Problem 100 - A Complete Resource Book in Chemistry for JEE Main

Question

If the half life period of a radioactive isotope is \(10 \mathrm{~s}\), then its average life will be (a) \(14.4 \mathrm{~s}\) (b) \(1.44 \mathrm{~s}\) (c) \(0.144 \mathrm{~s}\) (d) \(2.44 \mathrm{~s}\)

Step-by-Step Solution

Answer

The average life of the isotope is 14.4 s, corresponding to option (a).

Step 1: Understanding Half-Life and Average Life

The half-life of a radioactive isotope is the time it takes for half of the radioactive atoms in a sample to decay. The average life, also known as mean life, of the isotope can be calculated using the relationship between half-life and average life.

Step 2: Using the Relationship Between Half-Life and Average Life

The average life \( T \) of a radioactive isotope is related to its half-life \( T_{1/2} \) by the formula:\[ T = \frac{T_{1/2}}{\ln(2)} \]where \( \ln(2) \approx 0.693 \).

Step 3: Substituting the Given Half-Life

For the given problem, the half-life \( T_{1/2} = 10 \mathrm{~s} \). Substitute \( T_{1/2} \) into the formula:\[ T = \frac{10}{0.693} \]

Step 4: Calculating the Average Life

Perform the calculation:\[ \frac{10}{0.693} \approx 14.4 \mathrm{~s} \] Thus, the average life of the isotope is approximately \( 14.4 \mathrm{~s} \).

Step 5: Selecting the Correct Answer

The calculated average life of \( 14.4 \mathrm{~s} \) corresponds to option (a) \( 14.4 \mathrm{~s} \). Thus, option (a) is the correct answer.

Key Concepts

Understanding Half-Life

Half-life is a fundamental concept in the study of radioactive decay. It represents the time required for half of the atoms in a sample of a radioactive substance to decay into another form. This period of time is essential because it provides insight into how quickly a radioactive isotope will diminish and helps predict the duration of its activity.
Half-life is denoted by the symbol \( T_{1/2} \), and each isotope has its unique half-life, varying from fractions of a second to millions of years. This characteristic allows scientists to understand the behavior of radioisotopes in natural processes and applications such as dating archaeological finds or measuring geological time scales.
Key points to remember about half-life:

• Constant Rate: This process occurs at a constant rate regardless of environmental changes.
• Predictable Decay: It is predictable and helps in calculating the duration when the material will become safe or cease to be effective for its intended use.
• Exponential Decay: The decay follows an exponential pattern, meaning the quantity reduces by a consistent percentage over each half-life.

Explaining Average Life

Average life, also known as the mean life, is an important measure that complements the half-life understanding. It provides a broader view by considering the entire lifespan of the radioactive isotopes. While the half-life indicates the time for a 50% decay, the average life considers the total time all isotopes will remain before fully decaying.
The formula used to determine average life \( T \) is:\[ T = \frac{T_{1/2}}{\ln(2)} \]where \( \ln(2) \approx 0.693 \).This relationship provides a valuable connection between half-life and average life, allowing further calculations for more complex processes.
Highlights of average life:

• Slightly Longer than Half-Life: On average, isotopes last longer than the half-life period portrayed.
• Useful for Detailed Predictions: It is crucial for calculating doses and other detailed predictions in nuclear science.
• Contextual Understanding: Helps researchers and engineers design systems with better safety and efficiency.

Mean Life Explained

Mean life is the average time it takes for a given number of radioactive atoms to decay. It is essentially another term for average life, thus sometimes used interchangeably in scientific discussions about radioactive decay. Understanding mean life offers insights into practical applications such as medical treatments involving radioisotopes.
Reasons why mean life is important:

• Holistic View: Provides a complete understanding of the isotope’s activity over its entire decay period.
• Designing Treatments: In medical sciences, it helps design safer and more effective radiotherapy treatments.
• Key in Calculations: It’s essential for calculating everything from radiation exposure to decay heat in nuclear reactions.

In assessing mean life, it is critical to acknowledge that similar to average life, it is inherently linked to the half-life and is important for predicting how radioactive substances will behave over time.

Understanding Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are atoms that have excess nuclear energy, making them unstable. This energy is released by radioactive decay in the form of particles or electromagnetic waves. Because they vary in stability and half-life, radioisotopes are used in numerous fields, including medicine, archaeology, and power generation.
Characteristics of radioactive isotopes:

• Decay Types: They can emit alpha, beta or gamma radiation.
• Half-Life Variability: Each has a unique half-life, which dictates its utility for specific applications.
• Diagnostic & Treatment Uses: In medicine, they help diagnose and treat various conditions through imaging or the targeting of cancer cells.

Radioactive isotopes play pivotal roles in scientific exploration and technology. Such isotopes have transformed methods of age determination of historical artifacts and have become a backbone of modern diagnostic tools, showcasing their indispensability despite their inherent risks.