Carbon (Z = 6) with mass number 11 decays to boron (Z = 5). (a) Is it a β+-decay or a β−decay? (b) The half-life of the decay scheme is 20.3 minutes. How much time will elapse before a mixture of 90% carbon-11 and 10% boron-11 (by the number of atoms) converts itself into a mixture of 10% carbon-11 and 90% boron-11?
(a) The reaction is given by
`C_6 → B_5 + β^+ + "v"`
It is a β+ decay since atomic number is reduced by 1.
(b) Half-life of the decay scheme , `T_(1"/"2) = 20.3` minutes
Disintegration constant, `lambda = 0.693/T_(1/2) = 0.693/20.3 "min"^(-1)`
If t is the time taken by the mixture in converting, let the total no. of atoms be 100N0.
|Initial||90 N0||10 N0|
|Final||10 N0||90 N0|
N = N0e−λt
Here, N0 = Initial number of atoms
N = Number of atoms left undecayed
`10N_0 = 90N_0e^(-lambdat)` (For carbon)
⇒ `1/9 = e^(-0.693/20.3 xx t)`
⇒ `"In" 1/9 = (-0.693)/20.3 t`
⇒ t = 64.36 = 64 min