computer series, 132
edited bv JAMESP. BIRK Arizona State Uoivenily Tempe, AZ85281
Modeling Complex Kinetics Schemes A Computational Experiment Roger S. Macomber and loannis Constantinides University of Cincinnati, Cincinnati, OH 45221-0172
The creation of models desihmed to simulate thc behavior core ofscientific methodolorn -.Althouph scientific modeling is this important, most undergraduate chemistry majors get relatively little exposure to it. Chemical kinetics, the study of reaction rates, involves the development of mathematical models (rate laws) that describe the concentration dependence of reaction rates. When expressed in differential form, most rate laws do not appear too imposing. But as all chemistry majors eventually discover, the calculus required to generate many integrated rate laws is, to say the least, challenging. Our goal in this paper is to demonstrate that one can often "model the model". To show this, we will use a particularly complex, yet widely applicable, kinetics scheme to develop a simplified model of complex rate laws without resorting to integral calculus.' For some time our group has been examining reactions of the type depicted in the schematic diagram below. Molo.f -nature 1s~at - - ~ - ~-~ ~ the ~
ecule Xz, which is shown with two equivalent functional groups, is allowed to react with reagent R, which converts functional group X to product functional group P.28 According to the scheme, this reaction occurs in two discrete steps. Intermediate XP is formed in the first step, which has second-order rate constant kl. Then XP is converted to product Pz in the second step, which has seeondorder rate constant kg. One mole-equivalent of R is consumed in each step. The goal in working with such systems has been to deduce the ratio kdkl (hereaftercalled K) by monitoring the relative amounts of Xz, XP, and Pz present as a function of the amount of R c o n ~ u m e dSuch . ~ a kinetic scheme, which describes the reaction a s involvingconseeutive, competitive second-order steps, is of interest to chemists who devise synthetic strategies involving difunctional molecules. Such schemes are particularly interesting to polymer chemists studying the condensation polymerization of difunctional substrates. However, this type of scheme can also provide
students with a useful example that shows them how to devise simplified models of complex kinetic schemes. To the Student Problem 1. You plan to carry out the reactions depicted in the schematic diagram, using a solution that initially contains only R and Xz. Periodically analyze the reaction mixture for [Xzl, [XPI, and [Pzl. You will devise a model that predicts these concentrations as a function of the number of mole-equivalents (r) of R consumed for every mole of Xz originally present, given the initial concentration of Xz ( = KzlJ and K ( = kzlkl). You may assume that the reaction of R with functional group X occurs by an irreversible one-step mechanism.
Problem 2. Using the model(s) developed in Problem 1, deduce the value of K for the reaction below, given the experimental concentration data in Table 1.
To the Instructor The goal of this exercise is to let the students (individually or in groups) approach the problems in any way they select. Only when they become stuck should the instructor offerhelpful hints. However, they might first be encouraged to make some qualitative or semiquantitative estimates about the behavior of the system, with which their models can later be compared. For example, when Table 1. Experimental Concentration Data for the Reaction of 1,BDibromopropane with KCN in Methanol at 60 'C KCN
Br-(CHz)s-Br T,+Br-(CHz)*.CN
NC
