Problem 61 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive isotope is 3 hours. If the initial mass of the isotope was \(300 \mathrm{~g}\), the mass which remained undecayed in 18 hours would be (a) \(4.68 \mathrm{~g}\) (b) \(2.34 \mathrm{~g}\) (c) \(1.17 \mathrm{~g}\) (d) \(9.36 \mathrm{~g}\)

Step-by-Step Solution

Answer

The mass remaining after 18 hours is approximately 4.68 g (option a).

Step 1: Understand the Half-Life Concept

The half-life of a substance is the time it takes for half of the radioactive isotope to decay. In this problem, the half-life is given as 3 hours, meaning that every 3 hours, the mass of the isotope reduces by half.

Step 2: Calculate the Number of Half-Life Periods

To find out how many half-life periods have passed in 18 hours, we divide the total time by the half-life: \[\frac{18\, \text{hours}}{3\, \text{hours/half-life}} = 6\, \text{half-life periods}.\]

Step 3: Apply the Half-Life Formula

The remaining mass of the isotope after a certain number of half-lives can be calculated using the formula: \[ M = M_0 \times \left( \frac{1}{2} \right)^n \] where \( M_0 \) is the initial mass and \( n \) is the number of half-lives. Here, \( M_0 = 300 \) g and \( n = 6 \).

Step 4: Execute the Calculation

Plug the values into the formula: \[ M = 300 \times \left( \frac{1}{2} \right)^6 \] Compute \( \left( \frac{1}{2} \right)^6 = \frac{1}{64} \). Thus, \[ M = 300 \times \frac{1}{64} = 4.6875 \].

Step 5: Determine the Correct Answer

Round the result to two decimal places to find the closest answer choice: 4.68 g. This means that option (a) is the correct answer.

Key Concepts

Radioactive Decay

Radioactive decay is a fascinating natural process where unstable atomic nuclei lose energy by emitting radiation. This can happen through different types of decay such as alpha, beta, or gamma decay. During radioactive decay, atoms transform into different elements or different isotopes of the same element. Over time, this leads to a reduction in the number of radioactive particles within a given sample.

A crucial aspect of radioactive decay is that it follows a predictable pattern defined by the half-life. The half-life is the time it takes for half of the radioactive particles to decay. This is crucial in calculating how long the substance remains active or dangerous.

• Decay follows an exponential function.
• Commonly represented in formulas to predict future decay.
• Used in dating methods like carbon dating.

Isotopes

Isotopes are variants of a given chemical element, differing in neutron number, while retaining the same number of protons. Because neutrons vary, isotopes of a single element have different atomic masses but the same chemical properties.

The real charm of isotopes in nuclear chemistry comes from their stability. Some isotopes are stable, while others are radioactive, meaning they decay over time. Understanding and identifying isotopes is important for various scientific disciplines, from archeology to medicine.

• Isotopes behave identically in chemical reactions.
• Radioactive isotopes are used in medical imaging and cancer treatment.
• Stable isotopes help trace chemical pathways in the environment.

Nuclear Chemistry

Nuclear chemistry is a subfield of chemistry dealing with radioactivity, nuclear processes, and properties. This field revolves around the atom's nucleus and its structure and transformation. It plays a significant role in energy generation, medical treatments, and understanding the natural processes of the Earth.

Key concepts include nuclear stability, reactions, and the energy yield from these processes. Nuclear chemistry requires an understanding of both the physical and chemical aspects of nuclear reactions, as they significantly differ from typical chemical reactions.

• Nuclear power plants use principles from nuclear chemistry to fuel electricity generation.
• Advances in nuclear chemistry have led to the development of new medical treatments.
• Nuclear reactions can release vastly more energy than chemical reactions due to changes in the nucleus.