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If ' f ' denotes the ratio of the number of nuclei decayed

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]

  1. $-\lambda\left(1-e^{-\lambda t}\right)$

  2. $\lambda\left(1-e^{-\lambda t}\right)$

  3. $\lambda \mathrm{e}^{-\lambda \mathrm{t}}$

  4. $-\lambda \mathrm{e}^{-\lambda t}$

Correct Option: , 3

Solution:

$\mathrm{N}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}}$

$\mathrm{N}_{\mathrm{d}}=\mathrm{N}_{0}-\mathrm{N}$

$\mathrm{N}_{\mathrm{d}}=\mathrm{N}_{0}\left(1-\mathrm{e}^{-\lambda \mathrm{t}}\right)$

$\frac{N_{d}}{N_{0}}=f=1-e^{-\lambda . t}$

$\frac{\mathrm{df}}{\mathrm{dt}}=\lambda \mathrm{e}^{-\lambda t}$

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