What is the half-life period of a radioactive material if its activity drops to 1 / 16ᵗʰ of its initial value in 30 years?

Solution

Physics sir · 3 steps

Turn 1/16 into a power of one half: that tells you how many half-lives fit into 30 years.

1. Activity halves too. Activity is proportional to the number of undecayed nuclei, so it halves every half-life. After n half-lives it is (1/2)ⁿ of its starting value.

   A/A₀ = (1/2)ⁿ = 1/16 = (1/2)⁴

2. Count half-lives. So n = 4: the activity goes 1 → 1/2 → 1/4 → 1/8 → 1/16. Four half-lives fit into 30 years.
3. So. T_(1/2) = 30/4 = 7.5 years, which is option (c).

The idea to keep. If a quantity falls to (1/2)ⁿ of its start, exactly n half-lives have passed. Divide the total time by n to get the half-life.

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