Question

If N is the initial number of atoms in a radioactive sample, then the number of atoms decayed after the five half life periods is

• A. \(\frac{31}{32} N\)
• B. \(\frac{15}{16} N\)
• C. \(\frac{7}{8} N\)
• D. \(\frac{N}{32}\)
• E. \(\frac{N}{16}\)

Hint:

Fraction decayed = \(1 - (1/2)^n\). For n=5, it's \(1 - 1/32 = 31/32\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
After \(n\) half-lives, the remaining fraction is \((\frac{1}{2})^n\). Atoms decayed = Initial - Remaining.

Step 2: Detailed Explanation:
After \(n=5\) half-lives, remaining = \(N \times (\frac{1}{2})^5 = \frac{N}{32}\).
Atoms decayed = \(N - \frac{N}{32} = N(1 - \frac{1}{32}) = \frac{31}{32}N\).

Step 3: Final Answer:
The number of atoms decayed is \(\frac{31}{32} N\).