Chemical Kinetics and the Rössler System. 1 Introduction. 2 The NH 3 - HCl reaction. Dynamics at the Horsetooth Volume 2, 2010.

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1 Dynamics at the Horsetooth Volume 2, 21. Chemical Kinetics and the Rössler System Department of Mathematics Colorado State University Report submitted to Prof. P. Shipman for Math 4, Fall 21 Abstract. Chemical oscillations present questions about both the chemical mechanism behind the oscillations and the potential for mathematical models. The reaction of ammonia with hydrocloric acid, which is know to produce Liesegang ring patterns [1], exhibits oscillation patterns of mathematical interest [6]. This paper describes a mathematical model for this chemical system, as well as a comparison of experimental data to the well-known Rössler System. Keywords: Diffusion Equation, Law of Mass Action, Rössler System, Periodicity, Chaos 1 Introduction During the summer of 21 at Colorado State University, I was awarded a CIMS grant and worked with Dr. Patrick Shipman of the Department of Mathematics and Dr. Stephen Thompson of the Department of Chemistry. We studied the reaction of ammonia with hyrdrocloric acid, a reaction that forms an interesting pattern which falls under the broad classification of Liesegang Rings [1]. My interest in this problem continued into the fall of 21, where I took Math 4, Dynamical Systems with Dr. Patrick Shipman, and I realized that interesting connections could be made from my summer work to concepts learned in class. In this paper, I will summarize some of the results obtained during my summer research which includes a mathematical model of the NH 3 -HCl system. Then I will discuss the Rössler System, a well-known system of equations which exhibits very nice periodic patterns. I will conclude with a comparison of actual experimental data from Tim Lenczycki s work [6] to both the mathematical model and the Rössler System. 2 The NH 3 - HCl reaction For the modeling of this system, we considered a tube with ammonia (NH 3 ) on one end of the tube and hydrocloric acid (HCl) at the other end, as was the case in the data which we were trying to model [6]. We will denote [HCl] = a and [NH 3 ]=b. Since these two chemicals will react while in their gas phase, we certainly know that they will both obey the diffusion equation a t = D 2 a a t 2

2 b t = D 2 b b t 2 where D a and D b are diffusion coefficients. Now we consider the forward reaction of the system, where NH 3 (g) + HCl(g) NH 4 Cl(s) with reaction rate k 1. Denote [NH 4 Cl] = s. Then, by the Law of Mass Action, we have the following set of equations: a t = D 2 a a t 2 k 1ab b t = D 2 b b t 2 k 1ab s t = k 1ab. These equations are the basic building blocks of our model. We now proceed to include more of the details of this system. We know that both heterogeneous and homogeneous nucleation are important in this chemical reaction [6]. Mathematically, we needed to model the large jump in the reaction rate that occurs once ab exceeds a certain amount η. In Figure 1, we see an example of this where η = k 1 H(ab ) ab Figure 1: Jump in the reation rate as a result of nucleation. We modeled this behavior using the following equations a t = D 2 a a t 2 k 1H(ab η)ab b t = D 2 b b t 2 k 1H(ab η)ab s t = k 1H(ab η)ab. Dynamics at the Horsetooth 2 Vol. 2, 21

3 Using these equations, we were able to see oscillations in our solutions as seen in Figures 2, 3, and 4. Figure 2: The reaction front of the NH 3 - HCl system as a function of position and time. Figure 3: A top view of the reaction front; the movement of the reaction front. Figure 4: Diffusion of NH 3 and HCl from either sides of the tube. The oscillations are a promising sign that we are at least heading in the correct direction, Dynamics at the Horsetooth 3 Vol. 2, 21

4 although we should certainly test our model against actual experimental data. Future modifications to this model are discussed in the conclusion of this paper. 3 The Rössler System We now turn to a well-known system of equations, called the Rössler System. The Rössler equations are as follows dx = (y + z) dt dy dt = x + ay dz = b + xz cz dt where a, b, and c are parameters. If we let a = b =.2 and let c vary and we choose initial conditions x() = y() = z() = 1, we will see that we get very interesting patterns. When c =2.3, we can see that we get period-one behavior as shown in Figures and 6. Note that this period-one behavior, which begins after about t = 2, could be represented by a simple oscillatory function such as sin(x). The Rossler System 4 3 z(t) 2 1 y(t) Figure : Period-one limit cycle when c = 2.3. Dynamics at the Horsetooth 4 Vol. 2, 21

5 t Figure 6: Period-one behavior with respect to. When c =3.3, as shown in Figures 7 and 8, we see period-two behavior here. Note that we get two distinct amplitudes here as shown in Figure 8. The Rossler System 8 6 z(t) 4 2 y(t) Figure 7: Period-two limit cycle when c = 3.3. Dynamics at the Horsetooth Vol. 2, 21

6 t Figure 8: Period-two behavior with respect to. In the same way, we can let c =.3 and get period-three behavior, as shown in Figures 9 and 1. The Rossler System 2 1 z(t) 1 1 y(t) Figure 9: Period-three limit cycle when c =.3. Dynamics at the Horsetooth 6 Vol. 2, 21

7 t Figure 1: Period-three behavior with respect to. Finally, if we consider the case where c = 6.3, as in Figure 11, we can see that a slight change in the initial conditions has caused a great change in the solution. In the top part of Figure 11, we have initial conditions x() = y() = z() = 1 while in the bottom part, we have x() = 1.1,y() = z() = 1. Notice that the difference in the two solutions begins right around t =. z(t) z(t) y(t) y(t) The Rossler System The Rossler System t t Figure 11: The system exhibits chaotic behavior when c = 6.3. Dynamics at the Horsetooth 7 Vol. 2, 21

8 4 Highlights of the Experimental Data Experimental Data was collected for the NH 3 - HCl reaction which measures the position of the reaction front as time progressed [6]. It was noted that the reaction front of this system oscillates and oftentimes in a very periodic manner. We now discuss some specific examples of this. During a specific time segment of the NH 3 - HCl reaction, the reaction front oscillated as shown in Figure 12, which appears to exhibit period-one behavior. This is confirmed by the Fourier Analysis in Figure 13 since we obtain one clear peak.. Figure 12:. Figure 13: In a different time segment of the reaction, we find more periodic behavior. According to the Fourier Analysis, it appears that during this time segment, we have period-two behavior as shown in Figures 14 and 1. Dynamics at the Horsetooth 8 Vol. 2, 21

9 . Figure 14:. Figure 1: Finally, chaotic behavior of the oscillation front is apparent during another time segment of the reaction. This is shown in Figures 16 and 17. Dynamics at the Horsetooth 9 Vol. 2, 21

10 . Figure 16:. Figure 17: Conclusion This project certainly has many avenues of possiblity for future work. First, we would like to add a fourth equation to our NH 3 - HCl model to account for the fact that the NH 4 Cl is, at first, believed to be in the gas phase until enough NH 4 Cl gas molecules form a solid. If we denote [NH 4 Cl (g)] = m (for monomer) and [NH 4 Cl (s)] = s (for solid), these equations should look something like a t = D 2 a a t 2 k 1ab b t = D 2 b b t 2 k 1ab Dynamics at the Horsetooth 1 Vol. 2, 21

11 m t = D 2 m m t 2 + k 1ab k 2 H(m η) s t = k 2H(m η) k 3 s. We would also like to consider the idea of autocatalysis within our model which, simply put, is the idea that the more NH 4 Cl (g) there is, the faster the reaction rate will be. We have also done simple experiment of this NH 3 - HCl Reaction but in a vertical tube. This has resulted in a tornado-like reaction where oscillations are visible but a mathematical model for this would have to take into account things such as fluid flow and the effect that it has on these oscillations. Dynamics at the Horsetooth 11 Vol. 2, 21

12 References [1] Dee, G.T. Patterns Produced by Precipitation at a Moving Reaction Front. Physical Review Letters 7.3 (1986): [2] Hirsch, Morris W., Stephen Smale, and Robert L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos. Elsevier, 24. [3] Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. New York: Springer, 29. [4] Howard, P. Partial Differential Equations in MATLAB 7.. Spring 2. [] Lebedeva, M.I., D.G. Vlachos, and M. Tsapatsis. Bifurcation Analysis of Liesegang Ring Pattern Formation. Physical Review Letters 92.8 (24): 8831(1)-8831(4). [6] Lenczycki, Timothy T. (Thesis) Oscillations in Gas Phase Periodic Precipitation Systems: The NH 3 -HCl Story. Spring 23. [7] Lynch, Stephen. Dynamical Systems with Applications using MATLAB. Boston: Birkhauser, 24. [8] Strogatz, Steven H. Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering. Addison-Wesley, [9] Verhulst, Ferdinand. Nonlinear Differential Equations and Dynamical Systems. Second edition. New York: Springer-Verlag Berlin Heidelberg, Dynamics at the Horsetooth 12 Vol. 2, 21