Introduction 315568 years. Radioactive carbon is constantly being produced in the earth's
upper atmosphere by incoming cosmic rays. These rays produce neutrons,
which in turn collide with nitrogen 14 to produce carbon 14. The radioactive
carbon is oxidized and forms radioactive carbon dioxide, which circulates in
the earth's atmosphere. Plants which "breathe" carbon dioxide also breathe
radioactive carbon dioxide and through their life processes absorb radiocarbon
in their tissue. Likewise, animals which eat these plants absorb radiocarbon in
their tissue. The rate of absorption of radiocarbon by living tissue is in equi-
librium with the rate of disintegration. However, when a plant or animal dies,
it ceases to absorb radiocarbon and only the process of disintegration contin-
ues. The age of a substance of organic origin can be estimated by measuring
the radioactivity of carbon 14 of a sample of that substance. For example,
a piece of charcoal that has one-half the radioactivity of a living tree died
approximately 5568 years ago, and a piece of charcoal that has one-fourth the
radioactivity of a living tree died approximately 11,136 years ago.
Solving equation (9), kT = - ln 2, for the decay constant k, and substituting
5568 for the h alf-life, T, we find the decay constant for radioactive carbon,
(^14) C, is
ln2
k = - = -.00012449/years.
5568 years
From equation (6) we see that the amount, Q(t), of^14 C present at time t ::'.: to
in some organic substance is
Q(t) = Qoek(t-to)
where Q 0 is the amount that was present at the time to when the substance
died. Differentiating this equation, we find that the rate of disintegration,
R( t) , of^14 C at any time t ::'.: to is
R(t) = kQoek(t-to).
At time t = t 0 the rate of disintegration is
R(to) = kQo.So the ratio of the disintegration rate at time t to the disintegration rate at
time to is
R(t) = ek(t-to).
R(to)
Solving for the time since the death of the substance, t -to, we find
(10) t - to = k^1 ln ( R( R(t) to) ).
Assuming that for any particular living substance the rate of disintegration of
(^14) C is a constant (that is , the rate of disintegration is the same now as it was