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 is 48 minutes.

Step 1: Understanding Half-Life Concept

The half-life of a radioactive substance is the time it takes for half of the initial amount of the substance to decay. The amount remaining after multiple half-lives can be calculated using this concept.

Step 2: General Decay Equation

We know that after 192 minutes, only \(\frac{1}{16}\) of the substance remains. Since \(\frac{1}{16} = 2^{-4}\), we can infer that four half-lives have passed, as each half-life reduces the amount to half.

Step 3: Calculating the Half-Life

Given that four half-lives correspond to 192 minutes, we divide 192 minutes by 4 to find the duration of one half-life: \(192 \div 4 = 48\) minutes.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. It occurs in unstable atoms, also known as parent nuclei, and results in the transformation into a more stable nucleus, which may be a different element or a different isotope of the same element. This decay process continues until a stable nucleus is formed.

Key characteristics of radioactive decay include:

• It is a random process at the level of single atoms, but predictable in large numbers due to probabilistic nature.
• The transformation is spontaneous without being affected by external conditions like temperature or pressure.
• Radioactive decay emits different types of radiation, such as alpha particles, beta particles, or gamma rays, contributing to the change in the nucleus.

Understanding radioactive decay is crucial in fields ranging from nuclear physics and medicine to archaeology and climatology. It's a key concept for calculating the half-life of a substance, which measures how quickly a radioactive isotope will decay.

Radioactive Isotope

A radioactive isotope, also known as a radioisotope, is an atom that has excess nuclear energy, making it unstable. This instability causes the isotope to undergo radioactive decay. During this process, the isotope may transform into a different element or a more stable isotope.

Radioactive isotopes are useful in various scientific and industrial fields due to their unique properties. Here are some interesting points:

• They are used in medicine, particularly in diagnostic imaging and radiation therapy, helping to treat and identify diseases.
• Radioisotopes play a significant role in archaeological dating by measuring the isotopic ratios in ancient artifacts.
• In environmental science, they help track and analyze pollution and soil composition.

As these isotopes decay, understanding their behavior helps in determining the half-life, an essential factor in working with radioactive materials safely and efficiently.

Decay Equation

The decay equation is a mathematical representation that describes the rate at which a radioactive isotope decays over time. This formula helps us understand how much of a radioactive material will remain after a given period. The decay equation typically looks like this:\[ N(t) = N_0 \times \left( \frac{1}{2} \right)^{t/T} \]Where:

• \(N(t)\): the remaining quantity of the substance after time \(t\).
• \(N_0\): the original quantity of the substance.
• \(t\): the time that has elapsed, often measured in units like minutes, hours, or years.
• \(T\): the half-life of the substance.

Using this decay equation, you can calculate the remaining amount of material given any specified time. This equation is crucial in scientific work involving radioactive materials and enables precise calculation of a sample’s decay over time. It allows scientists to predict when a radioactive isotope will reduce to a specific amount or how long it will take for a material to become safe for handling.