Fig. 10-1

image via: nobelprize.org

Some natural processes show an exponential decrease, rather than an increase. An important type of exponential decrease is shown by the decay rate of radioactive materials.

Element number 84, the first radioactive element, was discovered by Marie Curie, a Polish chemist working in Paris. She named it Polonium. All of the elements from 84 to 92 are radioactive, which means that they are unstable, and break down spontaneously, emitting particles and rays. The atoms disintegrate on a predictable schedule, not according to the age of individual atoms, but according to the total present at any moment. The decay process is a statistical process, which can be described by equations like the population growth equations, but with a negative rate constant.

The first radioactive element isolated was radium, separated from uranium ore by Marie (Fig. 10-1) and Pierre Curie in 1898. (Pierre taught her French, and she taught him Polish.) The decay process is symbolized by:

 ₈₈Ra²²⁶ -----> ₈₆Rn²²² + α

 The products are a radon atom and an alpha particle, which consists of 2 protons and 2 neutrons, thus accounting for both the decrease of 2 in atomic number and the decrease of 4 in atomic weight.The rate of decay is:

dN/dt = λN

N is number of atoms; λ is a rate constant that is negative.

The same rearrangement and integration used in Chapter 7 yields:

λ = (ln N_t- ln N₀) / Δt

The decay constant can be calculated from the weight of the Curie sample, which was originally 200.0 mg, and 100 years later weighed 191.62 mg.

= (ln 191.62 - ln 200.0) / 100y = - 0.042803122 / 100y = -0.00042803122/y =  - 4.28 X 10^-4/y

The decay constant is an important factor in the use of radioactivity in  tracer experiments, radiation therapy, and dating of ancient artifacts. Another  useful constant is called the half-life, which applies to a chemist who needs to get a life. No, not really. It is the time to lose one-half of the original radioactivity, and is related to the decay constant. For the radium sample, the calculation is:

t_1/2 = ln 0.5/ λ =  - 0.69314718 / - 4.28 X 10^-4 = 1620 years

All of the atoms of a single element are not identical. They all have the same number of protons and electrons, and therefore the same atomic number, and the same chemistry. However, variation in the number of neutrons results in different kinds of atoms of the same element, called isotopes. For example, most carbon atoms are C-12, but there are three other isotopes. The abundances are ¹² = 98.9%, the heavy isotope C¹³ = 1.1 %, and very small  quantities of C¹¹ and C¹⁴, both radioisotopes.

C-11 has been produced from boron by bombardment with deuterons in a cyclotron. The reaction is symbolized by:

₅B¹⁰ (d,n) ₆C¹¹

C¹¹ decays back to boron, and the emitted positron can be detected by the appropriate radiation counter. Until the atoms disintegrate, they behave chemically just like ordinary C-12 carbon.

C¹¹ ----->  B¹¹ + β⁺

This result indicates that experiments with C¹¹ must be completed quickly, or the radioactivity will decay to the point where there is too little to measure. The general rule is a limit of 8 times the half-life, in this case only 2 hours 43 minutes. On the other hand, C¹⁴

Small amounts of C¹⁴ are continuously produced in the atmosphere by cosmic-ray bombardment of nitrogen. C-14 is produced in the laboratory by irradiation of N-14 with neutrons in a nuclear reactor, by the same n,p reaction.

₇N ¹⁴ (n,p) ₆C14

(The subscript numbers 7 and 6 are superfluous, because every chemist knows that nitrogen is element 7 and carbon is element 6.) The decay is by emission of beta particles, which are fast-moving electrons:

C¹⁴-----> N¹⁴+ β^-

Carbon-14 has been extremely useful in tracing pathways of metabolism, especially in photosynthesis studies, and in some aspects of medical research.

Important to archeologists was the use of C-14 by Willard F. Libby to determine the age of prehistoric artifacts. This radioactive isotope is present at very low concentration in the CO₂ of the air. Carbon dioxide is used by plants to make sugars, starches, and cellulose. C-14 enters the bodies of animals that eat plants. When living things die, they are removed from the carbon cycle, the C¹⁴ is no longer replenished, and it continues to decay. Thus, the samples of ancient wood, charcoal, and bone have less radioactivity than modern materials. Special apparatus, consisting of very sensitive detectors and heavy shielding, were built to record these extremely low levels of radiation.

Fig. 10-2

photo credit: vgm8383

The decay constant λ can be used in the same way as the growth constant r in the population-growth equation, to calculate age of the sample counted:

The basic equation describing the process is:

Δt = (ln Nt - ln  N₀) / λ

For example, a sample of charcoal taken in 2000 from the late Neolithic Stonehenge site (Fig. 10-2) had C¹⁴ radioactivity = 7.96 counts per minute (c/m) per gram carbon. Modern samples of material from living organisms average 13.56 c/m per g C.

Δt = (ln 7.96 - ln 13.56) / - 0.000120968/ y = -0.532695282/ - 0.000120968/y

= 4 403.6 years before present

This makes the age of the Stonehenge site:

2000 - 4 404 = 2404 BC.

The Lascaux cave in the Dordogne region of France has many paintings made by a Paleolithic artist, including the long-extinct bison.

Fig. 10-3

photo credit: Wikimedia Commons

Civilization had developed in Mesopotamia and Egypt long before Stonehenge, but some of the artifacts are less ancient. The funerary boat of Sesostris III (Fig. 10-3) was found in his burial chamber inside the pyramid near Dahshur, Egypt. A piece of wood, probably cedar, from the boat was obtained from the Chicago Natural History Museum by Dr. Lilly in 1949, and was analyzed for C-14 age.

Willard F. Libby received a Nobel prize in 1960, published a book on radiocarbon dating in 1965, and died in 1980.

The C-14 method cannot be used for dating fossils, remains of organisms that lived millions of years ago. Some of these have been dated by the K⁴⁰ method. Many minerals contain potassium, and are found in the sedimentary rocks in which fossils are embedded. K⁴⁰ has two decay products, Ca⁴⁰ and Ar⁴⁰ , so the analysis is complicated, requiring assumptions about the original quantity of K. Data from mass spectrometer analysis of a sample of mica from the Cambrian indicates N₀ = 0.0741mg of K⁴⁰, and present Nt = 0.0541mg. The decay constant is: λ = 5.501 X 10^-10 / y.

Δt = (ln 0.0541 - ln 0.0741) /  5.501 X 10^-10/ y = 0.0571862 X 10¹⁰ y

= 570  X 10⁶ = 570 million years

Geologists agree that this is the correct age for the Cambrian, the earliest period with abundant animal fossils. About half of the known Cambrian fossils are trilobites, beautiful marine animals that became totally extinct in the Permian period.

The time scale of the geological periods and epochs has been established through the use of radioactive elements of known half-life.

The decrease in sunlight with ocean depth also proceeds exponentially, as if each successive tiny layer were absorbing the same small fraction of light reaching it from above. Here are measured fractions of blue-green light at various depths:

Survival of  humans decreases rapidly after age 75. A college alumni directory contains the names of all male alumni from each class. From this list, and the number of those whose deaths have been reported, an average fraction of those surviving to each year can be calculated.

Note the similarity between the graph of light in the ocean and the graph of survival after age 76.