BZ reaction1 - An Analysis of the Belousov-Zhabotinskii...

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An Analysis of the Belousov-Zhabotinskii Reaction Casey R. Gray Calhoun High School Port Lavaca, TX 77979 and The High School Summer Science Research Program Department of Mathematics Baylor University Waco, TX 76798 [email protected] Abstract We begin with a brief history of the celebrated Belousov-Zhabotinskii (BZ) reaction. In particular, we consider the BZ reaction in a continuously stirred, closed vessel in the presence of a ferroin indicator. We examine the underlying chemical kinetics of the most significant reactions involved. This leads to the Oregonator model and an associated 3 × 3 system of non- linear ordinary differential equations. We nondimensionalize this system and further reduce it to a 2 × 2 stiff system. Relaxation oscillations are expected and an analysis of the phase plane confirms this. Finally, we solve the system numerically for a certain set of system parameters and compare our computations with experimental results. 1 Background The Belousov-Zhabotinskii (BZ) reaction is an intriguing experiment that dis- plays unexpected behavior. When certain reactants are combined, an “induc- tion” period of inactivity is followed by sudden oscillations in color from red to blue. In spatially nonhomogeneous systems (such as a simple petri dish), the red and blue oscillations propogate as spiral wave fronts. The oscillations last about one minute and are repeated over a long period of time. Eventually, the reaction stops oscillating and approaches an equilibrium state. We now know Faculty Advisor: John M. Davis, Department of Mathematics, Baylor University, Waco, TX 76798. Email: John M [email protected] 1
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that the color changes are caused by alternating oxidation-reductions in which cerium changes its oxidation state from Ce(III) (producing a magenta solution) to Ce(IV) (producing a blue solution) or vice versa. Because of this, we call the BZ reaction an “oscillating reaction”; this simply means a reaction in which there is a regular, periodic change in the concentration of one or more reac- tants. Because this reaction is well understood, it is considered the prototype oscillator. An excellent, accessible reference for this material is [15]. Although the BZ reaction is a chemical rather than biological oscillator, un- derstanding its mechanics will also help us understand biological oscillations such as the beating of the heart. The chemical traveling waves observed in the BZ reaction are very similar to the electromagnetic traveling waves in muscle tissue. Also, the system of differential equations derived for the BZ reaction (the Oreganator model [4]) is similar to the system of differential equations modeling electromagnetic waves in heart tissue (the Beeler-Reuter model [1]). Parallel in- vestigations have been made studying such diverse phenomena as nerve tissues (the Hodgkin-Huxley model [8] and the Fitzhugh-Nagumo equations [6, 13]) and slime mold aggregation (the Martiel-Goldbeter equations [11]). The trademark spiral waves of the BZ reaction have also been observed in other types of media
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