A nuclide decays by two independent parallel modes: alpha emission with partial half-life 20 h and beta^- emission with partial half-life 30 h. Starting from N_0 atoms, the number of alpha-decay daughter atoms formed in the first 24 h is

JEE Advanced levelSingle correct optionDifficulty 3 of 3as183

A nuclide decays by two independent parallel modes: $α$ emission with partial half-life $20 h$ and $β^{-}$ emission with partial half-life $30 h$ . Starting from $N_{0}$ atoms, the number of $α$ -decay daughter atoms formed in the first $24 h$ is

Partial rate constants add:

λ = ln 2 (1/20 + 1/30) = ln 2/12 h^{- 1}

, so the effective half-life is 12 h. In 24 h (two half-lives) the fraction decayed by any route is

1 - 0.25 = 0.75

. The

α

share is

λ_{α} / λ = (1/20) / (1/12) = 0.60

, so

α

daughters

= 0.60 \times 0.75 N_{0} = 0.45 N_{0}