Example 27-7 Ötzi the Iceman
In 1991, two German hikers discovered a human corpse in the Ötztal Alps on the border between Austria and Italy. The
remains were not those of the victim of a climbing accident, but rather a well-preserved natural mummy of a man who
lived during the last Ice Age. The rate of radioactive decay of 14C in the mummy of “Ötzi the Iceman” was measured to
be 0.121 Bq per gram. In a living organism, the rate of radioactive decay of 14C is 0.231 Bq per gram. How long ago did
Ötzi the Iceman live?
Set Up
Once Ötzi died, his body stopped taking in 14C.
After that time the number N(t) of 14C nuclei
in his body decreased due to beta-minus decay.
The 14C decay rate R(t), which is proportional
to N(t), decreased in the same manner. We are
given R1t2 = 0.121 Bq>g for the present-day
decay rate, and R 0 = 0.231 Bq>g (the decay
rate for a living organism and hence the decay
rate at time t = 0, the last date on which Ötzi
was still alive). We’ll solve Equation 27-6
for the present time t (the elapsed time since
Ötzi died), using Equation 27-8 to find the
decay constant l from the known half-life
t1>2 = 5730 y of 14C.
Solve
Rearrange Equations 27-6 and 27-8 to find an
expression for the time t since Ötzi died.
Decay rate as a function of time:
R1t2 = R 0e -lt
(27-6)
Half-life of a radioactive substance:
t1>2 =
ln 2
l
(27-8)
We know the present-day decay rate R(t) and the initial decay rate
in a living organism R0. We want to find the time t since Ötzi died,
so we rearrange Equation 27-6. Divide both sides by R0:
R1t2
= e -lt
R0
Take the natural logarithm of both sides and recall that ln ex = x:
ln a
R1t2
R0
b = ln e -lt = -lt
Divide both sides by 2l:
R1t2
1
b
t = - ln a
l
R0
To get an expression for 1>l, divide both sides of Equation 27-8
by ln 2:
t1>2
1
=
l
ln 2
Putting everything together, the time t since Ötzi died is
t = Substitute the given values into the expression
for t.
Ch27_example.indd 8
t1>2
ln 2
ln a
R1t2
R0
b
We are given t1>2 = 5730 y for 14C, R1t2 = 0.121 Bq>g, and
R 0 = 0.231 Bq>g:
t = -
15730 y2
= -
15730 y2
ln 2
0.693
ln a
0.121 Bq>g
0.231 Bq>g
b = -
15730 y2
0.693
ln 0.524
1 -0.6472 = 5350 y
9/4/13 12:54 PM
Reflect
Our result shows that Ötzi died 5350 y ago. His well-preserved mummy thus gives us a unique look into life in prehistoric Europe.
Carbon-14 dating can only be used on objects less than about 50,000 years old, or about 8 to 10 half-lives of 14C.
For older objects the decay rate of 14C has decreased to such a small value that it is difficult to measure accurately,
and so any determination of age becomes difficult with this technique. For much older objects such as rocks, a similar
approach is used but with isotopes with much longer half-lives. For example, the age of meteorites that fall to Earth is
determined by looking at the ratio of uranium to lead (the endpoint of a series of alpha decays that starts with uranium);
the oldest of these is more than 4.5 * 109 years old.
Ch27_example.indd 9
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