Solution: If 4 years is one half-life, then 12 years is three half-lives. During the first half-life—the
first 4 years—half of the sample will have decayed. During the second half-life (years 5
to 8), half of the remaining half will decay, leaving one-fourth of the original. During
the third and final period (years 9 to 12), half of the remaining fourth will decay,
leaving one-eighth of the original sample. Thus the fraction remaining after 3 half-lives
is (1/2)^3 or (1/8).
BASIC CONCEPT
Fraction of original nuclei remaining after n half-lives = Fraction of nuclei that has decayed away after n half-lives = The fact that different radioactive species have different characteristic half-lives is what enables
scientists to determine the age of organic materials. ^14 C, for example, is generated from nuclear
reactions induced by high-energy cosmic rays from outer space. There is therefore always a certain
fraction of this isotope in the carbon found on Earth. Living things, like trees and animals, are
constantly exchanging carbon with the environment, and thus will have the same ratio of carbon-14
to carbon-12 within them as the atmosphere. Once they die, however, they stop incorporating
carbon from the environment, and start to lose carbon-14 because of its radioactivity. It undergoes
a β-decay mechanism:
The longer the species has been dead, the less carbon-14 it will still have: for example, if the ratio of
(^14) C to (^12) C is half of that of the atmosphere, then we would conclude that the species existed about
one half-life of ^14 C ago.
Exponential Decay
Let n be the number of radioactive nuclei that have not yet decayed in a sample. It turns out that the
rate at which the nuclei decay (∆n/∆t) is proportional to the number that remain (n). This suggests
the equation: