question_answer

State the law of radioactive decay. If \[{{N}_{0}}\] is the number of radioactive nuclei at some initial time \[{{t}_{0}},\] find out the relation to determine the number N present at a subsequent time. Draw a plot of N as a function of time.

Answer:

Radioactive decay law The rate of decay of radioactive nuclei is directly proportional to the number of undecayed nuclei at that time. i.e.        \[-\frac{dN}{dt}\propto N\] or         \[\frac{dN}{dt}=-\lambda N\] where, \[\lambda \]  is the decay constant.             \[\frac{dN}{dt}=-\lambda N\Rightarrow \frac{dN}{N}=-\lambda t\] On integrating both sides, we get             \[{{\log }_{e}}N=-\lambda t+C\] where, C is a constant of integration. If \[{{N}_{0}}\] is initial number of radioactive nuclei at t = 0, So,       \[{{\log }_{e}}{{N}_{0}}=0+C\Rightarrow C={{\log }_{e}}{{N}_{0}}\] Substituting this in Eq. (ii), we get \[{{\log }_{e}}N=-\lambda t+{{\log }_{e}}{{N}_{0}}\] \[\Rightarrow \]   \[{{\log }_{e}}N-{{\log }_{e}}{{N}_{0}}=-\lambda t\] \[\Rightarrow \]   \[{{\log }_{e}}\frac{N}{N}=-\lambda t\] \[\Rightarrow \]   \[\frac{N}{{{N}_{0}}}={{e}^{-\lambda t}}\] \[\Rightarrow \]   \[N={{N}_{0}}{{e}^{-\lambda t}}.\] The graph of N as a function of time is shown in figure below.