Activity R = $R_0 e^{-\lambda t}$ Also instantaneous activity, R = – λN R = - $\frac{0.693}{T}$ N Initial activity, $R_0$ = – $\lambda N_0$ So, $R_0$ = - $\frac{0.693}{T} N_0$ (a) $\frac{R}{R_0}$ = $\frac{N}{N_0}$ = $\frac{3.125}{100}$ = $\frac{1}{32}$ or $\frac{N}{N_0}$ = $\left(\frac{1}{2}\right)^n$ = $\left(\frac{1}{2}\right)^5$ or n = 5 ∴ t = nT = 5T years. (b) $\frac{R}{R_0}$ = $\frac{N}{N_0}$ = $e^{-\lambda t}$ = $\frac{1}{100}$ Required time, as cannot be solved by direct calculation as in part (a). t = $\frac{2.303}{\lambda}$ log $\frac{N_0}{N}$ = $\frac{2.303 T}{0.693}$ log 100 = $\frac{2.303 \times 2 \times T}{0.693}$ = 6.65 T years