Chapter 2.5, Problem 9P

Chemical Reactions. A second order chemical reaction involves the interaction (collision) of one molecule of a substance $P$ with one molecule of a substance $Q$ to produce one molecule of a new substance $X$; this is denoted by $P+Q\to X$. Suppose that $p$ and $q$, where $p\ne q$, are the initial concentrations of $P$ and $Q$, respectively, and let $x(t)$ be the concentration of $X$ at time $t$. Then $p-x(t)$ and $q-x(t)$ are the concentrations of $P$ and $Q$ at time $t$, and the rate at which the reaction occurs is given by the equation

$dx/dt=\alpha \left(p-x\right)\left(q-x\right)$, (i)

where $\alpha$ is positive constant.

If $x\left(0\right)=0$, determine the limiting value of $x\left(t\right)$ as $t\to \infty$ without solving the differential equation. Then solve the initial value problem and find $x\left(t\right)$ for any $t$.

If the substances $P$ and $Q$ are the same, then $p=q$ and Eq. (i) is replaced by

$dx/dt=\alpha {\left(p-x\right)}^2$ (ii)

If $x\left(0\right)=0$, determine the limiting value of $x\left(t\right)$ as $t\to \infty$ without solving the differential equation. Then solve the initial value problem and determine $x\left(t\right)$ for any $t$.