Half-life of radioactive sample, when activity of material initially was 8 counts and after 3 hours it becomes 1 count is

Options

A. 2 h
B. 1 h
C. 3 h
D. 4 h

Answer

To solve the problem, we need to find the half-life of a radioactive sample given its initial and final activity over a specific time period.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Initial activity, \( n_0 = 8 \) counts
- Final activity after 3 hours, \( n_t = 1 \) count
- Time elapsed, \( t = 3 \) hours

2. **Use the Decay Formula:**
The relationship between the initial activity, final activity, and the number of half-lives is given by the formula:
\[ n_t = n_0 \left(\frac{1}{2}\right)^n \]
where \( n \) is the number of half-lives.

3. **Set Up the Equation:**
Plugging in the values we have:
\[ 1 = 8 \left(\frac{1}{2}\right)^n \]

4. **Solve for \( n \):**
Rearranging the equation gives:
\[ \left(\frac{1}{2}\right)^n = \frac{1}{8} \]
We know that \( \frac{1}{8} = \left(\frac{1}{2}\right)^3 \), so:
\[ n = 3 \]

5. **Relate Number of Half-Lives to Time:**
The total time elapsed is equal to the number of half-lives multiplied by the half-life period:
\[ t = n \cdot t_{1/2} \]
Substituting the known values:
\[ 3 \text{ hours} = 3 \cdot t_{1/2} \]

6. **Solve for Half-Life \( t_{1/2} \):**
Dividing both sides by 3 gives:
\[ t_{1/2} = 1 \text{ hour} \]

### Final Answer:
The half-life of the radioactive sample is **1 hour**.

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