MA 226: LAB 4
Oscillating Chemical Reactions Lab
This lab is due Tuesday, December 4, 2017 in class. Late labs will not be accepted.
Please write your name and discussion section day and time on the first page.
You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. It is perfectly acceptable to turn in hand-drawn figures. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.
In this lab you will investigate a system of differential equations governing a certain chemical reaction. The system depends on a parameter B and you are to describe qualitatively what happens when B passes through a certain threshold value. There are two parts to this lab. The first should be done before using the computer as the computations will take quite some time. If you make a mistake in these computations you will not be able to proceed, so check your work carefully. After making the appropriate computations, you may then use the Chemical Oscillator tool to view the phase plane for the system and report on the qualitative behavior of the system.
IMPORTANT: The work you submit should be your own and nobody else's. Any exceptions to this will be dealt with harshly. We have had problems in the past with several students submitting nearly duplicate lab reports. We forward any suspected collaborations to the Dean's office.
One of the most interesting recent results in chemistry is the fact that certain chemical reactions may tend to oscillate rather than tend to equilibrium. Here you will study one such reaction.
Iodide and chlorine dioxide are each poured into a reaction vessel at a certain rate, are well mixed together causing a certain reaction, and then the excess solution is removed so that the amount of reactant in the vessel remains constant. A mathematical model for a chlorine dioxide-iodide open system reaction is provided by the following system of (nonlinear) differential equations:
Here x is the amount of iodide (purple stuff) and y is the amount of chlorine dioxide (yellow stuff) in a tank at a given moment, so each of these quantities are non-negative. The quantity B is a parameter that we will vary between 1 and 6.
Part 1: The mathematics. Do this part before using the computer as the computations can be difficult and the numbers pretty bad.
1. First find all of the equilibrium points for this system for all values of the parameter B. Write these equilibrium points as functions of the parameter B. Free gift from the math dept: You should find exactly one equilibrium point for each value of B, and, in particular, this equilibrium point should not depend on B. If this is not the case, your ship is sinking and you had better call a life-guard. Or maybe a second-grader....
2. Compute the Jacobian matrix of the system at a general point (x, y). Be careful here: If your matrix is wrong, your ship has sunk and you can't go on.
3. Now compute the Jacobian matrix at the equilibrium point you found in 1. This matrix should depend on B. Then find the eigenvalues of this matrix. These should also depend on B.
4. Now, classify these equilibria as spiral sinks, real sources, saddles, whatever. Your answer should include a range of B values for each type that occurs. Remember, we are assuming that 1 < B < 6.
5. Next, answer the question: Where does a bifurcation occur, and what happens at this bifurcation.
6. Now sketch the curve in the trace-determinant plane given by this matrix as B varies.
7. Finally, sketch the nullclines for this system and indicate the direction of the vector field in the regions between these nullclines. How do these nullclines depend on the parameter B? It suffices to restrict attention here to the region 0 < x < 5, 3 < y < 8 as in the Chemical Oscillator applet.
Part B: The observations. Now use the Chemical Oscillator applet that comes with the book to view the phase planes for various values of B. Again, you should look only in the region 0 < x < 5, 3 < y < 8.
8. Discuss the behavior of solutions of the system for the parameter value B = 6. Answer the question: What happens to the chemicals in the vessel as time goes forward?
9. Discuss the behavior of solutions of the system for the B = 1. Answer the question: What happens to the chemicals in the vessel as time goes forward?
10. As the parameter B varies between 1 and 6, we know that there is a bifurcation (a change) at some B value, and from the linearization above, we know what to expect there. What else happens as B passes through this value? Use the computer to help you answer this question. In an essay, explain what this bifurcation entails, both mathematically and chemically. That is, something else should occur as your parameter passes through the bifurcation value of B. Explain what appears to happen in the phase plane and then what this means in terms of the chemicals involved. It is here that the little bowl of chemicals in the Chemical Oscillator applet should help. (I don't mean that you should drink the stuff in the bowl; just look at it!)