Problem 63 - A Complete Resource Book in Chemistry for JEE Main

Question

A human body required \(0.01 \mu\) activity of radioactive substance after 24 hours. Half-life of radioactive substane is 6 hours. Then injection of maximum activity of a radioactive substance that can be injected is (a) \(0.08\) (b) \(0.04\) (c) \(0.16\) (d) \(0.32\)

Step-by-Step Solution

Answer

(c) 0.16

Step 1: Understand the question

We need to determine the initial quantity (maximum activity initially) of a radioactive substance which will ensure that after 24 hours, the activity reduces to 0.01 μ. The substance has a half-life of 6 hours.

Step 2: Use the radioactive decay formula

The formula to calculate the remaining activity after a certain period, given its initial activity, is:\[A(t) = A_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}\]where \(A(t)\) is the remaining activity after time \(t\), \(A_0\) is the initial activity, and \(T_{1/2}\) is the half-life of the substance.

Step 3: Substitute known values into the formula

We know that \(A(t) = 0.01 \mu\), \(t = 24\) hours, and \(T_{1/2} = 6\) hours. Substitute these into the formula to find \(A_0\):\[0.01 = A_0 \left(\frac{1}{2}\right)^{24/6}\]Simplify the exponent:\[24/6 = 4\]Thus the equation becomes:\[0.01 = A_0 \left(\frac{1}{2}\right)^{4}\]

Step 4: Solve for initial activity \(A_0\)

The equation simplifies to:\[0.01 = A_0 \times \frac{1}{16}\]Multiply both sides by 16 to solve for \(A_0\):\[A_0 = 0.01 \times 16\]Calculate the result:\[A_0 = 0.16\]

Step 5: Conclude the answer based on calculations

From our calculations, the initial activity required for the substance to reach 0.01 μ after 24 hours is 0.16 μ.

Key Concepts

Half-life Explained

The concept of half-life is essential when discussing radioactive decay. It refers to the time it takes for half of a given amount of a radioactive substance to decay, or lose its radioactive properties. For example, if you start with 1 gram of a radioactive material, after one half-life, only 0.5 grams of that material will remain radioactive.

A half-life is unique to each radioactive substance and is a constant value. This means it doesn't change regardless of how much of the substance you have initially. In our exercise, the material has a half-life of 6 hours. So, every 6 hours, half of the radioactive particles decay and lose their radioactivity, leading to a reduction in the substance's activity level.

• Predictable Decay: The time it takes is consistent per substance.
• Measurement Consistency: Fundamental for calculating decay over time.
• Exponential Process: Decay happens exponentially, halving periodically.

Understanding half-life allows scientists and medical professionals to determine how much of a substance is left after a given time, which is crucial for applications involving radioactive materials.

Understanding Radioactive Substances

Radioactive substances are materials that emit radiation as they decay. This decay occurs at a predictable rate over time, characterized by the substance's half-life. Radiation comes in different forms, primarily alpha particles, beta particles, and gamma rays, each with different properties and levels of penetration.

These substances are used in various applications, including medical treatments, power generation, and scientific research. However, handling them requires strict safety protocols due to the potential risks posed by exposure to radiation.

• Types of Radiation: Alpha, beta, and gamma.
• Applications: Medicine, industry, and research.
• Safety Precautions: Critical to minimize radiation exposure.

By understanding what radioactive substances are and how they behave, we can use them safely and effectively in different fields.

Decay Formula and Its Application

The decay formula is a mathematical expression that helps calculate the remaining activity of a radioactive substance after a given time period. This formula is written as:
\[A(t) = A_0 \left(\frac{1}{2}\right)^{t/T_{1/2}}\]
where:

• \(A(t)\) is the remaining activity after time \(t\),
• \(A_0\) is the initial activity,
• \(T_{1/2}\) is the half-life of the substance.

This formula is crucial in calculating how much of a substance remains radioactive after a specific period.

In our problem, we used this formula to determine the initial activity needed for a radioactive substance to reduce to a final activity of 0.01 μ after 24 hours. By substituting the known values --- a 6-hour half-life and a 24-hour duration --- into the formula, we solved for the initial activity \(A_0\). This example shows how critical the formula is to practical applications like medicine, where precise dosages of radioactive substances are necessary for patient safety and treatment effectiveness.