Question

The half-life of radioactive element is 100 minutes. The time interval between the stages to \(50 \%\) and \(87.5 \%\) decay will be : (a) \(100 \mathrm{~min}\) (b) \(50 \mathrm{~min}\) (c) \(200 \mathrm{~min}\) (d) \(25 \mathrm{~min}\)

Short answer

The time interval between the stages to 50% and 87.5% decay will be 200 minutes.

Step-by-step solution

Understanding Half-Life

The half-life of a radioactive element is the time taken for half of the radioactive nuclei in a sample to decay. After one half-life, the quantity of the substance will become half of the initial amount.

Calculating Time Interval for Decaying to 50%

Given that the half-life is 100 minutes, 50% decay would mean that one half-life has elapsed. So the time taken to decay to 50% of the original amount is exactly one half-life, which is 100 minutes.

Determining Remaining Amount after 50% Decay

After 50% decay, only 50% of the radioactive substance remains. This will be our new starting amount for the next stage of decay.

Calculating Decay to 87.5%

To reach 87.5% decay, we need to lose another 37.5% of the radioactive substance. This means we need to go from 50% still remaining to 12.5% remaining (as 100% - 87.5% = 12.5%).

Finding the Number of Half-Lives for Additional Decay

To go from 50% to 12.5%, we need to halve the remaining amount twice (50% to 25% to 12.5%). Each halving is one half-life, so it takes two half-lives to go from 50% to 12.5%.

Calculating Total Time for Reaching 87.5% Decay

As each half-life is 100 minutes, two half-lives would be 200 minutes. The total time to reach 87.5% decay is the time to go from 100% to 50% (100 minutes) plus the time to go from 50% to 12.5% (200 minutes).

Determining the Time Interval

The time interval between the stages of 50% and 87.5% decay is the additional time needed after the first 100 minutes have passed, which is 200 minutes.

Key concepts

Radioactive Decay

Radioactive decay is a fundamental process whereby unstable nuclei lose energy by emitting radiation. The nature of radioactive decay is stochastic, meaning that we cannot predict when a particular atom will decay, but we can predict the behavior of a large number of atoms with high precision using statistical methods.

There are several types of radioactive decay, including alpha, beta, and gamma decay, each involving the release of different particles or electromagnetic radiation. The decay process transmutes the original element into a different element or a different isotope of the same element.

In the given exercise, understanding the concept of half-life is crucial to solve the problem. The half-life is consistent for a given isotope, which is why it is a valuable tool for dating archaeological samples and understanding nuclear reactions.

Nuclear Chemistry

Nuclear chemistry explores the changes in the nucleus of atoms, which encompasses not only radioactive decay but also nuclear fusion and fission. Here we're focusing on how unstable nuclei release energy to become more stable, a process that's both natural and can be harnessed for various applications.

In the context of the given exercise, it's important to grasp the implications of nuclear chemistry in real-world scenarios. For instance, medical diagnostics and treatments often employ radioactive tracers, and nuclear reactors utilize controlled fission reactions to generate electricity. Understanding the rate of decay, as expressed by half-life, is imperative in these applications to ensure safety and efficacy. The problem provided is a simplified representation of the complex calculations that nuclear chemists make on a regular basis.

Exponential Decay

Exponential decay is a process that decreases at a rate proportional to its current value. This is expressed mathematically by a negative exponential function. In the case of radioactive decay, this function describes how the quantity of a radioactive isotope decreases over time.

To solve half-life problems, it's crucial to understand that each half-life interval reduces the remaining amount of substance by half, no matter how much material you started with. That's the core of exponential decay’s consistent proportionality, and it's why knowing the half-life of a substance allows us to calculate future amounts of that substance even without knowing the initial amount.

For instance, in our exercise, we see that after one half-life (100 minutes), the substance will have 50% remaining, and after an additional two half-lives (200 minutes more), it will have decreased to 12.5%, showcasing the exponential nature of the decay process.