A radioactive sample `S_1` having an activity of `5muCi` has twice the number of nuclei as another sample `S_2` which has an activity of `10muCi`. The half-lives of `S_1` and `S_2` can be

Options

A. (a) 20 yr and 5 yr, respectively
B. (b) 20 yr and 10 yr, respectively
C. (c) 10 yr each
D. (d) 5 yr each

Explanation

To solve the problem, we need to analyze the relationship between the activities, the number of nuclei, and the half-lives of the two radioactive samples \( S_1 \) and \( S_2 \).

### Step-by-Step Solution:

1. **Understanding Activity**:
The activity \( A \) of a radioactive sample is given by the formula:
\[ A = \lambda N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive nuclei.

2. **Given Data**:
- Activity of sample \( S_1 \): \( A_1 = 5 \, \mu Ci \)
- Activity of sample \( S_2 \): \( A_2 = 10 \, \mu Ci \)
- Number of nuclei in sample \( S_1 \) is twice that of sample \( S_2 \): \( N_1 = 2N_2 \)

3. **Setting Up the Equations**:
From the activity formula, we can write:
\[ A_1 = \lambda_1 N_1 \quad \text{(1)} \]
\[ A_2 = \lambda_2 N_2 \quad \text{(2)} \]

4. **Substituting \( N_1 \)**:
Substitute \( N_1 = 2N_2 \) into equation (1):
\[ A_1 = \lambda_1 (2N_2) \]
This simplifies to:
\[ 5 = 2\lambda_1 N_2 \quad \text{(3)} \]

5. **Using Equation (2)**:
From equation (2), we have:
\[ 10 = \lambda_2 N_2 \quad \text{(4)} \]

6. **Finding the Ratio of Decay Constants**:
Now, we can express \( N_2 \) from equation (4):
\[ N_2 = \frac{10}{\lambda_2} \]
Substitute \( N_2 \) back into equation (3):
\[ 5 = 2\lambda_1 \left(\frac{10}{\lambda_2}\right) \]
Simplifying gives:
\[ 5 = \frac{20\lambda_1}{\lambda_2} \]
Rearranging leads to:
\[ \frac{\lambda_1}{\lambda_2} = \frac{5}{20} = \frac{1}{4} \]

7. **Relating Half-Lives**:
The half-life \( T_{1/2} \) is related to the decay constant \( \lambda \) by:
\[ T_{1/2} = \frac{\ln 2}{\lambda} \]
Thus, we can write the ratio of half-lives:
\[ \frac{T_{1/2,1}}{T_{1/2,2}} = \frac{\lambda_2}{\lambda_1} \]
Since \( \frac{\lambda_1}{\lambda_2} = \frac{1}{4} \), we have:
\[ \frac{T_{1/2,1}}{T_{1/2,2}} = 4 \]
This means:
\[ T_{1/2,1} = 4 T_{1/2,2} \]

8. **Conclusion**:
If we assume \( T_{1/2,2} = x \), then \( T_{1/2,1} = 4x \). If we check the options provided (not given here), we can find the correct half-lives.