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: d[C]/dt = k[A][B]. Thus, if the initial concentrations are [A] = a moles/L and [B] = b moles/L and we write x= [C], then we have dx/dt= k(a-x)(b-x).

(a) Assuming that a does not equal 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 [C] = (1/2)a after 20 seconds?