A block having 12 g of an element is placed in a room. This element is a radioactive element with a half-life of 15 years. After how many years will there be just 1.5 g of the element in the box ?

Options

A. 40 year
B. 45 year
C. 20 year
D. 15 year

Answer

To solve the problem of how many years it will take for a block of 12 g of a radioactive element with a half-life of 15 years to decay to 1.5 g, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the concept of half-life**: The half-life of a radioactive substance is the time required for half of the substance to decay. In this case, the half-life is given as 15 years.

2. **Determine the initial and final amounts**:
- Initial amount (A₀) = 12 g
- Final amount (A) = 1.5 g

3. **Calculate how many times the substance has halved**:
We need to find out how many half-lives it takes for 12 g to reduce to 1.5 g.
\[ A = \frac{A₀}{2^n} \]
Rearranging gives:
\[ 1.5 = \frac{12}{2^n} \]
To find n, we can rewrite this as:
\[ 2^n = \frac{12}{1.5} \]
Simplifying the right side:
\[ 2^n = 8 \]
Since \(8 = 2^3\), we find:
\[ n = 3 \]

4. **Calculate the total time elapsed**:
Since each half-life is 15 years, the total time (t) for 3 half-lives is:
\[ t = n \times \text{half-life} = 3 \times 15 \text{ years} = 45 \text{ years} \]

### Final Answer:
After 45 years, there will be just 1.5 g of the element left in the box.

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