Question

1600 atoms of a radioactive substance with a half life period \(T\) start decaying at \(t=0\). The time during which 1550 atoms would have decayed is

• A. \(t = 3T\)
• B. \(t = 6T\)
• C. \(t = 2T\)
• D. \(t = 5T\)

Hint:

Instead of formulas, just keep halving the number: \(1600 \to 800 \to 400 \to 200 \to 100 \to 50\). It took 5 steps, so 5 half-lives.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Radioactive decay follows an exponential law. After one half-life, the number of undecayed atoms reduces to half the original number.

Step 2: Key Formula or Approach:
\(N = N_0 \left( \frac{1}{2} \right)^n\), where \(n\) is the number of half-lives.
Undecayed atoms \(N = N_0 - \text{Decayed atoms}\).

Step 3: Detailed Explanation:
Given:
Initial atoms \(N_0 = 1600\)
Decayed atoms = 1550
Undecayed atoms \(N = 1600 - 1550 = 50\)
Now use the formula:
\[ 50 = 1600 \left( \frac{1}{2} \right)^n \]
\[ \frac{50}{1600} = \left( \frac{1}{2} \right)^n \implies \frac{1}{32} = \left( \frac{1}{2} \right)^n \]
Since \(32 = 2^5\), we have:
\[ \left( \frac{1}{2} \right)^5 = \left( \frac{1}{2} \right)^n \implies n = 5 \]
Total time \(t = nT = 5T\).

Step 4: Final Answer:
The time during which 1550 atoms decay is 5T.