Question

Carbon (Z = 6) with mass number 11 decays to boron (Z = 5). (a) Is it a ${\beta }^{+}$ -decay or a ${\beta }^{-}$ -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

$\left(a\right)P\to n+e^{-}+v$

$\text{Hence it is a}{\beta }^{-}\text{decay}$

(b)$\text{Let the total no. of atoms be}100{\text{N}}_1$

$\text{Carbon}\text{Boron}$

$\text{Initially}90\text{N}10{\text{N}}_0$

$\text{Fimlly}10{\text{N}}_{0}90N_0$

$\text{Now,}10{\text{N}}_0=90N_0{e}^{-\lambda t}$

$\Rightarrow \frac{1}{9}=e^{\frac{-0.698}{20.8}\times t}$

$\because t_{\frac{1}{2}}=20.3\text{min}$

$\Rightarrow ln\frac{1}{9}=\frac{-0.693}{20.3}t$

$\Rightarrow t=\frac{2.1972\times 20.3}{0.693}$

= $64.36=64\text{min}$