# A radioactive element has a half-life of 5 years. If you leave a 2 g sample of this element under your chemistry desk for 15 years, what mass will remain undecayed?

Dec 25, 2015

$\text{0.25 g}$

#### Explanation:

So, you know that your radioactive element has a nuclear half-life of $5$ years.

As you know, a radioactive isotope's half-life tells you the time needed for half of an initial sample to undergo radioactive decay.

If you start with an initial sample ${A}_{0}$, you can say that you'll be left with

• ${A}_{0} \cdot \frac{1}{2} \to$ after one half-life
• ${A}_{0} / 2 \cdot \frac{1}{2} = {A}_{0} / 4 \to$ after two half-lives
• ${A}_{0} / 4 \cdot \frac{1}{2} = {A}_{0} / 8 \to$ after three half-lives
$\vdots$

and so on. This means that you can express a relationship between the Initial sample of the radioactive isotope, ${A}_{0}$, and the amount that remains undecayed, $A$, in terms of how many half-lives pass in a given period of time

$\textcolor{b l u e}{A = {A}_{0} \cdot \frac{1}{2} ^ n} \text{ }$, where

$n$ - the number of half-lives

In your case, you can say that

$n = \left(15 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{years"))))/(5color(red)(cancel(color(black)("years}}}}\right) = 3$

This means that you'll be left with

$A = {A}_{0} \cdot \frac{1}{2} ^ 3 = \frac{1}{8} \cdot {A}_{0}$

Therefore, your original sample will be down to $\frac{1}{8} \text{th}$ of its initial value after the passing of $15$ years

A = 1/8 * "2 g" = color(green)("0.25 g")

I'll leave the answer rounded to two sig figs.