# Math 224--Differential Equations: Project 3, Spring 1997

## A bit of applied Chemistry...

### by Gavin LaRose (glarose@umich.edu), Nebraska Wesleyan University, January 1997

permission granted to use and distribute free in an academic setting

As your tenure at \RiMaC, Inc. continues, you find yourself basking in accolades and contemplating the expansion of your career in alternate venues. However, even as this possibility clouds the horizon, you find a glimmer of brightness in the arrival of a final contract. This comes from a company with several local plants, Chemproc, Inc.---a chemical producing and reprocessing company.

DVI file of project
PostScript version of project

## The letter...

### Chemproc, Inc.

20000 Ryan-ears Blvd.
Lonlinc, SK
04685

18 April 1997

Rigorous Mathematical Contractors (RiMaC), Inc.
Lonlinc, SK 04685

Dear RiMaC:

Chemproc, Inc., is a leader, through efficiency and shrewd market analysis, in the rapidly expanding field of the reprocessing of ``waste'' chemicals. As part of our continued effort to minimize the expense of setting up and maintaining chemical processing lines we routinely obtain theoretical analyses of their functioning, and it is for such an analysis we are contacting your company.

We are currently in the process of setting up a ``degradation'' line, in which an essentially toxic waste chemical is introduced and is reduced through a series of reactions to a more benign product. In this line the waste chemical being introduced, c1, reacts with the other reagents present to produce two products, c2 and c3. The rates at which these reactions proceed (thus reducing the amount of c1 present and increasing both c2 and c3) are proportional to the amount of c1 in the system, and proceed with constants of proportionality k1=1.21 and k3=.13. The first of these reactions may proceed in either direction, however, so that c1 is also produced (and c2 reduced) at a rate proportional (with k2=.70) to the amount of c2 present. The chemical c2 also reacts with the existant reagents to produce c3, with a rate characterized by the constant of proportionality k4=.05. Finally, the chemical c3 is degraded through the introduction of a catalyst to the benign product c4, which reaction proceeds at a rate proportional to the amount of c3 and proportionality constant k5=3.20.

Before beginning large scale tests and production with this line, we need a mathematical analysis of how the concentrations of the different chemicals will vary with time as the reaction(s) proceed. It is to this end that we have contracted with your company.

A good friend of ours, Dr. Gavin LaRose, has agreed in his copious free time to field your questions on this project. You should additionally contact him by the 24th of April and by the 1st of May with an indication of your progress on the problem. In order that the line be in full operation by its target date this summer, we need your final report on or before the 9th of May.

Sincerely,
E. Idu Pont
President, Chemproc, Inc.

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