# The half-life of #""^131"I"# is 8.07 days. What fraction of a sample of #""^131"I"# remains after 24.21 days?

##### 1 Answer

#### Explanation:

As you know, an isotope's nuclear half-life tells you how much time must pass in order for **half** of an initial sample of this isotope to undergo radioactive decay.

In other words, an isotope's half-life tells you how much must pass in order for a sample to be reduced to **half** of its initial value.

If you take **remaining** after a period of time

#A = A_0 * 1/2 -># *after***one**half-life#A = A_0/2 * 1/2 = A_0/4 -># *after***two**half-lives#A = A_0/4 * 1/2 = A_0/8 -># *after***three**half-lives

#vdots#

and so on. This means that you can express **number of half-lives** that pass using the equation

#color(blue)(A = A_0 * 1/2^n)" "# , where

**number of half-lives** that pass in a given period of time

#color(blue)(n = "period of time"/"half-life")#

So, you want to know what fraction of an initial sample of

*How many* half-lives do you get in that period of time, knowing that one half-life is equal to

#n = (24.21 color(red)(cancel(color(black)("days"))))/(8.07color(red)(cancel(color(black)("days")))) = 3#

This means that you have

#A = A_0 * 1/2^3#

#A = A_0 * 1/8#

Therefore, your initial sample of