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QUESTION

# 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.