• # 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.

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