Half-life

Half-life, usually written $t_{1/2}$ or $T_{1/2}$, is the time required for a decreasing quantity to reach half its initial value. It is especially associated with radioactivity, where it describes the disappearance of unstable nuclei, and with chemical kinetics, where it describes the consumption of reactants. For exponential decay, the half-life is constant: the same fraction disappears in each equal time interval, regardless of the amount remaining. Other processes can have half-lives that depend on their starting conditions. (openstax.org)

Mathematical definition

An exponentially decreasing quantity $N(t)$ satisfies the differential equation

where $\lambda>0$ is the decay constant, with units of inverse time. Its solution is

where $N_0$ is the initial quantity. Setting $N(t_{1/2})=N_0/2$ gives

A larger decay constant therefore corresponds to a shorter half-life. (ocw.mit.edu)

After one, two, and three half-lives, the remaining fractions are respectively $1/2$, $1/4$, and $1/8$. After $n$ half-lives, the fraction is $2^{-n}$. This differs from linear depletion, which removes equal amounts rather than equal fractions. The exponential expression approaches zero without reaching it at a finite time; for a finite collection of nuclei, however, the actual number remaining is discrete rather than continuous. (openstax.org)

The mean lifetime $\tau$ is related to, but distinct from, the half-life:

Thus an exponential mean lifetime is approximately 1.443 times the half-life. Half-lives may be expressed in seconds, days, years, or other time units, provided these are consistent with the decay constant. (media.iupac.org)

Radioactive half-life

In nuclear physics, half-life characterizes the decay of a particular radioactive isotope or nuclear state. Decay transforms an unstable atomic nucleus into another nucleus or a lower-energy state. Processes include alpha decay, beta decay, and emission of gamma radiation. A daughter nucleus may itself be radioactive, producing a decay chain rather than an immediately stable product. (openstax.org)

The activity $A$ of a sample is its decay rate:

For a single radionuclide undergoing decay without replenishment, its activity decreases with the same half-life as its population. For equal numbers of nuclei, a shorter half-life means a greater initial activity. If radioactive daughters accumulate, the total activity of the mixture need not follow the parent’s simple exponential curve. (media.iupac.org)

Half-life does not mean that each nucleus survives for a fixed time and then decays. In the constant-rate probabilistic model, an individual nucleus has a 50% probability of surviving one half-life. A nucleus that has already survived retains the same conditional probability of surviving the next half-life. Consequently, exactly half of a small sample need not decay during that interval; the halving law describes an expected value and becomes a reliable population description for large samples. These statements follow mathematically from the exponential survival model. (ocw.mit.edu)

Chemical half-life

For a chemical reaction obeying the first-order consumption law $-dc/dt=kc$, the concentration $c$ decreases exponentially and

Its half-life is independent of initial concentration as long as the rate constant $k$ remains unchanged. Constant successive halving times are therefore characteristic of first-order kinetics. (openstax.org)

Other reaction orders behave differently. For zero-order consumption, $-dc/dt=k$, the initial half-life is $c_0/(2k)$. For second-order consumption, $-dc/dt=kc^2$, it is $1/(kc_0)$. Both depend on the initial concentration $c_0$; successive halving intervals shorten for zero-order consumption and lengthen for second-order consumption. (openstax.org)

The chemical definition also accommodates reactions approaching a nonzero equilibrium concentration. In that setting, half-life is the time required to reach the arithmetic mean of the initial and final concentrations, not necessarily half the initial concentration. When several reactants are present outside their stoichiometric proportions, they can have different half-lives, so a single “half-life of the reaction” may be inappropriate. (goldbook.iupac.org)

Biological and effective half-life

A biological half-life describes reduction of a substance in an organism through biological elimination, separately from radioactive decay. The exponential description assumes first-order removal; saturation of elimination mechanisms can invalidate a constant-half-life model. Biological and physical half-lives therefore represent different processes. (energy.gov)

For a radionuclide undergoing independent exponential physical decay and biological removal, the effective half-life $T_{\mathrm{eff}}$ combines both:

For finite positive component half-lives, it is shorter than either component because both mechanisms reduce the retained radionuclide population. (energy.gov)

Dating and measurement

Radiometric dating uses known half-lives and measured isotope abundances to infer elapsed time. Carbon-14 has a half-life of approximately 5,730 years and supports dating of formerly living material. Uranium-238 has a half-life of approximately 4.5 billion years and is used in geological dating through its decay toward stable lead-206. These contrasting timescales make different isotopes suitable for different age ranges. (pubs.usgs.gov)

For first-order decay, plotting the natural logarithm of the measured quantity against time produces a straight line with slope $-\lambda$. This allows estimation of the decay constant and hence the half-life without waiting for a complete halving interval. The interpretation depends on the quantity actually measured: parent abundance, parent activity, and combined parent–daughter activity are not interchangeable. (openstax.org)