Question

A radioactive sample of half-life \( T \) releases an energy of 8 MeV per disintegration. If the initial number of atoms is \( 16 \times 10^{20} \), the total energy released in a time of \( 3T \) is:

• A. \( 17.92 \times 10^8 \, {J} \)
• B. \( 8.96 \times 10^8 \, {J} \)
• C. \( 35.84 \times 10^8 \, {J} \)
• D. \( 4.48 \times 10^8 \, {J} \)

Hint:

In radioactive decay problems, remember that the number of disintegrations is the initial minus the remaining number of nuclei. The total energy is then calculated by multiplying this number by the energy per disintegration.

Answer 1

The Correct Option is A

Step 1: Calculate the number of disintegrations in \( 3T \).

Using the formula for radioactive decay, the remaining atoms after \( 3T \) are:

\[ N = N_0 \left( \frac{1}{2} \right)^3 = 16 \times 10^{20} \times \frac{1}{8} = 2 \times 10^{20} \]

Total disintegrations = \( N_0 - N = 14 \times 10^{20} \)

Total energy released = \( 14 \times 10^{20} \times 8 \, \text{MeV} \times 1.6 \times 10^{-13} \, \text{J/MeV} = 17.92 \times 10^8 \, \text{J} \).