What is the half-life of a radioactive substance if 75% of a given amount of the substance disintegrates in 30 minutes

Options

A. 7.5 minutes
B. 25 minutes
C. 20 minutes
D. 15 minutes

Explanation

To find the half-life of a radioactive substance when 75% of it disintegrates in 30 minutes, we can follow these steps:

### Step 1: Understand the Disintegration
If 75% of the substance has disintegrated, that means 25% of the original amount remains.

### Step 2: Relate Remaining Amount to Half-Lives
In terms of half-lives:
- After 1 half-life, 50% of the substance remains.
- After 2 half-lives, 25% of the substance remains.

Thus, if 75% has disintegrated, it corresponds to 2 half-lives.

### Step 3: Set Up the Equation
Since 2 half-lives correspond to 30 minutes, we can express this mathematically:
\[ 2 \times t_{1/2} = 30 \text{ minutes} \]

### Step 4: Solve for Half-Life
To find the half-life (\(t_{1/2}\)), we divide both sides of the equation by 2:
\[ t_{1/2} = \frac{30 \text{ minutes}}{2} = 15 \text{ minutes} \]

### Conclusion
The half-life of the radioactive substance is **15 minutes**.

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