Question

What are the formulas for half-life in exponential decay?

Answer 1

The formulas for half-life are `t_½ = ln2/λ` and `t_½ = tln2/ln(N_0/N_t)`.

The equation for exponential decay is

(1) `N_t =N_0e^(-λt)`, where

`N_0` is the initial quantity
`N_t` is the quantity at time `t`
`λ` is the exponential decay constant

We can solve this for `λ`:

(2) `λ = 1/tln(N_0/N_t)`

And the formulas for half-life `t_½` are

(3) `t_½ = ln2/λ`

and

(4) `t_½ = tln2/ln(N_0/N_t)`

If you know the value of `λ`, you can use Equation (3). If you don't, you can use Equation (4).

EXAMPLE

Technetium-99m is used for brain scans. If a laboratory receives a shipment of
200 g of this isotope and only 12.5 g of this isotope remain after 24 h, what is the half-life of technetium-99m?

Solution

`t_½ = tln2/ln(N_0/N_t) = 24ln2/ln("200 g"/"12.5 g")` = 6.0 h