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 material, we can follow these steps:

### Step 1: Identify the initial and final masses
- The initial mass (I) of the radioactive material is 2 mg.
- The final mass (F) after 3 hours is 0.25 mg.

### Step 2: Calculate the fraction of the remaining mass
- The fraction of the remaining mass can be calculated as:
\[ \frac{F}{I} = \frac{0.25 \text{ mg}}{2 \text{ mg}} = \frac{1}{8} \]

### Step 3: Understand the relationship of half-lives
- The fraction \(\frac{1}{8}\) indicates how many half-lives have passed.
- Each half-life reduces the mass to half of its previous value:
- After 1 half-life: \(\frac{I}{2} = \frac{2 \text{ mg}}{2} = 1 \text{ mg}\)
- After 2 half-lives: \(\frac{I}{4} = \frac{2 \text{ mg}}{4} = 0.5 \text{ mg}\)
- After 3 half-lives: \(\frac{I}{8} = \frac{2 \text{ mg}}{8} = 0.25 \text{ mg}\)

### Step 4: Relate the number of half-lives to the total time
- We know that it took 3 hours for the mass to reduce to \(\frac{1}{8}\).
- Since this corresponds to 3 half-lives, we can find the duration of one half-life:
\[ \text{Total time} = \text{Number of half-lives} \times \text{Half-life duration} \]
\[ 3 \text{ hours} = 3 \times \text{Half-life duration} \]
\[ \text{Half-life duration} = \frac{3 \text{ hours}}{3} = 1 \text{ hour} \]

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

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