Question

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?

• A.
• B.
• C.
• D.

Hint:

The decay of a radioactive material follows an exponential decay law, and the mass of the decay products can be determined using the decay constants for each material.

Answer 1

The Correct Option is B

For a sequential decay process where material P decays into Q, and subsequently Q decays into R, the time-dependent masses of P, Q, and R are determined as follows:
1. Decay of P to Q: With an initial mass \( m_0 \) for P, its mass at time \( t \) follows exponential decay: \[ m_P(t) = m_0 e^{-\lambda_1 t} \] where \( \lambda_1 \) is the decay constant for P. This equation models the diminishing mass of P over time. 
2. Decay of Q to R: The mass of Q at time \( t \) is the cumulative mass produced from P's decay minus the mass that has subsequently decayed into R. 
The mass of Q at any given time is expressed 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 quantifies the evolution of Q's mass during its transformation from P to R. 
3. Mass of R: The mass of R at time \( t \) represents the total accumulated mass from the decay of P and Q. It can be calculated as the initial mass minus the masses of P and Q: \[ m_R(t) = m_0 - m_P(t) - m_Q(t) \] Substituting the expressions for \( m_P(t) \) and \( m_Q(t) \) yields: \[ m_R(t) = m_0 \left(1 - e^{-\lambda_1 t} - \frac{\lambda_1}{\lambda_2 - \lambda_1} \left(e^{-\lambda_1 t} - e^{-\lambda_2 t}\right)\right) \] This equation illustrates the increase in R's mass as P and Q decay. 
The appropriate representation of the time-dependent masses of P, Q, and R corresponds to (2), characterized by: 
- The mass of P exhibiting an exponential decrease. 
- The mass of Q initially rising and then falling as it transforms into R. 
- The mass of R demonstrating a steady increase due to the decay of Q.