Question

What is the half life decay rate formula?

Answer 1

If we start from an initial concentration of `A`, it is `[A]_0`. Then, it approaches a final concentration `[A]` that is half in quantity.

Therefore, the first half-life is written as `[A] = 1/2[A]_0`.

To find further half-lives, keep halving the concentration `n` times. So, we would have:

`[A] = [A]_0cdot1/2cdot1/2cdotcdotcdot`

or

`color(green)([A]= [A]_0(1/2)^n)`

If we want to determine the number of half-lives `n`, then we can use the total time passed `t` and divide by the half-life `t_"1/2"`.

So, we could write this in a more convenient form as

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

Or, in a more universal form, since `[A]` and `[A]_0` have the same units, we could easily just call the quantity of the decaying substance as a function of time `N(t)` to get

`color(blue)(N(t)= N_0(1/2)^(t"/"t_"1/2"))`

where `N` can be atoms, grams, mols, whatever.