JEE Main 2020 — Nuclear Physics Question with Solution

Using the equation of radioactive decay law,

$-\frac{dN}{dt}={\lambda }_{1}N+{\lambda }_{2}N ...\left(1\right)$

$-\frac{dN}{dt}=\left({\lambda }_1+{\lambda }_2\right)N ...\left(2\right)$

Where,

$-\frac{dN}{dt}$ is the rate of decay

${\lambda }_{1, }{\lambda }_2$ are decay constant for two different processes

$N$ is the number of atoms.

Now applying the decay law for the combine process, we get

$-\frac{dN}{dt}={\lambda }_{eff}N ...\left(3\right)$

Equating equation (1) and (2) we get

${\lambda }_{eff}=\left({\lambda }_1+{\lambda }_2\right) ...\left(4\right)$

Now the half life of the radioactive nucleus is given by,

$\lambda =\frac{\ln \left(2\right)}{T} ...\left(5\right)$

Where, $T$ is the half life of the radioactive nucleus

Using equation (5) equation (4) becomes,

$\frac{\ln \left(2\right)}{T}=\frac{\ln \left(2\right)}{T_1}+\frac{\ln \left(2\right)}{T_2}$

$\frac{1}{T}=\frac{1}{T_1}+\frac{1}{T_2}$

$\frac{1}{T}=\frac{1}{10}+\frac{1}{100}=\frac{11}{100}$

$\Rightarrow T=\frac{100}{11}\approx 9 \mathrm{s}$