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 correct answer is (d) 0.231 per hour.

Step 1: Understand the Relationship

The disintegration constant, often denoted as \( \lambda \), is related to the half-life (\( t_{1/2} \)) of a substance via the formula \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( \ln(2) \approx 0.693 \).

Step 2: Plug in the Half-life Value

Insert the given half-life of the isotope, which is 3 hours, into the formula. This gives us \( \lambda = \frac{0.693}{3} \).

Step 3: Calculate the Disintegration Constant

Perform the division: \( \lambda = \frac{0.693}{3} = 0.231 \) per hour.

Step 4: Match with Given Options

Compare the calculated disintegration constant with the provided options: (a) 1.57 per hour, (b) 1.92 per hour, (c) 1.04 per hour, and (d) 0.231 per hour. Our calculated value matches option (d), which is \(0.231\) per hour.

Key Concepts

Disintegration Constant

The disintegration constant, represented as \( \lambda \), is a fundamental concept in radioactive decay that describes how quickly a radioactive isotope undergoes disintegration. It shows how fast the atoms in a radioactive sample are breaking down over time. The larger the value of \( \lambda \), the faster the decay process is occurring.

• It's a rate that tells us the probability of a decay event occurring in a unit of time, typically an hour or a second.
• Mathematically, it's defined as the inverse of the decay constant: \[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
• Here, \( \ln(2) \approx 0.693 \) and \( t_{1/2} \) is the half-life of the material.

Understanding this constant helps us predict the behavior of radioactive substances in various fields, including medicine, archaeology, and nuclear physics.

Half-Life

Half-life, denoted as \( t_{1/2} \), is the amount of time it takes for half of a radioactive isotope's nuclei to decay. It's a crucial parameter because it gives us a measurable way to assess the stability and longevity of an isotope.

• The shorter the half-life, the faster the isotope decays, releasing energy rapidly.
• A long half-life indicates a stable isotope that decays slowly over time.
• Knowing the half-life of an isotope allows scientists to date artifacts, understand nuclear reactions, and safely manage nuclear materials.

The half-life value interacts with the disintegration constant to help us calculate how quickly a sample will lose its radioactivity. For example, in the original problem, a half-life of 3 hours suggests rapid decay, as seen in the calculated disintegration constant of 0.231 per hour.

Radioactive Isotope

A radioactive isotope, sometimes known as a radioisotope, is an unstable variant of an element that exhibits radioactive decay. This means it spontaneously emits energy in the form of particles or electromagnetic waves as it decays into a more stable state.

• These isotopes have an unstable combination of protons and neutrons in their nucleus.
• As they decay, they can transform into different elements or isotopes.
• Each radioactive isotope has a unique half-life reflecting how quickly or slowly it decays.

Radioactive isotopes are incredibly useful in a wide variety of fields:

• In medicine, they are used in imaging and treating diseases.
• In archaeology, they allow us to date ancient artifacts and fossils.
• In energy production, certain isotopes provide a source of power.

The study of radioactive isotopes encompasses their disintegration constants and half-lives, giving us deep insights into the elemental transformations occurring around us constantly.