Question

Plot a graph showing the variation of undecayed nuclei N versus time t.From the graph,find out how one can determine the half life and average life of the radioactive nuclei

Answer 1

according to radioactive decay law the rate of disintegration of a radioactive substance at an instant is directly proportional to number of nuclei in the radioactive substance at that time.

e.g., $\bf{N=N_0e^{-\lambda t}}$,

where, $N_0$ is number of nuclei in the radioactive substance at that time, $\lambda$ is radioactive decay constant.

from above expression it is clearly that graph between number of undecayed nuclei and time is $\textbf{strictly decreasing exponential}$.

for half life :

$N=N_0/2$ , $t=T_{1/2}$

$N_0/2=N_0e^{-\lambda T_{1/2}}\\\\1/2=e^{-\lambda T_{1/2}}\\\\ln2=\lambda T_{1/2}\\\\T_{1/2}=\frac{ln2}{\lambda}$

for average life :

average life of radioactive sample is the amount of time required for it to get decayed to 36.8% of its original amount.

so, $N=0.368N_0$, $t=\tau$

$0.368N_0=N_0e^{-\lambda\tau}$

or, $0.368=e^{-\lambda\tau}$

or, $ln(0.368)=-\lambda\tau$

or, $-0.999672341\approx-1=-\lambda\tau$

hence, $\bf{\tau=\frac{1}{\lambda}}$