Question

Initially \(8\times 10^{10}\) atoms are present in a radioactive sample. The half-life of the sample is 1 hour. The number of atoms that will decay in the time interval \(t=2\) hour to \(t=5\) hour, is

• A. \(0.25\times 10^{10}\)
• B. \(1.5\times 10^{10}\)
• C. \(1.75\times 10^{10}\)
• D. \(2\times 10^{10}\)

Hint:

Find the number of undecayed atoms at 2 hours and at 5 hours using the half-life, then subtract.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The number of atoms left after time \(t\) is \(N = N_0\left(\dfrac{1}{2}\right)^{t/T}\), where \(T\) is the half-life.

Step 2: Atoms at 2 hours
\(N_2 = 8\times10^{10} \times \left(\dfrac{1}{2}\right)^2 = 2\times10^{10}\)

Step 3: Atoms at 5 hours
\(N_5 = 8\times10^{10} \times \left(\dfrac{1}{2}\right)^5 = 0.25\times10^{10}\)

Step 4: Atoms Decayed
\(N_2 - N_5 = (2 - 0.25)\times10^{10} = 1.75\times10^{10}\)

Final Answer:
The number of atoms decayed in the interval is \(1.75\times10^{10}\). \[ \boxed{\text{(C) }1.75\times10^{10}} \]