Two nuclei P, Q have equal no.of atoms at t = 0. Their half-life are 3 hours, 9 hours. Compare their rates of disintegration after 18 hrs from
the start.

$$\begin{gathered} The number of half-lives of P in 18hours is given as, \\ \begin{array}{r}{\mathrm{n}}_1=\frac{\mathrm{t}}{{\mathrm{T}}_1}\\ =\frac{18}{3}\\ =6\end{array} \\ The number of nuclei P left after 6 half-lives is given as, \\ {\mathrm{N}}_1=\mathrm{N}{\left(\frac{1}{2}\right)}^6 \\ \mathrm{The} \mathrm{number} \mathrm{of} \mathrm{half}-\mathrm{lives} \mathrm{of} \mathrm{Q} \mathrm{in} 18\mathrm{hours} \mathrm{is} \mathrm{given} \mathrm{as}, \\ \begin{array}{r}{\mathrm{n}}_2=\frac{\mathrm{t}}{{\mathrm{T}}_2}\\ =\frac{18}{9}\\ =2\end{array} \\ \mathrm{The} \mathrm{number} \mathrm{of} \mathrm{nuclei} \mathrm{Q} \mathrm{left} \mathrm{after} 2 \mathrm{half}-\mathrm{lives} \mathrm{is} \mathrm{given} \mathrm{as}, \\ {\mathrm{N}}_2=\mathrm{N}{\left(\frac{1}{2}\right)}^2 \\ \mathrm{The} \mathrm{ratio} \mathrm{of} \mathrm{their} \mathrm{decay} \mathrm{rate} \mathrm{is} \mathrm{given} \mathrm{as}, \\ \begin{array}{r}\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{{\mathrm{T}}_2{\mathrm{N}}_1}{{\mathrm{T}}_1{\mathrm{N}}_2}\\ \frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{9\times \mathrm{N}{\left(\frac{1}{2}\right)}^6}{3\times \mathrm{N}{\left(\frac{1}{2}\right)}^2}\\ \frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=\frac{3}{16}\end{array} \end{gathered}$$