Question

How can you model half life decay?

Answer 1

The equation would be:

`[A] = 1/(2^(t"/"t_"1/2"))[A]_0`

Read on to know what it means.

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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.