In an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C: A + B rightarrow C. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B. The rate of change of concentration of C with respect to time is given by the equation 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 the equation dx/dt = k(a - x)(b - x). Find x(t) assuming that a = b. How does this expression for x(t) simplify if it is known that [C] = 1/2a after 20 seconds?
