Initial ratio of active nuclei in two different sample is 2: 3 their half lives are 2 hr and 3hr respectively . Ratio of their activities at the end of 12 hr is :

Options

A. `1: 6`
B. `6 : 1`
C. ` 1 : 4`
D. ` 4 :1`

Answer

To solve the problem, we need to find the ratio of activities of two radioactive samples A and B after 12 hours, given their initial ratio of active nuclei and their half-lives.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Let the initial number of active nuclei in sample A be \( A_0 = 2x \).
- Let the initial number of active nuclei in sample B be \( B_0 = 3x \).
- Here, \( x \) is a common factor.

2. **Determine the Half-Lives:**
- The half-life of sample A, \( t_{1/2A} = 2 \) hours.
- The half-life of sample B, \( t_{1/2B} = 3 \) hours.

3. **Calculate the Number of Half-Lives in 12 Hours:**
- For sample A:
\[ n_A = \frac{T}{t_{1/2A}} = \frac{12 \text{ hours}}{2 \text{ hours}} = 6 \]
- For sample B:
\[ n_B = \frac{T}{t_{1/2B}} = \frac{12 \text{ hours}}{3 \text{ hours}} = 4 \]

4. **Calculate Remaining Nuclei After 12 Hours:**
- The remaining active nuclei in sample A after 12 hours:
\[ N_A = A_0 \left( \frac{1}{2} \right)^{n_A} = 2x \left( \frac{1}{2} \right)^{6} = 2x \cdot \frac{1}{64} = \frac{2x}{64} = \frac{x}{32} \]
- The remaining active nuclei in sample B after 12 hours:
\[ N_B = B_0 \left( \frac{1}{2} \right)^{n_B} = 3x \left( \frac{1}{2} \right)^{4} = 3x \cdot \frac{1}{16} = \frac{3x}{16} \]

5. **Calculate the Ratio of Remaining Nuclei:**
- The ratio of remaining nuclei \( \frac{N_A}{N_B} \):
\[ \frac{N_A}{N_B} = \frac{\frac{x}{32}}{\frac{3x}{16}} = \frac{x}{32} \cdot \frac{16}{3x} = \frac{16}{96} = \frac{1}{6} \]

6. **Calculate the Ratio of Activities:**
- The activity \( A \) is proportional to the number of remaining nuclei. Thus, the ratio of activities \( \frac{A_A}{A_B} \) is the same as the ratio of remaining nuclei:
\[ \frac{A_A}{A_B} = \frac{N_A}{N_B} = \frac{1}{6} \]

### Final Answer:
The ratio of their activities at the end of 12 hours is \( \frac{1}{6} \).