Question

How do you derive first order half life?

Answer 1

You can always start from the first-order rate law:

`r(t) = k[A] = -(d[A])/(dt)`

for the reaction:

`A -> B`

Separate variables:

`-kdt = 1/([A])d[A]`

Integrate from the initial to the final state. It is convenient to set `t_0 = 0`.

`-int_(0)^(t)kdt = int_([A]_0)^([A])d[A]`

`-(kt - k*0) = ln[A] - ln[A]_0`

`=> ln[A] - ln[A]_0 = -kt`

`=> ln\frac([A])([A]_0) = -kt`

For the half-life, at time `t = t_"1/2"`, the concentration of `A` dropped to `1/2[A]_0`. Therefore:

`ln\frac(1/2cancel([A]_0))(cancel([A]_0)) = -kt_"1/2"`

`ln (1/2) = -kt_"1/2"`

`-ln (1/2) = kt_"1/2"`

`ln (1/2)^(-1) = kt_"1/2"`

`ln 2 = kt_"1/2"`

Therefore:

`color(blue)(t_"1/2" = ln2/k)`