In their paper on Optimization in Chemical Kinetics, Benjamin and Hogenson (2003, 1-7) present an approach to a mathematical solution to a chemical engineering problem. Given an abstract initial chemical solution containing X-X molecules, they choose how many Y molecules to add to the solution. In this situation, a Y molecule may attach permanently to either end of an X-X molecule, resulting in an X-X-Y molecule, with X-X-Y and Y-X-X molecules being identical. A Y molecule can also attach permanently to and "open" X on an X-X-Y molecule to form a Y-X-X-Y molecule. Their hypothesis is that a Y molecule is r times more likely to be attracted to a particular X-X molecule than a particular X-X-Y molecule, where r = k1/k2 is the ratio of the reaction rate constants. The probability that the t+ 1st Y molecule attaches itself to an X-X molecule is
rx1 + z1
Using this hypothesis, the researchers then used a simulation approach which they repeated several times, observing the t values where the number of X-X-Y molecules was maximized (Benjamin and Hogenson, 2003, 2). Each of these simulations suggested how many Y molecules should be added to the solution. By performing a sufficient number of simulation runs, they hoped to get an idea of the ideal number of Y molecules to add relative to the number of X molecules present initially. They then went on to determine what would happen when dealing with larger volumes of molecules such as moles of each. They developed two equations by which they could determine if they began with 1000 moles of X-X, maximizing the t-value suggested the number of moles of Y to use foe every 1000 moles of X-X. The equations were;
To satisfy the questions of chemists, they showed that the same differential equations they d
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