# How can you model half life decay?

Jul 3, 2016

The equation would be:

$\left[A\right] = \frac{1}{{2}^{t \text{/"t_"1/2}}} {\left[A\right]}_{0}$

Read on to know what it means.

Just focus on the main principle:

The upcoming concentration of reactant $A$ after half-life time ${t}_{\text{1/2}}$ becomes half of the current concentration.

So, if we define the current concentration as ${\left[A\right]}_{n}$ and the upcoming concentration as ${\left[A\right]}_{n + 1}$, then...

${\left[A\right]}_{n + 1} = \frac{1}{2} {\left[A\right]}_{n}$ $\text{ } \setminus m a t h b f \left(\left(1\right)\right)$

We call the (1) the recursive half-life decay equation for one half-life occurrence, i.e. when ${t}_{\text{1/2}}$ has passed by only once. This isn't very useful though, because half-lives can range from very slow (thousands of years) to very fast (milliseconds!).

Let's go through another half-life, until we've gone through $\setminus m a t h b f \left(n\right)$ half-lives. For this, we rewrite ${\left[A\right]}_{n}$ as ${\left[A\right]}_{0}$ (the initial concentration), and ${\left[A\right]}_{n + 1}$ as $\left[A\right]$ (the upcoming concentration).

Notice how ${\left[A\right]}_{0}$ will always be the same, but $\left[A\right]$ will keep changing over time.

$\left[A\right] = \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) \cdots \left(\frac{1}{2}\right) {\left[A\right]}_{0}$

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

$\implies \left[A\right] = \frac{1}{{2}^{n}} {\left[A\right]}_{0}$ $\text{ } \setminus m a t h b f \left(\left(2\right)\right)$

Now we have (2), the equation for any number of half-life decays... once we know how many half-lives passed by.

However, (2) can be made more convenient since we know that each half-life takes ${t}_{\text{1/2}}$ time to occur. When $n$ half-lives occur, each one taking ${t}_{\text{1/2}}$ to occur, it must occur over a set amount of time $t$. So:

$n {t}_{\text{1/2}} = t$ $\text{ } \setminus m a t h b f \left(\left(3\right)\right)$

That means $n = \frac{t}{t} _ \text{1/2}$, which is saying that we can divide the total time passed during the process by the time it takes to lose half of $A$ again to get the number of half-lives that passed by.

Therefore:

$\textcolor{b l u e}{\left[A\right] = \frac{1}{{2}^{t \text{/"t_"1/2}}} {\left[A\right]}_{0}}$ $\text{ } \setminus m a t h b f \left(\left(4\right)\right)$

So, we can use (4) to determine half-lives of any typical radioactive element for which we know $t$, the time passed during the half-life decay(s) AND:

• ${\left[A\right]}_{0}$, the initial concentration, and $\left[A\right]$, the upcoming concentration, OR
• $\frac{\left[A\right]}{A} _ 0$, the fraction of the element left after time $t$ passes.