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 (d) 0.231 per hour.

Step 1: Understanding Half-Life and Disintegration Constant

A radioactive isotope's half-life is the time it takes for half of the isotope to decay. The disintegration constant, denoted as \( \lambda \), represents the rate of decay per unit time.

Step 2: Using the Half-Life Formula

The relationship between the half-life (\( t_{1/2} \)) and the disintegration constant is given by the formula: \( t_{1/2} = \frac{0.693}{\lambda} \). We are given that \( t_{1/2} = 3 \) hours.

Step 3: Solving for the Disintegration Constant

Rearrange the half-life formula to solve for \( \lambda \): \( \lambda = \frac{0.693}{t_{1/2}} \). Substitute \( t_{1/2} = 3 \) into the equation: \( \lambda = \frac{0.693}{3} \).

Step 4: Calculating the Value

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

Step 5: Choosing the Correct Option

Compare the calculated disintegration constant, 0.231 per hour, with the given options. The correct option is (d) 0.231 per hour.

Key Concepts

Half-life

The concept of half-life is crucial in understanding radioactive decay. It is defined as the time required for half of a given quantity of a radioactive isotope to decay into another element or isotope. In simple terms, if you start with 100 grams of a radioactive isotope with a half-life of 3 hours, in 3 hours, you'll have 50 grams of the original isotope remaining. This characteristic time period is a constant for any given substance, meaning it does not change over time or with quantity.

Half-life helps scientists determine how quickly a radioactive isotope undergoes decay. For instance, in the case of medical isotopes used for imaging or treatment, knowing the half-life ensures that the isotope remains active for the duration needed for diagnosis or therapy but not longer. This concept also finds application in carbon dating, where scientists determine the age of an archaeological find by observing the remaining carbon-14 isotope, which has a half-life of about 5730 years.

Disintegration Constant

The disintegration constant, represented by the symbol \( \lambda \), is a value that indicates the rate at which a radioactive isotope decays per unit of time.

Mathematically, it is related to the half-life of the substance through the formula:

• \( t_{1/2} = \frac{0.693}{\lambda} \)

This equation shows the inverse relationship between half-life and the disintegration constant. A larger \( \lambda \) indicates a faster decay process, and thus a shorter half-life.

To calculate \( \lambda \), one would rearrange the half-life equation to \( \lambda = \frac{0.693}{t_{1/2}} \). Using this relationship, if you know the half-life of a radioactive substance, you can easily determine its disintegration constant, just as shown in the original exercise where the half-life was 3 hours, leading to a \( \lambda \) value of 0.231 per hour.

This constant helps in predicting how fast the atoms of a radioactive material decay, which is crucial for various scientific and industrial applications, including nuclear power generation and medical treatments.

Radioactive Isotopes

Radioactive isotopes, often called radioisotopes, are unstable forms of elements that spontaneously release energy in the form of radiation. This process, known as radioactive decay, occurs as these isotopes transform into more stable forms, often resulting in different elements.

Each radioactive isotope has unique properties, such as its half-life and disintegration constant, that make it suitable for specific applications.
Some common uses include:

• Medical imaging and treatments, where isotopes provide critical functions in diagnosing and treating conditions.
• Determining the ages of archaeological finds and geological samples through radiocarbon dating.
• Using isotopes like uranium-235 in nuclear reactors to generate electricity.

Radioactive isotopes play a vital role in scientific research as well, allowing for the tracing of processes that occur too slowly or at scales too small for direct observation. Understanding their behavior, including how they decay and transform, is essential for harnessing their potential safely and effectively.