Two radioactive sources A and B initially contain equal number of radioactive atoms. Source A has a half-life of 1 hour and source B has a half-life of 2 hours. At the end of 2 hours, the ratio of the rate of disintegration of A to that of B is:

The rate of disintegration is proportional to the number of radioactive atoms present initially in the source.

The number nuclei of radioactive sample of A left after 2 hours,

${\mathrm{N}}_A={\left(\frac{1}{2}\right)}^{2/1}{\mathrm{N}}_0=\frac{1}{4}{\mathrm{N}}_0$

The number of nuclei of radioactive sample of B left after 2 hours,

${\mathrm{N}}_B={\left(\frac{1}{2}\right)}^{2/2}{\mathrm{N}}_0=\frac{1}{2}{\mathrm{N}}_0$

The rate of disintegration at any time t where number of nuclei is N is given by:

$R = \frac{\mathrm{dN}}{\mathrm{dt}} = -\mathrm{\lambda N}$

$$\begin{gathered} \frac{{\mathrm{R}}_{\mathrm{A}}}{{\mathrm{R}}_{\mathrm{B}}} = \frac{{\mathrm{\lambda }}_{\mathrm{A}}{\displaystyle \frac{{\mathrm{N}}_0}{4}}}{{\mathrm{\lambda }}_{\mathrm{B}}\frac{{\mathrm{N}}_0}{2}} = \frac{{\mathrm{\lambda }}_{\mathrm{A}}}{2{\mathrm{\lambda }}_{\mathrm{B}}} = \frac{{\mathrm{T}}_{\mathrm{B}}}{2{\mathrm{T}}_{\mathrm{A}}} = \frac{2}{2} = 1 \\ (\because \mathrm{\lambda } = \frac{0.693}{\mathrm{T}}, \mathrm{where} \mathrm{T} \mathrm{is} \mathrm{half} \mathrm{life}) \end{gathered}$$