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