A radioactive material P first decays into Q and then Q decays to non-radioactive material R. Which of the following figure represents time dependent mass of P, Q and R?
Show Hint
The Correct Option is B
Approach Solution - 1
Given that material P decays into Q, and then Q decays into R, we need to determine the time-dependent mass of P, Q, and R.
1. Decay of P to Q: Let the initial mass of P be \( m_0 \). The mass of P at time \( t \) is given by the exponential decay equation: \[ m_P(t) = m_0 e^{-\lambda_1 t} \] where \( \lambda_1 \) is the decay constant for P. This describes how P decreases over time.
2. Decay of Q to R: The mass of Q at time \( t \) is determined by the difference between the mass of P that has decayed and the mass of Q that has decayed into R.
The mass of Q at any given time is given by: \[ m_Q(t) = \frac{\lambda_1}{\lambda_2 - \lambda_1} m_0 \left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right) \] where \( \lambda_2 \) is the decay constant for Q.
This equation represents how Q evolves as it decays from P to R.
3. Mass of R: The mass of R at time \( t \) is the remaining mass, which can be expressed as the sum of the mass lost from P and Q: \[ m_R(t) = m_0 - m_P(t) - m_Q(t) \] Substituting the expressions for \( m_P(t) \) and \( m_Q(t) \): \[ m_R(t) = m_0 \left(1 - e^{-\lambda_1 t} - \frac{\lambda_1}{\lambda_2 - \lambda_1} \left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right)\right) \] This shows how R increases as both P and Q decay.
The correct figure representing the time-dependent mass of P, Q, and R is (2), where:
- The mass of P decreases exponentially.
- The mass of Q initially increases and then decreases as it decays into R.
- The mass of R increases as Q decays.
Approach Solution -2
Step 1: Understanding the problem.
We are given a radioactive material \( P \), which first decays into material \( Q \), and then \( Q \) decays into a non-radioactive material \( R \). We need to determine the correct graph that represents the time-dependent masses of \( P \), \( Q \), and \( R \).
Step 2: Analyze the decay process.
- The mass of \( P \) decreases over time as it decays into \( Q \). Therefore, the graph for \( P \) should show a decreasing function over time.
The mass of \( Q \) increases as \( P \) decays into \( Q \), then decreases as \( Q \) decays into \( R \). Therefore, the graph for \( Q \) should show an initial increase followed by a decrease.
The mass of \( R \) increases over time as \( Q \) decays into \( R \), so the graph for \( R \) should show a gradual increase.
Step 3: Identifying the correct graph.
Option 1: Does not show the correct decay behavior for \( P \), \( Q \), and \( R \).
Option 2: Correctly shows \( P \) decreasing, \( Q \) increasing and then decreasing, and \( R \) increasing. This matches the description of the decay process.
Option 3: Does not show the correct increase and decrease patterns for \( Q \) and \( R \).
Option 4: Shows the wrong patterns for the masses of \( P \), \( Q \), and \( R \).
Step 4: Conclusion.
The correct option is (2), as it represents the time-dependent masses of \( P \), \( Q \), and \( R \) correctly.
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