Tin Pest When metallic tin is kept below \(13.2^{\circ} \mathrm{C}\) it slowly
becomes brittle and crumbles to a gray powder. Tin objects
eventually crumble to this gray powder spontaneously if kept in
a cold climate for years. The Europeans who saw tin organ pipes
in their churches crumble away years ago called the change tin
pest because it seemed to be contagious. And indeed it was, for
the gray powder is a catalyst for its own formation.
A catalyst for a chemical reaction is a substance that controls the
rate of reaction without undergoing any permanent change in
itself. An autocatalytic reaction is one whose product is a catalyst
for its own formation. Such a reaction may proceed slowly at first
if the amount of catalyst present is small and slowly again at the
end, when most of the original substance is used up. But in
between, when both the substance and its catalyst product are
abundant, the reaction proceeds at a faster pace. In some cases it is
reasonable to assume that the rate \(v=d x / d t\) of the reaction is
proportional both to the amount of the original substance present and to the
amount of product.
That is, \(v\) may be considered to be a function of \(x\) alone, and
$$v=k x(a-x)=k a x-k x^{2}$$
where
\(\begin{aligned} x &=\text { the amount of product, } \\ a &=\text { the
amount of substance at the beginning, } \\ k &=\text { a positive constant. }
\end{aligned}\)
At what value of \(x\) does the rate \(v\) have a maximum? What is the
maximum value of \(v ?\)
