How can you model half life decay?

The equation would be:

##[A] = 1/(2^(t"/"t_"1/2"))[A]_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_"1/2"## becomes half of the current concentration.

So, if we define the current concentration as ##[A]_n## and the upcoming concentration as ##[A]_(n+1)##, then...

##[A]_(n+1) = 1/2[A]_n## ##" "\mathbf((1))##

We call the (1) the recursive half-life decay equation for one half-life occurrence, i.e. when ##t_"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 ##\mathbf(n)## half-lives. For this, we rewrite ##[A]_n## as ##[A]_0## (the initial concentration), and ##[A]_(n+1)## as ##[A]## (the upcoming concentration).

Notice how ##[A]_0## will always be the same, but ##[A]## will keep changing over time.

##[A] = (1/2)(1/2)cdots(1/2)[A]_0##

##= (1/2)^n[A]_0##

##=> [A] = 1/(2^n)[A]_0## ##" "\mathbf((2))##

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_"1/2"## time to occur. When ##n## half-lives occur, each one taking ##t_"1/2"## to occur, it must occur over a set amount of time ##t##. So:

##nt_"1/2" = t## ##" "\mathbf((3))##

That means ##n = t/t_"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:

##color(blue)([A] = 1/(2^(t"/"t_"1/2"))[A]_0)## ##" "\mathbf((4))##

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:

• ##[A]_0##, the initial concentration, and ##[A]##, the upcoming concentration, OR
• ##([A])/[A]_0##, the fraction of the element left after time ##t## passes.