Examples Belousov-Zhabotinsky

Existing models were starting from Flory s equation and added other equations to meet the numbers of variables. This is the reason why they are complicated. On the other hand, reverse approach has not been investigated, in order to reduce the numbers of variables by focusing only on major variables. This approach is called reductive method[167], which is commonly used in the field of nonequilibrium thermodynamics. For example, Belousov-Zhabotinsky (BZ)... 

An example of the application of J2-weighted imaging is afforded by the imaging of the dynamics of chemical waves in the Belousov-Zhabotinsky reaction shown in figure B 1.14.5 [16]. In these images, bright... 

This subject has been reviewed by Noyes and Field,8 who give reference to the original formulation as well as a more explicit treatment. The presentation here will be given not in general terms but by means of one striking example, the oxidation of malonic acid by bromate ions catalyzed by cerium(IV). It is called the Belousov-Zhabotinsky (or BZ) reaction, after its discoverers.9 The stoichiometry of the reaction with excess malonic acid is... 

How relevant are these phenomena First, many oscillating reactions exist and play an important role in living matter. Biochemical oscillations and also the inorganic oscillatory Belousov-Zhabotinsky system are very complex reaction networks. Oscillating surface reactions though are much simpler and so offer convenient model systems to investigate the realm of non-equilibrium reactions on a fundamental level. Secondly, as mentioned above, the conditions under which nonlinear effects such as those caused by autocatalytic steps lead to uncontrollable situations, which should be avoided in practice. Hence, some knowledge about the subject is desired. Finally, the application of forced oscillations in some reactions may lead to better performance in favorable situations for example, when a catalytic system alternates between conditions where the catalyst deactivates due to carbon deposition and conditions where this deposit is reacted away. 

The Runge-Kutta algorithm cannot handle so-called stiff problems. Computation times are astronomical and thus the algorithm is useless, for that class of ordinary differential equations, specialised stiff solvers have been developed. In our context, a system of ODEs sometimes becomes stiff if it comprises very fast and also very slow steps and/or very high and very low concentrations. As a typical example we model an oscillating reaction in The Belousov-Zhabotinsky (BZ) Reaction (p.95). 

The 1970s saw an explosion of theoretical and experimental studies devoted to oscillating reactions. This domain continues to expand as more and more complex phenomena are observed in the experiments or predicted theoretically. The initial impetus for the smdy of oscillations owes much to the concomitance of several factors. The discovery of temporal and spatiotemporal organization in the Belousov-Zhabotinsky reaction [22], which has remained the most important example of a chemical reaction giving rise to oscillations and waves. 

Of special interest is the so-called Belousov-Zhabotinsky class of similar reactions [4, 6-12], This system can serve as an extremely successful example of self-organisation proper mixing of several liquids in a given proportion and at certain temperature demonstrates practically all kinds of the autowave processes just mentioned. 

Numerous versions of the Belousov-Zhabotinsky system differ by chemical compounds used. The typical reaction involves oxidation of some organic compound by bromate ion (BrOj ) occurring in acid medium with metal catalyst (Ce3+, Mn2+, as well as complexes of Fe2+, Ru2+). As an example, a particular reaction [4] could be mentioned, where an organic reductor is malonic acid CH2(COOH)2 and Ce3+ ions serve as a catalyst. In this reaction a solution changes periodically its colour due to oscillations in Ce3+ concentration. Generally speaking, the reaction consists of two stages. At the first one metal is oxidized... 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Another aspect of a very different nature also merits attention. For complex reaction schemes, it can be very cumbersome to write the appropriate set of differential equations and their translation into computer code. As an example, consider the task of coding the set of differential equations for the Belousov-Zhabotinsky reaction (see Section 7.5.2.4). It is too easy to make mistakes and, more importantly, those mistakes can be difficult to detect. For any user-friendly software, it is imperative to have an automatic equation parser that compiles the conventionally written kinetic model into the correct computer code of the appropriate language [37-39],... 

The Belousov-Zhabotinsky reaction system is one example leading to such chemical oscillations. One of the interesting phenomena is the effect of the very narrow range of controlling parameter /x on the stability of the Belousov-Zhabotinsky reaction system. The following reactions represent the Belousov-Zhabotinsky reaction scheme ... 

Example 13.5 The Belousov-Zhabotinsky reaction scheme Field et al. (1972) explained the qualitative behavior of the Belousov-Zhabotinsky reaction, using the principles of kinetics and thermodynamics. A simplified model with three variable concentrations producing all the essential features of the Belousov-Zhabotinsky reaction was published by Field and Noyes (1974). Some new models of Belousov-Zhabotinsky reaction scheme consist of as main as 22 reaction steps. With the defined symbols X = HBr02, Y = Br, Z = Ce4+, B = organic, A = B1O3 (the rate constant contains H+), FKN Model (Field et al., 1972) consists of the following steps summarized by Kondepudi and Priogogine (1999) ... 

The easily visualized Belousov Zhabotinsky (BZ) reaction is now a classic example of the emergence of temporal and spatio temporal dissipative struc tures in homogeneous chemical systems. The reaction was discovered by Soviet military chemist B. P. Belousov in 1951 when he was studying homo geneous oxidation of citric acid by potassium bromide, KBrOq, in the presence of cerium sulfate Ce(S04)2 as the catalyst for redox processes. In the dissolved mixture of these compounds under certain process conditions, Belousov discovered a time-oscillating synchronous reduction of cerium(4+) ions ... 

In this section, we consider the breakdown of the condition of normal hyperbolicity. First, we explain a simple example where breakdown of normal hyperbolicity leads to a bifurcation in reaction processes. In the Belousov-Zhabotinsky (BZ) reaction [40], the bifurcation from the stable fixed point to the limit cycle takes place through the breakdown of normal hyperbolicity. This is the simplest case where mathematical analyses are in progress [41]. 

An incomparable fascination is the magic of the colorful oscillation reactions which are known today to every chemist, although only a few decades ago they were considered to be completely atypical. Here we shall present an example which, in contrast to the variety of colors often occurring in the course of these processes, demonstrates the principle of a Belousov-Zhabotinsky reaction by means of a single color change. Perhaps we can follow Nietzsche here There are many things which, once and for all, I do not wish to know. Wisdom also sets limits to knowledge. .. ... 

Example 14.3. The Belousov-Zhabotinsky reaction [22,27-29], The reaction is an oxidation of malonic acid by bromate ion in sulfuric acid, catalyzed by a Ce(III)/Ce(IV) redox couple. Many variations with other organic acids and transition-metal ions are possible [22] (Belousov used citric acid, and manganese, ruthenium, or iron can replace cerium). The color of the solution alternates between clear [Ce(III)] and pale yellow [Ce(IV)], and more dramatically between red and blue if ferroin is added as indicator. 

A reaction may be periodic if its network provides for restoration of a reactant or intermediate that has been depleted, while conversion of main reactants to products continues. Periodic behavior often results from competition of two or more contending mechanisms. Predator-prey fluctuations in ecology (Lotka-Volterra mechanisms) provide an easily visualized example. The Belousov-Zhabotinsky reaction—catalyzed oxidation of malonic acid by bromate—involves a similar competition between two pathways. 

Examples include multiple steady states in isothermal CSTRs, predator-prey fluctuations, the Belousov-Zhabotinsky reaction, and a test for stability of quasi-stationary states in reactions with a self-accelerating intermediate steps. 

Spiral waves also arise in the oxidation of carbon-monoxide on platinum surfaces [10]. In 1972 they have been discovered by Winfree [79] in the photosensitive Belousov-Zhabotinsky (BZ) reaction, see for recent investigations for example [83, 84, 87]. Both reactions are studied in the SFB 555. The classical BZ reaction is a catalytic oxidation of malonic acid, using bromate in an acidic environment. Experimentally it exhibits well reproducible drift, meander and chaotic motions of the spiral wave and its tip. 

Examples of self-sustained oscillatory systems are electronic circuits used for the generation of radio-frequency power, lasers, Belousov-Zhabotinsky and other oscillatory chemical reactions, pacemakers (sinoatrial nodes) of human hearts or artificial pacemakers that are used in cardiac pathologies, and many other natural and artificial systems. An outstanding common feature of such systems is their ability to be synchronized. 

The most studied and cited example of complex dynamic behavior in homogeneous catalysis is the Belousov - Zhabotinsky (BZ) reaction named after B. P. Belousov who discovered the reaction and A. M. Zhabotinsky who continued Belousov s early work. The reaction is theoretically important in that it shows that chemical reactions do not have to be dominated by equilibrium. These reactions are far from equilibrium and remain so for a length of time. In this sense, they provide an interesting chemical model of nonequilibrium biological phenomena, and the mathematical model of the BZ reactions themselves are of theoretical interest. 

The spiral or concentric waves observed for the spatial distribution of cAMP (fig. 5.6) present a striking analogy with similar wavelike phenomena found in oscillatory chemical systems, of which the Belousov-Zhabotinsky reaction (fig. 5.7) provides the best-known example (Winfree, 1972a). 

As an example, let us examine the well-known Belousov-Zhabotinsky (BZ) reaction [15,16] of bromate with malonic acid catalyzed by cerium ions in acidic solution. 

A common example is the Belousov - Zhabotinsky reaction [24], Beautiful patterns of chemical wave propagation can be created in a chemical reaction - diffusion system with a spatiotemporal feedback. The wave behavior can be controlled by feedback-regulated excitability gradients that guide propagation in the specified directions [25, 26]. 

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