# What is the half life decay rate formula?

Apr 4, 2016

If we start from an initial concentration of $A$, it is ${\left[A\right]}_{0}$. Then, it approaches a final concentration $\left[A\right]$ that is half in quantity.

Therefore, the first half-life is written as $\left[A\right] = \frac{1}{2} {\left[A\right]}_{0}$.

To find further half-lives, keep halving the concentration $n$ times. So, we would have:

$\left[A\right] = {\left[A\right]}_{0} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \cdot \cdot$

or

$\textcolor{g r e e n}{\left[A\right] = {\left[A\right]}_{0} {\left(\frac{1}{2}\right)}^{n}}$

If we want to determine the number of half-lives $n$, then we can use the total time passed $t$ and divide by the half-life ${t}_{\text{1/2}}$.

So, we could write this in a more convenient form as

$\textcolor{g r e e n}{\left[A\right] = {\left[A\right]}_{0} {\left(\frac{1}{2}\right)}^{t \text{/"t_"1/2}}}$

Or, in a more universal form, since $\left[A\right]$ and ${\left[A\right]}_{0}$ have the same units, we could easily just call the quantity of the decaying substance as a function of time $N \left(t\right)$ to get

$\textcolor{b l u e}{N \left(t\right) = {N}_{0} {\left(\frac{1}{2}\right)}^{t \text{/"t_"1/2}}}$

where $N$ can be atoms, grams, mols, whatever.