Show that the decay rate 'R' of a sample of a radionuclide is related to the number of radioactive nuclei 'N' at the same instant by the expression R = λ N.

(a) For, a radioactive sample, rate of disintegration is given by, 

                           $\boxed{\mathrm{R} = -\frac{\mathrm{dN}}{\mathrm{dt}}}$                           ...(i) 

According to radioactive decay law,
                        $-\frac{\mathrm{dN}}{\mathrm{dt}}\propto \mathrm{N}$ 

$\Rightarrow$                   $\boxed{-\frac{\mathrm{dN}}{\mathrm{dt}} = \mathrm{\lambda N}}$                             ...(ii) 

$\therefore$                           $\mathrm{R} = \mathrm{\lambda N}$                  [By equation (i) and (ii)] 

(b) Activity of a sample is, 

                        $\mathrm{R} = \mathrm{\lambda N} = \frac{0.693}{{\mathrm{T}}_{1/2}}\times \mathrm{N}$        $\left(\therefore \mathrm{\lambda } = \frac{0.693}{{\mathrm{T}}_{1/2}}\right)$
                           $= \frac{0.693}{1.5\times {10}^{17}}\times 25\times {10}^{20}$ 

                           $= 11550 \mathrm{disintegrations}/\mathrm{second}.$