Question

Carbon $(Z=6)$ with mass number $11$ decays to boron $(Z=5).$ The half-life of the decay scheme is $20.3\text{minutes}$. How much time will elapse before a mixture of $90\text{\%}\text{carbon}-11$ and $10\text{\%}\text{boron}-11$ (by the number of atoms) converts itself into a mixture of $10\text{\%}\text{carbon}-11$ and $90\%\text{boron}-11?$
$(\ln \frac{1}{9}=-2.1972)$

• A. $20.3\text{minutes}$
• B. $40.6\text{minutes}$
• C. $64\text{minutes}$
• D. $90\text{minutes}$

Answer 1

The correct option is C $64\text{minutes}$

Let the total number of atoms be $100N_0.$

	$\text{Carbon}$	$\text{Boron}$
Initially	$90N_0$	$10N_0$
Finally	$10N_0$	$90N_0$

Now, applying law of radioactive decay for $\text{Carbon}$,

$N=N_0{e}^{-\lambda t}$

$\Rightarrow 10N_0=90N_0{e}^{-\lambda t}$

$\Rightarrow \frac{1}{9}=e^{\frac{-0.693}{20.3}\times t}$

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

$\Rightarrow t=\frac{2.1972\times 20.3}{0.693}=64.36$

$\Rightarrow t\approx 64\text{min}$

Hence, option $\left(C\right)$ is correct.