Determining the Time for a Radioactive Isotope's Decay Rate to Reach One-Quarter of Its Initial Value

The decay of a radioactive isotope follows an exponential decay model, where the amount of the isotope decreases by half after each half-life. This article explains how to determine the amount of time required for the decay rate of a sample to decrease to one-fourth of its initial decay rate.

Understanding Exponential Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by radiation.

The Exponential Decay Model

The decay of a radioactive isotope can be modeled exponentially, where the decay rate is proportional to the current amount of the isotope. The key parameter in this model is the half-life, which is the time it takes for the quantity of the isotope to decrease by half.

Determining the Time for One-Quarter Decay Rate

Given that a certain radioactive isotope has a half-life of 140 days, we need to determine the time it takes for the decay rate to decrease to one-fourth of its initial value. Let's denote the initial decay rate as R0.

After one half-life (140 days), the amount of isotope remaining is 1/2 of the initial amount. Consequently, the decay rate decreases to 1/2R0.

After two half-lives (280 days), the amount of isotope remaining is 1/4 of the initial amount, leading to a decay rate of 1/4R0.

Therefore, the decay rate decreases to one-fourth of its initial value after 280 days. This is achieved by two half-lives, as one half-life halved the initial rate, and the second half-life halved that intermediate rate.

Why Two Half-Lives?

Radioactive decay is a function of the remaining amount of the isotope. This means that the decay rate is directly proportional to the remaining isotope quantity. When the quantity is halved (first half-life), the decay rate is halved. Halving this half-rate (second half-life) reduces the decay rate to one-quarter of its initial value. This principle is consistent with the exponential nature of radioactive decay.

Intuitive Explanation

An intuitive approach to understanding this is to consider the 'half life' period. If the half-life is 140 days, after the second half-life period you would have "one half of one half" of the initial quantity, resulting in one-quarter of the initial decay rate.

Final Consideration

It is important to note that the decay rate itself does not change. It remains constant in terms of the fraction of the sample decaying over a unit time. What changes is the amount of the isotope left over time, which decreases exponentially.

In conclusion, a radioactive isotope with a half-life of 140 days will have a decay rate that decreases to one-quarter of its initial value after 280 days, or two half-lives.

Key Points:

Exponential decay model Half-life of radioactive isotope Decay rate decreases over time Two half-lives required for one-quarter decay rate