Problem 36 - Master Resource Book in JEE Main Physics

Question

The half-life of \({ }^{215}\) At is \(100 \mu \mathrm{s}\). The time taken for the cadioactivity of a sample of \({ }^{215}\) At to decay to \(\frac{1}{16}\) th of its initial value is (a) \(400 \mu \mathrm{s}\) (b) \(6.3 \mu \mathrm{s}\) (c) \(40 \mu \mathrm{s}\) (d) \(300 \mu \mathrm{s}\)

Step-by-Step Solution

Answer

The time taken is 400 microseconds.

Step 1: Understanding Half-life Concept

The half-life of a substance is the time taken for its radioactivity to decrease to half of its original value. This means that every 100 microseconds, the radioactive sample of \(^{215}\)At will reduce to half of its activity.

Step 2: Determine the Reduction of Sample

We need the sample to reduce to \(\frac{1}{16}\) of its original value. We know that \(\frac{1}{16}\) can be expressed as \(\left(\frac{1}{2}\right)^4\), meaning that the activity must go through 4 half-lives to reduce to this amount.

Step 3: Calculate the Total Time

Each half-life takes \(100 \mu \text{s}\). For the radioactivity to become \(\frac{1}{16}\)th of its initial value, it will take 4 half-lives, as \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\). Thus, the time taken is \(4 \times 100 = 400 \mu \text{s}\).

Key Concepts

Understanding Half-life

The half-life of a radioactive substance is a fundamental concept that describes the time required for half of the radioactive atoms in a sample to decay.
This is a constant time period unique to each radioactive isotope. For example, the half-life of actinium-215 is specifically 100 microseconds, meaning that every 100 microseconds, the activity of the sample reduces to half.
This concept is crucial in several fields such as nuclear medicine and carbon dating, as it helps predict the behavior of radioactive substances over time. Understanding this allows scientists and researchers to determine how long a sample remains radioactive and how it can be safely managed or used. By knowing the half-life, you can calculate how long it takes for a substance to decay to a certain level.

Radioactivity Basics

Radioactivity refers to the process by which unstable atomic nuclei lose energy by emitting radiation. This process can result in the emission of alpha particles, beta particles, or gamma rays.
This natural phenomenon occurs because certain isotopes have an imbalance in the number of protons and neutrons. To attain stability, these isotopes release particles and energy, transforming into different isotopes or elements in the process.

• Alpha decay: Involves the emission of an alpha particle (2 protons and 2 neutrons), causing the atomic number to decrease by 2 and mass number by 4.
• Beta decay: Occurs when a neutron converts into a proton, emitting a beta particle (electron), increasing the atomic number by 1.
• Gamma decay: Typically follows alpha or beta decay and involves the release of energy without changing the number of protons or neutrons.

Understanding radioactivity is crucial for safely using radioactive materials in various applications, from medical imaging to energy production.

Actinium-215 Decay

Actinium-215 is one of the numerous isotopes displaying radioactive decay. Its decay involves the emission of alpha particles, leading to a reduction in its atomic number by 2 each time an alpha particle is emitted.
With a half-life of just 100 microseconds, actinium-215 undergoes rapid decay, making it a challenging element to study in detailed experiments aimed at measuring its properties.
Actinium-215 and its decay products play a role in understanding the processes of nuclear reactions and contribute to scientific research into the physics of alpha decay. Due to its short half-life, it's only used in a few specialized industrial and scientific applications.

Decay Calculations

Performing decay calculations is essential for predicting how long a radioactive sample will remain active. To calculate the decay of a sample, you often start by understanding how many half-lives are required to achieve a desired level of decay.
For instance, if you need a sample to reduce to \(\frac{1}{16}\) of its original activity, you can express this decrease as \(\left(\frac{1}{2}\right)^4\). This indicates that 4 half-lives are necessary for such a reduction.
Given that each half-life for actinium-215 is 100 microseconds, multiplying the number of half-lives (4) by the half-life duration gives a total time of 400 microseconds. This systematic approach is helpful for accurately predicting timeframes in scenarios involving radioactive decay, ensuring safe practices and informed decision-making in scientific experiments and applications.