# How can you model half life decay?

##### 1 Answer

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 *upcoming* concentration as

#[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 ** 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 **half-lives**. For this, we rewrite **initial** concentration), and **upcoming** concentration).

Notice how ** always** be the same, but

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

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

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

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

That means

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

**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.