Question

A radioactive element of mass \(1\ \text{kg}\) after \(N\) years is left with only \(125\ \text{g}\). If the half-life of the element is \(12.5\) years, then the value of \(N\) is

• A. \(37.5\ \text{years}\)
• B. \(25.0\ \text{years}\)
• C. \(50.0\ \text{years}\)
• D. \(75.0\ \text{years}\)

Hint:

After \(n\) half-lives, the remaining mass is \[ m=m_0\left(\frac{1}{2}\right)^n. \] If the remaining fraction is \(\frac{1}{8}\), then \[ \frac{1}{8}=\left(\frac{1}{2}\right)^3, \] so \(3\) half-lives have passed.

Answer 1

The Correct Option is A

Step 1: Find number of half-lives.
Starting mass \( = 1\text{ kg} \), remaining \( = 125\text{ g} = \frac{1}{8}\text{ kg} \).
\( \left(\frac{1}{2}\right)^n = \frac{1}{8} \Rightarrow n = 3 \) half-lives.

Step 2: Calculate total time.
\( N = n\times T_{1/2} = 3\times12.5 = 37.5\text{ years} \)

\[ \boxed{N = 37.5\text{ years}} \]