Question

What is the half-life of a radioisotope if 1/16 of it remains undecayed after 26.4 days?

Answer 1

The half-life of your radioisotope is `"6.6 days"`.

When numbers allow it, the quickest way to determine the half-life of a radioisotope is to use the fraction left undecayed as a measure of how many half-lives have passed.

You know that the mass of a radioactive isotope gets halved with the passing of every half-life, which means that

`"1 half-life" -> 1/2 " left undecayed"`

`"2 half-lives" -> 1/4 " left undecayed"`

`"3 half-lives" -> 1/8 " left undecayed"`

`"4 half-lives" -> 1/16 " left undecayed"`

As you can see, 4 half-lives must pass until you have 1/16 of the original sample. Mathematically, this means that

`t/t_("1/2") = 4`

Since you know that 26.4 days have passed, the isotope's half-life will be

`"26.4"/t_("1/2") = 4 => t_("1.2") = 26.4/4 = "6.6 days"`