`._(84)Po^(218) (t_(1//2) = 3.05 min)` decays to `._(82)Pb^(214) (t_(1//2) = 2.68 min)` by `alpha` emisison while `Pb^(214)` is `beta`-emitter. In an experiment starting with `1 g` atom of pure `Po^(218)`, how much time would be required for the concentration of `Pb^(214)` to reach maximum?

Explanation

`._(84)Po^(218)overset(-alpha)rarr._(82)Pb^(214)overset(-beta)rarr ._(83)Bi^(214)`
`m_(0) = 1 g atom = 1 mol`
Time at which `._(82)Pb^(214)` is maximum
`t_(max) = (1)/(k_(1) - k_(2))ln(k_(1))/(k_(2)) = (2.303)/(k_(1)-k_(2))log.(k_(1))/(k_(2))`
`k_(1) = (0.693)/(3.05) = 0.2287`
`k_(2) = (0.693)/(3.05) = 0.258`
`t = (2.303)/(0.2287-0.258) log.(0.2287)/(0.258)`
`= 4.1 min`