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?

Answer 1

(a) The reaction is given by
${\mathrm{C}}_6\to {\mathrm{B}}_5+{\mathrm{\beta }}^{+}+\mathrm{v}$
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 =\frac{0.693}{T_{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$ = $\frac{0.693}{20.3}{\min }^{-1}$

If t is the time taken by the mixture in converting, let the total no. of atoms be 100N_0.

	Carbon	Boron
Initial	90 N0	10 N0
Final	10 N0	90 N0

N = N₀e^−λt
Here, N₀ = Initial number of atoms
N = Number of atoms left undecayed
10N₀ = 90N₀e^−λt ( For carbon)
$$\begin{gathered} \Rightarrow \frac{1}{9}=e^{-\frac{0.693}{20.3}\times t} \\ \Rightarrow \mathrm{In}\frac{1}{9}=\frac{-0.693}{20.3}t \\ \Rightarrow t=64.36=64\min \end{gathered}$$