Problem 34 - A Complete Resource Book in Chemistry for JEE Main

Question

The half-life of a radioactive isotope is \(1.5\) hours. The mass of it that remains undecayed after 6 hours is (the initial mass of the isotope is \(64 \mathrm{~g}\) ) (a) \(32 \mathrm{~g}\) (b) \(16 \mathrm{~g}\) (c) \(8 \mathrm{~g}\) (d) \(4 \mathrm{~g}\)

Step-by-Step Solution

Answer

After 6 hours, the mass of the radioactive isotope remaining is 4 g (option d).

Step 1: Understanding Half-Life

The half-life of a radioactive isotope is the time required for half of the isotope to decay. Here, the half-life is given as 1.5 hours.

Step 2: Determine Number of Half-Lives Passed

Since 6 hours is the total time passed, we divide this by the half-life duration to find out how many half-lives have passed: Number of half-lives = \( \frac{6}{1.5} = 4 \). Thus, 4 half-lives have elapsed.

Step 3: Calculate Remaining Mass After Each Half-Life

Starting with an initial mass of \(64\, g\), each half-life reduces the remaining mass by half. We perform this operation for 4 half-lives: - After 1st half-life (1.5 hours): \(64/2 = 32\, g\)- After 2nd half-life (3 hours): \(32/2 = 16\, g\)- After 3rd half-life (4.5 hours): \(16/2 = 8\, g\)- After 4th half-life (6 hours): \(8/2 = 4\, g\)

Step 4: Conclude Remaining Mass

The mass remaining after 6 hours is \(4\, g\), which matches the choice (d).

Key Concepts

Half-Life Calculation

In nuclear chemistry, half-life refers to the amount of time it takes for half of a substance's radioactive atoms to decay. This concept is crucial for understanding the decay process and the potential hazards of radioactive materials. In the exercise, the half-life is given as 1.5 hours. This means, after 1.5 hours, only half of the initial mass remains. For example, if you start with 64 grams, half of it—32 grams—remains after the first 1.5 hours.

• After the total of 6 hours, which is equal to four half-life periods, the decay process is applied repeatedly.
• With each passing half-life, the substance reduces to half of its previous amount.
• This results in a mass of 4 grams after 6 hours, demonstrating the exponential nature of radioactive decay.

Understanding half-life calculations is vital for scientific fields that work with radioisotopes, from medical treatments to nuclear power generation. It ensures proper application and safety procedures are followed.

Isotopes in Chemistry

Isotopes are variants of elements that have the same number of protons but different numbers of neutrons. This means they occupy the same position on the periodic table and share similar chemical properties, yet they differ in atomic weight. Isotopes are essential in various applications, particularly in nuclear chemistry.

• Stable isotopes do not change over time and can exist indefinitely.
• Radioactive isotopes, on the other hand, undergo decay over time, releasing energy in the form of radiation.

Understanding isotopes aids chemists and physicists in the exploration of elemental properties and reactions. In the context of our exercise, the focus was on a radioactive isotope. By understanding its half-life, we could compute the amount of isotope remaining after a given period of decay, employing the specific property that distinguishes it from its stable counterpart.

Nuclear Chemistry

Nuclear chemistry is a branch of chemistry that deals with radioactivity, nuclear processes, and properties of isotopes. Its core focus is on reactions that change the nucleus of an atom, such as fission, where a nucleus splits, and fusion, where nuclei combine. This discipline plays a significant role in numerous fields:

• In energy production through nuclear power plants, where controlled nuclear reactions generate electricity.
• In medicine, especially in diagnostics and treatments, such as PET scans and cancer radiotherapy.
• In environmental sciences, tracking isotopic signatures can help in understanding climate changes and pollution sources.

Nuclear chemistry is crucial for advancements we often take for granted. By understanding atomic nuclei and decay processes, scientists can harness powerful and often sustainable energy sources, improve health care, and monitor environmental benefits. Our exercise showcases one aspect of nuclear chemistry—how we predict the decay of a radioactive isotope over time.