## Lab 6: Simulating an Ionic ReactionMaths Lab 6 is available at http://metric.ma.ic.ac.uk/chem/1998/Mathslab6.nb.
Here the main focus is on the idea of differential equations and their solutions. The Lab is built around a reaction simulation that is effectively a Newtonian three-body problem, and therefore not exactly integrable in general. The solution process, which is numerical, is hidden inside The assignment involves simulating a reaction between a free chloride ion and a sodium chloride dipole. For simplicity, the ions are treated as Newtonian point charges and their movement is restricted to one dimension. Again, setting things up is rather involved. First the students need to set some physical constants: the permittivity of the vacuum, Avogadro's number, and so on. Then they define a couple of force functions, representing empirically derived classical approximations to what is actually, of course, a quantum phenomenon. These cover interactions between particles of like charge
and unlike charge
respectively. After setting a duration for the "reaction," and a set of initial positions and velocities for the three ions, the student is finally ready to solve the equations of motion of the ions.
The solutions that come from this call to
Some frames from this animation are shown in Figure 2. Here, the starkness of the representation and its inferiority to what could be achieved with more sophisticated
After typing in our code, the students have four things to do. First, they're asked to write some code of their own for viewing the system's behavior by means of a graph instead of an animation. Next, they have to vary the initial positions and velocities of the particles systematically in order to make the system behave--apparently, anyway--in at least three ways. 1. The chloride ion that was dissociated becomes bound, and vice versa (a reaction occurs, in other words); 2. The dissociated chloride ion interacts with the dipole but is repelled, and becomes dissociated once more; 3. The interaction between the three ions continues forever, with neither chloride ion becoming dissociated from the other two.
The third task is to test the model itself, specifically to check that energy appears to be conserved. This involves converting the forces into potentials, and sampling (or graphing) the total mechanical energy of the system during the reaction. Although the details of the numerical solution method are kept hidden behind Mathematica to make sure that the dynamical behavior that comes out of is physically plausible. In other words, the very system that generated the data can be used in a natural way to examine that data critically. With `NDSolve` Mathematica, a black box needn't be completely black.
Finally, students are asked to devise from scratch a
We were, in general, very pleased by the standard of work on this rather challenging task, which increased our confidence that we had been right to use Converted by Mathematica
September 30, 1999
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