The halflife of a radioactive isotope is 20.0 minutes. How much of a 1.00 gram sample of this isotope remains after 1.00 hour?

May 28, 2018

$0.125 \textcolor{w h i t e}{l} \text{g}$

Explanation:

There are

1.00color(white)(l)color(red)(cancel(color(black)("hours")))*(60.0color(white)(l)color(purple)(cancel(color(black)("minutes"))))/(1.00color(white)(l)color(red)(cancel(color(black)("hours")))) cdot (1color(white)(l) "halflife")/(20.0color(white)(l)color(purple)(cancel(color(black)("minutes"))))=color(navy)(3)color(white)(l) "halflives"

in $1.00 \textcolor{w h i t e}{l} \text{hours}$ of time.

The mass of the sample halves every halflife of the decay. That is: given an initial mass of ${m}_{0}$, the remaining sample will have mass

$m = {\left(\frac{1}{2}\right)}^{n} \cdot {m}_{0}$

after $n$ halflives.

Therefore, the sample will have a mass of

$m = {\left(\frac{1}{2}\right)}^{\textcolor{n a v y}{3}} \cdot 1.00 \textcolor{w h i t e}{l} \text{g}$
$\textcolor{w h i t e}{m} = \frac{1}{8} \cdot 1.00 \textcolor{w h i t e}{l} \text{g}$
$\textcolor{w h i t e}{m} = 0.125 \textcolor{w h i t e}{l} \text{g}$

after an hour.