Question

The disintegration rate of a certain radioactive sample at any instant is 5400 dpm. After 5 min the rate becomes \(2700 \mathrm{dpm}\). The half life of the sample in min is approximately

Short answer

The half-life of the sample is 5 minutes.

Step-by-step solution

Understand the Disintegration Rate

A radioactive disintegration rate is the number of disintegrations per minute (dpm). We are given two disintegration rates: the initial rate is 5400 dpm, and the rate after 5 minutes is 2700 dpm.

Recognize the Half-Life Concept

The half-life of a radioactive sample is the time required for its disintegration rate to fall to half of its initial value. We are told initially the rate is 5400 dpm and after 5 minutes it is 2700 dpm, which is exactly half of 5400 dpm.

Calculate the Half-Life

Since the disintegration rate after 5 minutes is exactly half of the initial rate, we can conclude that the half-life of the sample is 5 minutes.

Key concepts

Disintegration Rate

The disintegration rate in radioactivity refers to the number of times a radioactive atom's nucleus decays per minute. It's commonly measured in disintegrations per minute (dpm). Every radioactive sample emits radiation as their nuclei disintegrate over time. In our problem, you might have noticed two disintegration rates: initially 5400 dpm and after 5 minutes, it was noted as 2700 dpm.

The decrease in the disintegration rate over time is a direct indicator of radioactive decay. Over time, the actual number of radioactive nuclei decreases, and hence, the disintegration rate lowers. This is vital in fields where understanding radioactivity is necessary, such as nuclear medicine and archaeology.

• Initial disintegration rate: Defines how active a sample initially is.
• Reduced disintegration rate: Shows how the activity declines over time, key in determining how fast or slow this change happens.

This understanding of the disintegration rate guides us in anticipating how quickly a sample will lose its radioactivity.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the material's radioactive atoms to decay, reducing its disintegration rate by half. It's a fixed time characteristic for each radioactive element, allowing scientists to predict the duration it will remain active. In our exercise, we had an initial disintegration rate of 5400 dpm, which halved to 2700 dpm after 5 minutes.

Here's how you can reason about the half-life:
* If a sample's disintegration rate has fallen to half its initial value within a certain time span, that time span corresponds to its half-life.
* For this particular case, since it took 5 minutes for the disintegration rate to halve, the half-life is exactly 5 minutes.

Ultimately, knowing a sample's half-life is crucial in various domains. It can inform how long a sample will remain hazardous or active.

Radioactive Sample

A radioactive sample consists of unstable nuclei that release energy in the form of radiation as they decay into more stable configurations. These samples contain isotopes, which are atoms that have the same number of protons but different numbers of neutrons in their nuclei. Each isotope of an element will have its own unique half-life, affecting how the sample behaves over time.

This particular sample we are examining has a half-life of 5 minutes, which is determined by observing how its disintegration rate changes. Understanding these concepts:

• How the isotopic composition of a sample impacts its radioactive behavior.
• Why monitoring a sample's disintegration rate informs us about its safety and utility over time.

Radioactive samples are utilized not only in scientific research but also in medical treatments and energy production, hence knowing their properties is immensely valuable.