## Kinetics of Nuclear Decay

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/N0) = -kt.

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

ln(N/N0) = (ln 0.5)t/t1/2, or

log N = log N0 - 0.301...t/t1/2

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 N0, the logarithm of the ratio of count rates can be used to obtain the time t when the half-life t1/2 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 14C, 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 12C, does not decay, but 14C decays to 14N 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 14C 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 14C 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 14C 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 14C 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 14C activity is so low that accurate measurements can no longer be made. The ages of ancient trees calculated from their 14C 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 14C dating for these fluctuations.

Older objects can be dated by potassium-argon dating or uranium-lead dating. The decay of 40K to 40Ar has a very long half-life and analysis of the 40K/40Ar 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 238U, which ends with the stable isotope 206Pb after following several decay steps with much shorter-lived isotopes. If it is assumed that all of the 206Pb in a mineral originated from 238U, then a date at which the uranium was formed can be obtained from the 238U/206Pb ratio.

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

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

log N = log N0- 0.301t/t1/2

log 10.67 = log 13.92 - 0.301t/(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.