# Polonium-214 has a relatively short half life of 164 s. How many seconds would it take for 8.0 g of this isotope to decay to 0.25 g?

Jun 5, 2016

$820 \setminus \sec$

#### Explanation:

${m}_{i} = {m}_{r} \cdot {2}^{n}$

${m}_{i} \text{ is the initial mass present of the radioisotope.}$

${m}_{r} \text{ is the mass remaining after a certain time }$

$n \text{ is the number of periods," = (time)/T" where T is the half-life.}$

${m}_{i} = {m}_{r} \cdot {2}^{n}$

${m}_{i} / {m}_{r} = {2}^{n}$

${2}^{n} = \frac{8.0 \setminus g}{0.25 \setminus g}$

${2}^{n} = 32$

${2}^{n} = {2}^{5} \text{ } \implies$ n = 5

$n = \frac{t i m e}{T}$

$t i m e = n \times T$

$t i m e = 5 \times 164 \setminus \sec$

$t i m e = 820 \setminus \sec$

A quick approach:

$\underbrace{8 \setminus g \to 4 \setminus g \to 2 \setminus g \to 1 \setminus g \to 0.5 \setminus g \to 0.25 g}$
$\text{ } T i m e$

$T i m e = 5 \setminus p e r i o \mathrm{ds}$ ...