Water$-$treatment processes that generate hydroxyl radicals are often

used to remove organic chemicals that contaminate groundwater. The

hydroxyl radical $(*OH)$ is a very reactive free$-$radical species that can be

thought of as a piece of a molecule. In an attempt to again become a

whole molecule, $*OH$ will attack almost any organic molecule, either

adding to a double bond or pulling off a hydrogen atom to become water.

Because of this property, the hydroxyl radical can be used to degrade

organic contaminants completely to carbon dioxide and water, providing

a remediation tool that leaves no harmful by$-$products.

In some applications, a small amount of toxic material like trichloro$-$

ethylene $($TCE$)$ must be destroyed in the presence of a much larger

quantity of "nontarget" material such as natural organic material $($Exer$-$

cise $9\mathrm{.}3),$ or$,$ in the case of industrial wastewater, some organic material

that is either environmentally benign or can subsequently be treated

more cheaply by biological methods. This material, usually characterized

instrumentally by the measurement of dissolved organic carbon $($DOC$),$

may compete for hydroxyl radical with the contaminant to the extent

that the economic feasibility of the process may depend on the outcome

of this competition.

$($d$)$ Can you make any statement regarding the apparent reaction order

at very small and very large times?