Question: Many organ pipes in old European churches are made of tin. In cold climates such pipes can be affected with tin pest, when the tin becomes brittle and crumbles into a gray powder. This transformation can appear to take place very suddenly because the presence of the gray powder encourages the reaction to proceed. The rate of the reaction is proportional to the product of the amount of tin left and the quantity of gray powder, p, present at time t. Assume that when metallic tin is converted to gray powder, its mass does not change.
(a) Write a differential equation for p. Let the total quantity of metallic tin present originally be B.
(b) Sketch a graph of the solution p = f(t) if there is a small quantity of powder initially. How much metallic tin has crumbled when it is crumbling fastest?
(c) Suppose there is no gray powder initially. (For example, suppose the tin is completely new.) What does this model predict will happen? How do you reconcile this with the fact that many organ pipes do get tin pest?