Problem 74 - A Complete Resource Book in Chemistry for JEE Main

Question

The disintegration constant of a radioactive isotope whose half-life is 3 hours is (a) \(1.57\) per hour (b) \(1.92\) per hour (c) \(1.04\) per hour (d) \(0.231\) per hour

Step-by-Step Solution

Answer

The disintegration constant is 0.231 per hour, which corresponds to option (d).

Step 1: Understanding the Concept

The disintegration constant, often denoted by \(\lambda\), is related to the half-life \(t_{1/2}\) of a radioactive substance by the formula:\[\lambda = \frac{\ln(2)}{t_{1/2}}.\] This relationship helps us determine how quickly a radioactive isotope decays over time.

Step 2: Identify Given Information

We are given that the half-life \(t_{1/2}\) of the radioactive isotope is 3 hours. This is the time it takes for half of the radioactive material to decay.

Step 3: Apply the Formula

Substitute the half-life into the formula:\[\lambda = \frac{\ln(2)}{3}.\]Use the approximate value \(\ln(2) \approx 0.693\).

Step 4: Calculate the Disintegration Constant

Calculating the disintegration constant:\[\lambda = \frac{0.693}{3} = 0.231\text{ per hour}.\]

Step 5: Verify Against Options

Compare the calculated disintegration constant \(0.231\) with the given options:- (a) \(1.57\) per hour- (b) \(1.92\) per hour- (c) \(1.04\) per hour- (d) \(0.231\) per hourThe correct answer is (d) \(0.231\) per hour.

Key Concepts

Disintegration Constant

The disintegration constant, symbolized by \( \lambda \), is a fundamental concept in the study of radioactive decay. It gives us an idea of how fast a particular radioactive isotope decays over time. The formula to calculate the disintegration constant is \[ \lambda = \frac{\ln(2)}{t_{1/2}} \] where \( t_{1/2} \) stands for the half-life of the material. This constant is expressed in inverse time units, like per second or per hour, depending on how the half-life is measured.

• A higher \( \lambda \) value means the substance is decaying rapidly.
• A lower \( \lambda \) value indicates slower decay.

The disintegration constant is crucial because it allows scientists to predict how a substance will behave as it loses its radioactivity over time.

Half-Life

Half-life is an essential parameter when it comes to understanding radioactivity. It is defined as the time required for half of the radioactive atoms in a sample to decay. For example, if you start with 100 grams of a radioactive isotope, in one half-life, you'll have 50 grams of the original isotope remaining. The half-life is a constant for each specific substance and does not change regardless of the initial amount of material.

• It can range from fractions of a second to millions of years depending on the isotope.
• It provides a sense of how long a radioactive element will remain active in the environment.

By knowing the half-life, we can calculate the disintegration constant \( \lambda \) using the given formula, which directly links the two concepts and helps us understand the decay process better.

Radioactive Isotope

Radioactive isotopes, also known as radioisotopes, are atoms with an unstable nucleus that lose energy by emitting radiation. This spontaneous process can involve the emission of alpha particles, beta particles, or gamma rays, leading to the transformation of the original isotope into a different element or a more stable isotope. Radioactive isotopes are omnipresent both naturally and artificially.

• Natural radioisotopes, like Uranium-238, occur in the environment and are primordial.
• Artificial or man-made radioisotopes, like Cobalt-60, are produced in reactors or laboratories.

These isotopes have wide applications in medicine, industry, and research. They are used in radiometric dating, medical diagnostics, treatment of diseases like cancer, and more. Understanding the properties of these isotopes, including their half-life and disintegration constant, is crucial for safely and effectively utilizing their potential.

Exponential Decay

Exponential decay is a fundamental process that describes how the quantity of a radioactive isotope decreases over time. It’s characterized by a rate of loss that is proportional to the current value, making the decay process steady but disproportionately faster as time increases. In mathematical representation, this process follows the formula:\[ N(t) = N_0 e^{-\lambda t} \] where:

• \( N(t) \) is the quantity of the substance that still remains after time \( t \).
• \( N_0 \) is the original amount of the substance.
• \( e \) is the base of natural logarithms.
• \( \lambda \) is the disintegration constant.

This formula illustrates how the number of radioactive atoms decreases exponentially over time. Understanding this concept helps us predict how a substance will behave and allows scientists to calculate how long it will take for a given substance to decay to a desired level.