Chapter 8.5, Problem 50E

Chemical Reaction In a chemical reaction, one unit of compound Y and one unit of compound Z are converted into a single unit of compound X. Let x be the amount of compound X formed. The rate of formation of X is proportional to the product of the amounts of unconverted compounds Y and Z. So, $dx/dt=k\left(y_0-x\right)\left(z_0-x\right)$, where $y_0$ and $z_0$ are the initial amounts of compounds Y and Z. From this equation, you obtain

${\displaystyle \int \frac{1}{\left(y_0-x\right)\left(z_0-x\right)}dx={\displaystyle \int kdt.}}$

(a) Solve for x as a function of t.

(b) Use the result of part (a) to find x as $t\to \infty$ for

$\left(1\right)y_0<z_0,\left(2\right)y_0>z_0,\text{and}\left(3\right)y_0=z_0.$