Question

The half-life of radioactive nucleus is $100$ years. The time interval between $20\%$ and $80\%$ decay of the parent nucleus is

• A. 100 years
• B. 200 years
• C. 300 years
• D. 400 years

Answer 1

Answer: (B)

Half - life, $T=100$ years
We have, $\frac{M}{M_0}=(\frac{1}{2}{)}^{t/T}$
$\left(\frac{M_0\times \frac{20}{100}}{M_0}\right)={\left(\frac{1}{2}\right)}^{t_1/T}...\left(i\right)$
and $\left(\frac{M_0\times \frac{80}{100}}{M_0}\right)={\left(\frac{1}{2}\right)}^{t_2/T}...\left(ii\right)$
From Eqs. $(i)$ and $(ii)$, we get
$\frac{\frac{1}{5}}{\frac{4}{5}}=\frac{{\left(\frac{1}{2}\right)}^{t_1/T}}{{\left(\frac{1}{2}\right)}^{t_2/T}}$
$\frac{1}{4}={\left(\frac{1}{2}\right)}^{\left(t-t_2\right)/T}$
${\left(\frac{1}{2}\right)}^2={\left(\frac{1}{2}\right)}^{\Delta t/T}$
where, $\Delta t=t_1-t_2$
$\therefore 2=\frac{\Delta t}{T}$
Time, $\Delta t=2\times T$
$=2\times 100=200$ years