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Question
Show that for radioactive decay N(t) = `"N"_"O" "e"^{-λ"t"}`, where symbols have their usual meaning.
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Solution
If ‘N(t)’ 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’, then,
dN ∝ −N(t) dt
dN = −λ N(t) dt
where λ 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")} "dN"/("N"("t")) = int_0^"t" -lambda "dt"`
where N0 is a number of parent atoms at time t = 0.
∴ loge `("N"("t"))/"N"_0 = -lambda"t"`
∴ N(t) = N0 `"e"^{-lambda"t"}`
This is the required relation.
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