The following primer was written by Donald McFarlane, Ph.D. (Claremont Colleges, Claremont, California) and is used with his permission. Diagram of the difference between straight-line depletion and radioactive decay. In blue: Uniform straight-line depletion of most everyday processes. In orange: By contrast, the radioactive decay curve approaches zero line asymptotically. The end of one half-life interval is the beginning of a new one.
Elements differ according to the number of protons in their nuclei. Thus hydrogen has one, helium two, and lithium three protons in their respective nuclei. The proton number is often given the notation 'Z'. The number of neutrons in an atomic nucleus does not change the element, but it does change the atomic mass 'M'. Since the mass of a proton and a neutron are equal, the atomic mass of an element is determined by the number of protons plus the number of neutrons. The units are atomic mass units, or 'amu.' Thus ordinary hydrogen has one proton, no neutrons, and an atomic mass of 1. A second form of hydrogen has one proton (obviously) and one neutron, so its atomic mass is 2. Two or more forms of the same element with differing masses are termed 'isotopes.' There are about 1,700 isotopes in the periodic table, of which about 260 are stable (= not radioactive). Carbon (Z=6) exists in three isotopes, with 6, 7, and 8 neutrons respectively. These are termed 12C, 13C, and 14C by reference to their atomic masses. 12C and 13C are stable isotopes, whereas 14C is unstable and subject to radioactive decay. 14C decays by a process called 'beta decay,' in which one of the 8 neutrons breaks apart and ejects a high-speed beta particle, converting itself into a proton. Thus, the atom loses a neutron but gains a proton. The atomic weight does not change, but the atom is now Z=7 which is nitrogen. In summary, one 14C atom has decayed into one 14N atom by the emission of a beta particle. Any 14C atom has a small but finite probability of undergoing spontaneous decay at any time. That probability is fixed for any specific radioactive isotope, and is independent of the amount of that element or its physical environment. For 14C, the probability that any atom will decay spontaneously in 5,730 years is 50%. This is usually expressed as the 'half-life' of the isotope. In summary, if you start with any amount of 14C -- from a teaspoonful to a truckload -- you will have exactly half of that amount left 5,730 years later. When plotted graphically, these numbers would give an exponentially declining curve. The mathematics of exponential decay are such that the amount of 14C in a fossil is related to the amount of 14C present at the death of an animal. This is expressed by the following equation:
A=A0e-ht where A is the amount of 14C in the fossil; A0 is the amount present at the time of death; e is the mathematical constant equal to 2.718; h is the half life; t is the age of the fossil. By algebraic rearrangement we can solve for t: t = 19.035 x 103 log (A0/A) 14C is constantly created in the upper atmosphere by cosmic ray bombardment of nitrogen. If, as a first approximation, we assume that this production rate is constant then we can use the modern level of 14C as a proxy for the amount of 14C in the fossil at the time of death. This value is 13.56 decays per minute (dpm) per gram of carbon. If we measured the radioactivity of the carbon in a fossil mammal bone and found it to be 6.78 dpm, the equation becomes: t = 19.035 x 103 log (13.56/6.78), which is… t = 19.035 x 103 log (2), which is… t = 19.035 x 103 (0.3010), which is… t = 19.035 x (301), which is… t = 5,730 In other words, the fossil age is 5,730 years relative to the present, or 5,730 yrbp. Complications:
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