Chapter 9.3, Problem 42E

In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B → C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B:

$\frac{d\left[C\right]}{dt}=k\left[\text{A}\right]\left[\text{B}\right]$

(See Example 3.7.4.) Thus, if the initial concentrations are [A] = a moles/L and [B] = b moles/L and we write x = [C], then we have

$\frac{dx}{dt}=k\left(a-x\right)\left(b-x\right)$

(a) Assuming that a ≠ b, find x as a function of t. Use the fact that the initial concentration of C is 0.

(b) Find x(t) assuming that a = b. How does this expression for x(t) simplify if it is known that $\left[C\right]=\frac{1}{2}a$ after 20 seconds?