Question

The half-life of the isotope \(_{11}\text{Na}^{24}\) is 15 hr. How much time does it take for \((\frac{7}{8})^{\text{th}}\) of a sample of this isotope to decay ?

• A. \(45\) hour
• B. \(60\) hour
• C. \(75\) hour
• D. \(90\) hour

Hint:

If \(\frac78\) decays, \(\frac18\) remains, which takes three half-lives.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
After each half-life, half of the remaining nuclei are left: \(N=N_0\left(\tfrac12\right)^n\).

Step 2: Key Formula or Approach
\(\dfrac78\) decayed means \(\dfrac N{N_0}=\dfrac18=\left(\dfrac12\right)^3\), so \(n=3\).

Step 3: Detailed Explanation
\[ t=n\,t_{1/2}=3\times15=45\ \text{hours} \]

Final Answer:
The time required is 45 hours, option (A). \[ \boxed{45\ \text{hours (A)}} \]