The half life of radium (226) is 1620 years.
The time takend to convert `10 g` of radium to `1.25 g` is
a. 810 years b. 1620 years
c. 3240 years d. 4860 years

Options

A. a. 810 years
B. b. 1620 years
C. c. 3240 years
D. d. 4860 years

Explanation

To solve the problem of determining the time taken to convert 10 g of radium (Ra-226) to 1.25 g, we can use the concept of half-life and the formula for radioactive decay.

### Step-by-Step Solution:

1. **Understand the Half-Life Concept**: The half-life of a radioactive substance is the time required for half of the substance to decay. For radium-226, the half-life is given as 1620 years.

2. **Determine the Initial and Final Amounts**:
- Initial amount of radium (N₀) = 10 g
- Final amount of radium (N) = 1.25 g

3. **Calculate the Reduction Factor**:
To find out how many half-lives it takes to go from 10 g to 1.25 g, we need to determine how many times the initial amount has been halved.
- 10 g → 5 g (after 1 half-life)
- 5 g → 2.5 g (after 2 half-lives)
- 2.5 g → 1.25 g (after 3 half-lives)

Thus, the amount of radium has reduced by a factor of \( \frac{10}{1.25} = 8 \), which is equivalent to \( 2^3 \). This means it takes 3 half-lives to reach from 10 g to 1.25 g.

4. **Calculate the Total Time**:
Since we have determined that it takes 3 half-lives to reduce the amount of radium from 10 g to 1.25 g, we can calculate the total time taken.
- Total time = Number of half-lives × Half-life duration
- Total time = 3 × 1620 years = 4860 years

5. **Final Answer**:
Therefore, the time taken to convert 10 g of radium to 1.25 g is **4860 years**.

### Answer:
d. 4860 years

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