Problem 78 - Master Resource Book in JEE Main Physics

Question

A radioactive nucleus can decay simultaneously by two different processes which have decay constant \(\lambda_{1}\) and \(\lambda_{2} .\) The effective decay constant of the nuclide is \(\lambda\), where (a) \(\lambda=\lambda_{1}+\lambda_{2}\) (b) \(\lambda=2\left(\lambda_{1}+\lambda\right)\) (c) \(\frac{1}{\lambda}=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\) (d) \(\lambda=\sqrt{\lambda_{1} \lambda_{2}}\)

Step-by-Step Solution

Answer

The effective decay constant is \(\lambda = \lambda_1 + \lambda_2\) (option a).

Step 1: Understand the Problem

We are given a problem involving a radioactive nucleus that can decay via two processes with decay constants \(\lambda_1\) and \(\lambda_2\). We need to determine the correct formula for the effective decay constant \(\lambda\).

Step 2: Decay Process Analysis

In radioactive decay, when two decay processes can occur simultaneously, the effective decay constant is the sum of the individual decay constants. This is because the decay events are independent and can add up to give the total rate of decay.

Step 3: Formulate the Solution

From our understanding of decay processes, the correct formula for the effective decay constant \(\lambda\) is \(\lambda = \lambda_1 + \lambda_2\).

Step 4: Verify the Options

Review the given options: (a) \(\lambda = \lambda_1 + \lambda_2\) matches our formulation. (b) \(\lambda = 2(\lambda_1 + \lambda)\) does not match. (c) \(\frac{1}{\lambda} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2}\) is incorrect. (d) \(\lambda = \sqrt{\lambda_1 \lambda_2}\) is not suitable for this problem.

Step 5: Conclusion

The correct option is (a) \(\lambda = \lambda_1 + \lambda_2\).

Key Concepts

Decay Constant

The decay constant is a fundamental concept in the study of radioactive decay. It represents the probability per unit time that an individual nucleus will decay. For any given radioactive isotope, the decay constant is a fixed value, symbolized by \( \lambda \). It plays a key role in determining the rate at which a sample of radioactive material will decay. Understanding the decay constant helps us comprehend how quickly or slowly a radioactive substance decreases in quantity. A larger decay constant means the substance decays faster, while a smaller one indicates a slower decay process. This parameter is not influenced by the amount of substance; it is an intrinsic property of the radioactive isotope. In equations related to radioactive decay, such as the radioactive decay law, the decay constant is central. It helps describe the exponential decrease in the quantity of a radioactive sample over time.

Simultaneous Decay Processes

Simultaneous decay processes occur when a single radioactive nucleus has more than one path to decay. This can happen when different decay modes are available for the nuclide, each characterized by its own decay constant. For example, if a nucleus can decay through two distinct processes, each with its own decay constant \( \lambda_1 \) and \( \lambda_2 \), these processes are considered to happen independently of each other.

• The possibility of multiple decay channels provides a greater total probability for decay per unit time because the events can happen concurrently.
• It is crucial to realize that the total probability of a decay event is not merely dominated by one process but the summation of probabilities associated with all concurrent processes.

These simultaneous processes result in an overall effective decay constant that combines the effects of each separate decay path.

Effective Decay Constant

The effective decay constant \( \lambda \) provides a comprehensive measure of the total probability of decay when simultaneous decay processes are present. When a radioactive substance can decay via multiple independent methods, the effective decay constant is found by summing the individual decay constants of each process.Mathematically, the effective decay constant in such scenarios is expressed as:\[ \lambda = \lambda_1 + \lambda_2 \]This formula highlights the cumulative nature of the decay processes. Each independent decay path contributes to the overall rate at which the radioactive material separates or transforms. Understanding the effective decay constant is essential when analyzing complex decay scenarios as it allows prediction of the total rate of decay without having to track each path separately. This simplification makes it easier to model and predict the behavior of radioactive materials over time, an important aspect of nuclear physics and engineering.