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

1 Answer
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