10g of a radioactive substance is reduced to 1.25 g after 15 days. Its 1kg mass will reduce to 500g in

Options

A. 500 days
B. 125 days
C. 25 days
D. 5 days

Explanation

To solve the problem, we need to determine how long it will take for a 1 kg (1000 g) mass of a radioactive substance to reduce to 500 g, given that a 10 g sample reduces to 1.25 g in 15 days.

### Step-by-Step Solution:

1. **Identify the initial and final amounts for the first scenario:**
- Initial amount = 10 g
- Final amount = 1.25 g
- Time taken = 15 days

2. **Determine the number of half-lives that have passed:**
- The substance reduces from 10 g to 1.25 g.
- The sequence of reductions is:
- 10 g → 5 g (1 half-life)
- 5 g → 2.5 g (2 half-lives)
- 2.5 g → 1.25 g (3 half-lives)
- Therefore, it takes 3 half-lives to go from 10 g to 1.25 g.

3. **Calculate the half-life (T_half):**
- Since 3 half-lives correspond to 15 days:
\[ 3 \times T_{half} = 15 \text{ days} \]
- To find T_half:
\[ T_{half} = \frac{15 \text{ days}}{3} = 5 \text{ days} \]

4. **Apply the half-life to the second scenario:**
- Now, we need to find out how long it takes for 1 kg (1000 g) to reduce to 500 g.
- Since 500 g is half of 1000 g, it will take exactly 1 half-life to reach this amount.

5. **Determine the time for the second scenario:**
- Since T_half is 5 days, the time taken for 1000 g to reduce to 500 g is:
\[ \text{Time} = T_{half} = 5 \text{ days} \]

### Final Answer:
The time it will take for 1 kg of the radioactive substance to reduce to 500 g is **5 days**.

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