In the radioactive decay of an element it is found that the count rate reduces from 1024 to 128 in 3 minutes. Its half life will be

Options

A. 1 minute
B. 2 minutes
C. 3 minute
D. 5 minute

Answer

To find the half-life of the radioactive element based on the given decay information, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial and Final Count Rates:**
- The initial count rate (n₀) is 1024.
- The final count rate (n) after 3 minutes is 128.

2. **Determine the Number of Half-Lives:**
- We know that after each half-life, the number of radioactive nuclei is halved.
- Let's calculate how many times the count rate of 1024 has been halved to reach 128:
- 1st half-life: 1024 → 512
- 2nd half-life: 512 → 256
- 3rd half-life: 256 → 128
- Therefore, it takes 3 half-lives to go from 1024 to 128.

3. **Calculate the Total Time for the Decay:**
- The total time for this decay is given as 3 minutes.

4. **Calculate the Duration of One Half-Life:**
- Since 3 half-lives occur in 3 minutes, we can find the duration of one half-life by dividing the total time by the number of half-lives:
\[ \text{Half-life} = \frac{\text{Total Time}}{\text{Number of Half-lives}} = \frac{3 \text{ minutes}}{3} = 1 \text{ minute} \]

5. **Conclusion:**
- The half-life of the radioactive element is 1 minute.

### Final Answer:
The half-life of the radioactive nuclei is **1 minute**.

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