## Key Concepts

**The time required for one-half of a given material to undergo chemical reactions**. As a term, half-life is often used in nuclear physics to quantify the average time interval required for one-half of any quantity of identical radioactive atoms to undergo radioactive decay. For example, the half-life of uranium-238 is approximately 4.5 billion years, meaning that a given quantity *X* will be reduced to approximately ½ *X* after that time interval (see illustration). The term half-life is also more broadly used in chemistry and medicine to indicate the time interval within which half of a substance will have decayed or changed in some manner. An example is the biological half-life of a drug, which refers to how many days (typically) it tends to take for biological processes to have removed half of an administered dose from the body. * See also: ***Atom**; **Chemistry**; **Nuclear physics**; **Pharmacology**; **Radioactivity**; **Time**; **Uranium**

### Chemical reactions

The concept of the time required for all of a given material to react is meaningless, because the reaction goes very slowly when only a small amount of the reacting material is left and theoretically an infinite time would be required. The time for half-completion of the reaction, in contrast, is a definite and useful way of describing the rate of a reaction. * See also: ***Chemical kinetics**

The specific rate constant *k* provides another way of describing the rate of a chemical reaction. This is shown in a first-order reaction (1),

where *c*_{0} is the initial concentration and *c* is the concentration at time *t*. The relation between specific rate constant and period of half-life, *t*, in a first-order reaction is given by Eq. (2).

In a first-order reaction, the period of half-life is independent of the initial concentration, but in a second-order reaction it does depend on the initial concentration according to Eq. (3).

### Radioactive decay

As the time it takes for one-half of the original atoms of a radioactive isotope to decay, half-life is expressed as *T*_{½} and is also called a half-period. If starting with *A*_{0} atoms at time *t* = 0, then the number of atoms present at a later time is given by Eq. (4),

where λ is called the decay constant. Each radioactive isotope has a unique characteristic λ. The number of atoms and the activity of the sample, given by *A*λ, decrease exponentially with time. After one half-life, when *A*/*A*_{0} = ½, the time is given by Eq. (5), which relates the half-life to the decay constant.

After two half-lives, *A*/*A*_{0} = ¼; after three half-lives, *A*/*A*_{0} = ⅛; and so on.

Radioactive decay follows a statistical probability. The probability is exactly ½ that the actual life span of one individual radioactive atom will exceed *T*_{½}. When the number of atoms is very large, then one-half that number will have decayed in *T*_{½}. For a small number of initial atoms, however, the number remaining after the *T*_{½} can vary considerably around ½. * See also: ***Probability (physics)**

One common application of radioactive decay with regard to half-lives is radiometric dating, which utilizes the radioactive decay of certain long-lived, naturally occurring parent isotopes to stable daughter isotopes to measure the age of an object or sample. For example, the isotope ^{14}C is produced in the Earth's atmosphere and has a half-life of 5700 years. Living plants take in ^{14}C along with stable ^{12}C. When the plant dies, the ratio decreases and can be used to tell the time when the plant died up to 10 *T*_{½} ≈ 57,000 years, but not older. Other elements and isotopes allow for dating samples back billions of years. * See also: ***Atmosphere**; **Carbon**; **Dating methods**; **Isotope**; **Plant**; **Radiocarbon dating**