• # question_answer 56) The radioactivity of a sample is ${{R}_{1}}$ at a time ${{T}_{1}}$ and ${{R}_{2}}$at a time${{T}_{2}}$. If the half-life of the specimen is T, the number of atoms that have disintegrated in the time $({{T}_{2}}-{{T}_{1}})$ is proportional to A)  $({{R}_{1}}{{T}_{1}}-{{R}_{2}}{{T}_{2}})$       B)  $({{R}_{1}}-{{R}_{2}})$C)  $({{R}_{1}}-{{R}_{2}})/T$         D)  $({{R}_{1}}-{{R}_{2}})\times T$

$1.\,\lambda =\frac{0.693}{{{t}^{1/2}}}$                 2. $R=\lambda {{N}_{t}}$ Radioactivity at ${{T}_{1}}$ is ${{R}_{1}}=\lambda {{N}_{1}},$ Radioactivity at ${{T}_{2}}$ is ${{R}_{2}}=\lambda {{N}_{2}}$ $\therefore$Number of atoms decayed in time $({{T}_{1}}-{{T}_{2}})=({{N}_{1}}-{{N}_{2}})$ or $\frac{{{R}_{1}}-{{R}_{2}}}{\lambda }=\frac{({{R}_{1}}-{{R}_{2}})T}{0.693}$i.e., $\alpha ({{R}_{1}}-{{R}_{2}})T$