Plot a graph showing the variation of number of undecayed nuclei in a radioisotope as a function of time. Hence define half life of a nucleus and obtain its relation with average life.

Answers (1)

According to the law of radioactive decay, the number of nuclei disintegrated per unit time is proportional to the number of nuclei present in the sample
That is, $\frac{dN}{dt}\, \alpha -N$
$\frac{dN}{dt}\,= -\lambda N$
Negative sign indicates the rate of disintegration where $\lambda$ = radioactive decay instant or disintegration constant

This is the graph showing the variation of undecayed nuclei in a radioisotope as a function of time.
we get $\frac{dN}{dt}= -\lambda N$
or $\frac{dN}{N}= -\lambda dt$ integrating both sides give
$lnN-lnNo= -\lambda \left ( t-t_{0} \right )$
$t_{0}= 0$ so, $ln\frac{N}{N_{0}}= -\lambda t$
which gives $N\left ( t \right )= No\,\: e^{-\lambda t}$
Half-life of a nucleus means $T= T_{\frac{1}{2}}$ at that time $N_{0}$ is reduced to $\frac{N_{0}}{2}$
putting $N= \frac{N_{0}}{2}$ and $t= T_{\frac{1}{2}}$ we get
$T_{\frac{1}{2}}= \frac{ln2}{\lambda }= \frac{0\cdot 693}{\lambda }---(1)$
Next to find average life or mean life ' $\tau$ '.
Number of nuclei which decay in time $t$ to $t+\Delta t$ is
$= \lambda N_{0}e^{-xt}\Delta t$
Total time of all these nuclei is $t\lambda N_{0}e^{-\lambda t}dt$
The average life is
$\tau= \frac{\lambda N_{0}\int_{0}^{d}te^{-\lambda t}dt}{N_{0}}= \lambda \int_{0}^{d}te^{-\lambda t}dt$
$\tau=\frac{1}{\lambda }$
By comparing eq (1)
$T_{\frac{1}{2}}= \frac{ln2}{\lambda }=\tau ln2$
$T_{\frac{1}{2}}= \tau\, ln2$