A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be _________ hours.

Solution

Physics sir · 3 steps

64 is 2⁶, so you just wait six half-lives and convert the half-life to hours properly.

1. Activity halves. Activity is proportional to the number of undecayed nuclei, so it halves every half-life. After n half-lives the activity is A₀/2ⁿ.

   A₀/2ⁿ = A₀/64 ⇒ 2ⁿ = 64 = 2⁶ ⇒ n = 6

2. Half-life in hours. 2 hours 30 minutes is 2.5 h. Six half-lives take 6 × 2.5 h = 15 h.
3. So. The radiation falls to the safe level after a minimum of 15 hours.

The common slip: 13.8. Writing 2 hours 30 minutes as 2.3 h gives 6 × 2.3 = 13.8, not 15. Thirty minutes is half an hour, so the half-life is 2.5 h.

The idea to keep. When the reduction factor is a power of 2, count the half-lives directly: a factor of 2ⁿ needs n half-lives. Always convert the half-life into the units the answer asks for.

Answer from the NTA final answer key. Solution by PrepWiser.