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 (option d).

Step 1: Understanding decay

Radioactive decay follows an exponential pattern where the amount remaining can be described by the equation \( N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \), where \( N(t) \) is the remaining amount after time \( t \), \( N_0 \) is the initial amount, and \( T_{1/2} \) is the half-life.

Step 2: Setting up the equation

We know only \( \frac{1}{16} \) of the original amount remains after 192 minutes. In terms of the formula, \( N(t) \) would be \( \frac{1}{16}N_0 \). Thus, \( \frac{1}{16} = \left( \frac{1}{2} \right)^{\frac{192}{T_{1/2}}} \).

Step 3: Expressing \( \frac{1}{16} \) as a power of \( \frac{1}{2} \)

We know \( \frac{1}{16} = \left( \frac{1}{2} \right)^4 \), meaning we can equate the exponents of the base \( \frac{1}{2} \), giving us \( \frac{192}{T_{1/2}} = 4 \).

Step 4: Solving for the half-life

From the equation \( \frac{192}{T_{1/2}} = 4 \), we solve for \( T_{1/2} \) by multiplying both sides by \( T_{1/2} \) and then dividing by 4: \( 192 = 4 \times T_{1/2} \). Hence, \( T_{1/2} = \frac{192}{4} = 48 \).

Key Concepts

Understanding Half-Life

Half-life is the time required for half of a radioactive isotope to decay and transform into a different element or isotope. It's a measure of the rate of decay.
When analyzing a radioactive material, knowing its half-life helps predict how quickly the substance will reduce by half. This time frame remains constant, regardless of the initial quantity of the substance.

• For example, if a radioactive isotope has a half-life of 48 minutes, it means that every 48 minutes, the quantity of the substance will halve.
• If we start with 100 grams, after one half-life (48 minutes), we'd have 50 grams remaining.
• After another half-life (another 48 minutes), only 25 grams would remain, and so on.

The process continues, and each period of the half-life will see the quantity reduce by half again.

Exploring Exponential Decay

Exponential decay describes how quantities decrease gradually at a rate proportional to their current value. In radioactive decay, this means the amount of material decreases exponentially over time.
The formula used to represent exponential decay in radioactive materials is \[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where:

• \( N(t) \) is the amount left after time \( t \).
• \( N_0 \) represents the initial amount.
• \( T_{1/2} \) is the half-life of the isotope.

Exponential decay often results in a rapid decrease in the beginning, which slows down over time. This pattern is common in natural processes and not just limited to radioactive decay, showing up in contexts like population decline and heat dissipation.

The Role of Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are variants of chemical elements with unstable nuclei that release energy in the form of radiation as they decay into more stable forms.
These isotopes play essential roles in various fields, including:

• Medicine: Used for diagnostic imaging and cancer treatments, such as with isotopes like Technetium-99m.
• Archaeology: Carbon-14 dating helps estimate the age of ancient artifacts.
• Energy: Uranium-235 is critical in nuclear power generation.

The decay process of these isotopes is pivotal in many scientific and industrial applications. Understanding their half-life and decay patterns provides valuable insights for predicting behavior, managing risks, and harnessing their potential benefits.