# The half life of the radioactive element Strontium-90 is 37 years. In 1950, 15 kilograms of this element released accidentally. How do you determine the formula which shows the mass remaining after t years?

Apr 4, 2017

$N \left(t\right) = 15 K g \times {\left(\frac{1}{2}\right)}^{\frac{t}{\textcolor{red}{37}}}$

Note: The half life of Strontium-90 is now published as 28.8 years. This will require adjustments of all the following values in $\textcolor{red}{red}$.

#### Explanation:

The formula for the half life of an exponentially decaying substance is:

$N \left(t\right) = \left(N o\right) \times {\left(\frac{1}{2}\right)}^{\frac{t}{t \frac{1}{2}}}$

$N \left(t\right)$ ... is how much is still here.
$N o$ ... is how much we started with.
$t$ ...... is the time we have measured since the start of the decay.
$t \frac{1}{2}$ ... is the already calculated half life of the specific substance.

To calculate the mass of Strontium-90 remaining after $t$ years, plug in the given values into the formula:

$N \left(t\right) = 15 K g \times {\left(\frac{1}{2}\right)}^{\frac{t}{\textcolor{red}{37}}}$

For example $\textcolor{red}{37}$ years after 1950, the remainder of the $15 K g$ of Strontium-90 released would be:

$N \left(67\right) = 15 K g \times {\left(\frac{1}{2}\right)}^{\textcolor{red}{\frac{37}{37}}} = 15 K g \times \left(\frac{1}{2}\right) = 7.5 K g$

In 2017 (now) the amount left is $15 K g \times {\left(\frac{1}{2}\right)}^{\textcolor{red}{1.8}} = \textcolor{red}{4.3} K g$