Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsQuestion Bank [ 2022 ]3 Marks
Question
Show that for radioactive decay $N ( t )= N _{ O } e ^{-\lambda t }$, where symbols have their usual meaning.
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Answer
If ' $N ( t )^{\prime}$ is the number of parent nuclei present at any instant ' $t$ ', ' $dN$ ' is the number of nuclei disintegrated in the short interval of time ' $dt ^{\prime}$ ', then,
$ d N \propto-N(t) d t$
$d N=-\lambda N(t) d t $
where $\lambda$ is known as decay constant or disintegration constant.
The negative sign indicates the disintegration of atoms.
Integrating both sides of the equation,
$\int_{ N _0}^{ N ( t )} \frac{ dN }{ N ( t )}=\int_0^{ t }-\lambda dt$
where $N_0$ is a number of parent atoms at time $t=0$.
$\therefore \log _e \frac{ N ( t )}{ N _0}=-\lambda t$
$ \therefore \log _{ e } \frac{ N ( t )}{ N _0}=-\lambda t$
$\therefore N ( t )= N _0 e ^{-\lambda t } $
This is the required relation.
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