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 determine the half-life of the radioactive element, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Initial and Final Count Rates**:
- Initial count rate (I₀) = 1024 counts/min
- Final count rate (I) = 128 counts/min

2. **Calculate the Ratio of Final to Initial Count Rates**:
\[ \frac{I}{I₀} = \frac{128}{1024} = \frac{1}{8} \]

3. **Understand the Decay Process**:
- The ratio \(\frac{I}{I₀} = \frac{1}{8}\) indicates that the count rate has reduced to \( \frac{1}{8} \) of its original value.

4. **Determine the Number of Half-Lives**:
- We know that after each half-life, the remaining quantity of the substance is halved.
- To go from \(I₀\) to \(I\), we can express this in terms of half-lives:
- After 1 half-life: \(I₀ \to \frac{I₀}{2}\)
- After 2 half-lives: \(I₀ \to \frac{I₀}{4}\)
- After 3 half-lives: \(I₀ \to \frac{I₀}{8}\)
- Therefore, it takes 3 half-lives to reduce the count rate from 1024 to 128.

5. **Calculate the Total Time for the Decay**:
- The total time given for the decay is 3 minutes.
- Since this time corresponds to 3 half-lives, we can find the half-life (T₁/₂):
\[ T_{1/2} = \frac{\text{Total Time}}{\text{Number of Half-Lives}} = \frac{3 \text{ minutes}}{3} = 1 \text{ minute} \]

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

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