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Mechanism for the color transition of the Belousov-Zhabotinsky reaction catalyzed by cerium ions and ferroin
Laboratory of Mathematics, Faculty of Regional Environment Science, Tokyo University of Agriculture, 1-1-1 Sakuragaoka, Setagaya-ku, Tokyo 156-8502, Japan |
The oscillation property of the Belousov-Zhabotinsky reaction and the color transition of its solution depend on the catalytic action of the metal ions. The solution of the reaction system catalyzed by both cerium ions and ferroin has a more complicated effect on the color than either the cerium-catalyzed case or the ferroin-catalyzed case. To theoretically elucidate the color transition of the case catalyzed by these two ions, a reduced model consisting of three differential equations is proposed, incorporating both the Rovinsky-Zhabotinsky scheme and the Field-Körös-Noyes scheme simplified by Tyson [Ann. N.Y. Acad. Sci., 316 (1979), pp.279-295]. The presented model can have a limit cycle under reasonable conditions through a Hopf bifurcation, and its existence theorem is proven by employing the bifurcation criterion established by Liu [J. Math. Anal. Appl., 182 (1994), pp.250-256].
References:
[1] |
C. Egami,
Bifurcation analysis of the Nowak-Bangham model in CTL dynamics, Math. Biosci., 221 (2009), 33-42.
doi: 10.1016/j.mbs.2009.06.005. |
[2] |
R. J. Field and H.-D. Försterling, On the oxybromine chemistry rate constants with cerium ions in the Field-Körös-Noyes mechanism of the Belousov-Zhabotinskii reaction: The equilibrium $\rm HBrO_2 + BrO_3^{-} + H^{+} \rightleftharpoons 2BrO_2{·} + H_2O$, J. Phys. Chem., 90 (1986), 5400-5407. |
[3] |
R. J. Field, E. Körös and R. M. Noyes,
Oscillations in chemical systems. Ⅱ. Thorough analysis of temporal oscillation in the Bromate-Cerium-Malonic acid system, J. Am. Chem. Soc., 94 (1972), 8649-8664.
doi: 10.1021/ja00780a001. |
[4] |
R. J. Field and R. M. Noyes,
Oscillation in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884.
doi: 10.1063/1.1681288. |
[5] |
H.-D. Försterling, L. Stuk, A. Barr and W. D. McCormick, Stoichiometry of bromide production from ceric oxidation of bromomalonic acid in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 97 (1993), 7578-7584. |
[6] |
L. Györgyi and R. J. Field,
Aperiodicity resulting from two-cycle coupling in the Belousov-Zhabotinskii reaction. ⅲ. Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous-flow, stirred tank reactor, J. Chem. Phys., 91 (1989), 6131-6141.
doi: 10.1063/1.457432. |
[7] |
L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602. |
[8] |
L. Györgyi, T. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170. |
[9] |
J. P. Keener and J. J. Tyson,
Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, Phys. D, 21 (1986), 307-324.
doi: 10.1016/0167-2789(86)90007-2. |
[10] |
G. Kshirsagar and R. J. Field, A kinetic and thermodynamic study of component processes in the equilibrium $\rm 5HOBr \rightleftharpoons 2BR_2+BrO_3^{-}+2H_2O+H^{+}$, J. Phys. Chem., 92 (1988), 7074-7079. |
[11] |
W. M. Liu,
Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.
doi: 10.1006/jmaa.1994.1079. |
[12] |
A. B. Rovinsky and A. M. Zhabotinsky,
Mechanism and mathematical model of the oscillating Bromate-Ferroin-Bromomalonic acid reaction, J. Phys. Chem.(25), 88 (), 6081-6084,1984.
doi: 10.1021/j150669a001. |
[13] |
C. E. Sjogren, S. Kolboe and P. Ruoff, Transitions between two oscillatory states in a closed malonic acid Belousov-Zhabotinsky reaction simultaneously catalyzed by ferroin and cerium ions, Chem. Phys. Lett., 130 (1986), 72-75. |
[14] |
A. F. Taylor, V. Gáspár, B. R. Johnson and S. K. Scott,
Analysis of reaction-diffusion waves in the ferroin-catalysed Belousov-Zhabotinsky reaction, Phys. Chem. Chem. Phys., 1 (1999), 4595-4599.
doi: 10.1039/a904994k. |
[15] |
J. J. Tyson,
Oscillations, bistability, and echo waves in models of the Belusov-Zhabotinskii reaction, Ann. N.Y. Acad. Sci., 316 (1979), 279-295.
|
[16] |
J. J. Tyson,
Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082.
doi: 10.1063/1.446690. |
[17] |
J. J. Tyson and P. C. Fife,
Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237.
doi: 10.1063/1.440418. |
[18] |
A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin and I. R. Epstein,
Oscillations and waves in metal-ion-catalyzed Bromate oscillating reactions in highly oxidized states, J. Phys. Chem., 97 (1993), 7578-7584.
doi: 10.1021/j100131a030. |
show all references
References:
[1] |
C. Egami,
Bifurcation analysis of the Nowak-Bangham model in CTL dynamics, Math. Biosci., 221 (2009), 33-42.
doi: 10.1016/j.mbs.2009.06.005. |
[2] |
R. J. Field and H.-D. Försterling, On the oxybromine chemistry rate constants with cerium ions in the Field-Körös-Noyes mechanism of the Belousov-Zhabotinskii reaction: The equilibrium $\rm HBrO_2 + BrO_3^{-} + H^{+} \rightleftharpoons 2BrO_2{·} + H_2O$, J. Phys. Chem., 90 (1986), 5400-5407. |
[3] |
R. J. Field, E. Körös and R. M. Noyes,
Oscillations in chemical systems. Ⅱ. Thorough analysis of temporal oscillation in the Bromate-Cerium-Malonic acid system, J. Am. Chem. Soc., 94 (1972), 8649-8664.
doi: 10.1021/ja00780a001. |
[4] |
R. J. Field and R. M. Noyes,
Oscillation in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884.
doi: 10.1063/1.1681288. |
[5] |
H.-D. Försterling, L. Stuk, A. Barr and W. D. McCormick, Stoichiometry of bromide production from ceric oxidation of bromomalonic acid in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 97 (1993), 7578-7584. |
[6] |
L. Györgyi and R. J. Field,
Aperiodicity resulting from two-cycle coupling in the Belousov-Zhabotinskii reaction. ⅲ. Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous-flow, stirred tank reactor, J. Chem. Phys., 91 (1989), 6131-6141.
doi: 10.1063/1.457432. |
[7] |
L. Györgyi and R. J. Field, Simple models of deterministic chaos in the Belousov-Zhabotinskii reaction, J. Phys. Chem., 95 (1991), 6594-6602. |
[8] |
L. Györgyi, T. Turányi and R. J. Field, Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction, J. Phys. Chem., 94 (1990), 7162-7170. |
[9] |
J. P. Keener and J. J. Tyson,
Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, Phys. D, 21 (1986), 307-324.
doi: 10.1016/0167-2789(86)90007-2. |
[10] |
G. Kshirsagar and R. J. Field, A kinetic and thermodynamic study of component processes in the equilibrium $\rm 5HOBr \rightleftharpoons 2BR_2+BrO_3^{-}+2H_2O+H^{+}$, J. Phys. Chem., 92 (1988), 7074-7079. |
[11] |
W. M. Liu,
Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl., 182 (1994), 250-256.
doi: 10.1006/jmaa.1994.1079. |
[12] |
A. B. Rovinsky and A. M. Zhabotinsky,
Mechanism and mathematical model of the oscillating Bromate-Ferroin-Bromomalonic acid reaction, J. Phys. Chem.(25), 88 (), 6081-6084,1984.
doi: 10.1021/j150669a001. |
[13] |
C. E. Sjogren, S. Kolboe and P. Ruoff, Transitions between two oscillatory states in a closed malonic acid Belousov-Zhabotinsky reaction simultaneously catalyzed by ferroin and cerium ions, Chem. Phys. Lett., 130 (1986), 72-75. |
[14] |
A. F. Taylor, V. Gáspár, B. R. Johnson and S. K. Scott,
Analysis of reaction-diffusion waves in the ferroin-catalysed Belousov-Zhabotinsky reaction, Phys. Chem. Chem. Phys., 1 (1999), 4595-4599.
doi: 10.1039/a904994k. |
[15] |
J. J. Tyson,
Oscillations, bistability, and echo waves in models of the Belusov-Zhabotinskii reaction, Ann. N.Y. Acad. Sci., 316 (1979), 279-295.
|
[16] |
J. J. Tyson,
Relaxation oscillations in the revised Oregonator, J. Chem. Phys., 80 (1984), 6079-6082.
doi: 10.1063/1.446690. |
[17] |
J. J. Tyson and P. C. Fife,
Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73 (1980), 2224-2237.
doi: 10.1063/1.440418. |
[18] |
A. M. Zhabotinsky, F. Buchholtz, A. B. Kiyatkin and I. R. Epstein,
Oscillations and waves in metal-ion-catalyzed Bromate oscillating reactions in highly oxidized states, J. Phys. Chem., 97 (1993), 7578-7584.
doi: 10.1021/j100131a030. |







No. | Contents and concentrations |
(S1) | |
(S2) | |
(S3) | |
(S4) | |
(S5) | |
No. | Contents and concentrations |
(S1) | |
(S2) | |
(S3) | |
(S4) | |
(S5) | |
Exp. | Solution number and volume [mL] | Type of catalyst |
Ⅰ | (S1) 20, (S2) 20, (S3) 20 | cerium-catalyzed |
Ⅱ | (S1) 20, (S2) 20, (S4) 15, (S5) 5.0 | ferroin-catalyzed |
Ⅲ | (S1) 20, (S2) 20, (S3) 20, (S5) 1.0 | cerium-ferroin-catalyzed |
Exp. | Solution number and volume [mL] | Type of catalyst |
Ⅰ | (S1) 20, (S2) 20, (S3) 20 | cerium-catalyzed |
Ⅱ | (S1) 20, (S2) 20, (S4) 15, (S5) 5.0 | ferroin-catalyzed |
Ⅲ | (S1) 20, (S2) 20, (S3) 20, (S5) 1.0 | cerium-ferroin-catalyzed |
Concentration | Rate constant | ||||
0.04 M | |||||
0.10 M | |||||
6.3 mM | |||||
0.13 mM | |||||
0.65 mM | |||||
1.29 M |
Concentration | Rate constant | ||||
0.04 M | |||||
0.10 M | |||||
6.3 mM | |||||
0.13 mM | |||||
0.65 mM | |||||
1.29 M |
| | type | |
+ | + | + | T1 |
+ | + | 0 | T5 |
+ | + | - | T3 |
+ | 0 | + | T6 |
+ | 0 | 0 | T3 |
+ | 0 | - | T3 |
+ | - | + | T3 |
+ | - | 0 | T3 |
+ | - | - | T3 |
0 | + | + | T7 |
0 | + | 0 | T0 |
0 | + | - | T0 |
0 | 0 | + | T9 |
0 | 0 | 0 | T0 |
0 | 0 | - | T0 |
0 | - | + | T8 |
0 | - | 0 | T0 |
0 | - | - | T0 |
- | + | + | T4 |
- | + | 0 | T0 |
- | + | - | T0 |
- | 0 | + | T4 |
- | 0 | 0 | T0 |
- | 0 | - | T0 |
- | - | + | T2 |
- | - | 0 | T0 |
- | - | - | T0 |
| | type | |
+ | + | + | T1 |
+ | + | 0 | T5 |
+ | + | - | T3 |
+ | 0 | + | T6 |
+ | 0 | 0 | T3 |
+ | 0 | - | T3 |
+ | - | + | T3 |
+ | - | 0 | T3 |
+ | - | - | T3 |
0 | + | + | T7 |
0 | + | 0 | T0 |
0 | + | - | T0 |
0 | 0 | + | T9 |
0 | 0 | 0 | T0 |
0 | 0 | - | T0 |
0 | - | + | T8 |
0 | - | 0 | T0 |
0 | - | - | T0 |
- | + | + | T4 |
- | + | 0 | T0 |
- | + | - | T0 |
- | 0 | + | T4 |
- | 0 | 0 | T0 |
- | 0 | - | T0 |
- | - | + | T2 |
- | - | 0 | T0 |
- | - | - | T0 |
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