Chemical Kinetics and the Roessler System

Dynamics at the Horsetooth Volume 2, 2010.

Chemical Kinetics and the Rossler SystemJaime ShinnDepartment of Mathematics

Colorado State University [email protected]

Report submitted to Prof. P. Shipman for Math 540, Fall 2010

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 Rossler System.

Keywords: Diffusion Equation, Law of Mass Action, Rossler System, Periodicity, Chaos

1 Introduction

During the summer of 2010 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 2010, where I took Math 540, 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 NH3-HCl system. Then I will discuss the Rossler 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 Rossler System.

2 The NH3 - HCl reaction

For the modeling of this system, we considered a tube with ammonia (NH3) 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 [NH3] = 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= Da

∂2a

∂t2

Chemical Kinetics and the Rossler System Jaime Shinn

∂b

∂t= Db

∂2b

∂t2

where Da and Db are diffusion coefficients.

Now we consider the forward reaction of the system, where

NH3(g) + HCl(g) → NH4Cl(s)

with reaction rate k1. Denote [NH4Cl] = s. Then, by the Law of Mass Action, we have the

following set of equations:

∂a

∂t= Da

∂2a

∂t2− k1ab

∂b

∂t= Db

∂2b

∂t2− k1ab

∂s

∂t= k1ab.

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 = 5.

0 2 4 6 8 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

ab

k 1H

(ab

−5)

Figure 1: Jump in the reation rate as a result of nucleation.

We modeled this behavior using the following equations

∂a

∂t= Da

∂2a

∂t2− k1H(ab− η)ab

∂b

∂t= Db

∂2b

∂t2− k1H(ab− η)ab

∂s

∂t= k1H(ab− η)ab.

Dynamics at the Horsetooth 2 Vol. 2, 2010


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 NH3 - 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 NH3 and HCl from either sides of the tube.

The oscillations are a promising sign that we are at least heading in the correct direction,



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 Rossler System

We now turn to a well-known system of equations, called the Rossler System. The Rossler equations

are as follows

dx

dt= −(y + z)

dy

dt= x+ ay

dz

dt= b+ xz − cz

where a, b, and c are parameters. If we let a = b = 0.2 and let c vary and we choose initial

conditions x(0) = y(0) = z(0) = 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 5 and 6. Note

that this period-one behavior, which begins after about t = 20, could be represented by a simple

oscillatory function such as sin(x).

−4−2

02

46

−5

0

50

1

2

3

4

x(t)

The Rossler System

y(t)

z(t)

Figure 5: Period-one limit cycle when c = 2.3.



0 20 40 60 80 100−4

−3

−2

−1

0

1

2

3

4

5

t

x(t)

Figure 6: Period-one behavior with respect to x(t).

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.

−10−5

05

10

−10

−5

0

50

2

4

6

8

x(t)

The Rossler System

y(t)

z(t)

Figure 7: Period-two limit cycle when c = 3.3.



0 20 40 60 80 100−6

−4

−2

0

2

4

6

8

t

x(t)

Figure 8: Period-two behavior with respect to x(t).

In the same way, we can let c = 5.3 and get period-three behavior, as shown in Figures 9 and 10.

−10−5

05

1015

−20

−10

0

100

5

10

15

20

x(t)

The Rossler System

y(t)

z(t)

Figure 9: Period-three limit cycle when c = 5.3.



0 20 40 60 80 100−10

−5

0

5

10

15

t

x(t)

Figure 10: Period-three behavior with respect to x(t).

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(0) = y(0) = z(0) = 1 while in the bottom part, we have

x(0) = 1.01, y(0) = z(0) = 1. Notice that the difference in the two solutions begins right around

t = 50.

−200

20

−20

0

200

20

40

x(t)

The Rossler System

y(t)

z(t)

0 50 100−10

0

10

20

t

x(t)

−200

20

−20

0

200

20

40

x(t)

The Rossler System

y(t)

z(t)

0 50 100−10

0

10

20

t

x(t)

Figure 11: The system exhibits chaotic behavior when c = 6.3.



4 Highlights of the Experimental Data

Experimental Data was collected for the NH3 - 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 NH3 - 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 15.



Figure 14:

.

Figure 15:

.

Finally, chaotic behavior of the oscillation front is apparent during another time segment of the

reaction. This is shown in Figures 16 and 17.



Figure 16:

.

Figure 17:

.

5 Conclusion

This project certainly has many avenues of possiblity for future work. First, we would like to add a

fourth equation to our NH3 - HCl model to account for the fact that the NH4Cl is, at first, believed

to be in the gas phase until enough NH4Cl gas molecules form a solid. If we denote [NH4Cl (g)] = m

(for monomer) and [NH4Cl (s)] = s (for solid), these equations should look something like

∂a

∂t= Da

∂2a

∂t2− k1ab

∂b

∂t= Db

∂2b

∂t2− k1ab



∂m

∂t= Dm

∂2m

∂t2+ k1ab− k2H(m− η)

∂s

∂t= k2H(m− η)− k3s.

We would also like to consider the idea of autocatalysis within our model which, simply put, is

the idea that the more NH4Cl (g) there is, the faster the reaction rate will be.

We have also done simple experiment of this NH3 - 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.



References

[1] Dee, G.T. ”Patterns Produced by Precipitation at a Moving Reaction Front.” Physical ReviewLetters 57.3 (1986): 275-278.

[2] Hirsch, Morris W., Stephen Smale, and Robert L. Devaney. Differential Equations, DynamicalSystems, and an Introduction to Chaos. Elsevier, 2004.

[3] Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. New York:

Springer, 2009.

[4] Howard, P. Partial Differential Equations in MATLAB 7.0. Spring 2005.

[5] Lebedeva, M.I., D.G. Vlachos, and M. Tsapatsis. ”Bifurcation Analysis of Liesegang Ring

Pattern Formation.” Physical Review Letters 92.8 (2004): 088301(1)-088301(4).

[6] Lenczycki, Timothy T. (Thesis) Oscillations in Gas Phase Periodic Precipitation Systems:The NH3-HCl Story. Spring 2003.

[7] Lynch, Stephen. Dynamical Systems with Applications using MATLAB. Boston: Birkhauser,

2004.

[8] Strogatz, Steven H. Nonlinear Dynamics and Chaos with Applications to Physics, Biology,Chemistry, and Engineering. Addison-Wesley, 1994.

[9] Verhulst, Ferdinand. Nonlinear Differential Equations and Dynamical Systems. Second edition.

New York: Springer-Verlag Berlin Heidelberg, 1996.


Chemical Kinetics and the Roessler System

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