Problem 67 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive substance having a half-life of 3 days was received in 12 days. It was found that there was only \(3 \mathrm{~g}\) of the isotope in the container. The initial weight of the isotope when packed was (a) \(12 \mathrm{~g}\) (b) \(24 \mathrm{~g}\) (c) \(48 \mathrm{~g}\) (d) \(96 \mathrm{~g}\)

Step-by-Step Solution

Answer

(c) \(48 \mathrm{~g}\)

Step 1: Understanding the Concept of Half-Life

The half-life of a substance is the time it takes for half of it to decay. Here, the half-life of this radioactive isotope is given as 3 days.

Step 2: Determining the Number of Half-Life Periods

The isotope was packed and received in 12 days. In that time span, the number of half-life periods is calculated by dividing the total time by the half-life: \( \frac{12}{3} = 4 \). This means that 4 half-lives have passed.

Step 3: Applying the Decay Formula

The mathematical formula to find the remaining mass after a given number of half-lives is given by: \[ m = m_0 \left( \frac{1}{2} \right)^n \]where \( m \) is the remaining mass, \( m_0 \) is the initial mass, and \( n \) is the number of half-lives. Here, \( m = 3 \text{ g} \) and \( n = 4 \).

Step 4: Calculate the Initial Mass

Rearrange the decay formula to solve for \( m_0 \): \[ m_0 = m \left( \frac{1}{2} \right)^{-4} \]Substituting the values, we have:\[ m_0 = 3 \text{ g} \times \left( \frac{1}{2} \right)^{-4} = 3 \text{ g} \times 16 = 48 \text{ g} \].

Step 5: Conclusion

The initial weight of the isotope when it was packed was 48 g.

Key Concepts

Half-Life

The concept of half-life is important in understanding how radioactive materials decay over time. When we talk about the half-life of a substance, we refer to the time required for half of the material to decay into a different substance, often a more stable form.
For example, if you have a radioactive isotope with a half-life of 3 days, as in the exercise, then half of the initial amount will remain after 3 days. This means if you start with 100 grams, only 50 grams will be left after this period. After another 3 days, only 25 grams will remain, demonstrating an exponential decrease.
This principle is vital in fields ranging from archaeology, through radiocarbon dating, to medicine, where it helps in determining the dosage and timing of radioactive materials in treatments.

Radioactive Isotopes

Radioactive isotopes are atoms with an unstable nucleus that emit radiation during their decay to a stable form. These isotopes are also known as radioisotopes. Each element may have a number of different isotopes, some of which may be stable while others are radioactive.
For instance, carbon-14 is a radioactive isotope used in carbon dating. It decays over time, allowing scientists to date objects that are thousands of years old. Similarly, in the exercise, a radioactive isotope with a certain half-life is used to show how its quantity decreases over 12 days.
The properties of radioactive isotopes make them useful tools in research, medical imaging, treatment and energy production. The decay they undergo can be detected and measured, providing vital information for various scientific and practical processes.

Decay Formula

The decay formula is crucial for calculating how much of a radioactive substance remains after a period of time. This formula involves exponential decay and is expressed as: \[ m = m_0 \left( \frac{1}{2} \right)^n \]where:

• \( m \) is the remaining mass of the substance,
• \( m_0 \) is the initial mass,
• \( n \) is the number of half-life periods that have elapsed.

Using this formula, you can easily determine how much material remains after any given number of half-lives. In the provided exercise, this formula helps find the initial mass of a radioactive isotope by rearranging it to solve for \( m_0 \):\[ m_0 = m \left( \frac{1}{2} \right)^{-n} \]Understanding this equation is fundamental for scientists and students as it allows the calculation of how long a substance can remain active or how much remains after a specific time. Such calculations have real-world applications in nuclear energy, medicine, and radiological safety.