Question:
If ' $f$ ' denotes the ratio of the number of nuclei decayed $\left(\mathrm{N}_{\mathrm{d}}\right)$ to the number of nuclei at $t=0\left(\mathrm{~N}_{0}\right)$ then for a collection of radioactive nuclei, the rate of change of ' $f$ ' with respect to time is given as :
$[\lambda$ is the radioactive decay constant]
Correct Option: , 3
Solution:
$\mathrm{N}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}}$
$\mathrm{N}_{\mathrm{d}}=\mathrm{N}_{0}-\mathrm{N}$
$\mathrm{N}_{\mathrm{d}}=\mathrm{N}_{0}\left(1-\mathrm{e}^{-\lambda \mathrm{t}}\right)$
$\frac{N_{d}}{N_{0}}=f=1-e^{-\lambda . t}$
$\frac{\mathrm{df}}{\mathrm{dt}}=\lambda \mathrm{e}^{-\lambda t}$