Question 3.9.4: Modeling the Rate of a Chemical Reaction In an autocatalytic...
Modeling the Rate of a Chemical Reaction
In an autocatalytic chemical reaction, the reactant and the product are the same. The reaction continues until some saturation level is reached. From experimental evidence, chemists know that the reaction rate is jointly proportional to the amount of the product present and the difference between the saturation level and the amount of the product. If the initial concentration of the chemical is 0 and the saturation level is 1 (corresponding to 100%), this means that the concentration x(t) of the chemical satisfies the equation
x^{\prime}(t)=r x(t)[1-x(t)],
where r > 0 is a constant.
Find the concentration of chemical for which the reaction rate x′(t) is a maximum.
To clarify the problem, we write the reaction rate as
f(x)=r x(1-x).
Our aim is then to find x ≥ 0 that maximizes f(x). From the graph of y = f(x) shown in Figure 3.99, the maximum appears to occur at about x=\frac{1}{2}. We have
\begin{aligned}f^{\prime}(x) & =r(1)(1-x)+r x(-1) \\& =r(1-2 x)\end{aligned}and so, the only critical number is x=\frac{1}{2}. Notice that the graph of y = f(x) is a parabola opening downward and hence, the critical number must correspond to the absolute maximum. Although the mathematical problem here was easy to solve, the result gives a chemist some precise information. At the time the reaction rate reaches a maximum, the concentration of chemical equals exactly half of the saturation level.
