Question

Half life of a radioactive element is $5\text{min. and}10\text{sec}$. The time taken for $90\%$ of it to disintegrate is nearly,
(Given: ${\log }_{10}2=0.301$)

• A. ${10}^4\text{min}$
• B. ${10}^4\text{sec}$
• C. ${10}^3\text{sec}$
• D. ${10}^3\text{min}$

Answer 1

The correct option is C ${10}^3\text{sec}$

$T=5\text{min. and}10\text{sec}=310\text{sec}$

If $90$ of the sample disintegrates, it means that $10$ has remained undecayed.
Now, $N=\frac{N_0}{2^n}$

$\frac{N_0}{10}=\frac{N_0}{2^n}$

$2^n=10$

taking log of both sides,

$n\log 2=\log 10$

$\therefore n=\frac{1}{0.301}$

Now, $t=nT=\frac{1}{0.301}\times 310=1029$
$\therefore t\approx 1000\text{sec.}$

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