Question

For a radioactive material, half-life is $10$ minutes. If initially there are $600$ number of nuclei, the time taken (in minutes) for. the disintegration of $450$ nuclei is

• A. 15
• B. 20
• C. 30
• D. 10

Answer 1

The Correct Option is B

To determine the time taken for the disintegration of 450 nuclei from an initial 600 nuclei, we apply the concept of radioactive decay. The half-life of the material is given as 10 minutes.

Let's solve the problem step-by-step:

1. The half-life (\( T_{1/2} \)) is the time required for half of the radioactive nuclei to decay. For this material, \( T_{1/2} = 10 \) minutes.
2. The formula for radioactive decay is given by: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] where \( N(t) \) is the number of remaining nuclei at time \( t \), and \( N_0 \) is the initial number of nuclei.
3. In this case, initially, \( N_0 = 600 \) nuclei. We need to determine the time \( t \) when 450 nuclei have disintegrated. Hence, the remaining nuclei \( N(t) = 600 - 450 = 150 \).
4. Substitute the values into the decay formula: \[ 150 = 600 \left( \frac{1}{2} \right)^{\frac{t}{10}} \]
5. Solving for \( t \): \[ \frac{150}{600} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \] \[ \frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]
6. We know that \( \frac{1}{4} = \left( \frac{1}{2} \right)^2 \). Therefore, \[ \left( \frac{1}{2} \right)^2 = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]
7. This implies that: \[ \frac{t}{10} = 2 \] Therefore, \( t = 2 \times 10 = 20 \) minutes.

Thus, the time taken for the disintegration of 450 nuclei is 20 minutes.