Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

A radioactive isotope decays at such a rate that after 192 minutes only \(1 / 16\) of the origin amount remains. The half-life of the radioactive isotope is (a) \(12 \mathrm{~min}\) (b) \(24 \mathrm{~min}\) (c) \(32 \mathrm{~min}\) (d) \(48 \mathrm{~min}\)

Step-by-Step Solution

Answer

The half-life of the radioactive isotope is 48 minutes.

Step 1: Understanding the Problem

We are given that a radioactive isotope decays such that after 192 minutes, only \(\frac{1}{16}\) of the original amount remains. We need to find out the half-life of the isotope.

Step 2: Expressing Remaining Amount Mathematically

We know that the remaining amount of a substance after time \(t\) is given by the function \( A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}, \) where \( A_0 \) is the original amount and \( T_{1/2} \) is the half-life. In this problem, after 192 minutes, \( A(t) = \frac{A_0}{16} \).

Step 3: Set up the Equation

We have \( \frac{A_0}{16} = A_0 \left( \frac{1}{2} \right)^{\frac{192}{T_{1/2}}} \). This reduces to the equation \( \left( \frac{1}{2} \right)^{\frac{192}{T_{1/2}}} = \frac{1}{16} \).

Step 4: Simplify the Equation

The equation \( \frac{1}{16} \) can be rewritten as \( \left( \frac{1}{2} \right)^4 \). Therefore, equating the exponents gives us \( \frac{192}{T_{1/2}} = 4 \).

Step 5: Solve for the Half-Life

Rearrange the equation to solve for \( T_{1/2} \): \( T_{1/2} = \frac{192}{4} \). Calculate this to find \( T_{1/2} = 48 \) minutes.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. This is a path to reach a more stable state.
In this decay process, the isotope transforms into a different element or a different isotope of the same element. Common types of decay include:

• Alpha decay (emitting helium nuclei)
• Beta decay (emitting beta particles/electrons or positrons)
• Gamma decay (emitting electromagnetic rays)

Understanding radioactive decay is essential for fields such as archaeology, geology, and medical sciences. This process allows scientists to date ancient artifacts or analyze the structural changes in materials over time. It is also the foundation of many nuclear medicine treatments and diagnostics.

Exponential Decay

Exponential decay describes a process where the quantity reduces at a rate proportional to its current value, resulting in a rapid decrease initially that slows over time. It's represented by the mathematical function:
\[ A(t) = A_0 imes e^{-kt} \]
where:

• \( A(t) \) is the amount remaining at time \( t \).
• \( A_0 \) is the initial amount.
• \( k \) is the decay constant, a positive number.

In the context of radioactive decay, this concept implies that with each passing half-life, the amount of radioactive material decreases by half. It reflects how rapidly the substance disintegrates. Even complex systems and processes such as population decline, chemical reactions, and charging and discharging capacitors in circuits can be modeled using exponential decay.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are variants of elements that have unstable nuclei and exhibit radioactive decay. These isotopes emit radiation as they transform into more stable isotopes.
The decay rate and the type of radiation emitted are characteristic of each radioisotope, making them useful in a variety of applications:

• **Carbon-14** is widely used in radiocarbon dating to determine the age of archaeological artifacts.
• **Iodine-131** and **Technetium-99m** have significant uses in medical diagnostics and treatment.
• **Uranium-238** and **Plutonium-239** are critical in nuclear power generation and nuclear weaponry.

The use of radioactive isotopes in scientific research, medical applications, and energy production helps push the boundaries of technology and improves our understanding of natural processes.