Question

Why do radioactive decay series obey first-order kinetics?

Short answer

Radioactive decay series follow first-order kinetics because the rate of decay is directly proportional to the number of parent nuclides present at any given time, which fits the definition of first-order kinetics. This means that each decay event is independent of all others and does not depend on the history of the sample, but only on the number of parent nuclides present.

Step-by-step solution

Understanding Radioactive Decay

Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. However, the chance that a given atom will decay is constant over time. For a large number of atoms, the overall decay can be seen to follow a predictable pattern. This pattern forms the basis for the principles of radioactivity and notably the first-order kinetics.

Understanding First-Order Kinetics

First-order kinetics is a concept in physical chemistry where the rate of reaction is directly proportional to the concentration of one of the products. It's called 'first order' because only the first power of the concentration affects the reaction rate. The mathematical representation of first-order kinetics is \(-\frac{d[A]}{dt} = k[A]\), where [A] represents the concentration of the material, k is the rate constant, and t represents time.

Applying First-Order Kinetics to Radioactive Decay

In a radioactive decay series, each parent nuclide spontaneously decays into a daughter nuclide. The decay rate (which is the change in the number of parent nuclides over time) is directly proportional to the number of parent nuclides present. This fits the definition of first-order kinetics, as the rate of decay depends on the first power of the number of parent nuclides present and remains constant over time. As such, we can describe radioactive decay using the mathematical model of first-order kinetics.