If `3//4` quantity of a radioactive substance disintegrates in 2 hours, its half`-` life period will be

Options

A. `15 mi n`
B. `30 mi n`
C. `60 mi n`
D. `90 mi n`

Explanation

To find the half-life period of a radioactive substance given that \( \frac{3}{4} \) of it disintegrates in 2 hours, we can follow these steps:

### Step 1: Understand the Disintegration
When \( \frac{3}{4} \) of the substance disintegrates, it means that \( \frac{1}{4} \) of the substance remains. In percentage terms, this is equivalent to 25% remaining.

### Step 2: Relate Remaining Quantity to Half-Life
The half-life (\( T_{1/2} \)) is the time required for half of the substance to disintegrate. We need to determine how many half-lives it takes for the substance to reduce from 100% to 25%.

- From 100% to 50%: This takes 1 half-life.
- From 50% to 25%: This takes another half-life.

Thus, it takes 2 half-lives to go from 100% to 25%.

### Step 3: Calculate the Half-Life
Since we know that it takes 2 hours for the substance to decrease to 25%, and this corresponds to 2 half-lives, we can express this mathematically:

\[ \text{Time for 2 half-lives} = 2 \text{ hours} \]

Let \( T_{1/2} \) be the half-life. Then:

\[ 2 \cdot T_{1/2} = 2 \text{ hours} \]

### Step 4: Solve for Half-Life
To find \( T_{1/2} \), we divide both sides of the equation by 2:

\[ T_{1/2} = \frac{2 \text{ hours}}{2} = 1 \text{ hour} \]

### Final Answer
The half-life period of the radioactive substance is **1 hour**.

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