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

Sep 15, 2014

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