The rate of radioactive disintegration at an instant for a...

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 $$\begin{aligned}\frac{\mathrm{dN}}{\mathrm{dt}}=\lambda \mathrm{N} \quad(\lambda=\text{ decay constant })\\ \Rightarrow \quad \mathrm{R}=\lambda \mathrm{N} \quad \text{ or } \mathrm{N}=\frac{\mathrm{R}}{\lambda} \quad \ldots \text{ (i) } \end{aligned}$$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}