid ferroin
0.400 0.500 0.199 0.250 0.310 0.043 0.250 0.500 0.00294 0.00350 0.00400
1 I5 110 f 8
116 f 27 140i 21 86 f 5 116 f 27 126 f 17 118 f 18 1 I6 f 27 162 f 60 82 f 7
204 161 f 14 155 f 31 187 f 17 129 f 8 155 f 31 163 f 21 156 f 24 155 f 31 225 f 66 117 f 8
282 f 63 311 f 27 352 f 34 406 f 83 311 f 27 165 f 23 336 f 73 311 f 27 183 f 84 298 f 12
Series B: Oscillatory Medium, [bromomalonic acidlo = 0.000 M 88 f IO 139 f 23 110 f 34
0.333 0.500 0.199 0.250 0.3IO 0.250 0.500 0.00294 0.00350
94 f 26 99f 13 16 f 7 94 f 26 87 f 6 94 f 26 128 f 52 94 f 22
upper bound of 2nd width, pm 408 f 37 402 f 80 408 f 37 476 f 29 558 f 58 408 f 37 215 f 31 421 f 96 408 f 37 255 f 101 381 f 13 171 f 36
137 f 27 141 f 18 132 f 38 137 & 27 145 f 22 137 f 21 210 i 85 138 f 21
‘The first two columns give the reactant varied and its concentration; the concentration of the other reactants are those of the corresponding standard set of concentrationsgiven in the experimental details. When only a first width is listed, the waves formed at that set of concentrations had a straight front whose width is reported as the first width. When both a first and a second width is given, the waves formed at that set of concentrations had a bent front: the first width is then the width of the first steep segment and the second width is the combined width of the steep and less steep segment.
or concave down. The bent front’s typical full width, Le., the distance between points x and 2, is 300-400 pm, which is much larger than the width of the straight front. The bent front was observed in all but one of the waves propagating in an excitatory medium, while the straight front was observed in all but one of the waves travelling through an oscillatory medium. We estimate by eye the point in space a t which the wave front begins, ends, and perhaps bends. For each wave, the scatter in the asymptotic width measured for that wave is given by a standard deviation of approximately 40 pm for the first widths and by a standard deviation of 100 pm for the second widths. The average reproducibility in the determined asymptotic width for many waves with a particular set of initial reactant concentration is 25 pm for the first width and 50 pm for the second width. For waves with a straight front, we estimate both an upper and lower limit on the width. For waves with a bent front, four estimates are made: the width of the first steep segment and an upper bound to this first width, and the combined width of both the steep and less steep segments with an upper bound to this full width. These results are listed in Table I1 for the various concentrations of reactants studied. For series A (B), the standard concentrations are those for an excitatory (oscillatory) medium, and the reactant concentration varied is listed in the first column. Within the scatter of the measured widths of the wave fronts, the width and structure of the front have no notable dependence on the initial concentrations of sulfuric acid or malonic acid within the concentration ranges studied. However, for waves travelling through an excitatory medium, the full width of the front tends to widen as the concentration of sodium bromate is increased, see Table 11. We further note the ferriin concentration of the medium where the wave has not reached and the ferriin concentration of the maximum of the wave. It is convenient to speak of the conversion level, which is defined here as the percent of catalyst in the solution that has been converted to its oxidized state at a particular point in space. In the standard excitatory medium, the baseline level of ferriin through which the wave travels corresponds to approximately 82% conversion of catalyst; and approximately 87% of the catalyst has been oxidized in the standard oscillating solution. This conversion level along the more reduced baseline is sensitive to the initial reactant concentrations as it increases with
increasing [H,SO,] or [ Br03-] and decreases with increasing [MA], as can be seen in both series of profiles. At the maximum of the wave, close to 100% of the catalyst present is converted to its oxidized state, and this maximum is less sensitive to the initial concentrations of the reactants. The lowest conversion for the maximum of the wave is observed in Figure 1, series B, profile b, and it is 90%. All other conversions at the maximum of the wave fall between 95% and 100%. The velocity of waves measured in the standard excitatory medium is 0.08 mm/s, while the velocity of waves measured in the standard oscillatory medium is 0.10 mm/s. The velocities were calculated by plotting the position of the maximum of the wave versus time and performing a least-squares fit to determine the slope of the line.
IV. Model We initially modeled the waves with the extended Oregonator model which Nagy-Ungvarai et al. proposed in ref 14. We refer to this model as NTH. Numerical solutions of the pair I1 of equations in ref 14 predict a much lower level of conversion of ferroin to its oxidized state along the baseline than that of 85% observed experimentally and predict velocities about 10 times greater than found in experiments when K - ~is kept at its nonzero value. We have found that when modeling ferriin waves with the NTH model, it is important to keep K - ~at its nonzero value even when the value of K - ~is approximately 10” or even smaller. If one sets K - ~= 0, then the calculated conversion to ferriin at the maximum of the wave far exceeds 100% of the catalyst at that point in space. This result is neither physically expected nor experimentally observed. Appropriate parameter values could not be found to attain agreement between experiments and calculations with respect to ferriin levels or wave velocities. The same equations along with the suggested rate constants of ref 14 also predict a maximum conversion of 100% at the top of the wave, which instead of falling off as the wave progresses is maintained over a macroscopic distance before falling off. Hence, the calculated profile has the appearance of a saturated flat top, which is not experimentally observed. This shortcoming, though, could be remedied by choosing a smaller rate constant for the step in which ferroin is oxidized.
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9363
Ferroin-Catalyzed Belousov-Zhabotinskii Reaction
2
To attain better agreement with the experiments, we targeted the organic step of the NTH model. The nonelementary organic process modeled by a single irreversible step referred to as process J by NTH org
+ Fe3+
-
hBr-
+ Fe2+ + oxidation products
(J)
is a known shortcoming of the Oregonator model. org represents organic species and h is a stoichiometric factor. We elaborate this process by breaking it into two reactions, one of which is reversible. Consider the reactions (8) and (9) of Rovinsky and ZhabotinskiiI2listed as (RZ8) and (RZ9) and the reactions of Jwo and NoyesI6 listed as (8') and (9'). R Z take
+ BrMA Fe3+ + M A BrMA' + H 2 0 BrMA + MA' + H 2 0 Fe3+
--
+ BrMA' Fe2+ + H+ + MA' hBr- + H+ + product(s) Fe2+ + H+
hBr-
- k8BZ + I 8 H ( C - Z)R 0 = 2klHAX - 2k-lU2 - k,HU(C - Z ) - k _ J Z l? = k8BZ - k-BH(C - Z)R - kgR k3HU(C - Z ) - k - 3 2
The steady-state approximation for R yields k8BZ R= k-SH(C - Z)+ k9 Transformation to the dimensionless concentration variables is done as in NTH:
(RZ8) (8') (RZ9)
as well as the transformation to the dimensionless time variable 7
+ H+ + M A + product(s)
(9') reaction RZ9 to be irreversible and suggest the following relationship for the rate constants of their proposed mechanism:
7
The conversions between dimensional and dimensionless parameters are e'-
k-RZ0 >> kRZ9 >> kRZs If we assume that the rate constants kRZsand k8,are similar, that kmRZBand k-*, are similar, and that kRZ9and ky[BrMA] are similar, then we may add reactions RZ8 and 8' together to get a composite reaction 8 and also add reactions RZ9 and 9' together to get a composite reaction 9. A species B = [BrMA] [ M A ] can then be used instead of [MA]and [BrMA]separately; and one has reactions explicitly involving the additional organic radical R = [BrMA']+ [MA'],the kinetics of which are then considered in the kinetic equations. We also adopt RZ's relationship of the rate constants for the obtained reactions 8 and 9 k-8 >> k9 >> k8
-
-
Br03- + Br-
-
+ 2H+ HBr02 + HOBr Fe3+ + BrMA + MA Fe2+ + H+ + BrMA' + MA' BrMA' + MA' + H 2 0 hBr- + MA + p%uct(s)
-
Y = [Br-] Z = [Fe3+] R = [BrMA']
2k4e
klk3HA
k3H
2k4k7 klk5
2k4 k5H
g=-
2k4k8B
c=-
2k4c
( k l H A ) 2 C=
klHAC
k-8 (klHA)' (M3) 044) 045)
where
ksk9 k-8 The following equations in the dimensionless variables then result:
K8 = -
(M7)
CX,
(M8)
= U(C - Z ) - K 4 X Z - X
hz cay, = p(c - z )
(M9)
U = [BrO,']
k8C
p==2k,B=KgC
(Ml)
The numbering of the chemical reactions are chosen to resemble the numbering of the RZ model,I2 but the first five reactions are exactly the same as reactions R2-R6 in the Oregonator model as presented in ref 14. With the choice of symbols
X = [HBr02]
2k4k8B
-'4
and take reaction 9 to be irreversible. The final form of the reactions of the model we use is
+ H+ = 2Br0,' + H 2 0 Fe2+ + Br02' + H+ Fe3+ + HBr02 2HBr02 HOBr + Br03- + H+ HBr02 + Br- + H+ 2HOBr
k8B klHA
p=-=-
+
HBr02 + Br03-
= k8Bt
Z,
=
U(C
+ L 5 U 2 - X 2 - Xy + Qy + 1 - QY - XY
- Z ) - K+XZ -
Z
p(c
- z) + 1
where
and the constants
B = [MA]+ [BrMA] C = [Fe3+] + [Fe2+] H = [H+] h = stoichiometric factor the dimensional kinetic equations as given by the model are A = [Br03-]
kIk-3HA
K-6
The steady-state approximation for cuy, and
k = - k l H A X + k-lL/2+ k , H U ( C - Z ) - k - J Z - 2 k N k5HKY + kTH2AY
L = -ksHXY - k7H2AY + hkgR
u=
(C - Z)
+
pa,
gives
2x(2 + K d Z ) [(c - z)' + 8 ~ - g ~ + ( 2u+z)]'/'
~~
(16) Jwo,J. J.; Noyes. R. M.J . Am. Chrm. Soc. 1975, 76, 1392.
k-3
=2k4ksB = 2k4~
So, the set of equations reduces to
9364 The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 cx, = x
- x2 -
Mori et al. of k3 = lo7 was found to give a maximum conversion level of near 100% while allowing the ferriin level to decrease soon after. The parameter values used in all calculations are
hz
kl = 40 M-2 s-‘ k-l = 4 After adding the diffusion term, the dimensionless reaction diffusion equations are then p ( c -hz r)+1 z, = z,,
+ 2x - 2
(=) x+q
~- 4
p(c
-K-5U2]
X
IO7 M-l s-l
k3 = 1 X lo7 M-2 s-I k-3 = 40 M-’
(1)
k4 = 2
X
S-I
lo3 M-l s-l
Z
~
-z) + 1
where s is the dimensionless space coordinate related to the dimensional space coordinate S by s = (k8B/D)’/2S
where D is the diffusion coefficient of both ferriin and HBr02, which are presumed to have identical diffusion coefficients of D = 1.5 X cm2/s. Equations 1 and 2 with added periodic boundary conditions are integrated with the Runge-Kutta-Merson method. Since we are modeling one trigger wave that passes through a solution and not a train of waves, a long spatial wavelength of 10 cm is chosen to allow the medium to relax back to its baseline value before the next calculated wave passes through. The space grid contains 100 pints, and the distribution of these p i n t s is not uniform in space but changes as the calculation continues, in a manner developed by Dwyer.” A much denser distribution of points is used near the wavefront where the concentration is changing most rapidly. The time step chosen by the integrator is approximately 0.01 s. A semiimplicit Runge-Kutta method of integration for stiff systems was also tried and yielded similar results; the RungeKutta-Merson method was deemed sufficient. The values of the rate constants k l , k-l, k-3, k4, kS,and k7 are the same as those used by NTH for the analogous reactions. The value of ka was chosen to be 0.4 M-’ s-l, the same value N T H chose for the rate constant of the overall organic process J used in their model. This choice is motivated by RZ’s value of ka, which is of the order I . The actual values of k4 and k9 were not determined separately. k-8 and k9 appear in the equations only in the combination kak9/k4,which we have defined as K8. A value for the composite rate constant Ksonly is chosen. Changing the value of Ka alters the baseline concentration of ferriin in the calculated waves, and we choose the value of Ka to match the experiments of the standard solutions. However, the process for choosing Ksis dependent on the value of h, which controls whether a stirred solution has bulk oscillations or not. Below a critical value of h, which depends slightly on the values of other rate constants, a solution shows bulk oscillations, and above the critical value the mixture is excitatory. Although the solutions used in the experiments of series B have oscillations when continuously stirred, the solution does not undergo bulk oscillations while the wave is being measured. Hence, we choose the same h value corresponding to an excitatory medium for both the excitatory and the oscillatory solutions. The h value chosen is close to the critical value since, even in the excitatory solution, a single oxidative excursion of the entire medium, ending in the excitatory state, can be induced by stirring. The values of K8 and h were varied simultaneously, and we choose values of h and K8 for which the model reproduces the experimentally found lower conversion level of ferriin near the critical value of h for the standard concentrations used in the excitatory medium. The final values used were Ka = 3 X 10” M s-l and h = 1.3. The value of k3 was varied to obtain a more correct wave shape near the maximum of the wave. If one takes k3 = IO9 as suggested in NTH, the wave, instead of having a maximum of nearly 100% conversion which then falls off, has a maximal region of nearly 100% which extends over a macroscopic region in space before decreasing. While the value of k 3 has been measured,I7 the value
D = 1.5 X cm2 s-I wavelength = 10 cm The profiles obtained by the integration of eqs 1 and 2, with variation of the initial concentrations of sulfuric acid, bromate, and malonic acid, are shown in Figure 2. The first column of profiles are the computations for a wave travelling in the excitatory medium, while the second column of profiles is for a wave travelling in the oscillatory medium. The concentrations used in the simulations are identical with those used in the experiments displayed in Figure 1. Again, the first row of profiles is for the standard concentrations, and the following rows of profiles reflect the effect of the change of concentration of sodium bromate, sulfuric acid, and malonic acid, respectively.
V. Discussion Comparison of Experiments and Model of Ferriin Waves. (a) Qualitative Comparison. The calculations reproduce qualitatively the general shape and the width of the experimental results. However, the waves travelling in the excitatory mixture are experimentally found to have a bent wave-front structure; this bent structure is not seen in the calculated waves. Another difference between the experimental and the calculated profiles is that the ferriin concentration drops off more rapidly in space after the maximum is attained in the experimental profiles. (b) Quantitative Front Width. The waves in the excitatory medium have fronts with broader width than the waves in the oscillatory medium in the experiments. Experimentally, the width of the front is 30O-r400 pm for waves in the excitatory medium and about 95 hm for waves in the oscillatory medium. The calculations show the same trend with widths of about 300 pm for the excitatory concentrations and 170 pm for the oscillatory concentrations. The width of the front, somewhat surprisingly, varies little within the waves of series A and little within series B of the experiments. The calculated profiles possess the robustness of width as well. (c) Ferriin Concentration Levels. As stated earlier, the rate constants Ka and k3 and the parameter h were chosen so the model gives the proper ferriin concentrations for waves in the standard excitatory concentrations. The ferriin concentrations of the wave travelling in the standard oscillating solution are reproduced quite accurately as well. The baseline concentration of ferriin, however, is sensitive to changes in the concentrations of the various reactants. In both the excitatory and oscillatory cases, this baseline level drops if either the sodium bromate or the sulfuric acid concentration is decreased, while the baseline level of ferriin increases if the malonic acid concentration is decreased. The model shows this same qualitative behavior. Take for example the case of increasing the acid concentration of the standard excitatory (17) Field,
R. J.; Forsterling, H.-D. J . Phys. Chem. 1986, 90, 5400.
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solution. The baseline level experimentally increases from 2.4 to 2.65 mM (Figure I , series A, profiles a and c) while the baseline level of the calculated profiles increases from 2.45 to 2.7 mM (Figure 2, series A, profiles a and c). However, the agreement for the change of baseline level due to different malonic acid concentration is less quantitative. Comparing the profiles a and d in Figure 1 , series A, one sees an experimental decrease in the ferriin concentration from 2.4 to 1.9 mM, while profiles a and d in Figure 2, series A, show a calculated decrease from 2.45 to 1.6 mM. (d) Wave Velocity. The computed wave velocities are 0.25 mm/s for the standard excitatory solution and 0.30 mm/s for the standard oscillatory solution. These values are 3 times faster than the observed velocities of 0.08 mm/s for the standard excitatory mixture and 0.10 mm/s for the standard oscillatory case. However, the calculated velocities of ferriin waves we obtain using the NTH model, having set K - ~= 0, are about IO times faster than the observed velocities, so our version of the Oregonator represonts
an improvement. NTH reported difficulty in correctly modelling both the observed cerium concentration levels and the wave velocity. Likewise, we have modeled the concentration levels accurately, but the velocities less so. Comparison of Present Work with Previous Works on Ferriin Waves. In ref 6, measurements are done on a ferriin wave travelling through an oscillating mixture where all concentrations are identical with those used in the standard concentration for the oscillatory medium in this work. The reported width front of 88 f 16 Fm and wave velocity at room temperature of 100 pm/s coincide with our measurements of a front width of 90 pm and a velocity of 100 pm/s for the same system. In ref 5 , measurements were taken of ferriin waves travelling in an excitatory medium comprised of the same concentrations as the standard excitatory solution in this work. The resolution of the present work is higher, and the bent structure of the wave front is more evident. The temperature in the present work is also warmer: 22.8 OC versus 18 OC used in ref 5 ; our observed wave
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velocity of 80 pm/s is hence faster than the reported velocity of 67 pm/s in ref 5. Comparison of Ferriin and Cerium Waves. Cerium wave fronts tend to be broader than the ferriin wave fronts. The widths observed in the cerium waves7 range from about 200 to 1500 pm, while the ferriin waves observed have widths between about 90 and 500 pm. Both types of waves sometimes have a bent front. The bent front is far more prominent in the excitatory medium for ferroin waves, while the bent structure is seen in both in the excitatory and oscillatory mediums for cerium waves and is somewhat more prominent in the oscillatory case. It is interesting that some of the cerium waves calculated by the NTH model have a bent front, but the ferroin waves calculated by the NTH model (not shown) or by our model do not. A significantly higher fraction of catalyst is oxidized in the ferroin system compared to the cerium system. The conversion levels observed in the ferroin system are 55-90% along the baseline and 90-100% at the maximum, whereas only 6-30% of the cerium is in its oxidized state along the baseline and the maximum cerium conversion observed is 20-55%.' In both systems, however, there are seen the same qualitative effects of varying the initial reactant concentrations, in particular on the baseline conversion levels. A larger initial concentration of bromate or sulfuric acid raises the baseline conversion, while a larger initial concentration of malonic acid lowers the conversion.
VI. Conclusion The resolution of the wave images in this study is greater than that of the previous wave profiles obtained in this laboratory? and the different structure of the straight and bent wave fronts is noted in this ferroin-catalyzed BZ system. We observe no trends of the width of the wave front versus concentration of reactants for the concentration ranges studied, with the exception that the full width of a wave travelling through an excitatory medium increases as the initial concentration of sodium bromate is increased. We also observe that the baseline concentration of ferriin increases with increasing [ Br03-] or [H2S04] and decreases with increasing [MA], while the percent conversion of ferroin to ferriin at the wave maximum remains close to 100% for all concentrations studied. Finally, we present a modified Oregonator model in which the organic process is divided into two reactions. The widths and velocities of ferriin chemical waves predicted by this model agree still only qualitatively, but better, with those found in the experiments, and the predicted ferriin concentrations agree more quantitatively with experiments than in previous models.
Acknowledgment. We thank Dr. Benjamin Irvin for his valuable assistance. This work was supported in part by the National Science Foundation and the Air Force Office of Scientific Research. Registry No. MA, 141-82-2; BrO,, 15541-45-4; ferroin, 14708-99-7.
Kinetics of the Reactions between O('P) and 1-Butene from 335 to 1110 K Taeho KO,George Yaw Adusei, and Arthur Fontijn* High- Temperature Reaction Kinetics Laboratory, The Isermann Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180-3590 (Received: May 30, 1991; In Final Form: July 13, 1991)
Rate coefficients for consumption of ground-state 0 atoms by reaction with I-butene have been measured by using the high-temperature photochemistry (HTP) technique. The oxygen atoms were generated by flash photolysis of either O2or C 0 2 and their relative concentrations were monitored by resonance fluorescence. The data in the 335-1 110 K range are well fitted by the expression k(T) = 2.7 X IO-" exp(424 KIT) 2.4 X IO4 exp(-5657 K/T) cm3 molecule-l s-I with 2a precision limits of *I3 to f27%, depending upon temperature, and corresponding 2u accuracy limits of *24 and *34%. Good agreement exists among the rate coefficients measured here and those measured by other isolated elementary reaction techniques; we derive a combined fit expression for the 190-1 110 K range of k(T) = 9.0 X IO-'2 exp(-230 K/T)+ 2.2 X exp(-2612 KIT)cm3 molecule-l s-l, with a suggested 2a accuracy of f30%. A transition-state theory calculation for the addition reaction, which had been shown to fit the data well till 490 K, is extended to 1110 K and seen to reproduce the experimental data increasingly poorly. However, a good fit is obtained when the rate coefficients for H-atom abstraction from the alkyl group in 1-butene are added to those for addition. The conclusion that the interaction between O(3P) atoms and I -butene involves both addition and abstraction is reflected in the desirability for the double-exponential fit expressions given. Standard three-parameter fits, given in the text for the convenience of data compilers, approximate the observations well.
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Introduction The oxidation reactions of the c4hydrocarbons are of considerable inherent and applied interat. They can Serve as a model for reactions of higher paraffinic hydrocarbons, since they represent the simplest system that allows comparison of the behavior of straight-chain molecules to that of branched-chain molecules. Thus, n-butane has been found to be the smallest alkane to exhibit engine knock, while isobutane is considerably more knock-resistant.12 Butenes are important intermediates in the combustion of n - b ~ t a n e as , ~ well as higher hydrocarbons: and can be sig( I ) Pitz, W. J.; Westbrook, C. K.;Proscia, W. M.; Dryer, F. L. 20th Symposium (International) on Combustion [froceedings];The Combustion Institute: Pittsburgh, 1984; p 831. (2) Pitz. W. J.; Westbrook, C. K. Combust. Flame 1986, 63, 113. (3) Westbrook, C. K.;Dryer, F. L.frog. Energy Combusr. Sci. 1984, IO, I.
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nificant soot precursor^.^*^ From an abstract by Perry,' which gives k( T ) for 0-atom reactions with the four isomeric butenes Over the 260-860 K temperature range, a difference between 1-butene and the other three members is evident. While the latter show very little temperature dependence of the rate coefficients, the former has a strongly upwardly Curved Arrhenius Plot at higher temperatures. As the first part of a series of studies of the 0-atom reactions with the four butenes we report here on the reaction (4) Axelsson, E. I.; Brezinsky, K.;Dryer, F. L.;Pitz, W. J.; Westbrook, C. K. 2lst Symposium (International) on Combustion [Proceedings];The Combustion Institute: Pittsburgh, 1986; p 783. ( 5 ) Brezinsky, K.;Hura, H. S.;Glassman, 1. Energy Fuels 1988, 2,487. ( 6 ) Glassman, I. 22nd Symposium (International) on Combustion [froceedings]; The Combustion Institute: Pittsburgh, 1988; p 295. (7) Perry, R. A. 188th ACS National Meeting, Philadelphia, 1984; Div. Phys. Chem. Abstract No. 241.
0 1991 American Chemical Society
