Problem 80 - A Complete Resource Book in Chemistry for JEE Main

Question

The radioisotope, tritium \(\left({ }_{3}^{1} \mathrm{H}\right)\) has a half- life of \(12.3\) years. If the initial amount of tritium is \(32 \mathrm{mg}\), how many milligrams of it would remain after \(49.2\) years? (a) \(4 \mathrm{mg}\) (b) \(8 \mathrm{mg}\) (c) \(1 \mathrm{mg}\) (d) \(2 \mathrm{mg}\)

Step-by-Step Solution

Answer

2 mg of tritium would remain.

Step 1: Understanding the Half-Life Formula

The formula to calculate the remaining amount of a radioactive substance after a certain period, given its half-life, is: \ \[ A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \( A \) is the remaining amount, \( A_0 \) is the initial amount, \( t \) is the time elapsed, and \( T_{1/2} \) is the half-life of the substance.

Step 2: Identify the Known Values

From the problem, we know: initial amount \( A_0 = 32 \, \text{mg} \), half-life \( T_{1/2} = 12.3 \, \text{years} \), and time elapsed \( t = 49.2 \, \text{years} \).

Step 3: Calculate The Number of Half-Lives

Determine how many half-lives have passed using the formula: \ \[ n = \frac{t}{T_{1/2}} = \frac{49.2}{12.3} = 4 \] So, 49.2 years corresponds to 4 half-lives.

Step 4: Calculate the Remaining Amount of Tritium

Use the half-life formula to find the remaining amount \( A \): \ \[ A = 32 \times \left( \frac{1}{2} \right)^4 = 32 \times \frac{1}{16} = 2 \, \text{mg} \] After 4 half-lives, only \( 2 \, \text{mg} \) of tritium remains.

Key Concepts

Half-Life Calculation

Understanding the concept of half-life is crucial in nuclear chemistry. Half-life is the time required for half of a radioactive substance to decay. When you calculate how much of a substance remains, you use this principle. This concept is applied in various fields like carbon dating and nuclear medicine.

You can determine the remaining amount of a radioactive isotope using a simple formula:

• Let the initial quantity be represented by \( A_0 \).
• The remaining amount after time \( t \) is \( A \).
• \( T_{1/2} \) represents the half-life of the isotope.
• The formula is \( A = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \).

By plugging in known values for initial amount, elapsed time, and half-life into this formula, you can at any given point determine the amount of material remaining.

Tritium Decay

Tritium is a special kind of hydrogen known as a radioisotope, which means it is radioactive. This specific hydrogen atom, \({ }_{3}^{1} \mathrm{H}\), possesses one proton and two neutrons, making it heavier and unstable. Because of its instability, tritium undergoes radioactive decay to stabilize itself over time.

In the context of the problem, tritium's decay is evaluated over a span of 49.2 years. With each passing half-life, exactly half of the tritium nuclei decay.

This progression continues until significantly less remains, as shown in the provided calculation. Studying tritium decay is vital for practical applications such as dating former groundwater and understanding ecological impacts from nuclear weaponry or energy production.

Nuclear Chemistry

Nuclear chemistry explores the atomic level of chemistry, focusing on the reactions and changes that occur within the nucleus. These reactions happen at extraordinary energies compared to typical chemical reactions, resulting in significant transformations like the decay of radioisotopes.

Key applications of nuclear chemistry include:

• Radiometric dating: measuring the age of objects by examining isotopic decay.
• Medical imaging: using radioisotopes to visualize body organs for diagnoses.
• Energy generation: nuclear fission and fusion to create electricity.

Pioneering studies in nuclear chemistry, such as the study of tritium decay, advance our knowledge of both practical and theoretical aspects of radioactive substances.

Radioisotope

Radioisotopes are isotopes that have unstable nuclei and exhibit radioactive decay. Each radioisotope has a distinct half-life and decays at its own unique rate, transforming into a different element or isotope over time.

The identification and understanding of radioisotopes have numerous practical applications:

• Health sector: for cancer treatments and sterilization of medical equipment.
• Archaeology: for dating ancient artifacts using techniques like radiocarbon dating.
• Environmental studies: tracing and analyzing pollution sources.

In the given exercise scenario, tritium acts as a textbook example of a radioisotope undergoing decay, with a half-life of 12.3 years, transforming slowly into a stable isotope.