J. Phys. Chem. 1985, 89, 2855-2860 the perturbing frequencies. In contrast to these findings, we have found no resonance effects. In all cases depicted in Figures 1 and 4 the oscillations' amplitude is 3-4 orders of magnitude, in Br- concentration, regardless of the peristaltic period. Entrainment or periods which follow the peristaltic ones are obtained, but with no simple relationship with the natural frequency. The entrainment at large 7%was explained above, but this entrainment continues smoothly to 7"s smaller than T (Figure ld,e and right-hand side of Figure 3). On the other hand, when T becomes very small, the natural frequency and
2855
pattern take over and the oscillations resemble again the original ones. It seems that at such high frequencies the chemical rates follow their own course and "disregard" the fast changes in the feed. In previous ~ t u d i e s ' ~ the - ' ~ limit cycle was near the marginal stability, Le. near the Hopf bifurcation frequency can produce, therefore, entrainment and resonance. In the present study, the limit cycle contains many Fourier components so that the effect of a single perturbing frequency is expected to have a different effect.
The Minimal Bromate Oscillator Simplified K. Bar-Eli Department of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel- Aviv University, Tel- Aviv 69978, Israel (Received: December 19, 1984)
The full Noyes-Field-Thompson (NFT) mechanism with 10 species and 14 rate constants which describes quite adequately the oxidation of cerous ion by bromate is simplified by discarding slow and unimportant reactions and by taking certain species in comparatively large concentrations to be constant. The conditions considered are those obtaining in a continuously stirred tank reactor (CSTR). Good quantitative agreement with experiment can be obtained with only four species and five rate constants. A further simplification with only three species and four rate constants has all the striking qualitative features of the full scheme including bistability and oscillation regions.
Introduction The mechanism for ceric ion oxidation by bromate ions in sulfuric acid medium was suggested by Noyes, Field, and Thompson' (NFT). The mechanism compared very well with experimental results done in batch by Barkin et al.2 When the reaction is conducted in a CSTR (continuously stirred tank reactor) one obtains a region of bistability with hysteresis between one steady state and the other, and a small region of oscillations near the critical point, Le., the point of the coalescence of the steady stales. The NFT mechanism compared very well with the experimental data of both bistability regions and oscillation^.^-^^ Some of the rate constants used were criticized by Tyson,12but Bar-Eli and R ~ n k i nhave ' ~ shown that the best agreement with experimental data is achieved when the original set of rate constants suggested by Noyes'*2and collaborators is used. In this work we shall assume that the complete NFT mechanism is correct and compare it with computations done with simplified versions of it. (1) R. M. Noyes, R. J. Field, and R. C. Thompson, J . Am. Chem. Soc., 93, 7315 (1971). (2) S. Barkin, M. Bixon, R. M. Noyes, and K. Bar-Eli, Inf. J . Chem. Kinef., 11, 841 (1977). (3) K. Bar-Eli and R. M. Noyes, J . Phys. Chem., 81, 1988 (1977). (4) K. Bar-Eli and R. M. Noye$, J . Phys. Chem., 82, 1352 (1978). (5) W. Geiseler, and K. Ba'r.-Eli,J . Phys. Chem., 85, 908 (1981). (6) K. Bar-Eli and W. Geiseler, J. Phys. Chem., 85, 3461 (1981). (7) K. Bar-Eli and W. Geiseler, J . Phys. Chem., 89, 3769 (1983). (8) K. Bar-Eli in 'Nonlinear Phenomena in Chemical Dynamics", Vol. 12, C. Vidal and A. Pacault, Eds.,Springer Series in Synergetics, Springer-Verlag, Berlin, 1981, pp 228-239. (9) W. Geiseler, Ber. Bunsenges. Phys. Chem., 86, 721 (1982). (10) W. Geiseler, J . Phys. Chem., 86, 4394 (1982). (1 1) M. Orban, P. De Kepper, and I. R. Eptein, J. Am. Chem. Soc., 104, 2657 (1982). (12) J. J. Tyson in "Oscillations and Travelling Waves in Chemical Systems", R. J. Field and M. Burger, Eds., Wiley, New York, in press. (13) K. B a r 4 and J. Ronkin, J . Phys. Chem., 88, 2844 (1984).
0022-3654/85/2089-2855$01.50/0
The complete mechanism involves ten chemical species and seven reversible reactions, namely 14 rate constants. Out of the ten species, only five are independent, due to conservation laws, and out of the 14 rate constants only 12 are independent, due to thermodynamic restrictions on some of the reactions. The method of dealing with these dependencies and working with the independent species only is shown e l ~ e w h e r e . ~ J ~ - ' ~ This complicated mechanism can, in principle, be simplified as follows: (a) The number of involved species can be reduced assuming them to be constant. These simplifications reduce the amount and tediousness of all the dynamic calculations of solving the relevant differential equations. We realize, however, that some conservation laws may not be valid with these assumptions. Thus, for example, by assuming water concentration to be constant, we have to give up the conservation of H and 0 atoms. The number of independent species will not be changed by this p r o c e d ~ r e . ~ J ~ J ~ (b) The number of rate constants involved can be reduced by neglecting slow or unimportant reactions. The neglect of slow reactions is correct from a kinetic point of view, since they will only contribute a negligible amount to the chemical change. However, by neglecting them, we loose the thermodynamic completeness of the mechanism. Reactions which are not slow may also be neglected if they do not change appreciably the system behavior. (c) The number of independent species can be reduced, thus reducing the dimensions of the phase space in which the dynamics of the system occurs. This reduction provides us with an insight into the reactions which are really important for the main observations, namely, the bistability and oscillations. The computational results obtained for [BrO,-],-[Br-], subspace of constraints with constant constraints [Ce3+Io= 1.5 X (14) K. Bar-Eli and W. Geiseler, J . Phys. Chem., 87, 1352 (1983). (1 5) G. R. Gavalas, "Nonlinear Differential Equations of Chemically Reacting Systems", Springer-Verlag. Berlin, 1968.
0 1985 American Chemical Society
2856
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
Bar-Eli
TABLE I: Turning Points and Hopf’s Bifurcations for NFT and Several Simplified Versionsa 106[Br-],’ 106[Br-]o c 1O6 [ Br - J o b 103[Br0,-],
l o 6[Br -1
I
full mechanism 0.009711203 3.770804 0.4250427 38.48557 37.51 662 153.7028 (153.5816)d 229.3921 249.5644 (229.4 163) (233.1209) (321.0801 j (273.7169) 56.7283; 290.3700
0.2 2 20 50
60 criticale
stage d 0.003233288 0.3065 39 3 36.20456
3.867017 38.822 1 8 153.8610 (15 3.7 383) 249.7365 (2 33.2958) (273.9710)
228.1467 (228.1531) (320.9502) 57.05; 292.455
stage e 0.2 2 20
0.00 3233591 0.3065599 36.2071 1
50
228.1632 (228.6196) (320.97 66)
60 criticale
stage f 5.96606 22.84766 5 6.74840 (56.57915) 90.82632 85.67966 (85.6957 3) (79.8001 6) (119.2635) (92.75758) 55.25; 103.2454
6.617497 40.61978 154.1141 (15 4.01 09) 249.7744 (233.3552 j (273.9979)
0.002884152 0.1543989 13.93562
81.27 400 135.8835 205.7130 (205.3795) 280.5857 (259.4616) 336.9394 (291.6398) (360.4182)
0.002883850 0.1543849 13.92549
74.058 39 74.26004 79.10938
86.26779
110.8462
124.0550
133.3788
57.07; 292.90 stage g
0.2 2 20
0.00323 3401 0.3065611 36.21194
50
229.5475
60
329.6249 (329.6470) (42 1.3 162)
70 criticale
stage h
169.1265
63.05; 362.0
169.1484 70.1; 169.7
,
M, k, = 4 X s-’. The various stages are described in the text and in a Constant constraint: [ H i ] a = 1.5 M, [Ce”] = 1.5 X Turning point between SSI (the one with comparatively high bromide concentration) and SSIII (the unstable steady state). Table 111. Numbers in parentheses are Hopf bifurTurning point between SSII (the one with comparatively low bromide concentration) and SSIII. cation points, when only these exist, the system is in the oscillation region. e In the line “critical”, the first number is [BrO,‘] ,(critical) X l o 3 and the second is [Br-] ,(critical) X IO6.
lo4 M, [H+l0 = 1.5 M, and ko = 4 X s-l will be used as a reference for comparison of the simplified versions of the mechanism. Results The complete NFT mechanism is given by eq 1-7 together with the appropriate rate constants. The rate constants are given with the assumption of water being at constant unit concentration. Br03-
+ Br- + 2H+ s H B r 0 2 + HOBr k-, = 1
k , = 2.1 M-3 s-I HBr02
k, = 2
X
k, = 8 Br0,-
X
k5 = 6.5 Ce4+
lo9 M-2 s-’
M-I s-l
X
(3)
k-, = 110 s-’
+ HBrO, + H+ e 2 B r 0 2 - + H 2 0 k-4 = 2
X 10’
M-I
(4) S-I
+ BrO,. + H+ s Ce4+ + HBrO,
X lo5 M-2 s-,
k-5 = 2.4 X IO7 M-I
(5) SKI
+ Br02. + H 2 0 s Ce3+ + Br03- + 2H+
k , = 9.6 M-I SKI
2HBr0, k7 = 4
k-2 = 5
(2)
+ Br- + H+ s Br, + H,O
k4 = 1 X lo4 M-, s-l Ce3+
lo4 M-’ s-’
X
+ Br- + H + s 2HOBr
lo9 Me2 s-l
HOBr
(1)
X 10’
i=
M-I s-’
Br03- + HOBr k-, = 2.1
+ H+
(6)
X
(7) M-2s-l
where c is the concentration vector, co is the concentration vector of the inflowing reactants, R(c) is chemical mass action rate obtained from the above mechanism, and ko is the flow rate. The steady states, Le., the points where C = 0 are solved by Newton’s method, and thus the turning points (where the Jacobian matrix &lacj becomes singular) and the Hopf bif~rcations’~*’’ (where two conjugate eigenvalues have their real part zero) are found. Whenever necessary, the differential equations were solved by Gear’s method.]* In the region between the turning points the system has three steady states, two of which are stable, i.e., all the eigenvalues are negative. The turning points can thus be measured experimentally as the hysteresis limits. The third steady state is unstable since at least one eigenvalue is positive and is physically unattainable. At the critical point SSIII (the unstable one) disappears and only o n e SS remains. The results are shown in Figure lA, and a few points are also given in Table I. Bistability occurs in the region between the solid lines, while inside the dashed loop, beyond the critical point, there is only one unstable steady state. In this region, the system oscillates. Details ~~
M-3s-l
k-6 = 1.3 X
In this way the first simplification of the mechanism, Le., reducing the number of species from 10 to 9, is done, as in all previous ~ o r k . ~ - ’ ~ The resulting differential equations are of the form c = R(c) + ko(co - C) (8)
~~
~
~~~
~
~
~
(16) G. Ioos, and D. D. Joseph, “Elementary Stability and Bifurcation Theory”, Springer-Verlag. Berlin, 1980. (17) B. D. Hassard, N. D. Kazarinoff, and Y . H . Wan, “Theory and Applications of Hopf Bifurcation”, London Mathematical Society, Lecture Notes Series 41, Cambridge University Press, London, 1981. (18) (a) C. W. Gear, ‘Numerical Initial Value Problems in Ordinary Differential Equations”, Prentice-Hall, Englewood Cliffs, NJ, 1971, pp 209-229. (b) A. C. Hindmarsh, “Gear: Ordinary Differential Equations System Solver”, VCID-2001, rev. 3, University of California, Livermore, CA, Dec 1974.
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2857
The Minimal Bromate Oscillator TABLE Ik Steady-State Concentrations and Rates of Chemical Change of the Various Reactions for the Full Nm Mechanismu
SS concn, M species Br03HBrO, HOBr Br0, BrCe4+ Ce’+ Br2 H+
SSIII
SSI
SSII
0.1999 X lo-’ 0.4606 X 0.8271 X 10” 0.2627 X 0.3347 X 10” 0.7321 X 0.1427 X lo-’ 0.3020 X lo4 0.1500 X 10’
0.1996 X lo-’ 0.1525 X 0.5236 X IO4 0.4690 X 10” 0.3028 X 0.8291 X lo4 0.6709 X lo4 0.1729 X lo4 0.1500 X 10’ rates, M/s
0.1998 0.1641 0.2304 0.1617 0.1253 0.2493 0.1251 0.3150 0.1500
2.855 X 7.985 X 1.385 X 1.371 X 1.902 X 1.902 X 4.566 X 4.399 X 3.068 X 3.034 X 3.733 X 3.917 X 9.303 X 3.136 X
1.183 X 3.781 X 6.169 X 2.655 X 3.465 X 3.465 X 4.919 X 5.231 X 1.972 X 9.820 X 3.871 X 7.310 X 1.077 X 1.450 X
reaction 1 -1 2 -2 3 -3 4 -4 5 -5 6 -6 7 -7
3.161 X 3.810 X 4.625 X 3.421 X 3.322 X 3.322 X 1.381 X 1.380 X 3.654 X 8.093 X 1.846 X 8.341 X 8.486 X 5.208 X
lo-” lo-’’ lo-’ lo-’ lo-’*
lo-’
lo-’ lo-’ 10“ 10“
lo4
X lo-’ X lo4 X X lo-* X 10” X IO4 X lo-’ X lo4 X 10’
300
1
*
250.
O -
200
,’
,jl
j
P .
lo-* lo-’, 10“
lo-’ lo-’
lo-”
i ~ ~ o ; 1i 3, 3~
lo-’’
‘Constant constraints: [BrO,-], = 20 X lo-’ M, [Br-lo = 50 X 10“ X lo4 M, [HtIo = 1.5 M, ko = 4 X lo-’ 8.
M, [Ce’+lo = 1.5
of the figure very near the critical point are shown in previous These details will remain essentially the same throughout. In Table I, the bromide and bromate ion concentrations in the flow for some points, including the critical one, are given, for the full N l T mechanism and some of its simplified versions. Comparison with experimental data is given e l ~ e w h e r e . ~ -We ~ J ~shall consider these computational results as “correct” and make, in the following, comparisons to them. In Table I1 the steady-state concentrations and the rates of the various reactions, for typical input parameters are given. From this table we can immediately deduce some “obvious” simplified versions. A summary of the various simplified schemes with the relevant number of rate constants and species is shown in Table 111. (a) [I-I’] i= [H+Io= 1.5 M, with appropriate changes in rate constants. Changes in location of Hopff bifurcations and turning points are less than 0.1%. (b) [BrOc] i= [BrO,-], with appropriate changes in rate constants give essentially the same results as for the full mechanism. The greatest discrepancy, less than 2%, occurs for very low concentrations of bromate. These low bromate concentrations, lower than cerous ones, are, of course, impractical experimentally, since all the bromate will be consumed. (c) Reactions -2 and -7 are very slow and can be neglected with no change in the results.
Figure 1. Bistability (full lines) and oscillation ranges (dashed lines) for (A) a full NFT mechanism and (B) a NFT mechanism with reactions 1, 2, f 4 , and f 5 only; [BrOC] = [BrO,-], and [H+] = [H+], are included in the appropriate rate constants (stage fj. Constant constraints: [Ce3’lo = 1.5 X lo4 M, [HtIo = 1.5 M, and ko = 4 X IO-’ s-’,
(d) Omission of reactions 6 and -6, which are much slower than 5 and -5, leave the system nearly the same. The results (except for the lower most turning points), shown in Table I, deviate at most 3%. Reaction 5 is, therefore, the main channel of oxidizing and reducing cerium ions. These results are in agreement with those of Geiseler and Bar-Eli,5 where k6 did not affect the location of the hysteresis limits at all, while k-6 affected them, when increased only. (e) The results, shown in Table I, remain essentially the same if reactions -1 and +7 are also omitted. Since the sum of reactions 1, -2, and 7 is identically zero, removing reaction 7 means that the disproportionation of bromous acid is rather unimportant in removing it from the system. As shown in Table 11, bromous acid is removed faster by reaction -5. ( f ) Bar-Eli and Geisler14 have shown that the eigenvectors at the steady states are linear combinations of the columns of the stoichiometric matrix. When the eigenvectors are examined, both in the bistability region and the oscillating region, it is found that the eigenvector belonging to the most negative eigenvalue is always parallel to reaction 3. In other words, near the steady state, the dynamics of the system does not occupy the whole space but occurs mostly at a subspace which is perpendicular to reaction 3. Reaction 3, although the fastest reaction in the scheme, is always very near equilibrium (see Table 11) and therefore seems to be unimportant creating both bistability and the oscillations. Indeed this reaction was not included in the work of De Kepper and
TABLE III: Stages of Simplifying the NFT Mechanism
stage full mechanism a“ bb Cb
db eb fb
gb hb ib jb [H+] = [H’],.
particbating reactions all all all f l , 2, f 3 , 1 4 , f 5 , 1 6 , 7 f l , f 2 , f 3 , f4, f5, 7 1, 2, f 3 , f 4 , f 5 1, 2, f 4 , f 5 1, 2, f 4 , 5 1, 2, 4, 5 1, 2, f4’, 5’ 1, 2, f4‘!
[H’] = [Htl0; [BrOC] = [BrO3-Io.
no. of rate constants 14 14 14 12 10 8 6 5 4 5 4
no. of species hdeDendent)
remarks results in agreement with experiment results within few percent of full mechanism results within few percent of full mechanism results within few percent of full mechanism results within few percent of full mechanism results within few percent of full mechanism qualitative agreement with full mechanism qualitative agreement with full mechanism critical point; no oscillations no critical point; no oscillations qualitative agreement with full mechanism
2858
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
i 1
L
L
20
60
40
80 [Br0;],4G3
100
I20
I40
A
1
I60
Figure 2. Bistability and oscillation regions for (A) reactions 1, 2, f3, f4, and k5 only and (B) reactions 1, 2, f4, and +5 only. Constraints as in Figure 1.
Bar-EliIg and Showalter et aLzowho used the Field-Kor6s-Noyes2' mechanism to analyze experimental results of the BelousovZhabotinskii reaction.22 The results obtained when reaction 3 are also omitted and only reactions 1, 2, f 4 and f5 remain are shown in Table I and in Figure 1B. Apart from its location-at lower bromide valuesboth plots A and B look pretty much the same. The scheme at this point takes the form Br-
F!
HBr02
(9)
k , = 4.725[BrO3-Io HBr02
+ Br- F? 2HOBr
k, = 3
(10)
lo9
X
H B r 0 2 + 2Br02.
k4 = 1.5 X 1O4[BrO3-I0
(11)
k-, = 2 X lo7
Ce3+ + Br02- F! Ce4+ + H B r 0 2 k , = 9.75
X
lo5
k-, = 2.4
X
Bar-Eli ratio of ceric to cerous ions at [BrO,-], = 2 X M, [Br-1, = 10 X lod M is 6 X loe4at SSI regardless of the scheme, 0.33 at SSII with the full scheme, 0.15 when only reactions 1, 2, f 3 , f4, and f 5 participate, 0.16 when reactions 3 are omitted, and 2.3 X IO3 when reaction -5 is omitted. (h) When reaction -4 is omitted, the bistability region moves to a slightly lower bromide values (Table I). However, the oscillation region disappears completely. Thus this omission changes the system qualitatively and will not be used anymore. Up to this stage we have made progress by omitting certain reactions and by including hydrogen and bromate ions in the appropriate rate constants. The system includes now 4 independent species with 5 rate constants. Further reduction of the number of species must be done by excluding the radical BrO,., which appears only in reactions 4 and 5. Indeed summing (4) 2 X (5) gives
+
Br03- + 3H+ + 2Ce3+
+ HBr02 s 2Ce4+
This reaction produces bromous acid autocatalytically, while, at the same time, oxidizing cerous to ceric ions. In devicing the Oregonator model, Noyes and Field23approximated the kinetics of this reaction in the forward direction by
H+ + Br03-
+ H B r 0 2 (+ 2Ce3+)
lo7
(g) The effect of omitting reeaction -5, i.e., the reduction of ceric ions by bromous acid, depends heavily on whether reaction 3 is included in the scheme or not, as shown in Figure 2. Plot A shows the results when only reactions 1, 2, f 3 , f 4 , and 5 are included. Compared to plot A of Figure 1, which shows the full mechanism, or the scheme of stage (e) which is very similar to it, it is seen that the lower hysteresis limit remains nearly the same, while the upper limit is somewhat above that of stage (e). This results in a critical point located at higher values of [Br-1, and [BrO,-], (Table I), and somewhat larger oscillation region. When reaction 3 is omitted, plot B, the bistability region falls again to lower values of [Br-I,, the upper hysteresis limit becomes more flat, i.e., independent of bromate concentrations (especially at low values of bromate), and an appreciable enlargement of the oscillation region, which extends to [BrO,-], = 0.160 M. When reaction -5 is omitted, ceric ions cannot be reduced. At SSI their concentration is low since most of the cerium is in the cerous state; however, in SSII, the omission of reaction -5 results in nearly complete oxidation of the cerium, and not only in partial reduction as indicated by the full scheme. thus, for example, the (19) P. De Kepper and K.Bar-Eli, J . Phys. Chem., 87, 480 (1983). (20) K. Showalter, R. M. Noyes, and K.Bar-Eli, J. Chem. Phys., h9.2514 (1 978). (21) R. J. Field, E. Koros, and R. M. Noyes, J. Am. Chem. Soc., 94, 8649 (1972). (22) (a) B. P. Belousov, Sb. Re$ Radiors. Med., 145 (1959); (b) A. M. Zhabotinskii, Dokl. Akad. Nauk. SSSR, 157, 392 (1969). (c) A. N. Zaikin and A. M. Zhabotinskii, Nature (London), 225, 535 (1970).
F!
2HBr02 + 2Ce4+ (14)
and limited the production of bromous acid by its disproportionation described by reaction 7. Cerous ions are in parentheses since they do not influence the kinetics, i.e., their oxidation is zero order in their concentration. Geiseler and Fo11ner24have used the same approximation for the N F T model. The hysteresis limits obtained, in the bromate-bromide subspace, are two parallel lines (on a log scale). No critical point and no oscillations are obtained. In this work we have tested a few variations of the scheme; the variations include only reactions 4 and 5 while leaving 1 and 2 as before. (9 HBrO,
(12)
+ 2HBr0, + H 2 0 (1 3)
G
2HBr02
k4, = 1.5 X lo4 [Br03-], Ce3+
k-,, = 4
(40 X
lo7
+ H B r 0 2 + Ce4+ + H B r 0 2 k5, = 9.75
X
lo5
This version differs from the original Geiseler-Follner model in two points: (1) Two molecules of bromous acid do not disproportionate, but give one molecule as a product. This only changes,the effective value of k.+ (2) Cerous ions disappear by first-order reaction and not a zero-order one. One notes that a zero-order oxidation of cerous ions may result in negative concentrations of these ions. Bar-Eli and G e i ~ l e also r ~ ~describe the oxidation of cerous ions to be of first order. The hysteresis limits are located at [Br-lo (lower) = 0.036[BrOY]: and [Br-1, (upper) = (1.4 X 10*o/k,,)[Br03-]~. Both limits are proportional to the square of bromate ion concentration-with no critical point (Le., will give parallel lines on a log-log plot), do not change if k5,is proportional to [BrO,-], or not, and do not change if reaction 5' is made reversible by taking k-,, = k-,. The upper limit decreases with an increase of k-,,. The similarity between the results of Geiseler and Fo11ner24and those given here stems from the fact that in both schemes the equations for bromide ions and bromous acid are independent of cerous ions and can be solved independently. The analysis given by the above authors shows clearly that the steady states are either a single stable node or two stable nodes and a saddle. The addition of the third (cerium ions) equation does not change these results. (23) R. J. Field, and R. M. Noyes, J . Chem. Phys. 60, 1877 (1974). (24) W. Geiseler and H. H. Follner, Biophys. Chem., 6, 107 (1977). (25) K. Bar-Eli and W. Geiseler, Acta Chim. Acad. Sci. Hung.,110, 239 (1982).
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2859
The Minimal Bromate Oscillator TABLE I V Critical Point Calculated for the Various Alternatives of Stage j
kp k-4” 1O6 [Br-] o-
x 105 9.75 x 105 9.75 x 105. 9.75 x 105. tBrO3-10 [BrO,-Io 4X109 4 x 107 4X109 4X107 9.75
95.65
146.27
54.2
146.1
1065
485
1025
i0n
a 0
1
*,-.
__-----:‘! -- _ ---I -
/’
I
,I
/
I
I
,
,
I
X
0
(critical) 1O6[BrO3-]- 467 (critical)
m Y
1
1
0
200
I
1
400
600
800
1000
[ar0;]0x~~3
Figure 4. Bistability and oscillation region for stage 6). Upper plot: k4,, = 9.75 X 105k4,, = 4 X lo9. Lower plot: k4,, = 9.75 X 105[Br03]ok4,, = 4 X lo9. Constant constraints as in Figure 1.
to higher values of bromate and bromide ions. Less significant changes are obtained whether or not k4”depends on bromate ions. The extents of the bistability and oscillations region a: e shown in Figures 3 and 4. As k+ is lowered the oscillation region extends to very large values of bromate ion concentrations. At k+ = 4 X lo9 and k4,,independent of bromate ion the bistability region becomes very narrow. Although the results differ quite a bit from the experimental ones, qualitative agreement still exists, and better quantitative agreement can be achieved by adjusting the parameters. At this point the N F T mechanism is reduced to three variables with only four rate constants.
0
~
I
O
O
O I500
-
L”30
Thus, one obtains either a single steady state or bistability with hysteresis, but Hopf bifurcation and oscillations are impossible, regardless of the particular rate constants used. Both the original Geiseler-Follner model and stage (i) do not exclude critical points. In fact the latter authors obtained a low critical point at [BrO3-Io = 10“ M and [Br-lo = M, which is beyond experimental range and outside the validity of constant bromate approximation. An upper critical point may also be obtained at tremendously high values of bromate and bromide concentrations. A rough approximation for the location of this point is given by [BrO
