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This model demonstrates the kinetics of a simple reversible reaction. It demonstrates numerically that the application of the Principle of Stationary Concentrations is valid in this case.
In this model there are two kinds of molecules, green and red. Green turtles turn into red turtles bimolecularly whereas red turtles turn back into pairs of green turtles monomolecularly. You can control the rate at which this changes occur with sliders Kb and Ku.
The reaction here is a reversible reaction of the type:
text
Kb
A + A <=======> B
Ku
An example of such a reaction would be dimerization of acetic acid:
text
Kb
2 H CCOOH <=======> H CCOOH~...~HOOCC H
3 Ku 3 3
This reaction is an example of a complex reaction which consists of two elementary reactions. The forward bimolecular reaction
text
Kb
A + A > B
is characterized by the constant Kb and the reverse unimolecular reaction
text
Ku
B > A + A
is characterized by the constant Ku.
The system of ordinary differential equations (ODE) that describes the concentrations of A and B is given below:
```text dA 2  = 2Kb * A + 2Ku * B (1) dt
dB 2
 = Kb * A  Ku * B (2)
dt
```
The usual initial conditions are A(0) = Ao and B(0) = 0. While it is possible to solve this system of ODE analytically, chemists usually apply the Principle of Stationary Concentrations when they investigate the kinetics of reactions of this type. The Principle says that one can assume that the concentrations of the species stop changing from some point on after the system reaches equilibrium. If concentrations are stationary, the derivatives
text
dA dB
 and 
dt dt
are zero. Hence one can replace the system of ODE above with the system of algebraic equations below:
```text 2 0 = 2Kb * A* + 2Ku * B* (1')
2
0 = Kb * A*  Ku * B* (2')
```
where concentrations marked with * are stationary concentrations. The second equation (2') is linearly dependent on the first equation (1'). Luckily we also have another equation coming from the law of the conservation of mass:
text
A* + 2 * B* = Ao (3)
From equation (2') we can express B* in terms of A* :
text
Kb 2
B* =  A* (4)
Ku
We can now plug in expression (4) into (3) and then we will have a quadratic equation in terms of A*:
text
Kb 2
A* +  A* = Ao (5)
Ku
whose solution is:
text
_____________

 Kb
 1 + 4 *   1
\ Ku
 (6)
Kb
2 * 
Ku
One can now find the stationary concentration of B using equation (4).
Choose the values of Ku and Kb with appropriate sliders:  Kb controls the rate of the forward reaction by which two green turtles turn bimolecularly into a single red turtle.  Ku controls the rate of the reverse reaction, by which a red turtle turns unimolecularly into two green turtles.
Having chosen appropriate values of the constants, press SETUP to clear the world and create an initial number of green turtles. Note: we do not create red turtles initially, although this could be done in principle.
Press GO to start the simulation.
You will see turtles wandering around the world and changing color. Pay more attention to the plot of the concentrations. Do the plots soon reach stationary concentrations?
How do the stationary concentrations depend on the values of Kb and Ku? You can change Ku and Kb while the model is running. See if you can predict what the stationary concentrations will be with various combinations of Kb and Ku.
Try to implement the following reaction:
text
Kb K2
A + A <======> B > C
Ku
This reaction underlines a vast number of microbiological processes (e.g. fermentation). You can read about its kinetics in any book on Biochemistry. Look up the socalled MichaelesMenten equation. Does it check numerically?
Try to implement the following reaction:
text
Kb K2
A + B <======> C > D
Ku
Enzyme Kinetics Chemical Equilibrium Simple Kinetics 2 Simple Kinetics 3
Thanks to Mike Stieff for his work on this model.
If you mention this model or the NetLogo software in a publication, we ask that you include the citations below.
For the model itself:
Please cite the NetLogo software as:
Copyright 1998 Uri Wilensky.
This work is licensed under the Creative Commons AttributionNonCommercialShareAlike 3.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/byncsa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.
This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECTBASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program)  grant numbers RED #9552950 and REC #9632612.
This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORKBASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs)  grant numbers REC #9814682 and REC0126227. Converted from StarLogoT to NetLogo, 2001.
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