At a given instant, say \(t = 0,\) two radioactive substances \(A\) and \(B\) have equal activates. The ratio \(\frac{{{R_B}}}{{{R_A}}}\) of their activities. The ratio \(\frac{{{R_B}}}{{{R_A}}}\) of their activates after time \(t\) itself decays with time \(t\) as \(e^{-3t}.\) If the half-life of \(A\) is \(ln2,\) the half-life of \(B\) is

\(\therefore\) \(\frac{{{\text{R}}_{\text{B}}}}{{{\text{R}}_{\text{A}}}}=\frac{{{\text{R}}_{\text{o}}}{{\text{e}}^{-{{\lambda }_{\text{B}}}\text{t}}}}{{{\text{R}}_{\text{o}}}{{\text{e}}^{-{{\lambda }_{\text{B}}}t}}}\) \(={{\text{e}}^{-\left( {{\lambda }_{B}}-{{\lambda }_{\text{A}}} \right)\text{t}}}={{\text{e}}^{-3\text{t}}}\)

\({\Rightarrow}  {\lambda_{B}-\lambda_{A}=3} \)

\(\Rightarrow \) \(\frac{\ell {{n}^{2}}}{{{T}_{B}}}-\frac{\ell n2}{\ell n2}=3\)

\({\Rightarrow \quad T_{B}=\frac{\ell n 2}{4}}\)