Development of a Real-Time Dynamic Graphing Web Page

Development of the Theory

The theory involved relates to rates of chemical reactions. In the case illustrated here:

This really now stands for two standard differential equations:

Thus, we can relate the change in concentration of A (d[A]/dt) at any time to the concentrations of A and B in solution at that time. The same goes for B. These equations can be rewritten:

"Time" in a simulation can be adjusted to fit the circumstances. In our case, we will just consider "dt" to be "1 unit of time" (however long we allow for the JavaScript to go through its loop. Then, technically, we have D[A] and D[B] rather than d[A] and d[B], and we can write:

Furthermore, no one is interested really in k₁ or k₂ per se. So we can arbitrarily assign them in relation to an equilibrium constant, K, as follows:

Notice that there really isn't any assumption here. This is just a definition. The quotient of any two constants is a constant. What will be interesting to see is whether that constant ultimately, at equilibrium, is related to the concentrations of A and B:

That ratio, [B]/[A], is a meaningful quantity called the reaction quotient, Q. It can be measured anytime, but at equilibrium, it's value should be K.

Furthermore, the exact values of k₁ and k₂ do not matter. Only their ratio matters. So we will simplify things by adding one more relationship, where r is some constant that we will set and will determine how fast equilibrium is reached.

Now, we can put all these equations together and write:

So we now have k₁ and k₂ in terms of K. That's important, because it is K that will be input by the user.

This is all the theory we need. Time to implement the theory as code.

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