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

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 #A_0# to be the initial sample of an isotope, and #A# to be the sample remaining after a period of time #t#, then you can say that

• #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 #A# in terms of #A_0# and the number of half-lives that pass using the equation

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

#n# - the 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 #""^131"I"# remains after #"24.21 days"#.

How many half-lives do you get in that period of time, knowing that one half-life is equal to #"8.07 days"#?

#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 #""^131"I"# will be reduced to #1/8"th"# of its initial value after #"24.21 days"#.