# Radon-222 has a half-life of 92.hours. How long would it take for the activity of a sample of the gas to be reduced to about 3% of its initial value?

Oct 17, 2016

It would take 19. days for the radon to decay.

#### Explanation:

The formula for the half-life of a first-order process is

color(blue)(bar(ul(|color(white)(a/a) t_½ = (ln2)/kcolor(white)(a/a)|)))" "

k = (ln2)/t_½ = (ln2)/"92. h" = "0.007 53 h"^"-1"

The integrated formula for a first-order process is

color(blue)(bar(ul(|color(white)(a/a) ln("A"_0/"A") = ktcolor(white)(a/a)|)))" "

ln("A"_0/"A") = ln(stackrelcolor(blue)(1)(color(red)(cancel(color(black)("A"_0))))/(0.03color(red)(cancel(color(black)("A"_0)))))= "0.007 53 h"^"-1"t

$\ln \left(\frac{1}{0.03}\right) = \text{-ln0.03" = 3.51 = "0.007 53 h"^"-1} t$

$t = \frac{3.51}{\text{0.007 53 h"^"-1" = 4.7 × 10^2color(white)(l) "h" = "19. days}}$