Answer
(a) 3.32 half-lives must elapse.
(b) 6.64 half-lives must elapse.
Work Step by Step
(a) We can find the required number of half-lives:
$N = N_0~(0.5)^{t/(t_{1/2})}$
$0.10~N_0 = N_0~(0.5)^{t/(t_{1/2})}$
$(0.5)^{t/(t_{1/2})} = 0.10$
$\frac{t~ln(0.5)}{t_{1/2}} = ln(0.10)$
$t = \frac{ln(0.10)}{ln(0.5)}~t_{1/2}$
$t = 3.32~t_{1/2}$
3.32 half-lives must elapse.
(b) We can find the required number of half-lives:
$N = N_0~(0.5)^{t/(t_{1/2})}$
$0.01~N_0 = N_0~(0.5)^{t/(t_{1/2})}$
$(0.5)^{t/(t_{1/2})} = 0.01$
$\frac{t~ln(0.5)}{t_{1/2}} = ln(0.01)$
$t = \frac{ln(0.01)}{ln(0.5)}~t_{1/2}$
$t = 6.64~t_{1/2}$
6.64 half-lives must elapse.
