Question

Nucleus of a radioactive material can undergo beta decay with half life of 4 minutes. Suppose beta decay starts with 4096 nuclei at \( t = 0 \), the number of nuclei left after 20 minutes would be

• A. 1024
• B. 128
• C. 512
• D. 256

Hint:

In radioactive decay, the number of remaining nuclei is halved after each half-life. Use this to calculate the remaining nuclei at any time.

Answer 1

The Correct Option is B

Step 1: Understanding the decay process. 
The number of nuclei left after a certain time in a radioactive decay process is given by the formula: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T}} \] where: - \( N(t) \) is the number of nuclei remaining after time \( t \), - \( N_0 \) is the initial number of nuclei, - \( T \) is the half-life of the substance. 
 

Step 2: Substituting the values. 
Here, \( N_0 = 4096 \), \( T = 4 \) minutes, and \( t = 20 \) minutes. Substituting into the formula: \[ N(20) = 4096 \left( \frac{1}{2} \right)^{\frac{20}{4}} = 4096 \left( \frac{1}{2} \right)^5 = 4096 \times \frac{1}{32} = 128 \]

Step 3: Conclusion. 
The correct answer is (B) 128.