The CIMA Reaction
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Acknowledgement :

Governing equations can be found in:

I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics, Oscillations, Wavesn Patterns, and Chaos, Oxford University Press, Oxford, 1998.

G. Nicolis, Dynamique Chimique, Thermodynamique, cinétique et mécanique statistique, Dunod, Paris 2005.

Off[General::"spell1"]

Chlorite-iodide-malonic acid (CIMA) reaction 
gives rise to temporal oscillations.

The reaction scheme is the following:

A -> X
X -> Y
4 X + Y -> P

where X=I\(TraditionalForm\`\(\^-\)\) and Y=\(TraditionalForm\`\(ClO2\^-\)\)

eq1 = 
        D[u[τ], {τ, 1}] ==  
          a - u[τ] - 
            4 u[τ] v[τ]/(1 + u[τ]^2);

eq2 = 
        D[v[τ], {τ, 1}] ==  
          b (u[τ] - 
                u[τ] v[τ]/(1 + u[τ]^2));

init1 = u[0] == 4.29 10^-3;

init2 = v[0] == 5.71 10^-3;

a = k1p/(k2p (Sqrt[α])); 
    b = k3p/(k2p (Sqrt[α]));

k1p = k1a MA; MA = 1.0 10^-3; k1a = 7.5 10^-3;

k2p = k2 ClO2; ClO2 = 1.98 10^-4; k2 = 6 10^3;

k3p = k3b I2; I2 = 5.14 10^-4; k3b = 2.65 10^-3;

α = 10^-14;

sol = 
      NDSolve[{eq1, eq2, init1, init2}, {u[τ], v[τ]}, {τ, 0, 
          20}]

plt1 = 
      Plot[Evaluate[{u[τ], v[τ]} /. sol], {τ, 0, 20}, 
        PlotStyle -> {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]

We get oscillating patterns because:

			 b < ((3  a)/5) - (25/a)
			 
In fact, we have b = 11.4655 and ((3  a)/5) - (25/a) = 37.4828

3 a/5 - 25/a

b

We study the case where:

			 b > ((3  a)/5) - (25/a)
			 
In fact, we have b = 45 and ((3  a)/5) - (25/a) = 37.4828

b = 45;

sol = 
      NDSolve[{eq1, eq2, init1, init2}, {u[τ], v[τ]}, {τ, 0, 
          100}]

plt1 = 
      Plot[Evaluate[{u[τ], v[τ]} /. sol], {τ, 0, 20}, 
        PlotStyle -> {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]

We get a stable steady state because:

			 b > ((3  a)/5) - (25/a)
			 
In fact, we get the following steady state:
	
	 (Subscript[u, ss])= (a/5)=12.6263 and (Subscript[v, ss])=1+((a^2)/25)= 160.423

a/5

1 + a^2/25

Evaluate[{u[τ], v[τ]} /. sol] /. τ ->  
        100