The half-life (τ1/2) of a reaction is the time needed for the concentration of the radioactive substance
to decrease to one-half of its original value. In other words, the half-life is the value of t such that the
following equality holds:
We can substitute in the expression for [A]t:
Manipulation of the above equation leads to the following formula for half-life:
where k is the first-order rate constant.
After one half-life, half of the reactants will be left. After another half-life has elapsed, half of that
half will remain; in other words, the concentration of the reactants will be one-quarter the initial
value. More generally, the concentration of A is after the passage of n number of half-lives.