Chapter D1.1, Problem 55E

Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y′(t) = –kyⁿ(t), where y(t) is the concentration of the compound for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume that the initial concentration of the compound is y(0) = y₀ > 0.

a.    Consider a first-order reaction (n = 1) and show that the solution of the initial value problem is y(t) = y₀e^–kt.

b.    Consider a second-order reaction (n = 2) and show that the solution of the initial value problem is $y\left(t\right)=\frac{y_0}{y_{0}kt+1}.$

c.    Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions.