Question

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?

Answer 1

`"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

`color(blue)(A = A_0 * 1/2^n)" "`, 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 = (52.7 color(red)(cancel(color(black)("years"))))/(5.27 color(red)(cancel(color(black)("years")))) = 10`

This means that you have

`A = A_0 * 1/2^(10) = 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