Question

The half-life period of a radioactive element is 2 days. If \( \frac{1}{32} \) part of the initial amount remains undecayed after a time \( t \), then the value of \( t \) in days is

• A. 8
• B. 10
• C. 6
• D. 12
• E. 4

Hint:

The number of half-lives can be determined by the fraction of the substance remaining. Then, multiply the number of half-lives by the half-life duration to find the time.

Answer 1

The Correct Option is B

The amount of substance remaining after time \( t \) is given by the equation: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T}} \] where: - \( N(t) \) is the remaining quantity after time \( t \), - \( N_0 \) is the initial quantity, - \( T \) is the half-life of the substance. 
We are told that \( \frac{1}{32} \) of the substance remains undecayed, so: \[ \frac{N(t)}{N_0} = \frac{1}{32} \] This means the substance has undergone 5 half-lives because: \[ \frac{1}{32} = \left(\frac{1}{2}\right)^5 \] Therefore, the time \( t \) is given by: \[ t = 5 \times T = 5 \times 2 = 10 \, {days} \] Hence, the correct answer is (B).