A radioactive nucleus n₂ has 3 times the decay constant as compared to the decay constant of another radioactive nucleus n₁. If initial number of both nuclei are the same, what is the ratio of number of nuclei of n₂ to the number of nuclei of n₁, after one half-life of n₁ ?

Solution

Physics sir · 3 steps

One half-life of n₁ is three half-lives of n₂, because n₂ decays three times as fast.

1. Compare half-lives. Half-life is T = (ln 2)/λ. Since λ₂ = 3λ₁, the half-life of n₂ is one-third of the half-life of n₁. So in the time T₁, nucleus n₂ goes through 3 half-lives.
2. Count what remains. After one half-life, n₁ keeps 1/2 of N₀. After three half-lives, n₂ keeps (1/2)³ = 1/8 of N₀.

   N₂/N₁ = (N₀/8)/(N₀/2) = 1/4

3. So. The ratio of n₂ nuclei to n₁ nuclei is 1/4, option (a).

The idea to keep. Turn the time into a number of half-lives for each nucleus separately. Then the fraction left is (1/2)ⁿ, and the ratio follows at once.

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