The half-life of a radioactive substance is 3h and its activity is `1mu Ci`. Then the activity after 9h will be (in `mu Ci`)-

Options

A. `1/9`
B. `1/27`
C. `1/3`
D. `1/8`

Explanation

To solve the problem, we need to determine the activity of a radioactive substance after a certain period of time, given its half-life and initial activity.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Half-life (T_half) = 3 hours
- Initial activity (A_0) = 1 µCi
- Time elapsed (t) = 9 hours

2. **Calculate the Number of Half-Lives:**
- The number of half-lives (n) that have passed in 9 hours can be calculated as:
\[ n = \frac{t}{T_{half}} = \frac{9 \text{ hours}}{3 \text{ hours}} = 3 \]

3. **Determine the Remaining Activity:**
- The activity of a radioactive substance after n half-lives is given by the formula:
\[ A = A_0 \left( \frac{1}{2} \right)^n \]
- Substituting the values we have:
\[ A = 1 \text{ µCi} \left( \frac{1}{2} \right)^3 \]

4. **Calculate the Value:**
- Calculate \( \left( \frac{1}{2} \right)^3 \):
\[ \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]
- Now substitute this back into the activity formula:
\[ A = 1 \text{ µCi} \times \frac{1}{8} = \frac{1}{8} \text{ µCi} \]

5. **Final Answer:**
- Therefore, the activity after 9 hours will be:
\[ A = 0.125 \text{ µCi} \]

### Summary of the Steps:
- Identify the half-life and initial activity.
- Calculate the number of half-lives that have passed.
- Use the formula for remaining activity after n half-lives.
- Perform the calculations to find the final activity.