question_answer

Define the activity of a radionuclide. Write its SI unit. Give a plot of the activity of a radioactive species versus time. How long will a radioactive iostope, whose half life is T years, take for its activity to reduce to \[1/8\text{th}\] of its initial value?                                

Answer:

                The activity of a sample is defined as the number of radioactive disintegrations taking place per second at any instant in the sample. Its \[SI\]unit is becquerel. \[1\]becquerel \[=1Bq=1\]decay per second. \[R=\frac{dN}{dt}=\lambda N=\frac{0.693}{{{T}_{1/2}}}N\] For sample 1, \[{{R}_{1}}=\frac{0.693}{{{T}_{1}}}{{N}_{1}}\] For sample 2, \[{{R}_{2}}=\frac{0.693}{{{T}_{2}}}{{N}_{2}}\] \[\therefore \]  \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{{{N}_{1}}{{T}_{2}}}{{{N}_{2}}{{T}_{1}}}\] The activity R decreases exponentially with time t \[(R={{R}_{0}}{{e}^{{{-}^{\lambda t}}}})\] as shown in the graph. \[\frac{R}{{{R}_{0}}}={{\left( \frac{1}{2} \right)}^{n}}\] or            \[\frac{1}{8}={{\left( \frac{1}{2} \right)}^{t/T}}\] or            \[{{\left( \frac{1}{2} \right)}^{3}}={{\left( \frac{1}{2} \right)}^{t/T}}\] \[\therefore \]  \[t=\mathbf{3T}\mathbf{.}\]