Problem 46 - Master Resource Book in JEE Main Physics

Question

There are two radioactive substances \(A\) and \(B\). Decay constant of \(B\) is two times that of \(A\). Initially, both have equal number of nuclei. After \(n\) half-lives of \(A\), rate of disintegration of both are equal. The value of \(n\) is (a) 4 (b) 2 (c) 1 (d) 5

Step-by-Step Solution

Answer

n = 1

Step 1: Understand the Decay Law

The number of remaining nuclei in a radioactive substance after time \( t \) is given by the formula \( N(t) = N_0 \cdot e^{-\lambda t} \), where \( \lambda \) is the decay constant and \( N_0 \) is the initial number of nuclei.

Step 2: Identify Decay Constants

Let the decay constant of substance \( A \) be \( \lambda_A \). Since the decay constant of \( B \) is two times that of \( A \), we have \( \lambda_B = 2\lambda_A \).

Step 3: Calculate Rate of Disintegration

The rate of disintegration or activity for a substance at time \( t \) is given by \( R(t) = \lambda \cdot N(t) \). For \( A \) and \( B \), this becomes \( R_A(t) = \lambda_A \cdot N_A(t) \) and \( R_B(t) = \lambda_B \cdot N_B(t) \), respectively.

Step 4: Apply Initial Conditions

Initially, both substances have the same number of nuclei, \( N_0 \). Thus, \( N_A(0) = N_B(0) = N_0 \).

Step 5: Consider After n Half-Lives of A

The time after \( n \) half-lives of \( A \) is \( t = n \cdot \frac{\ln(2)}{\lambda_A} \). The number of remaining nuclei in \( A \) after \( n \) half-lives is \( N_A(t) = N_0 \cdot e^{-\lambda_A t} \). For \( B \), it's \( N_B(t) = N_0 \cdot e^{-2\lambda_A t} \).

Step 6: Set Equations for Equal Disintegration Rates

According to the problem, the rates of disintegration of \( A \) and \( B \) are equal after \( n \) half-lives of \( A \). Therefore, \( \lambda_A \cdot N_A(t) = \lambda_B \cdot N_B(t) \). Substitute \( \lambda_B = 2\lambda_A \) into the equation: \( \lambda_A \cdot N_0 \cdot e^{-\lambda_A t} = 2\lambda_A \cdot N_0 \cdot e^{-2\lambda_A t} \).

Step 7: Simplify and Solve for n

Cancel \( \lambda_A \cdot N_0 \) from both sides, resulting in \( e^{-\lambda_A t} = 2 \cdot e^{-2\lambda_A t} \). Rearranging gives \( e^{\lambda_A t} = 2 \), and taking the logarithm gives \( \lambda_A t = \ln(2) \). Substitute \( t = n \cdot \frac{\ln(2)}{\lambda_A} \) into the equation to solve for \( n \): \( n \cdot \ln(2) = \ln(2) \) which simplifies to \( n = 1 \).

Key Concepts

Decay Constant

The decay constant \( \lambda \) is a fundamental concept that defines the rate at which a radioactive substance undergoes decay. It is a measure of the probability per unit time that a single nucleus will decay. The larger the decay constant, the quicker the substance will decay.

• It is expressed in units of inverse time, such as \( \text{s}^{-1} \) or \( \text{year}^{-1}\).
• In mathematical terms, it is used in the equation \( N(t) = N_0 \cdot e^{-\lambda t} \), where \( N(t) \) is the number of remaining nuclei at time \( t \).

In our problem, radioactive substance \( B \) has a decay constant that is twice that of substance \( A \), meaning \( B \) decays faster than \( A \). This relationship allows us to set up equations for comparing their decay behavior.

Half-Life

The half-life of a radioactive substance is the time required for half of its nuclei to decay. It serves as an intuitive way to understand how quickly a substance undergoes radioactive decay.

• The formula for half-life \( T_{1/2} \) is \( T_{1/2} = \frac{\ln(2)}{\lambda} \).
• Each half-life reduces the number of undecayed nuclei by half, a situation that occurs repeatedly over the lifetime of a substance.

In the given exercise, we analyze the scenario after \( n \) half-lives of substance \( A \). If \( n = 1 \), it means half the nuclei of \( A \) have decayed during this time period. Knowing n allows you to calculate the time elapsed and how much of \( A \) remains, critical for understanding when the rate of disintegration becomes equal to different substances.

Rate of Disintegration

The rate of disintegration, also known as the activity, is the number of decays per unit time in a sample of radioactive material. It is directly related to the decay constant and the number of undecayed nuclei.

• The formula is \( R(t) = \lambda \cdot N(t) \).
• This means the activity changes as the number of radioactive nuclei decreases over time.

Our exercise involves calculating when the rate of disintegration of substances \( A \) and \( B \) are the same. Initially, \( A \) and \( B \) have equal numbers of nuclei. The problem states that after \( n \) half-lives of \( A \), they achieve an equal rate of disintegration. By applying equations with the given decay constants, you can derive the necessary time \( n \) for which the rates equate.