After 3 hours, only 0.25 mg of a pure radioactive material is left, if initial mass was 2 mg than the half life of the substance is

Options

A. 1.5 hr
B. 1 hr
C. 0.5 hr
D. 2 hr

Answer

To find the half-life of the radioactive substance, we can follow these steps:

### Step 1: Understand the problem
We know that the initial mass of the radioactive material is 2 mg, and after 3 hours, only 0.25 mg is left. We need to find the half-life of the substance.

### Step 2: Determine the fraction of the substance remaining
The initial mass is 2 mg, and the remaining mass after 3 hours is 0.25 mg. We can express the remaining mass as a fraction of the initial mass:
\[ \text{Fraction remaining} = \frac{\text{Remaining mass}}{\text{Initial mass}} = \frac{0.25 \text{ mg}}{2 \text{ mg}} = \frac{0.25}{2} = \frac{1}{8} \]

### Step 3: Relate the fraction remaining to half-lives
The fraction remaining after \( n \) half-lives can be expressed as:
\[ \frac{1}{2^n} \]
We found that the fraction remaining is \( \frac{1}{8} \), which can be written as:
\[ \frac{1}{8} = \frac{1}{2^3} \]
This means that \( n = 3 \). Therefore, 3 half-lives have passed in the 3 hours.

### Step 4: Calculate the half-life
Since 3 half-lives correspond to 3 hours, we can find the duration of one half-life:
\[ \text{Half-life} (t_{1/2}) = \frac{\text{Total time}}{n} = \frac{3 \text{ hours}}{3} = 1 \text{ hour} \]

### Conclusion
The half-life of the radioactive substance is 1 hour.