#### Question

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?

#### Solution

(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 100N_{0}_{.}

Carbon | Boron | |

Initial | 90 N_{0} |
10 N_{0} |

Final | 10 N_{0} |
90 N_{0} |

N = N_{0}e^{−λ}^{t}

Here, N_{0} = 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