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, $dx/dt=k(y_0-x)(x_0-x)$, where $y_0$ and $z_0$ are the initial amounts of compounds Y and Z. From this equation, you obtain

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

(a) Perform the two integrations and solve for x in terms of t.

(b) Use the result of part (a) to find x as $t\to \infty$ for $(1)y_0<z_0,(2)y_0>z_0,\text{and}(3)y_0=z_0$.