#### Problem 24E

24. If, in Example 4, one molecule of the product $\mathrm{C}$ is formed from one molecule of the reactant $\mathrm{A}$ and one molecule of the reactant $\mathrm{B}$, and the initial concentrations of $\mathrm{A}$ and $\mathrm{B}$ have a common value $[\mathrm{A}]=[\mathrm{B}]=a$ moles $/ \mathrm{L}$, then

$$[\mathrm{C}]=a^{2} k t /(a k t+1)$$

where $k$ is a constant.

(a) Find the rate of reaction at time $t$.

(b) Show that if $x=[\mathrm{C}]$, then

$$\frac{d x}{d t}=k(a-x)^{2}$$