Physics - Modern Physics Question with Solution | TestHub

Given, half life $T_{1/2}=2.2\times 10^9\mathrm{s}$ Rate of disintegration, $R=10^{10}{\mathrm{s}}^{-1}$ If $\mathrm{N}$ be the number of nuclei present, then the rate of disintegration is $$\begin{array}{r}\frac{\mathrm{dN}}{\mathrm{dt}}=\lambda \mathrm{N}(\lambda =\text{ decay constant })\\ \Rightarrow \mathrm{R}=\lambda \mathrm{N}\text{ or }\mathrm{N}=\frac{\mathrm{R}}{\lambda }\dots \text{ (i) }\end{array}$$ Also, the half life is given by, $${\mathrm{T}}_{1/2}=\frac{0.693}{\lambda }\Rightarrow \lambda =\frac{0.693}{{\mathrm{T}}_{1/2}}$$ From Eq. (i) and (ii), we get $$\mathrm{N}=\frac{\mathrm{R}}{0.693}\times {\mathrm{T}}_{1/2}=\frac{10^{10}\times 2.2\times 10^9}{0.693}=3.17\times 10^{19}$$