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

Apr 3, 2015

The half-life of your radioisotope is $\text{6.6 days}$.

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

$\text{1 half-life" -> 1/2 " left undecayed}$

$\text{2 half-lives" -> 1/4 " left undecayed}$

$\text{3 half-lives" -> 1/8 " left undecayed}$

$\text{4 half-lives" -> 1/16 " left undecayed}$

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

$\frac{t}{t} _ \left(\text{1/2}\right) = 4$

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

$\text{26.4"/t_("1/2") = 4 => t_("1.2") = 26.4/4 = "6.6 days}$