Problem 33 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

\({ }^{227}\) Ac has a half-life of \(21.8\) years with respect to radioactive decay. The decay follows two parallel paths, one leading to \({ }^{27} \mathrm{Th}\) and the other to \({ }^{223} \mathrm{Fr}\). The percentage yields of these two daughter nuclides are \(1.2\) and \(98.8\) respectively. What are the decay constants \((\lambda)\) for each of the separate paths?

Step-by-Step Solution

Answer

The decay constants are 0.00038 per year for \(^{223}\text{Th}\) and 0.03143 per year for \(^{223}\text{Fr}\).

Step 1: Calculate the Total Decay Constant

The formula for the decay constant (\( \lambda \)) in terms of half-life (\( t_{1/2} \)) is \( \lambda = \frac{\ln(2)}{t_{1/2}} \). For \( ^{227}\text{Ac} \), the half-life \( t_{1/2} \) is 21.8 years. \( \lambda = \frac{\ln(2)}{21.8} \approx 0.03181 \text{ per year}\). This is the total decay constant for \( ^{227}\text{Ac} \).

Step 2: Understanding Parallel Pathway Decay

Since \( ^{227}\text{Ac} \) decays via two parallel pathways, each path shares the total decay constant based on its percentage yield. We need to distribute the total decay constant by these percentage yields into two separate decay constants for \( ^{223}\text{Th} \) and \( ^{223}\text{Fr} \).

Step 3: Calculate the Decay Constant for \( ^{223}\text{Th} \)

The percentage yield to \( ^{223}\text{Th} \) is 1.2%. To find its decay constant:\[ \lambda_{\text{Th}} = \frac{1.2}{100} \times \lambda = \frac{1.2}{100} \times 0.03181 \approx 0.00038172 \text{ per year} \]

Step 4: Calculate the Decay Constant for \( ^{223}\text{Fr} \)

The percentage yield to \( ^{223}\text{Fr} \) is 98.8%. Hence, its decay constant is:\[ \lambda_{\text{Fr}} = \frac{98.8}{100} \times \lambda = \frac{98.8}{100} \times 0.03181 \approx 0.03142828 \text{ per year} \]

Key Concepts

Understanding Half-Life Calculations

Half-life is an essential concept when studying radioactive decay. It is defined as the time required for half of a radioactive substance to undergo decay. This is a constant property of each radioactive isotope and provides insights into its stability. To find the half-life of a radioactive substance, you use the formula: \[ t_{1/2} = \frac{\ln(2)}{\lambda} \]where:

• \( t_{1/2} \) is the half-life.
• \( \lambda \) is the decay constant.
• \( \ln(2) \approx 0.693 \), which is the natural logarithm of 2.

This relationship helps us calculate how quickly a radioactive isotope will decay, which is particularly useful in fields like archaeology (carbon dating) and medicine (radioactive tracers). For instance, if the half-life of \( ^{227}\text{Ac} \) is 21.8 years, it means after 21.8 years, only half of the original quantity will remain radioactive. Another 21.8 years later, half of what was left will decay, and so on.

Decay Constant Demystified

The decay constant \( \lambda \) is a key factor in understanding the rate of radioactive decay. It represents the probability per unit time that a nucleus will decay. The relationship between half-life and decay constant is straightforward using the formula:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]where:

• \( \lambda \) is the decay constant.
• \( t_{1/2} \) is the half-life.

For \( ^{227}\text{Ac} \), with a half-life of 21.8 years, the decay constant is calculated as:\[ \lambda = \frac{0.693}{21.8} \approx 0.03181 \text{ per year} \]This indicates the fraction of \( ^{227}\text{Ac} \) atoms decaying in one year. The decay constant provides a clearer picture of how quickly a substance is decaying in real-time, complementing the qualitative insight that half-life provides.

Exploring Parallel Decay Pathways

In certain radioactive decays, a parent isotope can decay into more than one type of daughter isotope through different paths. These are known as parallel decay pathways. For example, \( ^{227}\text{Ac} \) has two parallel decay paths:

• To \( ^{223}\text{Th} \) with a 1.2% yield.
• To \( ^{223}\text{Fr} \) with a 98.8% yield.

Each path uses a fraction of the total decay constant based on the percentage yield. Calculating the decay constant for each pathway involves distributing the total decay constant according to these percentage yields:- \( \lambda_{\text{Th}} = \frac{1.2}{100} \times 0.03181 \approx 0.00038172 \text{ per year} \)- \( \lambda_{\text{Fr}} = \frac{98.8}{100} \times 0.03181 \approx 0.03142828 \text{ per year} \)Understanding these pathways is crucial as it highlights how different decay modes can occur simultaneously, influencing how we interpret the behavior and final products of radioactive decay.