# Half-life

Last updated
Number of
half-lives
elapsed
Fraction
remaining
Percentage
remaining
011100
11250
21425
31812.5
41166.25
51323.125
61641.5625
711280.78125
n12n1002n

Half-life (symbol t½) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential (or, rarely, non-exponential) decay. For example, the medical sciences refer to the biological half-life of drugs and other chemicals in the human body. The converse of half-life (in exponential growth) is doubling time.

## Contents

The original term, half-life period, dating to Ernest Rutherford's discovery of the principle in 1907, was shortened to half-life in the early 1950s. [1] Rutherford applied the principle of a radioactive element's half-life in studies of age determination of rocks by measuring the decay period of radium to lead-206.

Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic unit for the exponential decay equation. The accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed.

## Probabilistic nature

A half-life often describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition that states "half-life is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom, and its half-life is one second, there will not be "half of an atom" left after one second.

Instead, the half-life is defined in terms of probability: "Half-life is the time required for exactly half of the entities to decay on average ". In other words, the probability of a radioactive atom decaying within its half-life is 50%. [2]

For example, the accompanying image is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not exactly one-half of the atoms remaining, only approximately, because of the random variation in the process. Nevertheless, when there are many identical atoms decaying (right boxes), the law of large numbers suggests that it is a very good approximation to say that half of the atoms remain after one half-life.

Various simple exercises can demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program. [3] [4] [5]

## Formulas for half-life in exponential decay

An exponential decay can be described by any of the following four equivalent formulas: [6] :109–112

{\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}2^{-{\frac {t}{t_{1/2}}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}}

where

• N0 is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
• N(t) is the quantity that still remains and has not yet decayed after a time t,
• t½ is the half-life of the decaying quantity,
• τ is a positive number called the mean lifetime of the decaying quantity,
• λ is a positive number called the decay constant of the decaying quantity.

The three parameters t½, τ, and λ are directly related in the following way:

${\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)}$

where ln(2) is the natural logarithm of 2 (approximately 0.693). [6] :112

### Half-life and reaction orders

In chemical kinetics, the value of the half-life depends on the reaction order:

#### Zero order kinetics

The rate of this kind of reaction does not depend on the substrate concentration, [A]. Thus the concentration decreases linearly.

${\displaystyle d[{\ce {A}}]/dt=-k}$
The integrated rate law of zero order kinetics is:
${\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}-kt}$

In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2:

${\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}-kt_{1/2}}$

and isolate the time:

${\displaystyle t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}}$

This t½ formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.

#### First order kinetics

In first order reactions, the rate of reaction will be proportional to the concentration of the reactant. Thus the concentration will decrease exponentially.

${\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}\exp(-kt)}$

as time progresses until it reaches zero, and the half-life will be constant, independent of concentration. The time t½ for [A] to decrease from [A]0 to 1/2[A]0 in a first-order reaction is given by the following equation:

${\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}\exp(-kt_{1/2})}$

It can be solved for

${\displaystyle kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2}$

For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of A at some arbitrary stage of the reaction is [A], then it will have fallen to 1/2[A] after a further interval of ${\displaystyle {\tfrac {\ln 2}{k}}.}$ Hence, the half-life of a first order reaction is given as the following:

${\displaystyle t_{1/2}={\frac {\ln 2}{k}}}$

The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, k.

#### Second order kinetics

In second order reactions, the rate of reaction is proportional to the square of the concentration. By integrating this rate, it can be shown that the concentration [A] of the reactant decreases following this formula:

${\displaystyle {\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}}$

We replace [A] for 1/2[A]0 in order to calculate the half-life of the reactant A

${\displaystyle {\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}}$

and isolate the time of the half-life (t½):

${\displaystyle t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}}$

This shows that the half-life of second order reactions depends on the initial concentration and rate constant.

### Decay by two or more processes

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T½ can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:

${\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}}$

For three or more processes, the analogous formula is:

${\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots }$

For a proof of these formulas, see Exponential decay § Decay by two or more processes.

### Examples

There is a half-life describing any exponential-decay process. For example:

• As noted above, in radioactive decay the half-life is the length of time after which there is a 50% chance that an atom will have undergone nuclear decay. It varies depending on the atom type and isotope, and is usually determined experimentally. See List of nuclides.
• The current flowing through an RC circuit or RL circuit decays with a half-life of ln(2)RC or ln(2)L/R, respectively. For this example the term half time tends to be used rather than "half-life", but they mean the same thing.
• In a chemical reaction, the half-life of a species is the time it takes for the concentration of that substance to fall to half of its initial value. In a first-order reaction the half-life of the reactant is ln(2)/λ, where λ (also denoted as k) is the reaction rate constant.

## In non-exponential decay

The term "half-life" is almost exclusively used for decay processes that are exponential (such as radioactive decay or the other examples above), or approximately exponential (such as biological half-life discussed below). In a decay process that is not even close to exponential, the half-life will change dramatically while the decay is happening. In this situation it is generally uncommon to talk about half-life in the first place, but sometimes people will describe the decay in terms of its "first half-life", "second half-life", etc., where the first half-life is defined as the time required for decay from the initial value to 50%, the second half-life is from 50% to 25%, and so on. [7]

## In biology and pharmacology

A biological half-life or elimination half-life is the time it takes for a substance (drug, radioactive nuclide, or other) to lose one-half of its pharmacologic, physiologic, or radiological activity. In a medical context, the half-life may also describe the time that it takes for the concentration of a substance in blood plasma to reach one-half of its steady-state value (the "plasma half-life").

The relationship between the biological and plasma half-lives of a substance can be complex, due to factors including accumulation in tissues, active metabolites, and receptor interactions. [8]

While a radioactive isotope decays almost perfectly according to so-called "first order kinetics" where the rate constant is a fixed number, the elimination of a substance from a living organism usually follows more complex chemical kinetics.

For example, the biological half-life of water in a human being is about 9 to 10 days, [9] though this can be altered by behavior and other conditions. The biological half-life of caesium in human beings is between one and four months.

The concept of a half-life has also been utilized for pesticides in plants, [10] and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants. [11]

In epidemiology, the concept of half-life can refer to the length of time for the number of incident cases in a disease outbreak to drop by half, particularly if the dynamics of the outbreak can be modeled exponentially. [12] [13]

## Related Research Articles

In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. This state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as dynamic equilibrium.

In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in 1884 that the van 't Hoff equation for the temperature dependence of equilibrium constants suggests such a formula for the rates of both forward and reverse reactions. This equation has a vast and important application in determining the rate of chemical reactions and for calculation of energy of activation. Arrhenius provided a physical justification and interpretation for the formula. Currently, it is best seen as an empirical relationship. It can be used to model the temperature variation of diffusion coefficients, population of crystal vacancies, creep rates, and many other thermally induced processes and reactions. The Eyring equation, developed in 1935, also expresses the relationship between rate and energy.

Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent. Exponential growth is the inverse of logarithmic growth.

Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the most common types of decay are alpha, beta, and gamma decay. The weak force is the mechanism that is responsible for beta decay, while the other two are governed by the electromagnetism and nuclear force.

The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit time. Reaction rates can vary dramatically. For example, the oxidative rusting of iron under Earth's atmosphere is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. A reaction's rate can be determined by measuring the changes in concentration over time.

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant:

In the context of radioactivity, activity or total activity (symbol A) is a physical quantity defined as the number of radioactive transformations per second that occur in a particular radionuclide. The unit of activity is the becquerel (symbol Bq), which is defined equivalent to reciprocal seconds (symbol s-1). The older, non-SI unit of activity is the curie (Ci), which is 3.7×1010 radioactive decay per second. Another unit of activity is the rutherford, which is defined as 1×106 radioactive decay per second.

A reversible reaction is a reaction in which the conversion of reactants to products and the conversion of products to reactants occur simultaneously.

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

In chemical kinetics, a reaction rate constant or reaction rate coefficient is a proportionality constant which quantifies the rate and direction of a chemical reaction by relating it with the concentration of reactants.

In chemistry, the rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters only. For many reactions, the initial rate is given by a power law such as

In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid, surface friction, radiation, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes. Damping is not to be confused with friction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.

In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(−t/τ).

In chemistry, molecularity is the number of molecules that come together to react in an elementary (single-step) reaction and is equal to the sum of stoichiometric coefficients of reactants in the elementary reaction with effective collision and correct orientation. Depending on how many molecules come together, a reaction can be unimolecular, bimolecular or even trimolecular.

In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e; it is the base-e analog of doubling time. This term is often used in many areas of science, such as in atmospheric chemistry, medicine and theoretical physics, especially when cosmic inflation is investigated. Physicists and chemists often talk about the e-folding time scale that is determined by the proper time in which the length of a patch of space or spacetime increases by the factor e mentioned above.

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In chemistry, transition state theory (TST) explains the reaction rates of elementary chemical reactions. The theory assumes a special type of chemical equilibrium (quasi-equilibrium) between reactants and activated transition state complexes.

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In physical chemistry, there are numerous quantities associated with chemical compounds and reactions; notably in terms of amounts of substance, activity or concentration of a substance, and the rate of reaction. This article uses SI units.

Potassium–calcium dating, abbreviated K–Ca dating, is a radiometric dating method used in geochronology. It is based upon measuring the ratio of a parent isotope of potassium to a daughter isotope of calcium. This form of radioactive decay is accomplished through beta decay.

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