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

Feb 20, 2016

$\text{2 g}$

#### Explanation:

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

• ${A}_{0} \cdot \frac{1}{2} = \textcolor{p u r p \le}{{A}_{0} / 2} \to$ after one half-life

What about after the passing of another half-life?

• $\textcolor{p u r p \le}{{A}_{0} / 2} \cdot \frac{1}{2} = \textcolor{\mathmr{and} a n \ge}{{A}_{0} / 4} \to$ after two half-lives

What about after the passing of another half-life?

• $\textcolor{\mathmr{and} a n \ge}{{A}_{0} / 4} \cdot \frac{1}{2} = \textcolor{b r o w n}{{A}_{0} / 8} \to$ after three half-lives

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

Mathematically, you can express this as

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

$A = {A}_{0} \cdot \frac{1}{2} ^ 2 = {A}_{0} / 4$

Since you started with an $\text{8-g}$ sample, you will be left with

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