Nonlinear Chemical Reactions: The Role of A and B in Producing E, Lab Reports of Theatre

Download Nonlinear Chemical Reactions: The Role of A and B in Producing E and more Lab Reports Theatre in PDF only on Docsity! §2.5: NONLINEAR MODELS OF CHEMICAL REACTIONS Instead of just mixing two substance together (like water and salt) we can think about reactions that take place when chemicals are mixed. Suppose chemicals named A and B react to make a chem- ical named E. We will measure amounts of chemicals in moles, since a mole of a given chemical always contains the same number of molecules. Assume for now that one molecule of A and one molecule of B react to make one molecule of E. (We will change this later.) We write A + B → E This is chemistry notation, not mathematics. Let a(t), b(t), and y(t) denote the amount in moles of chemicals A, B, and E at time t in seconds. We may as well assume that there is no E at time t = 0, so y(0) = 0. Since each molecule of E produced requires one molecule of A and one molecule of B, we see that a(t) = a(0)− y(t) and b(t) = b(0)− y(t). Since it is not reasonable to expect we start out with exactly the same number of molecules of A and B, there will be more of one chemical. Suppose, for concreteness there is more of B. In terms of the functions, this says b(0) > a(0). So the A chemical will run out first; the reaction stops when A is used up. The simplest model we can think of is that the rate of the reaction is proportional to the amount of A left at time t. y′ = ka(t) = k(a(0)− y). This is an easy first order linear equation, also separable, with solu- tion y(t) = a(0)− Ce−kt and the initial condition y(0) = 0 gives C = a(0) and y(t) = a(0)(1− e−kt). Unfortunately, this answer does not agree with the data one gets by measuring concentrations in the laboratory. This model is wrong. In fact, the rate of reaction is also influenced by the amount of B left at time t. This makes sense if you think about it. For a given amount of A, the reaction will of course be faster if there is an ocean of B than if there is only a spoonful. So y′(t) should be proportional 1