Two radioactive elements A and B initially have same number of atoms. The half life of A is same as the average life of B. If λ_A and λ_B are decay constants of A and B respectively, then choose the correct relation from the given options.

Solution

Physics sir · 4 steps

Write half-life and mean life in terms of λ, set them equal, and the relation drops out in one line.

1. Half-life of A. The half-life is the time for half the atoms to decay, which gives T_(1/2) = (ln 2)/λ_A.
2. Mean life of B. The average (mean) life is the reciprocal of the decay constant, so τ_B = 1/λ_B.

   (ln 2)/λ_A = 1/λ_B ⇒ λ_A = λ_B ln 2

3. Sense check. Since ln 2 ≈ 0.693, λ_A is smaller than λ_B. That fits: a mean life is longer than a half-life, so A has the longer half-life and decays more slowly.
4. So. The correct relation is λ_A = λ_B ln 2, option (a). The equal initial number of atoms plays no part here.

The idea to keep. Keep these two at your fingertips: T_(1/2) = (ln 2)/λ and τ = 1/λ. Mean life is always longer, τ ≈ 1.44 T_(1/2).

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