Radioactive Decay

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 . The mass of P at time is given by the exponential decay equation: where 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 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: where 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 is the remaining mass, which can be expressed as the sum of the mass lost from P and Q: Substituting the expressions for and : 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.

To solve this problem, we need to understand the relationship between the decay constant and the number of nuclei remaining after a certain time period. The decay of a radioactive nucleus is described by the equation:

where:

• is the number of nuclei remaining at time .
• is the initial number of nuclei.
• is the decay constant.
• is the time elapsed.

Given that the decay constant of nucleus is 3 times that of nucleus , we can write:

The half-life of a radioactive sample is related to its decay constant by:

Since we are considering one half-life of nucleus , we have:

After one half-life, the number of nuclei remaining is half of the initial number:

For nucleus , which has been decaying for the same time period:

Substituting and , we have:

Hence, the ratio of the number of nuclei of to the number of nuclei of after one half-life of is:

Therefore, the correct answer is .