Oscillating reactions are reactions where there is regular but periodic concentration change in one or more of the reactants. BZ reactions are the simple to understand hence they are mainly used to understand the concepts of oscillating reactions. It is essential to note that the nature of BZ reactions is more chemical than biological. However, understanding its mechanism can help one understand the concepts of biological oscillators; for example, heart beat in living organisms. This is evident in observations that have been made in biological systems that are characterized by aspects of BZ reactions such as its spiral wave. The spiral waves of BZ reaction, for example, spiral waves of these oscillatory reactions have also been observed in intact and cultured cardiac tissues and cortical neural preparations and retinal tissues. This proves that the excitable media is always characterized by the existence of spiral pattern. This form the basis of the research which is the current information regarding pattern formation
1.3 History of the reaction
Chemical oscillators were recognized as mainstream science during the 1960s. Initially, it was believed that the progress of chemical reaction was always in one direction toward the equilibrium, that is; they were progressed monotonically. Boris Pavlovich became the first scientist discover oscillating reactions. He took a solution of citric acid in water with acidified bromate. In this solution, ceric ions oscillated for up to one hour from colorless to yellow. However, the work of Belousov Zhabotinskii was not received well by other scientist. Later on his works was adopted by Anatol Zhabotinskyin 1961. This new scientist succeeded in what Zhabotinskii did not achieve; he convinced the scientific community that chemical oscillator were in deed valid scientific works. The cerium-bromate reaction was therefore changed to the Belousov-Zhabotinskiis (BZ) reaction with regards to the discovering scientist (Belousov) and the scientist that continued with the work (Zhabotinskiis)
Presently, there are numerous chemical systems that are known to oscillate. Different scholars and scientists have come up with different mathematical models to describe the reaction. Two models are of key interest, especially in this study; Brusselator and Oregonator. The actual reaction mechanism, however, still remains a mystery up to date.
1.4 Thesis Outline
There are five main sections in this research work. The section includes the Introduction, the chemical rules, the Belousov-Zhabotinsky mechanism, the conclusion, and the reference page. Nevertheless, each section with an exception of the conclusion and the works cited page is subdivided into distinct subheadings to help facilitate the understanding of each part. The previous sections covered a brief introduction to the concept of oscillatory chemical reactions and a brief history and background of the BZ reaction.
The next part is a discussion of the chemical rules and kinetics. Chemical rules section would help in creating an understanding of mathematical derivations of the models through expressing the rates of reaction and the kinetics of the catalysts. Nevertheless, it entails simpler example to the more complex ones. The concept of the collision theory and the Law of Mass Action are also introduced in this section and the relevant applications illustrated.
The third section of this thesis looks at the modeling of BZ reaction, mathematical modeling of BZ reactions and mathematical analysis. Moreover, it looks at the convert the chaotic behavior of BZ reaction and algorithms that can be applied to these systems to control the behavior. Lastly is the reference page, which includes all the borrowed literature and all the cited information.
2.1 Chemical Rules
It is essential to create a familiarity, introduce basic concepts and the equilibrium constants that would be discussed in the mathematical section of this paper. In other words, typical examples and rules that must be observed during equilibrium reactions will be given. From the following typical example;
A + B = C (constant = K1)
It is clear that C can only be formed from the reaction of substance A and B. K1 is the constant parameter, which is associated with the rate of reaction. In order to bring this reaction to a time domain, one must define the Ordinary Differential Equations (ODE) (Gray & Scott, 1990). ODE is a chemical reaction that defines derivatives and ordinary reactions. The reaction rate in this case, is an indication of how fast the concentration of the reactants (A & B) and the products (C) are changing. A postulate that assists in obtaining the ODEs is the Law of Mass Action as explained previously. This law was put forward by Cato Maximilian (1836-1902) and Peter Wage (1833-1900).
A balanced chemical equation would thus, give the identities and molar ratios of the reactants and products during chemical conversion (Swanson & Alvord, 2003). This does not describe how the conversion occurs. A reaction can proceed in a single step but in most practical cases, they occur in more than one single step (Swanson & Alvord, 2003). The series of steps that complete a reaction is called the reaction mechanism. Since most thermodynamic processes consider point functions (concepts like the net change in internal energy will only consider the internal energy at the start of a process and the internal energy at the end of a process) it is understandable how the early scientists missed concepts such as those of the oscillatory reactions (Swanson & Alvord, 2003).
From rate equations it would thus, be possible to learn how a reaction goes from the initial state to the final state. Combining the mechanism and the reaction to study a process will constitute a full understanding of its kinetics. If we have a chemical equation that describes a single molecular event then we can describe such an equation a reaction step.
A good example is the reaction between Nitrogen (II) Oxide and Bromine.
2NO (g) + Br2 (g) = 2BrNO (g)
The equation above gives an avenue for various ways in trying to describe how the reaction actually occurred. All the molecules combine in one step as shown below;
NO (g) + NO + Br2 (g) = 2BrNO (g)
The second mechanism may involve a two-step process;
NO (g) + Br2 (g) = Br2NO (g)
NO (g) + Br2NO (g) = 2 BrNO (g)
The mechanism involved in breaking down the equation of a single line reaction into a stepwise reaction constitutes the description of a reaction mechanism if the steps satisfactorily describe the kinetic properties of the reaction. Each step can be described in terms of its reactant molecules (Swanson & Alvord, 2003). The number of reactant molecules /atoms/ions/radicals coming together in a step is called its molecularity. Unlike the order of a reaction, the molecularity of a step must be a whole number. When one molecule is the reactant in a step, the step is said to be unimolecular (Tyson, 1976). When two reactants molecules are written, the step is said to be bimolecular.
The term molecularity hence applies only to steps in a proposed mechanism and it is equal to the sum of the coefficients of the reactants written for the step. The molecularity is also defined as the number of molecules that must come together to form a transition state. Most mech...
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