J . Phys. Chem. 1985, 89. 4914-4921
4914 c
rR molecules
lC’*
8CC
1200
T(K)
Figure 10. Plot of rR vs. 7 for LH model (solid curve) and models with adsorption limit and coverage dependent activation energy. For Eco(6co), a,= 12000 cal/(gmol K) and E, = 42000 cal/(gmol K). For the adsorption limit, kaNO = 5 X I O l 9 molecules/(cm2 s), k , = 7 X 10’0e14050/Rrs-1 kdNo = 10”es25o/Rr s-1 and kco ,= 4 x lO+e242“/Rr,
Both models predict a lower and broader rate maximum than observed. a t low temperature and kR < kd a t high temperature. This can give a broad maximum as shown in Figure IO. W e do not claim that either of these mechanisms gives an accurate explanation of rate data, but merely note that they give good fits to data. Models could only be distinguished if extremely accurate adsorption and reaction rate data were available. Summary The kinetics of the NO CO reaction on polycrystalline Pt measured over a wide range of temperatures, pressures, and
+
compositions can be interpreted accurately using LangmuirHinshelwood models. While kinetics cannot prove mechanisms, the agreement with this rate expression indicates that the reaction steps are simple adsorption, desorption, and reaction in coadsorbed monolayers on the surface. AES confirms the absence of excess oxygen, nitrogen, or carbon and eliminates the possibility of contaminants. Each crystal plane of the Pt surface is known to have characteristic NO and CO adsorption and reaction parameters. This could lead to very complicated kinetics on a polycrystalline catalyst. None of these effects is evident in rate measurements in that LH models give adequate descriptions of data. This of course does not imply that they do not exist but only that the averaging process over crystal planes and binding states does not reveal their presence. In fact, the ( I 1 1 ) plane, the lowest free energy and probably most abundant plane of clean polycrystalline Pt, is not reactive at all. The unreactivity of Pt( 111) should be evident only in making the preexponential factor of kR slightly smaller than it would be if the (1 11) plane were not present. These results also indicate strongly that the rate-limiting step is a bimolecular reaction step between N O and C O rather than the unimolecular dissociation reaction. The magnitude of the rate and its dependences on temperature and on NO and CO partial pressures appear to be inconsistent with NO dissociation as the rate-limiting step. The low activity of Pt in the automotive catalytic converter is not simply explained by these results because the rate expression predicts adequate rates of reaction. Presumably, the inhibition by O,, CO, or other species causes the reduced activity of Pt compared to R h for NO reactions. Perhaps R h is a superior catalyst because the dissociation of NO occurs instantly upon adsorption, and therefore the overall reaction does not depend on a true bimolecular process which may more easily be inhibited by other species. Finally, we note that, while the steady-state reaction appears “simple”, behavior on single-crystal planes such as (100) is quite inconsistent with LH models, and the approach to and stability of steady states can be quite complex. We shall describe these phenomena in a later paper.
Acknowledgment. This work was partially supported by NST under Grant No. DMR8216729. Registry No. Pt, 7440-06-4; NO, 10102-43-9; CO. 630-08-0.
Measurement of Bromine Removal Rate in the Osclllatory BZ Reaction of Oxalic Acid. Transition from Limit Cycle Oscillations to Excitability via Saddle-Node Infinite Period Bifurcation Zoltin Noszticzius,* PBter Stirling, and Miria Wittmann Institute of Physics, Technical University of Budapest, H - 1521 Budapest, Hungary (Received: February 21, 1985: In Final Form: May 28, 1985) The BZ reaction with oxalic acid substrate is one of the simplest BZ oscillator which oscillates only if the produced bromine is removed by an inert carrier gas. In the present study an electrochemical method is reported to determine kBR,the “rate constant” for bromine removal. Oscillations can be found within a certain interval of the kgRparameter values only. Outside of that range two different excitable states were discovered. Transitions from the oscillatory to the excitable states go via saddle-node infinite period bifurcations. Different types of bifurcations leading to periodic orbits, a two-dimensional phase portrait of an excitable system, and a perturbation technique to reveral that portrait are also discussed. Introduction The mechanistic problems of the Belousov-Zhabotinsky (BZ) reactions were reinvestigated in a recent series of papers.’-* Now, (1) Noszticzius, Z . ; Farkas, H.: Schelly, Z. A . J . Chem. Phys. 1984, 80. 6062-6070.
0022-3654/85/2089-4914$01.50/0
in our opinion, the lack and uncertainity of the rate remain a major problem in the numerical simulation of the dif(2) Noszticzius, Z.: Farkas, H.; Schelly, Z . A. Reacf. Kinet. Catal. Lerr. 1984, 25. 305-311. (3) Noyes. R. M. J . Chem. Phys. 1984, 80. 6071-6078.
0 1985 American Chemical Society
Bromine Removal Rate in the Oscillatory BZ Reaction
The Journal of Physical Chemistry, Vol. 89, No. 23, 1985 4915
t
ferent BZ reactions. Therefore in this paper we start a systematic collection of the required rate constants for one of the simplest BZ systems where the substrate is oxalic acid.’s8 The oxalic acid substrate system is a rather special one: it requires an inert gas stream to remove the produced bromine, otherwise no oscillations can be dete~ted.’.~Formally, the physical removal of bromine can be regarded as a first-order chemical reaction
where Br2 is the elementary bromine dissolved in the liquid phase, (Br2Jgasis the bromine in the gas phase, and k g R is the “rate constant” of the reaction. The original aim of the present study was to determine k g R as a function of the gas flow rate and reaction volume. At the same time a special possibility emerged to perform experiments where k g R is a bifurcation parameter. Namely, the gas flow rate and the reaction volume have no effect on other rate constants or parameters of the system; thus k B R can be varied independently of the other parameters in a rather similar way like ko, the reciprocal residence time in a CSTR. (There is only one significant difference; while ko determines the input and output rate of several components simultaneously k g R determines the removal rate of a single component selectively. Thus, from a chemical point of view, the effect of k g R is more specific than that of ko.) These experiments led us to the discovery of saddle-node infinite period bifurcations in our system. As often happens, this “byproduct” of our work seems to be at least as important as its original aim. First, after the pioneering works of Smoes,lo,” De Kepper and B o i ~ s o n a d e , ~and ~ ~ ’especially ~ M a s e i k 0 ~ ~ and 3 ’ ~ Maseiko and Epstein,16 we could find such infinite period bifurcations in a system which is relatively simple from a chemical point of view. Second, a very straightforward explanation of the excitability emerges from our work. Also the detection technique of the excitability points in the direction of a general experimental procedure: a combination of the bifurcation14-16 and pert~rbationl’-~~ techniques which can reveal among others unstable steady states too. Finally, it is clear now that a new challenge appeared for the mechanistic models of oscillating reactions: to explain the experimentally observed bifurcations to periodic orbitals.
Experimental Section Measurement of k B R . Method and Apparatus. A simple electrochemical method was developed to measure k g R . The potential of a platinum electrode was recorded in a solution containing some initial bromine, potassium bromide, and sulfuric
(4) Tyson, J. J. J . Chem. Phys. 1984,80, 6079-6082. (5) Noszticzius, Z.; Gispir, V.; Forsterling, H. D. J. Am. Chem. Soc. 1985, 107, 2314-2315. (6) (a) Tyson, J. J. In “Oscillations and Travelling Waves in Chemical Systems”, Field, R. J.; Burger, M. Eds.; Wiley: New York, 1984. (b) Field, R. J. Ibid. (7) Noszticzius, 2.;Baiss, J. J . Am. Chem. SOC.1979, 101, 3177-3182. (8) SevEik, P.; AdamEikova, L. Collect. Czech. Chem. Cornmun. 1982, 47, 891-898. (9) Kasperek, G. J.; Bruice, T. C. Inorg. Chem. 1971, 10, 382-386. (10) Smoes, M. L. In ’Dynamics of Synergetic Systems”, Vol. 6 , Haken, H., Ed.; Springer-Verlag: West Berlin, 1980; Springer Series in Synergetics pp 80-96. (1 1) Smoes, M. L. J. Chem. Phys. 1979, 71, 4669-4679. (12) Boissonade, J.; De Kepper, P. J . f h y s . Chem. 1980, 84, 501-506. (13) De Kepper, P.; Boissonade, J. J . Chem. Phys. 1981, 75, 189-19s. (14) Maselko, J. Chem. Phys. 1982, 67, 17-26. (15) Maselko, J. Chem. Phys. 1983, 78, 381-389. (16) Maselko, J.; Epstein, I. R. J . Phys. Chem. 1984, 88, 5305-5308. (17) Heinrichs, M.; Schneider, F. W. J . Phys. Chem. 1981,85,2112-2116. (18) Ganapathisubramanian,N.; Showalter, K. J . Phys. Chem. 1983.87, 1098-1099. (19) Ganapathisubramanian, N.; Showalter, K. J . Chem. Phys. 1984,80, 41 77-41 84. (20) Ruoff, P. Chem. Phys. Letf. 1982, 90, 76-80. (21) Ruoff, P. Chem. Phys. Lett. 1982, 92, 239-244. (22) Ruoff, P. Chem. Phys. Lett. 1983, 96, 374-378.
N2
E /rnV
0 -
-20
.
-40 .
-60
.
0 2 4 6 8 t/rnin Figure 1. (a) Schematic diagram of the apparatus. The potential of the first platinum electrode, situated in half cell “ l ” , was measured vs. the second one. (b) “V”-shaped potential-time diagram as it was recorded by a potentiometric recorder. Gas flow rate, 150 cm3/min; volume, 5 cm3 (measurement of kgR for experiment 23 in Table I ) . acid while an inert carrier gas stream was purging out the volatile bromine. (All the chemicals were of reagent grade.) The volume of the solution and the velocity of the gas stream were the same as in the case of the oscillatory BZ reaction with oxalic acid. The rate of the bromine removal [Bf,] is (Since the stability constant of B r c
(3) is 16.3 M-’ at 25 0C23and in all of our experiments [Br-] was less than M any effect due to Br3- formation was negligible.) The potential of the platinum electrode at 25 0C24is (23) Bailar, J. C.; Emeleus, H. J.; Nyholm, R.; Trotman-Dickenson, A. F., Eds. “ComprehensiveInorganic Chemistry”, Vol. 2; Pergamon: Oxford, 1973; p 1540, Table 98.
4916
The Journal of Physical Chemistry, Vol, 89, No. 23, 1985 E = E,
+ 12.8 mV In ([Br,]/[Br-J2)
and its time derivative E is E = 12.8 mV [Bi2]/[Brz]
(4) (5)
because [BY] = 0 as [Br-] = constant in our system. Comparing ( 5 ) and (2) we get Ei‘ = -12.8 mV k B R (6) Thus we can calculate the rate constant for bromine removal k,, by simply measuring the slope of the potentiometric trace (which should be a straight line in this case) and dividing that slope by -12.8 mV. To carry out the measurements a simple apparatus was constructed as can be seen in Figure la. Two glass vessels containing built-in platinum wire electrodes were connected via a special “salt bridge” made of a Corning microporous glass disk. Both half-cells were filled with the same amount of the same electrolyte ([Br,] M, [H,SO,] = 1.5 M), and the voltage M, [Br-1 = =2 X of the resulting galvanic cell was measured by a Keithly 616 digital electrometer. The potentiometric traces were recorded by a Radelkis OH-814/ 1 potentiometric recorder. Initially no potential difference was detected as both half-cells contained the volatile bromine in equal concentrations. Then nitrogen was applied as an inert carrier gas in the first half-cell and the potential of the first platinum electrode began to fall linearly, just as predicted by theory (Figure lb). After some minutes the carrier gas was switched over to the second half-cell. Now the potential of the second platinum electrode started to decrease in a similar way and, as a result, a characteristic V-shaped curve could be recorded (Figure lb). The average slope of its two sides was measured to calculate kBR. The measured slopes were independent of the M > [Br2] > bromine concentration in the range of 2 X M. The reproducibility of the slopes was within 3%. The error is due to some difference in the geometric position of the elastic Teflon tubes in the different experiments. The carrier gas leaving the half-cells was freed of bromine in a third vessel (not shown in Figure I ) containing 1 M N a O H solution. The gas flow rate was measured after that third vessel by a soap-bubble flow meter. The instability of the flow rate was less than 1%. Good stability was achieved by a “flow generator” system consisting of a manostat and a narrow capillary of high flow resistance. This way the nitrogen gas flow was controlled exclusively by the regulated pressure of the manostat and it was independent of the hydrodynamic conditions in the reaction vessel. Monitoring Chemical Oscillations. The chemical oscillations were observed in the same “ H cell” (Figure l a ) where the determination of k g R was carried out. The first half-cell was filled with a 1:l mixture of two standard solutions A and B. The composition of the standard solutions was as follows: (A) [(C0 0 H ) 2 ] = 8 X lo-, M, [HzSO,] = 1.5 M; (B) [ N a B r 0 3 ] = 8 X M, [H2S0,] = 1.5 M. Thus the M, [Ce(SO,),] = initial conditions in the mixture were always the following: M, [BrO,-], = 4 X lo-, M, [Ce4+Io= [(COOH),], = 4 X 5 X lo-“ M, [H2SO4lo= 1.5 M. The second half-cell was filled with a reference solution ([Br,] M, [Br-1 = M, [H,SO,] = 1.5 M) in a volume -= 2 X equal to the oscillating system in the first half-cell. The experiment started by adding solution A to solution B and within 3-4 s the carrier gas was turned on to mix the solutions. A sudden change in the potential associated with that mixing was not recorded; thus the first 5-8 s of the reaction are not included in the potentiometric traces. kgR was measured before and after monitoring the chemical oscillations under the same experimental conditions; the same volume and gas flow rate were applied. All the experiments were carried out at 25 f 0.5 “C. Perturbation Experiments. The presence of an inert carrier gas stream in our system offers a very simple way to generate perturbations. If we apply a sudden change (decrease or increase: ‘‘jumpn in the following) in the gas flow rate the system starts (24) Bell, R. P.: Ramsden. E . N . J . Chem. Soc. 1958, 161-167.
Noszticzius et al. to evolve immediately toward a new attractor (toward a new steady state, or a new limit cycle, or, in general, it can be even a new strange attractor too). This evolution means a continuous change in the intermediate concentrations. Now, after a while, we can restore the original gas stream and we can observe our perturbed system relaxing back to the original attractor. In fact, the whole method is completely analogousZ5to the classical chemical relaxation technique^^^,^' where a chemical equilibrium is perturbed by different jump methods.,* As a consequence our method offers similar advantages: e.g., it is not necessary to inject and to mix homogeneously diifferent chemicals in our system to produce concentration changes. At the same time there is a special advantage offered by the relatively slow dynamics of our system compared to the time scale of the jumps in the gas flow rate. Namely, after the first jump, the intermediate concentrations start to change with a finite speed. Thus the more time elapsed between the first and second jumps the larger is the applied perturbation of our original system. Naturally, this does not mean a linear relationship between the time elapsed and the generated concentration changes. However, a linear relationship is not a prerequisite for the applicability of our method. The only important requirement is that different time intervals separating the first and second jump have to produce different perturbations. If the evolution of our system toward a new attractor is relatively slow it is easy to find such time intervals. Experimentally the jumps of the gas stream were produced with two bypass valves and two independent gas flow generators. To minimalize the transients in the gas stream the flow generators were working continuously and their nitrogen stream could be conducted through the reaction vessel or directly to the laboratory atmosphere depending on the state of their bypass valves. Thus four different gas velocities (0, 11,I,, I I + I,) could be applied and stepwise transitions were possible between them. (Iiis the gas stream from generator i, i = 1, 2.)
Results Observation of Oscillations and Infinite Period Bifurcations to Stable Steady States. Changing the gas flow rate or the volume of the reaction mixture, we found that there is a certain range within which oscillations appear. Outside of this range two different regions of monostability were observed: steady-state “LO” at low gas flow rates (associated with a relatively low electrode potential) and steady-state “HI” at high gas flow rates characterized by a relatively high electrode potential. There is no hysteresis in our system; the different regions do not overlap. First of all we wanted to get evidence that the real control parameter determining the course of the reaction is kBR, the bromine removal rate constant, exclusively: that is the gas flow rate or the reaction volume alone has no specific effect. To demonstrate this we performed parallel experiments where k B R was roughly the same but the gas flow rates and reaction volumes differed considerably; see, e.g., experiments 12 and 13 in Table I. (Actually the flow rate/reaction volume ratio was held constant as the mass transfer coefficient is determined mainly by that ratio.29) Since these experiments confirmed our expectations in the following we varied kBR, the control parameter, either by changing the gas flow rate or by changing the reaction volume whichever was more comfortable. The diverse behavior of our system at different kgR values is depicted in Figure 2. Representative examples of oscillations are shown in Figure 2a-d at low, medium, high, and very high k B R values, respectively. Table 1 contains numerical data for a series of similar experiments. To produce the quantitative data of Table I from a series of potentiometric traces we had to make some considerations. First, as the above systems are not open, in all cases a slow change in oxalic acid (25) Schelly, Z. A,, personal communication. (26) Schelly, Z. A,: Eyring, E. M . J . Chem. Educ. 1971, 48, A639-A643. (27) Schelly, Z. A,; Eyring, E. M . J . Chem. Educ. 1971,48, A695-A697. (28) Strehlow, H.; Knoche, W. “Fundamentals of Chemical Relaxation”; Verlag Chemie: Weinheim, 1977. (29) Danckwerts, P. V. ’Gas Liquid Reactions”: McGraw-Hill: New York. 1970.
Bromine Removal Rate in the Oscillatory BZ Reaction
70
-
60
-
The Journal of Physical Chemistry, Vol. 89, No. 23, 1985 4917
50 40
-
30
-
20
-
10
0
5
10
15
20
25
30
1 0
5
10
15
t/min
Figure 2. Potentiometric trace of (a) experiment 5 of Table I, k B R = 7.48 k,, = 18.7 x lO-’s-’; (d) experiment 29, k g = ~ 20.0 SC’.
concentration causes a slow drift in the shape and the period of the oscillations. (The acidic bromate is in stoichiometric excess compared to oxalic acid; thus its slow consumption causes fewer problems.) To calculate a time period consistent with the initial oxalic acid concentration we plotted the period of the first few (3-4) subsequent oscillations as a function of time and its value was determined by a linear regression-extrapolation to t = 0. On the other hand, it proved useful to split the whole time period of a n oscillation into two main parts: the time spent nearby steady-state HI denoted by T , and the time spent in the vicinity of steady-state LO denoted by TTRLo (TRHI and T R L O are transient states which are not too far from the related steady states, see also Figure 2d). These regions are separated by sharp and fast transition zones. The reason of the splitting is the following. The oscillating system studied here spends relatively more time in T R L O state. Thus, by splitting the time period into separate parts it is more easy to realize the infinite period bifurcation at the high kgR values. (The increase of the total time period is less striking.) In Figure 3 TTR,, and TTRLo is depicted as a function of kBR. As can be seen in Figure 3 nearing to the critical values of kgR (near the boundaries where the oscillations disappear) the oscillating system spends more and more time in the vicinity of the appropriate steady state, therefore the time period increases sharply in these regions. Another characteristic feature of the transitions from oscillatory to nonoscillatory regimes is the appearance and disappearance of full blown oscillations at the
50
50
40
40
30
30
20
20
10
10
0 X
t/min
E/mV
E /mV
lo
35
5
t/min
0
5 t/mii
lO-’s-’; (b) experiment 18, kBR= 12.5 X lO-’s-’; (c) experiment 23,
transition points. The whole phenomenon is usually called as “infinite period b i f ~ r c a t i o n ” . ~The ~ ~ ~following ~ perturbation experiments are aimed to make a distinction between the two possible types of infinite period bifurcations discussed in the theoretical section. Perturbation Experiments. Reveal of Excitable Steady States. The perturbations of steady-states LO and H I are depicted in parts a and b of Figure 4, respectively. The experimental details are given in the figure caption. What is important to realize is that there is a critical threshold in both cases above which relaxation to the steady state follows an entirely different course. E.g., in Figure 4b perturbations lasting 1-5 s give qualitatively the same response, but perturbations of 6 or 7 s give responses which differ dramatically from the previous ones. Thus the responses follow a characteristic “all-or-none” pattern showing that both steady states are excitable not too far from the bifurcation points. We were not able to disturb the oscillatory states in such a way that would permanently prevent the restoration of oscillations; this fact will be important later on.
Discussion Bifurcations Leading to Periodic Orbitals. Androno@ and (30)Keener, J. P. In ”Modelling of Chemical Reaction Systems”, Ebert, K. H., Deuflhard, P., Jager, W., Ed.; Springer: Berlin, 1981; pp 126-137. (31) Keener, J. P. Siam J . Appl. Math. 1981, 41, 127-144.
4918
The Journal of Physical Chemistry, Vol. 89, No. 23, 1985
TABLE I: Some Characteristic Features of the Oscillating System as a Function of the Bromine Removal RateD expt 1 2 3 4 5
6
7 8 9 IO 11
12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 276
28* 29h 30’ 3Ih 3 2h
1 0 ~ k , , / ~ - ~ v/cm3 10 6 51 6.56 6.17 7.42 7.48 7.55 7.65 7.8 I 9.87 10.9 11.3 11.5 11.5 11.9 12.2 12.2 12.4 12.5 13.4 15.1 16.3 16.5 18.7 18.9 19.4 19.4 20.0 20.0 20.0 20.0 20.0 20.2
10 10 IO IO IO 10 IO IO 10 10 10 5 IO 10
10 IO 9 8 7 6 6 5 5 4.8 4.8 6 6 6 6 6 5
w/cm’ min-’
71 70 75 83 85 88 80 83 I20 140 1 50 I56 79 156 160 160 I60 I50 I50 150 150 150 150 150 150 150 I40 140 140 140 140 140
Noszticzius et al.
7-1
TTRH1/
n
TTRLo/min
min
11 9 12 13 13 13 9 12 16 14 16 16 17 16 17 17 16 17 14 12 10 I1 8 2 2 2 2 2 3 2 2 2
7.4 7.5 6.2 4.5 6.0 4.2 4.9 4.6 2.4 0.70 0.90 0.80 0.70 0.65 0.75 0.66 0.50 0.57 0.45 0.43 0.33 0.32 0.32 0.30 0.25 0.30 0.35 0.30 0.30 0.30 0.35 0.30
0.07 0.08 0.09 0.05 0.07 0.10 0.10 0.07 0.09 0.20 0.20 0.12 0.1 5 0.18 0.20 0.18 0.20 0.13 0.20 0.21 0.20 0.23 0.23 0.90 0.90 0.85 0.80 0.80 0.80 1.10 0.75 1.50
5
4
oscillatory state
5
10
15
20
103k,, A-’
5
10
15
20 10’k,,/
