Show that the decay rate 'R' of a sample of a radionuclide is related to the number of radioactive nuclei 'N' at the same instant by the expression R = λ N.

Solution

(a) For, a radioactive sample, rate of disintegration is given by,

$R = - \frac{dN}{dt}$ ...(i)

According to radioactive decay law,
$- \frac{dN}{dt} \propto N$

$\Rightarrow$ $- \frac{dN}{dt} = λN$ ...(ii)

$∴$ $R = λN$ [By equation (i) and (ii)]

(b) Activity of a sample is,

$R = λN = \frac{0.693}{T_{1 / 2}} \times N$ $(∴ λ = \frac{0.693}{T_{1 / 2}})$
$= \frac{0.693}{1.5 \times 10^{17}} \times 25 \times 10^{20}$

$= 11550 disintegrations / second .$

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Wave Optics

Show that the decay rate 'R' of a sample of a radionuclide is related to the number of radioactive nuclei 'N' at the same instant by the expression R = λ N.

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