Radioactive nuclei are unstable and spontaneously transform into more stable nuclei through radioactive decay. However, not all radioactive isotopes decay at the same rate. Some, such as radium and polonium, decay relatively quickly, while others, such as uranium-238, remain radioactive for billions of years. Because different isotopes decay at different rates, scientists use quantitative measures such as half-life and activity to describe how long radioactive materials persist and how rapidly they decay.

Half-Life

Why do scientists use the term half-life instead of simply referring to the lifetime of a radioactive nucleus? The answer lies in the random nature of radioactive decay. As shown in Figure 120.1, the number of radioactive nuclei in a sample decreases gradually over time rather than all disappearing at once. The half-life, denoted by [latex]t_{1/2}[/latex], is defined as the time required for half of the radioactive nuclei in a sample to decay. After one half-life, one-half of the original nuclei remain. After a second half-life, half of those remaining nuclei decay, leaving one-quarter of the original sample. After a third half-life, one-eighth of the original nuclei remain, and so on. For a large collection of nuclei, this predictable pattern continues over many half-lives. The amount of radioactive material steadily decreases but never reaches exactly zero. This behavior is an example of exponential decay.

Although the behavior of a large sample is highly predictable, the decay of any individual nucleus is completely random. A nucleus does not "age" or become more likely to decay simply because it has existed for a long time. A more precise way to describe half-life is in terms of probability. During one half-life, every radioactive nucleus has a 50% chance of decaying and a 50% chance of surviving. If a nucleus survives one half-life, it again has a 50% chance of surviving the next half-life. Even if it survives many half-lives, its probability of decaying during the next half-life remains exactly the same. This idea is similar to repeatedly flipping a fair coin. Each flip has a 50% chance of landing heads, regardless of the outcomes of previous flips. Likewise, each radioactive nucleus behaves independently of its past history.

Half-lives span an enormous range of values, from approximately [latex]10^{-23}\text{ s}[/latex] for the most unstable nuclei to more than [latex]10^{16}\text{ y}[/latex] for the most stable radioactive isotopes—a range of nearly 46 orders of magnitude. The concept of half-life also applies to unstable subatomic particles and to excited atomic and nuclear states. The number of radioactive nuclei remaining after a time [latex]t[/latex] is described by the exponential decay equation

where [latex]N_0[/latex] is the initial number of radioactive nuclei, [latex]N[/latex] is the number remaining after time [latex]t[/latex], [latex]e=2.71828\ldots[/latex] is the base of the natural logarithm, and [latex]\lambda[/latex] is the decay constant. A larger decay constant corresponds to a faster rate of decay and therefore a shorter half-life. The decay constant and half-life are related by

This equation shows that the decay constant is inversely proportional to the half-life: isotopes with long half-lives have small decay constants, whereas isotopes with short half-lives have large decay constants. Substituting [latex]t=t_{1/2}[/latex] into the exponential decay equation gives

confirming that exactly one-half of the original nuclei remain after one half-life. When the elapsed time is an integer number of half-lives, the calculation is straightforward: simply divide the original number of nuclei by 2 for each half-life that has passed. For example, after ten half-lives,

For times that are not whole multiples of the half-life, the exponential equation must be used.

Radioactive Dating

One of the most important applications of radioactive decay is radioactive dating, which allows scientists to determine the age of archaeological artifacts, fossils, and geological materials. The best-known technique is radiocarbon (carbon-14) dating. Carbon-14 has a half-life of 5730 years and is continuously produced in Earth's upper atmosphere when neutrons generated by cosmic rays interact with nitrogen nuclei:

Once formed, radioactive carbon combines with oxygen to produce carbon dioxide, which mixes throughout the atmosphere. Plants absorb this carbon dioxide during photosynthesis, and animals obtain carbon by consuming plants or other animals. As a result, every living organism contains a small, nearly constant fraction of carbon-14 mixed with stable carbon isotopes. Carbon-14 accounts for approximately [latex]1.3\times10^{-12}[/latex] of all carbon atoms. Knowing the total number of carbon atoms in a sample therefore allows the initial number of carbon-14 nuclei to be estimated. When an organism dies, it no longer exchanges carbon with its environment. The carbon-14 already present continues to decay, but no new carbon-14 is incorporated into the tissue. By measuring how much carbon-14 remains and comparing it with the amount found in living organisms, scientists can determine how much time has passed since the organism died. Radiocarbon dating is most useful for biological materials between a few hundred and about 50,000 years old. Beyond this range, so little carbon-14 remains that accurate measurements become increasingly difficult. Modern radiocarbon dating also uses calibration curves to account for small historical variations in atmospheric carbon-14 concentration, improving the accuracy of age estimates. Since its development by American chemist Willard Libby (1908–1980), radiocarbon dating has revolutionized archaeology, paleontology, and environmental science. Libby received the 1960 Nobel Prize in Chemistry for this work. One of the best-known applications of radiocarbon dating involved the Shroud of Turin, a linen cloth traditionally associated with the burial of Jesus (see Figure 120.2). In 1988, small samples of the cloth were independently analyzed by three laboratories using carbon-14 dating. To minimize bias, each laboratory received several unidentified samples, only one of which came from the shroud. All three laboratories obtained consistent results showing that the cloth retained approximately 92% of the carbon-14 found in living tissue. As demonstrated in Example 120.1, this measurement corresponds to a medieval age for the fabric.

Carbon-14 dating is only one example of radioactive dating. Different isotopes are used to study materials of different ages. For example, ancient rocks can be dated using the decay of [latex]{}^{238}\mathrm{U}[/latex] into [latex]{}^{206}\mathrm{Pb}[/latex]. By measuring the ratio of uranium to lead and knowing the half-life of uranium-238, scientists can estimate when the rock solidified. Because uranium-238 has a very long half-life of approximately [latex]4.5\times10^9\ \text{y}[/latex], uranium–lead dating is useful for determining the ages of Earth's oldest rocks and even meteorites. In contrast, carbon-14 dating is limited to relatively recent biological materials because its much shorter half-life causes nearly all of the carbon-14 to decay after about 50,000 years.

Activity: The Rate of Radioactive Decay

When a radioactive source is described as highly active, this means that a large number of nuclei decay in a given amount of time. The activity, represented by [latex]R[/latex], is the rate at which radioactive decays occur:

where [latex]\Delta N[/latex] is the number of nuclei that decay during the time interval [latex]\Delta t[/latex]. Activity describes the number of decay events, not the mass of radioactive material or the amount of energy deposited in tissue. The SI unit of activity is the becquerel (Bq), named after Henri Becquerel, one of the discoverers of radioactivity. One becquerel corresponds to one radioactive decay per second:

Because one becquerel is a very small amount of activity, medical and laboratory sources are often labeled in kilobecquerels (kBq), megabecquerels (MBq), or gigabecquerels (GBq). Another unit of activity is the curie (Ci), named in recognition of the work of Marie and Pierre Curie. It was originally based approximately on the activity of 1 g of [latex]{}^{226}\mathrm{Ra}[/latex] and is now defined exactly as

Thus, one curie corresponds to [latex]3.70\times10^{10}[/latex] decays each second. The curie is a very large unit, so smaller quantities such as the millicurie (mCi) and microcurie ([latex]\mu\text{Ci}[/latex]) are commonly used:

The activity of a sample depends primarily on two quantities:

• the number [latex]N[/latex] of radioactive nuclei present, and
• the half-life [latex]t_{1/2}[/latex] of the isotope.

A sample containing more radioactive nuclei generally undergoes more decays per second. For the same number of nuclei, an isotope with a shorter half-life has a greater activity because its nuclei decay more rapidly. The activity is related to the decay constant by

Using

we obtain

This relationship shows that activity is directly proportional to the number of radioactive nuclei and inversely proportional to the isotope’s half-life. The units used for [latex]t_{1/2}[/latex] must match the desired time unit for the activity. To obtain activity in becquerels, the half-life must be expressed in seconds.

Human-Made Radioactivity

Radioactive isotopes can also be produced artificially in nuclear reactors, particle accelerators, and specialized isotope-production facilities. These isotopes have many beneficial applications, including medical imaging, cancer treatment, sterilization of medical equipment, scientific research, industrial measurements, and food irradiation. In nuclear medicine, isotopes are selected according to the type of radiation they emit, their chemical behavior in the body, and their half-lives. Diagnostic isotopes should generally remain active long enough to complete an examination but decay quickly enough to limit unnecessary exposure. Therapeutic isotopes must deliver sufficient radiation to the intended tissue while minimizing damage to surrounding healthy structures. Radioactive materials must nevertheless be handled carefully. Large accidental releases can contaminate air, soil, water, and food supplies. The 1986 accident at the Chernobyl nuclear power plant released several important radionuclides, including [latex]{}^{131}\mathrm{I}[/latex], [latex]{}^{90}\mathrm{Sr}[/latex], and [latex]{}^{137}\mathrm{Cs}[/latex]. These isotopes differ greatly in half-life, chemical behavior, and biological effects. For example, radioactive iodine can accumulate in the thyroid, whereas cesium can be distributed more broadly through soft tissues. The Chernobyl accident illustrates why both activity and isotope identity matter when evaluating radioactive contamination. A numerical activity alone does not fully describe the health risk unless the type of radiation, exposure pathway, distribution in the body, and duration of exposure are also known.

Human and Medical Applications

Radioactivity has become an indispensable tool in medicine, science, and industry. Carefully controlled radioactive sources are used to diagnose disease, treat cancer, sterilize medical equipment, preserve food, and inspect manufactured components without causing damage. At the same time, radioactive materials must be handled responsibly because accidental releases can have long-lasting environmental and health consequences. As the 1986 Chernobyl accident discussed above illustrates, understanding the activity and identity of released isotopes such as [latex]{}^{137}\mathrm{Cs}[/latex] is essential for assessing environmental contamination and radiation risk.

Like the number of radioactive nuclei, the activity of a radioactive source decreases exponentially with time. Since activity is proportional to the number of undecayed nuclei, it follows the same mathematical form:

where [latex]R_0[/latex] is the activity at [latex]t=0[/latex]. As time passes, the activity decreases by one-half during each half-life. For example, a source with an initial activity of 1.00 mCi will have an activity of 0.500 mCi after one half-life, 0.250 mCi after two half-lives, and 0.125 mCi after three half-lives. For elapsed times that are not exact multiples of the half-life, the exponential decay equation must be used.

Interactive Exploration: Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle. An alpha particle contains two protons and two neutrons, making it identical to the nucleus of a helium-4 atom. When alpha decay occurs, the parent nucleus loses two protons and two neutrons. Its atomic number therefore decreases by 2, and its mass number decreases by 4. Because the atomic number changes, the decay transforms the original nucleus into a different element. In this simulation, you will observe alpha particles escaping from polonium nuclei and compare the behavior of a single nucleus with that of a large collection of radioactive nuclei. Individual decay events occur randomly, but a large sample follows a predictable exponential pattern characterized by its half-life.