A sample of radioactive element has a mass of $$10\,g$$  at an instant $$t = 0.$$  The approximate mass of this element in the sample after two mean lives is

Solution :
Mean life of radioactive substance is given by
$$\tau = \frac{1}{\lambda },\,\,\left( {\lambda \,{\text{is}}\,{\text{decay constant}}} \right)$$
Also, it is given that $$t = 2\tau $$
So, $$t = 2 \times \frac{1}{\lambda } = \frac{2}{\lambda }$$
Thus, mass remained after time $$t$$ is
\[M = {M_0}{e^{ - \lambda t}}\,\,\left[ {\begin{array}{*{20}{c}} {M = {\rm{Final\,mass}}}\\ {{M_0} = {\rm{Inital\,mass}}}\\ {\lambda = {\rm{Decay\,constant}}} \end{array}} \right]\]
$$\eqalign{ & = 10{e^{ - \lambda \times \frac{2}{\lambda }}}\,\,\,\left( {\because {M_0} = 10\,g} \right) \cr & = 10{e^{ - 2}} \cr & = \frac{{10}}{{{e^2}}} \cr & = 1.35\,g \cr} $$