Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by the Curies, decay faster than uranium. This means they have shorter lifetimes, producing a greater rate of decay. In this section we explore half-life and activity, the quantitative terms for lifetime and rate of decay.

## Half-Life

Why use a term like half-life rather than lifetime? The answer can be found by examining this figure, which shows how the number of radioactive nuclei in a sample decreases with time. The *time in which half of the original number of nuclei decay* is defined as the **half-life**, \({t}_{1/2}\). Half of the remaining nuclei decay in the next half-life. Further, half of that amount decays in the following half-life. Therefore, the number of radioactive nuclei decreases from \(N\) to \(N/2\) in one half-life, then to \(N/4\) in the next, and to \(N/8\) in the next, and so on. If \(N\) is a large number, then *many* half-lives (not just two) pass before all of the nuclei decay.

Nuclear decay is an example of a purely statistical process. A more precise definition of half-life is that *each nucleus has a 50% chance of living for a time equal to one half-life \({t}_{1/2}\)*. Thus, if \(N\) is reasonably large, half of the original nuclei decay in a time of one half-life. If an individual nucleus makes it through that time, it still has a 50% chance of surviving through another half-life. Even if it happens to make it through hundreds of half-lives, it still has a 50% chance of surviving through one more. The probability of decay is the same no matter when you start counting. This is like random coin flipping. The chance of heads is 50%, no matter what has happened before.

There is a tremendous range in the half-lives of various nuclides, from as short as \({\text{10}}^{-\text{23}}\) s for the most unstable, to more than \({\text{10}}^{\text{16}}\) y for the least unstable, or about 46 orders of magnitude. Nuclides with the shortest half-lives are those for which the nuclear forces are least attractive, an indication of the extent to which the nuclear force can depend on the particular combination of neutrons and protons. The concept of half-life is applicable to other subatomic particles, as will be discussed in Particle Physics. It is also applicable to the decay of excited states in atoms and nuclei. The following equation gives the quantitative relationship between the original number of nuclei present at time zero (\({N}_{0}\)) and the number (\(N\)) at a later time \(t\):

\(N={N}_{0}{e}^{-\mathrm{\lambda t}}\text{,}\)

where \(e=\text{2.71828}\text{…}\) is the base of the natural logarithm, and \(\lambda \) is the **decay constant** for the nuclide. The shorter the half-life, the larger is the value of \(\lambda \), and the faster the exponential \({e}^{-\mathrm{\lambda t}}\) decreases with time. The relationship between the decay constant \(\lambda \) and the half-life \({t}_{1/2}\) is

\(\lambda =\cfrac{ln(2)}{{t}_{1/2}}\approx \cfrac{0\text{.}\text{693}}{{t}_{1/2}}\text{.}\)

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 \(N={N}_{0}{e}^{-\mathrm{\lambda t}}={N}_{0}{e}^{-0.693}=0.500{N}_{0}\). 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 \({}^{\text{14}}N\) 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 \(1\text{.}3×{\text{10}}^{-\text{12}}\) 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 this figure). 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 this example).

There are other forms of radioactive dating. Rocks, for example, can sometimes be dated based on the decay of \({}^{\text{238}}\text{U}\). The decay series for \({}^{\text{238}}\text{U}\) ends with \({}^{\text{206}}\text{Pb}\), so that the ratio of these nuclides in a rock is an indication of how long it has been since the rock solidified. The original composition of the rock, such as the absence of lead, must be known with some confidence. However, as with carbon-14 dating, the technique can be verified by a consistent body of knowledge. Since \({}^{\text{238}}\text{U}\) has a half-life of \(4\text{.}5×{\text{10}}^{9}\) y, it is useful for dating only very old materials, showing, for example, that the oldest rocks on Earth solidified about \(3\text{.}5×{\text{10}}^{9}\) years ago.