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To see how the number of nuclei declines to half its original value in one half-life, let $t={t}_{1/2}$ in the exponential in the equation $N={N}_{0}{e}^{-\mathrm{\lambda t}}$ . This gives $N={N}_{0}{e}^{-\mathrm{\lambda t}}={N}_{0}{e}^{\mathrm{-0.693}}=0.500{N}_{0}$ . For integral numbers of half-lives, you can just divide the original number by 2 over and over again, rather than using the exponential relationship. For example, if ten half-lives have passed, we divide $N$ by 2 ten times. This reduces it to $N/\text{1024}$ . For an arbitrary time, not just a multiple of the half-life, the exponential relationship must be used.
Radioactive dating is a clever use of naturally occurring radioactivity. Its most famous application is carbon-14 dating . Carbon-14 has a half-life of 5730 years and is produced in a nuclear reaction induced when solar neutrinos strike ${}^{\text{14}}\mathrm{N}$ in the atmosphere. Radioactive carbon has the same chemistry as stable carbon, and so it mixes into the ecosphere, where it is consumed and becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon. Thus, if you know the number of carbon nuclei in an object (perhaps determined by mass and Avogadro’s number), you multiply that number by $1\text{.}3\times {\text{10}}^{-\text{12}}$ to find the number of ${}^{\text{14}}\text{C}$ nuclei in the object. When an organism dies, carbon exchange with the environment ceases, and ${}^{\text{14}}\text{C}$ is not replenished as it decays. By comparing the abundance of ${}^{\text{14}}\text{C}$ in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the artifact’s age (or time since death). Carbon-14 dating can be used for biological tissues as old as 50 or 60 thousand years, but is most accurate for younger samples, since the abundance of ${}^{\text{14}}\text{C}$ nuclei in them is greater. Very old biological materials contain no ${}^{\text{14}}\text{C}$ at all. There are instances in which the date of an artifact can be determined by other means, such as historical knowledge or tree-ring counting. These cross-references have confirmed the validity of carbon-14 dating and permitted us to calibrate the technique as well. Carbon-14 dating revolutionized parts of archaeology and is of such importance that it earned the 1960 Nobel Prize in chemistry for its developer, the American chemist Willard Libby (1908–1980).
One of the most famous cases of carbon-14 dating involves the Shroud of Turin, a long piece of fabric purported to be the burial shroud of Jesus (see [link] ). This relic was first displayed in Turin in 1354 and was denounced as a fraud at that time by a French bishop. Its remarkable negative imprint of an apparently crucified body resembles the then-accepted image of Jesus, and so the shroud was never disregarded completely and remained controversial over the centuries. Carbon-14 dating was not performed on the shroud until 1988, when the process had been refined to the point where only a small amount of material needed to be destroyed. Samples were tested at three independent laboratories, each being given four pieces of cloth, with only one unidentified piece from the shroud, to avoid prejudice. All three laboratories found samples of the shroud contain 92% of the ${}^{\text{14}}\text{C}$ found in living tissues, allowing the shroud to be dated (see [link] ).
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