(From Borelli and Coleman (1996).) Olduvai Gorge, in Kenya, cuts through
volcanic flows, tuff (volcanic ash), and sedimentary deposits. It is the site
of bones and artefacts of early hominids, considered by some to be precursors
of man. In 1959, Mary and Louis Leakey uncovered a fossil hominid skull and
primitive stone tools of obviously great age, older by far than any hominid
remains found up to that time. Carbon-14 dating methods being inappropriate
for
a specimen of that age and nature, dating had to be based on the ages of the
underlying and overlying volcanic strata.
The method used was that of potassium-argon decay. The potassium-argon clock
is an accumulation clock, in contrast to the \({ }^{14} \mathrm{C}\) dating
method. The potassium-argon method depends on measuring the accumulation of
'daughter' argon atoms, which are decay products of radioactive potassium
atoms. Specifically, potassium- \(40\left({ }^{40} \mathrm{~K}\right)\) decays
to argon \(\left({ }^{40} \mathrm{Ar}\right)\) and to Calcium- 40 \(\left({
}^{40} \mathrm{Ca}\right)\) by the branching cascade illustrated below in
Figure 2.17. Potassium decays to calcium by emitting a \(\beta\) particle (i.e.
an electron). Some of the potassium atoms, however, decay to argon by
capturing an extra-nuclear electron and emitting a \(\gamma\) particle.
The rate equations for this decay process may be written in terms of \(K(t),
A(t)\) and \(C(t)\), the potassium, argon and calcium in the sample of rock:
$$
\begin{aligned}
&K^{\prime}=-\left(k_{1}+k_{2}\right) K \\
&A^{\prime}=k_{1} K \\
&C^{\prime}=k_{2} K
\end{aligned}
$$
where
$$
k_{1}=5.76 \times 10^{-11} \text { year }^{-1}, \quad k_{2}=4.85 \times
10^{-10} \text { year }^{-1} \text { . }
$$
(a) Solve the system to find \(K(t), A(t)\) and \(C(t)\) in terms of \(k_{1},
k_{2}\), and \(k_{3}=k_{1}+k_{2}\), using the initial conditions \(K(0)=k_{0},
A(0)=C(0)=0 .\)
(b) Show that \(K(t)+A(t)+C(t)=k_{0}\) for all \(t \geq 0 .\) Why would this be
the case?
(c) Show that \(K(t) \rightarrow 0, A(t) \rightarrow k_{1} k_{0} / k_{3}\) and
\(C(t) \rightarrow k_{2} k_{0} / k_{3}\) as \(t \rightarrow \infty\).
(d) The age of the volcanic strata is the current value of the time variable
\(t\) because the potassiumargon clock started when the volcanic material was
laid down. This age is estimated by measuring the ratio of argon to potassium
in a sample. Show that this ratio is
$$
\frac{A}{K}=\frac{k_{1}}{k_{3}}\left(e^{k_{3} t}-1\right)
$$
(e) Now show that the age of the sample in years is
$$
\frac{1}{k_{3}} \ell \mathrm{n}\left[\left(\frac{k_{3} A}{k_{1}
K}\right)+1\right]
$$
(f) When the actual measurements were made at the University of California at
Berkeley, the age of the volcanic material (and thus the age of the bones) was
estimated to be \(1.75\) million years. What was the value of the measured ratio
\(A / K ?\)
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