One microgram of radioactie sodium `Na^(24)` with a half life of 15 h was injected fin to a living system for a bio assay .How long will iot take for the radioactivity to fall to 25% of the initial value?

Options

A. 60 h
B. 22.5 h
C. 375 h
D. 30h

Explanation

To solve the problem of how long it will take for the radioactivity of sodium-24 (Na-24) to fall to 25% of its initial value, we can follow these steps:

### Step 1: Understand the Problem
We know that the half-life of Na-24 is 15 hours. We need to find the time it takes for the radioactivity to reduce to 25% of its initial value.

### Step 2: Identify Initial and Final Values
- Initial amount of Na-24 = 1 microgram
- Final amount after decay = 25% of initial = 0.25 microgram

### Step 3: Use the Concept of Half-Life
The half-life is the time required for a quantity to reduce to half its initial value. After one half-life (15 hours), the amount of Na-24 will be:
- After 15 hours: 0.5 microgram (50% of initial)
- After another half-life (30 hours total): 0.25 microgram (25% of initial)

### Step 4: Calculate the Total Time
Since we need to go from 1 microgram to 0.25 microgram, we can see that this requires two half-lives:
- First half-life (15 hours): 1 microgram → 0.5 microgram
- Second half-life (15 hours): 0.5 microgram → 0.25 microgram

Thus, the total time required is:
\[ \text{Total time} = 15 \text{ hours} + 15 \text{ hours} = 30 \text{ hours} \]

### Conclusion
It will take **30 hours** for the radioactivity of Na-24 to fall to 25% of its initial value.

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