The half life of a radioactive substance is 20 minutes. The approximate time interval (t₂–t₁) between the time t₂ when 2/3 of it has decayed and time t₁ when 1/3 of it has decayed is :

No.of undecayed atoms after time
${\mathrm{t}}_2={\mathrm{N}}_0-\frac{2}{3}{\mathrm{N}}_0=\frac{{\mathrm{N}}_0}{3}={\mathrm{N}}_0{\mathrm{e}}^{-{\mathrm{\lambda t}}_2}\to \left(1\right)$
No of undecayed atoms after time t₁
$={\mathrm{N}}_0-\frac{1}{3}{\mathrm{N}}_0=\frac{2}{3}{\mathrm{N}}_0={\mathrm{N}}_0{\mathrm{e}}^{-{\mathrm{\lambda t}}_1}...\left(2\right)$
$\begin{array}{l}\left(2\right)÷\left(1\right)\text{ we get };2={\mathrm{e}}^{\mathrm{\lambda }\left({\mathrm{t}}_2-{\mathrm{t}}_1\right)}\\ \Rightarrow {\log }_{\mathrm{e}}^{ }2=\mathrm{\lambda }\left({\mathrm{t}}_2-{\mathrm{t}}_1\right){\log }_{\mathrm{e}}^{ }\mathrm{e}\\ \Rightarrow {\mathrm{t}}_2-{\mathrm{t}}_1=\frac{{\log }_{\mathrm{e}}^{ }2}{\mathrm{\lambda }}=\frac{0.693}{\mathrm{\lambda }}=\mathrm{T}=20\min \end{array}$