Question

If \( N_0 \) be the number of nuclei present at time \( t = 0 \), then the number of undecayed nuclei, \( N \), present after \( n \) mean life is

• A. \( N = \left( \frac{1}{2} \right)^n N_0 \)
• B. \( N = \left( \frac{1}{2} \right)^{1/n} N_0 \)
• C. \( N = \left( \frac{1}{4} \right)^n N_0 \)
• D. \( N = \left( \frac{1}{4} \right)^{1/n} N_0 \)

Hint:

Radioactive decay follows an exponential law, and after each mean life, half of the remaining nuclei decay.

Answer 1

The Correct Option is A

The number of undecayed nuclei \( N \) after a time \( n \) mean lives is given by the decay law: \[ N = N_0 e^{- \lambda t} \] where \( \lambda \) is the decay constant and \( t = n \times \tau \) is the time, with \( \tau \) being the mean life. In terms of the fraction of undecayed nuclei, we can express the number of remaining nuclei after \( n \) mean lives as: \[ N = N_0 \left( \frac{1}{2} \right)^n \]
Thus, the correct answer is (a).