13.1 Chemical Reactions

Suppose two chemicals, (molecules or ions), A and B, have the property that if they bump into one another, they can rearrange themselves into molecules or ions of two other substances, C and D. This reaction may give off or absorb energy, making the objects created move either faster or slower than the reacting objects were moving before the collision. This is perhaps the simplest possible chemical reaction; typically lots more is going on and there are many more complicated situations that can be modeled using the approach we apply below.

A standard question is: if we start with given amounts or concentrations of A, B, C and D, at time t = 0, what can we expect these to be at time t?

The simplest situation occurs if the energy changes in the reaction are sufficiently small compared to the heat interchanges between the system and its surroundings that the temperature of the system remains constant. In that case, there will be a certain probability, p1, that A and B come together, and having done so, react to form C and D, and a certain probability, p2, that a C and D come together to do the opposite reaction.

The probability p1 will obviously be proportional to the concentrations both of A and of B in the system, and p2 will similarly be proportional to the concentrations of C and of D. If we denote these concentrations as A, B, C and D, we get the following model:

On the other hand, if the reactions that take place are sufficient to cause temperature changes, a model should take the effect of these changes into consideration.

The temperature of a substance is roughly proportional to the average amount of energy in each of its degrees, of freedom, and that includes the kinetic energy of motion of its constituent objects. This energy in turn is proportional to the square of the speeds of the objects. The probability that objects come together is proportional to the speed at which they are moving relative to one another.

We can model the changes in temperature caused by the reaction to be is a measure of how much energy is given off when A and B become C and D. There is another effect, and that is the leakage of energy from the system to the outside world which we assume has temperature T0. We can model this by a term (c4)(T0 - T).

Finally, we can model as a function of T by , to take into account that depends on the speed of the objects which is crudely proportional to the square root of the temperature. The same model can be applied to c2. (You might want to refine the model to allow for the fact that the reverse reaction cannot take place at all unless the kinetic energy of the reactants is enough to supply the energy required to create the rearrangement, but we won't bother doing this here.)

This leads us to the following coupled equations

This is probably not a very good model; we put it here to illustrate how you construct a model; if you want to go further, you must study chemistry. If you choose initial conditions and values of the various parameters here, you can solve these equations numberically, and observe the consequences.

The idea here is that you construct a model that is as simple as possible, that has a chance of describing your phenomenon. You then work out the consequences and compare them with the phenomenon itself.

If the model, with parameters chosen appropriately, describes the phenomenon perfectly, that is both good and bad. Its goodness lies in the fact that such a model can be used to predict behavior. Its badness lies in the fact that scientists are generally looking for new phenomena, for discovering them is how a scientist makes his or her name. And a new phenomenon is a place where accepted models break down and a new one is needed.

It is where models break down that there is scope for further scientific activity.