The half-life of Iodine-131 is 8 days. What mass of I-131 remains from an 8.0g sample after 2 half-lives?

The key to this problem lies with how the nuclear half-life of a radioactive isotope was defined.

For a given sample of a radioactive isotope, the time needed for half of the sample to undergo decay will give you that isotope's nuclear half-life.

This means that every passing of a half-life will leave you with half of the sample you started with.

Let's say that you start with a sample #A_0#. Using the definition of a nuclear half-life, you can say that you will be left with

• #A_0 * 1/2 = color(purple)(A_0/2) -># after one half-life

What about after the passing of another half-life?

• #color(purple)(A_0/2) * 1/2 = color(orange)(A_0/4) -># after two half-lives

What about after the passing of another half-life?

• #color(orange)(A_0/4) * 1/2 = color(brown)(A_0/8) -># after three half-lives

and so on. With every half-life that passes, your sample will be halved.

Mathematically, you can express this as

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

#A# - the mass of the sample that remains after a period of time
#n# - the number of half-lives that pass in that period of time

You know that your sample of iodine-131 has a half-life of #8# days. In your case, you are interested in figuring out how much iodine-131 will remain undecayed after the passing of #2# half-lives.

This means that here #n=2#. You will thus have

#A = A_0 * 1/2^2 = A_0/4#

Since you started with an #"8-g"# sample, you will be left with

#A = "8 g" * 1/4 = color(green)("2 g")#