Problem 141 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive isotope is three hours. If the initial mass of the isotope were \(256 \mathrm{~g}\), the mass of it remaining undecayed after 18 hours would be (a) \(4.0 \mathrm{~g}\) (b) \(8.0 \mathrm{~g}\) (c) \(12.0 \mathrm{~g}\) (d) \(16.0 \mathrm{~g}\)

Step-by-Step Solution

Answer

The remaining mass is 4.0 g. Option (a) is correct.

Step 1: Understanding Half-Life

The half-life of a substance is the time required for its quantity to reduce to half its initial amount. For this problem, the half-life is given as 3 hours.

Step 2: Determine Number of Half-Life Periods

Calculate how many half-life periods are in 18 hours by dividing the total time by the half-life: \[ \frac{18 \text{ hours}}{3 \text{ hours/half-life}} = 6 \text{ half-life periods} \]

Step 3: Calculate Remaining Mass

The remaining mass is found by halving the mass for each half-life period. Start with 256 g and halve it for each of the 6 periods:\[\text{After 1 half-life: } \frac{256}{2} = 128 \text{ g}\]\[\text{After 2 half-lives: } \frac{128}{2} = 64 \text{ g}\]\[\text{After 3 half-lives: } \frac{64}{2} = 32 \text{ g}\]\[\text{After 4 half-lives: } \frac{32}{2} = 16 \text{ g}\]\[\text{After 5 half-lives: } \frac{16}{2} = 8 \text{ g}\]\[\text{After 6 half-lives: } \frac{8}{2} = 4 \text{ g}\]

Step 4: Select the Correct Option

The remaining mass after 18 hours is 4 g, so the correct answer is (a) \(4.0 \text{ g}\).

Key Concepts

Half-life Calculation

Half-life is a fundamental concept when studying radioactive decay. Simply put, it refers to the amount of time it takes for half of the radioactive nuclei in a sample to decay. This decay process is random at the atomic level, but predictable across large numbers of atoms. Here, we're dealing with a radioactive isotope with a half-life of 3 hours. This means every 3 hours, the mass of the radioactive material is reduced by half. To determine how much of a substance remains after a certain time, we calculate the number of half-life periods that have passed. For example, in our exercise, 18 hours equate to 6 half-life periods because 18 divided by 3 equals 6. Once you know the number of half-life periods, you can repeatedly halve the initial mass to find the remaining mass. Starting with 256 g, halving it 6 times leaves 4 g.

Radioactive Isotopes

Radioactive isotopes, also known as radioisotopes, are variants of chemical elements that have unstable nuclei. This instability causes them to decay over time, emitting radiation. Every chemical element can have several isotopes, differing in the number of neutrons in the nucleus. Some isotopes are stable, while others are radioactive. The half-life of a radioisotope indicates how stable or unstable it is. Shorter half-lives mean the isotope decays more quickly. Radioactive isotopes have widespread applications:

• Medical Imaging and Treatment: Used in diagnosing and treating diseases.
• Carbon Dating: Helps determine the age of archaeological finds.
• Power Generation: Used as fuel in nuclear power plants.

Radioactive isotopes play a crucial role in modern science and technology.

Nuclear Chemistry

Nuclear chemistry is the branch of chemistry that deals with the chemical and physical properties of elements as influenced by changes in the structure of the atomic nucleus. It is crucial in understanding reactions and processes involving radioactive materials. There are several key areas involved: - **Radioactive Decay:** The transformation of an unstable nuclide into a more stable form, accompanied by emission of radiation. - **Nuclear Reactions:** Processes that involve changes in an atom's nucleus and can release substantial amounts of energy. These reactions are the basis for the energy produced in nuclear power plants and atomic bombs. Nuclear chemistry not only helps us understand the principles of radioactivity and radioactive decay but also enables us to harness these processes for practical uses, from energy production to medical applications. Specialists in this field work closely with physicists to explore new frontiers in science and technology.