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

(d) 2 mg

Step 1: Understanding Half-life

The half-life is the time it takes for a quantity to reduce to half its initial amount. For tritium, the half-life is 12.3 years.

Step 2: Calculate Number of Half-lives

First, find out how many half-lives have passed in 49.2 years by dividing the total time by the half-life of tritium: \( \text{Number of half-lives} = \frac{49.2}{12.3} = 4 \).

Step 3: Determine Amount Remaining After Each Half-life

With each half-life, the amount of tritium is halved. Starting with 32 mg:- After 1 half-life: \( \frac{32}{2} = 16 \) mg- After 2 half-lives: \( \frac{16}{2} = 8 \) mg- After 3 half-lives: \( \frac{8}{2} = 4 \) mg- After 4 half-lives: \( \frac{4}{2} = 2 \) mg.

Step 4: Conclusion

After 4 half-lives (49.2 years), the amount of tritium remaining is 2 mg.

Key Concepts

Tritium Decay

Tritium, denoted as \({ }_{3}^{1} \mathrm{H}\), is a radioactive isotope of hydrogen. It possesses one proton and two neutrons. Tritium is unique due to its radioactivity, which originates from its unstable nucleus. Over time, tritium undergoes a process called decay, where the unstable nucleus loses energy and transforms into a more stable form.
This radioactive decay process is often measured in terms of half-life, a concept we will explore further. When tritium decays, it releases a form of beta radiation, converting into a non-radioactive helium isotope.

• The lightweight and gaseous nature of tritium makes it valuable, particularly in scientific research and applications like self-illuminating devices.
• Its production primarily occurs through natural processes in the upper atmosphere or artificially in nuclear reactors and accelerators.
• By understanding its decay, scientists can predict how long tritium sources remain active and safe to use.

Half-life Concept

The half-life of a substance is a crucial measure in nuclear chemistry. It describes the time required for half of a radioactive sample to decay into its byproducts. This concept is key for predicting how long a substance will remain active and its eventual decrease over time. In the case of tritium decay, the half-life is known to be \(12.3\) years.
Understanding half-life allows us to determine how much of a radioactive material remains after a specific period. The process involves repeated halving of the material within each half-life period.

• The calculation is straightforward: simply count the number of half-lives that fit into the given time span.
• This helps illustrate that even after several half-lives, a small portion of the original amount will persist, emphasizing the importance of long-term decay calculations.
• For tritium, knowing its half-life helps in managing its applications and waste products effectively, ensuring safety and compliance with regulations.

Nuclear Chemistry Calculations

Nuclear chemistry often involves calculations to predict the behavior of radioactive materials. In the exercise, you're given the initial amount of tritium, which is \(32 \mathrm{mg}\), and asked how much remains after \(49.2\) years. Since you know tritium's half-life is \(12.3\) years, you can use this to calculate the remaining amount after the stated time.
The first step is to calculate the number of half-lives that have occurred by dividing the total time by the half-life: \[ \text{Number of half-lives} = \frac{49.2}{12.3} = 4 \]This calculation indicates that four half-lives have passed.

• Start with the initial \(32 \mathrm{mg}\) and halve it for each half-life to find how much remains.
• After one half-life: \( \frac{32}{2} = 16 \mathrm{mg} \)
• After two half-lives: \( \frac{16}{2} = 8 \mathrm{mg} \)
• After three half-lives: \( \frac{8}{2} = 4 \mathrm{mg} \)
• After four half-lives: \( \frac{4}{2} = 2 \mathrm{mg} \).

Ultimately, after \(49.2\) years, there remains \(2 \mathrm{mg}\) of tritium. This approach used in nuclear chemistry highlights the systematic yet simple nature of decay processes, allowing for clear predictions and planning.