# How do you find radioactive decay half life?

Jan 4, 2018

Use the radioactive decay half-life formula.

#### Explanation:

The radioactive decay half-life formula states that

$N \left(t\right) = {N}_{0} {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\frac{1}{2}\right)}$

where

$N \left(t\right)$ is the final amount of substance

${N}_{0}$ is the initial amount of a substance

$t$ is the time (usually in years or seconds)

${t}_{\frac{1}{2}}$ is the half-life of the substance

To solve for half-life of a substance, rearrange the formula in terms of ${t}_{\frac{1}{2}}$.

$N \left(t\right) = {N}_{0} {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\frac{1}{2}\right)}$
$\frac{N \left(t\right)}{N} _ 0 = {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\frac{1}{2}\right)}$
${\log}_{\frac{1}{2}} \left[\frac{N \left(t\right)}{N} _ 0\right] = \frac{t}{t} _ \left(\frac{1}{2}\right)$

$\therefore {t}_{\frac{1}{2}} = \frac{t}{\log} _ \left(\frac{1}{2}\right) \left[\frac{N \left(t\right)}{N} _ 0\right]$