# The half-life of cobalt—60 is 5.27 years. How many milligrams of cobalt-60 remain after 52.7 years if you start with 10.0 mg?

Nov 28, 2015

$\text{0.00977 mg}$

#### Explanation:

The important thing to recognize here is that the amount of time that passes is a whole number multiple of the isotope's nuclear half-life.

As you know, the equation for nuclear half-life calculations looks like this

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

$A$ - the mass of the substance that remains undecayed
${A}_{0}$ - the initial mass of the substance
$n$ - the ratio between the amount of time that passes and the half-life of the isotope.

In your case, you know that you're interested in finding out how much cobalt-60 would be left undecayed after $52.7$ years. This means that $n$ would be

$n = \left(52.7 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{years"))))/(5.27 color(red)(cancel(color(black)("years}}}}\right) = 10$

This means that you have

$A = {A}_{0} \cdot \frac{1}{2} ^ \left(10\right) = {A}_{0} / 1024$

Plug in the value you have for the initial mass of the sample to get

A = "10.0 mg"/1024 = color(green)("0.00977 mg") -> rounded to three sig figs