The Belousov-Zhabotinskii reaction

The Belousov-Zhabotinskii (BZ) reaction has been selected as an example illustrating diverse dynamical states observable in chemical systems. The BZ reagent is very convenient both for experimental and theoretical investigations, since the BZ reaction has many dynamical states of interest, which will be described below. In the BZ reaction one may observe the steady state, the time periodic state (concentration oscillations), the spatially periodic state, the stationary state (dissipative structures), the time and spatially periodic state (propagating chemical waves) and turbulent states (chaotic oscillations, stochastic spatial structures, stochastic chemical waves). 

It should be emphasized that there have been exceptions to this attitude. In 1910 and 1920 Lotka published his theory of chemical reactions in which the oscillations of reagent concentrations could appear. An essential feature of the Lotka models was nonlinearity. In mathematics and physics a trend has long persisted to examine linear systems and phenomena and to replace non-linear models by (approximate) linear models. The trend, originating from insufficient mathematical means, has turned into specific philosophy. The non-linear Lotka models thus constituted a deviation from a canon. Hence, general arguments of thermodynamic nature, lack of interest in non-linear models and commonness of observations of a monotonic attainment of the equilibrium in chemical reactions were the reasons for skepticism and disbelief which the results of Belousov have met with. 

The situation has begun to change on obtaining new results in thermodynamics by Prigogine and his school for the systems far from the state of thermodynamic equilibrium. In addition, two crucial experimental discoveries have taken place. The Belousov system has been modified by Zhabotinskii (the starting of research on the Belousov system by Zhabo-tinskii was in fact the turning-point until then all articles and reports on 

It has also been established that on changing the initial conditions the system may pass from the stationary state to any of the states mentioned above transitions between the other states of the system are also allowed. The behaviour of this type can be described in terms of the notions of catastrophe theory. A reaction mixture is the system whose state may be represented by a number of state variables, whereas other variables, called control parameters, are varied continuously and a change in the state of the system is examined. An abrupt change in the state of the system, a catastrophe, occurs for some values of control parameters. For instance, at a continuous change in concentration of one of components the system may pass from the stationary state to the oscillatory state. 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

While concentration oscillations have been observed in a variety of systems, only two homogeneous chemical oscillators have been unambiguously established. When malonic acid (HMa) is oxidized by BrOg- in sulfuric acid solution using a Ce(IV)/Ce(III) couple as catalyst, Belousov observed that the tCe(IV)]/[Ce(III)] ratio oscillated periodically. Further studies by Zhabotinskii demonstrated oscillatory behavior when Ce(IV)/Ce(III) was replaced by Fe(III)/Fe(II) or Mn(III)/Mn(II) and 

Both the Belousov-Zhabotinskii and the Bray-Liebhafsky systems have been characterized mechanistically. We consider the Belousov reaction in detail. Oscillatory behavior is not limited to a narrow range of concentrations. Initial conditions which lead to periodic concentration variation are given in Table 7.1. The period of oscillation is quite dependent upon initial concentrations, as suggested by the Lotka result (7.23) it varies from 15 to 200 sec in different experiments. Not only are there oscillations in the [Ce(IV)]/[Ce(III)] ratio but [Br ], which is present in trace quantities, oscillates as well. Typical results are shown in Fig. 7.6 the amplitudes diminish only very gradually over dozens of cycles. For the experiment il- 

SBrOa + 3H+ + 5CH2(COOH)2 = 3BrCH(COOH 2 + 4C02 

The key to understanding the phenomena is knowledge of the chemistry of BrOs which, while a powerful oxidizing agent, does not react directly with organic acids. The brominating agent in (7.26) is Br2 while production of CO2 and formic acid is due to attack by Ce(IV) on either HMa or BrMa. As is so often the case, kinetically important species do not appear in the overall stoichiometry. 

Two separate sequences account for (7.26). Schematically they may be summarized as 

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In the Belousov-Zhabotinskii reaction, beautiful regular patterns form spontaneously as the result of the oscillating concentrations of reactants and products due to competing reactions. 

Halc enation of Acids - The Belousov-Zhabotinskii Reaction... 

For the catalytic oxidation of malonic acid by bromate (the Belousov-Zhabotinskii reaction), fimdamental studies on the interplay of flow and reaction were made. By means of capillary-flow investigations, spatio-temporal concentration patterns were monitored which stem from the interaction of a specific complex reaction and transport of reaction species by molecular diffusion [68]. One prominent class of these patterns is propagating reaction fronts. By external electrical stimulus, electromigration of ionic species can be investigated. 

This reaction, widely known as the Belousov-Zhabotinskii reaction, can proceed in an oscillatory fashion [68]. For overall slow conversion, the concentrations of intermediates and the catalyst undergo cyclic changes. By this means, many pulselike reaction zones propagate in a spatially distributed system. Ferroin/ferriin can be applied as an optically detectable catalyst... 

Figure 4.97 Global changes of the Belousov-Zhabotinskii reaction behavior in the capillary reactor under electrical field. (A) Reversal of the direction of the reaction zone (white stripe) propagation (0.5 mm capillary reactor) ...

The Belousov-Zhabotinskii reaction is a typical oscillating chemical reaction. Spiral structures form periodically, disappear and reappear as the result of an autocatalytic reaction, the oxidation of Ce3+ and Mn2+ by bromate (lessen, 1978). 

J- J. Tyson. The Belousov-Zhabotinskii Reaction, Lecture Notes in Biomathematics, vol 10. Springer-Verlag, New York, 1976. 

Fig. 1.1. Typical experimental records of oscillatory behaviour in the Belousov-Zhabotinskii reaction (a) platinum electrode which responds primarily to the Ce3+/Ce4+ couple (b) bromide-sensitive electrode measuring In [Br ].

Fig. 1.19. Complex, but strictly periodic, oscillations in a chemical reaction showing bursting in a model of the Belousov-Zhabotinskii reaction. (Reprinted with permission from Bar-Eli, K and Noyes, R. M. (1988). J. Chem. Phys., 88, 3636-54. American Institute of Physics.)...

Tyson, J. J. (1976). The Belousov-Zhabotinskii reaction. Springer, Berlin. Field, R. J. and Burger, M. (eds) (1985). Oscillations and travelling waves in chemical systems. Wiley-Interscience, New York. 

Waves of chemical reaction may travel through a reaction medium, but the ideas of important stationary spatial patterns are due to Turing (1952). They were at first invoked to explain the slowly developing stripes that can be exhibited by reactions like the Belousov-Zhabotinskii reaction. This (rather mathematical) chapter sets out an analysis of the physically simplest circumstances but for a system (P - A - B + heat) with thermal feedback in which the internal transport of heat and matter are wholly controlled by molecular collision processes of thermal conductivity and diffusion. After a careful study the reader should be able to ... 

As we have already commented, mappings of the type discussed above are not in any way easily related to a given set of reaction rate equations. Such mappings have, however, been used for chemical systems in a slightly different way. A quadratic map has been used to help interpret the oscillatory behaviour observed in the Belousov-Zhabotinskii reaction in a CSTR. There, the variable x is not a concentration but the amplitude of a given oscillation. Thus the map correlates the amplitude of one peak in terms of the amplitude of the previous excursion. 

The Belousov-Zhabotinskii reaction is a cerium-catalyzed oxidation of malonic acid by bromate, in which the quotient [Ce3+]/[Ce4+] oscillates by a factor of 10 to 100.8... 

Kenney in England and Luss, Takoudis, Schmidt, Ray, Jensen over here spring to mind in addition to those already mentioned. The Belousov-Zhabotinskii reaction has of course generated a minor industry of experimental and theoretical studies, dissipative systems and locus-ators which it would be rash to rush into here. 

The Belousov-Zhabotinskii reaction in an isothermal CSTR can undergo a series of transitions among periodic and chaotic states. One segment of this series of transitions is investigated in detail. Liapunov characteristic exponents are calculated for both the periodic and chaotic regions. In addition, the effect of external disturbances on the periodic behavior is investigated with the aid of a mathematical model. 

In an open system such as a CSTR chemical reactions can undergo self-sustained oscillations even though all external conditions such as feed rate and concentrations are held constant. The Belousov-Zhabotinskii reaction can undergo such oscillations under isothermal conditions. As has been demonstrated both by experiments [1] and by calculations 12,3] this reaction can produce a variety of oscillation types from simple relaxation oscillations to complicated multipeaked periodic oscillations. Evidence has also been given that chaotic behavior, as opposed to periodic or quasi-periodic behavior, can take place with this reaction [4-12]. In addition, it has been shown in recent theoretical studies that chaos can occur in open chemical reactors [11,13-17]. 

We have investigated the transitions among the types of oscillations which occur with the Belousov-Zhabotinskii reaction in a CSTR. There is a sequence of well-defined, reproducible oscillatory states with variations of the residence time [5]. Similar transitions can also occur with variation of some other parameter such as temperature or feed concentration. Most of the oscillations are periodic but chaotic behavior has been observed in three reproducible bands. The chaos is an irregular mixture of the periodic oscillations which bound it e.g., between periodic two peak oscillations and periodic three peak oscillations, chaotic behavior can occur which is an irregular mixture of two and three peaks. More recently Roux, Turner et. al. 

We have therefore made a preliminary investigation of the effects of such disturbances using the model of Ganapathisubramanian and Noyes (3). This is a seventh order model for the Belousov-Zhabotinskii reaction in a CSTR. The equations and all necessary parameters are given in their paper. The model predicts a periodic 2-3 oscillatory region bracketed by a two peak and a three peak periodic oscillation (for constant feed rates). The transition points predicted by the model have been calculated to two or three significant figures by numerical simulation. The transition between 11(2) and n(2,3) occurs at... 

Tyson, J. J. (1985) A quantitative account of oscillations, bistability, and travelling waves in the Belousov-Zhabotinskii reaction. In R. J. Field and M. Burger, eds. Oscillations and Traveling Waves in Chemical Systems (Wiley, New York). 

IIIC) 1978 Wegmann, K., Rossler, O. E. Different Kinds of Chaotic Oscillations in the Belousov-Zhabotinskii Reaction, Z. Naturforschung A, vol. 33A, no. 10, 1170-1183 (III J) 1980 Willamowski, Rossler, O. E. Irregular Oscillations in a Realistic Abstract Quadratic Mass Action System, Z, Naturforsch., vol. 35a, 317-318 (IIIG) 1965 Yamazaki, L., Yokoya, K., Nakajima, R. Oscillatory Oxidations of Reduced Pyridine Nucleotide by Peroxidase, Biochim. Biophys. Res. Commun., vol. 21, 582-586 (IIIG) 1967 Yamazaki, I., Yokota, K. Analysis of the Conditions Causing the Oscillatory Oxidation of Reduced Nicotinamide-Adenine Dinucleotide by Biochem. Biophys. Acta, vol. 132, 310-320... 

Fig. III.C.2. A horatian solution of the Belousov-Zhabotinskii Reaction, Pt, Br, and Pt(t-10) as variables. (After Hudson Mankin (1981))...

IIIC) Handlirova, M., Tockstein, A. New Catalysts for the Belousov-Zhabotinskii Reaction. 

IIIC, D) Hudson, J. L., Mankin, J. C. Chaos in the Belousov-Zhabotinskii Reaction. J. Chem. 

Different Types of the Belousov-Zhabotinskii Reaction. Acta Chim. Acad. Sci. Hung. 

IIIC) Tomita, K., Tsuda, I. Chaos in the Belousov-Zhabotinskii Reaction in a Flow System. 1979-1 Phys. Lett. 71A, 489 192... 

Ions in Relation to the Belousov-Zhabotinskii Reaction, Collect. Czech. Chem. Commun. 47, 2831-2837... 

IIIC) Treindl, L. U., Fabian, P. Influence of Oxygen on the Belousov-Zhabotinskii Reaction. 

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