**James Richard Fromm**

All radioactive decompositions, or **decays**, follow first-order
kinetics. Indeed, radioactive decay is considered to be the classic example of first-order
reaction kinetics. This is not surprising because the energy of any molecular collision or
interaction is negligible in comparison to the energies involved in nuclear processes. As
a consequence the decay of radioactive isotopes is usually described in terms of
half-lives rather than in terms of rate constants. A table of half-lives of selected
radioactive isotopes is useful in discussing nuclear decay.

When dealing with radioactive isotopes the concentration is difficult to measure and so
concentration is less useful than the actual number N of nuclei of some type present. The
integrated form of the first-order rate law is then ln(N/N_{0}) = -*kt*.

Use of the equation for half-life given above to eliminate the rate constant yields

ln(N/N_{0}) = (ln 0.5)*t*/*t*_{1/2}, or

log N = log N_{0} - 0.301...*t*/*t*_{1/2}

This latter formula is often particularly convenient, because tables of the half-lives
of radioactive isotopes are available. Since the count rate is directly proportional to N,
and the count rate at initial time is directly proportional to N_{0}, the
logarithm of the ratio of count rates can be used to obtain the time *t* when the
half-life *t*_{1/2} is known.

One of the more useful applications of radiochemical kinetics is in the dating of rocks
and archaeological specimens. Dr. W.F. Libby was awarded the 1960 Nobel Prize in chemistry
for the development of the most well-known such technique, **radiocarbon dating**.

Cosmic radiation bombarding the atmosphere establishes a small but steady concentration
of the radioactive isotope ^{14}C, which through the carbon dioxide cycle is
spread uniformly through the Earth's biosphere. All living organisms exist in equilibrium
with this radioactive carbon, and this equilibrium amount of radioactive carbon yields
15.3 disintegrations per minute per gram of total carbon. The remainder of the carbon,
mostly ^{12}C, does not decay, but ^{14}C decays to ^{14}N with
emission of a beta particle. A beta particle, which is a very energetic electron, can be
detected by radiation counters. The half-life of ^{14}C is 5.76 x 10^{+3}
years.

The information given above provides the necessary background for an understanding of
radiocarbon dating. When an organism dies, it ceases to exchange carbon with its
environment and the equilibrium concentration of ^{14}C begins to decrease in
accordance with the first-order kinetic rate law. The date of death can be calculated from
the measured rate of decay of a carbon sample, assuming that no recent carbon contaminated
the sample.

Example. The papyrus wrappings of an Egyptian mummy are found to have a ^{14}C
disintegration rate of 7.48 disintegrations per minute per gram carbon. If the present
year is 2000, we can calculate in what year the papyrus (and presumably also the mummified
person) died.

Since count rate is directly proportional to the number of atoms of ^{14}C
present, N, log 7.48 = log 15.3 - 0.301 *t*/(5760 y). Then *t* is 5947 years
before the present (AD 2000), so the papyrus died in 3947 BC.

Radiocarbon dating is useful for archaeological specimens, but for materials which died
more than about 50,000 years ago the residual ^{14}C activity is so low that
accurate measurements can no longer be made. The ages of ancient trees calculated from
their ^{14}C content differs somewhat from their ages determined by counting
annual growth rings. This is believed to be due to slight fluctuations in cosmic ray
intensity throughout history, and the tree ring counts are being used by archaeologists to
correct ^{14}C dating for these fluctuations.

Older objects can be dated by potassium-argon dating or uranium-lead dating. The decay
of ^{40}K to ^{40}Ar has a very long half-life and analysis of the ^{40}K/^{40}Ar
ratio by mass spectrometry can give a date at which the rock was formed. However, the loss
of gaseous argon from rocks is comparatively easy and this method may not be reliable on
all samples. Uranium-lead dating is based on the natural decay chain of ^{238}U,
which ends with the stable isotope ^{206}Pb after following several decay steps
with much shorter-lived isotopes. If it is assumed that all of the ^{206}Pb in a
mineral originated from ^{238}U, then a date at which the uranium was formed can
be obtained from the ^{238}U/^{206}Pb ratio.

Example. In a sample of uranium ore the mass of ^{238}U found was 10.67 mg and
that of ^{206}Pb found was 2.81 mg. We can estimate the age of the mineral from
these data as follows.

The mass of ^{238}U which decayed to produce 2.81 mg of ^{206}Pb is
2.81(238/206) = 3.25 mg. The original sample therefore must have contained 3.25 + 10.67 =
13.92 mg of ^{238}U. Using the equation for half-life,

log N = log N_{0}- 0.301*t*/*t*_{1/2}

log 10.67 = log 13.92 - 0.301*t*/(4.51 x 10^{+9} y)

The age of the uranium ore is estimated as 1.73 x 10^{+9} years, which suggests
that the solar system was formed about two thousand million years ago.

Copyright 1997 James R. Fromm