The half-life of a quantity whose value decreases with time is the interval required for the quantity to decay to half of its initial value. The concept originated in the study of radioactive decay which is subject to exponential decay but applies to all phenomena including those which are described by non-exponential decays. Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value

The term half-life was coined in 1907, but it was always referred to as half-life period. It was not until the early 1950s that the word period was dropped from the name. [1]

Number of
half-lives
elapsed
Fraction
remaining
As
power
of 2
As %
01/11/20100
11/21/2150
21/41/2225
31/81/2312. 5
41/161/246. 25
51/321/253. 125
61/641/261. 562
71/1281/270. 781
. . . . . . . . . . . .
n1 / 2n1 / 2n100(1 / 2n)

The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.

It can be shown that, for exponential decay, the half-life t1 / 2 obeys this relation:

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

where

• ln(2) is the natural logarithm of 2 (approximately 0. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational 693), and
• λ is the decay constant, a positive constant used to describe the rate of exponential decay. Lambda (uppercase Λ, lowercase λ; Λάμβδα or el Λάμδα Lamda is the 11th letter of the Greek alphabet. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value A negative number is a Number that is less than zero, such as −2

The half-life is related to the mean lifetime τ by the following relation:

$t_{1/2} = \ln (2) \cdot \tau.$

## Examples

The constant λ can represent many different specific physical quantities, depending on what process is being described. Given an assembly of elements the number of which decreases ultimately to zero the lifetime (also called the mean lifetime) is a certain number that characterizes the rate A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value

• In an RC circuit or RL circuit, λ is the reciprocal of the circuit's time constant. A resistor–capacitor circuit (RC circuit, or RC filter or RC network, is an Electric circuit composed of resistors and capacitors driven by A resistor-inductor circuit (RL circuit, or RL filter or RL network, is one of the simplest analogue Infinite impulse response In Physics and Engineering, the time constant usually denoted by the Greek letter \tau, (tau characterizes the Frequency For simple RC and RL circuits, λ equals 1 / RC or R / L, respectively.
• In first-order chemical reactions, λ is the reaction rate constant. A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called In Chemical kinetics a reaction rate constant k or \lambda quantifies the speed of a Chemical reaction.
• In radioactive decay, it describes the probability of decay per unit time: dN = λNdt, where dN is the number of nuclei decayed during the time dt, and N is the quantity of radioactive nuclei. Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom
• In biology (specifically pharmacokinetics), from MeSH: Half-Life: The time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiologic activity. Foundations of modern biology There are five unifying principles Pharmacokinetics (in Greek: “pharmacon” meaning drug and “kinetikos” meaning putting in motion the study of time dependency sometimes abbreviated as “PK” is a Medical Subject Headings ( MeSH) is a huge Controlled vocabulary (or metadata system for the purpose of indexing journal articles and books Year introduced: 1974 (1971).

## Decay by two or more processes

Some quantities decay by two processes simultaneously (see Decay by two or more processes). A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value In a fashion similar to the previous section, we can calculate the new total half-life T1 / 2 and we'll find it to be:

$T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,$

or, in terms of the two half-lives t1 and t2

$T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,$

i. e. , half their harmonic mean. In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average.

## Simple Formula

m(t) mass left depending on time:

$m(t) = m(0) \cdot 0.5 ^ \frac{t}{t_{1/2}}\,$

m(0) = initial mass
t = time passed
t1 / 2 = half-life of the object.

## Derivation

Quantities that are subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items. This interpretation is valid in many, but not all, cases of exponential decay. ) If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:

$N(t) = N_0 e^{-\lambda t} \,$

where N0 is the initial value of N (at t = 0).

When t = 0, the exponential is equal to 1, and N(t) is equal to N0. As t approaches infinity, the exponential approaches zero. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In particular, there is a time $t_{1/2} \,$ such that

$N(t_{1/2}) = N_0\cdot\frac{1}{2}.$

Substituting into the formula above, we have

$N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}}, \,$
$e^{-\lambda t_{1/2}} = \frac{1}{2}, \,$
$- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2}, \,$
$t_{1/2} = \frac{\ln 2}{\lambda}. \,$

## Experimental determination

The half-life of a process can be determined easily by experiment. In fact, some methods do not require advance knowledge of the law governing the decay rate, be it exponential decay or another pattern.

Most appropriate to validate the concept of half-life for radioactive decay, in particular when dealing with a small number of atoms, is to perform experiments and correct computer simulations. Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation. See in [1] how to test the behavior of the last atoms. Validation of physics-math models consists in comparing the model's behavior with experimental observations of real physical systems or valid simulations (physical and/or computer). The references given here describe how to test the validity of the exponential formula for small number of atoms with simple simulations, experiments, and computer code.

In radioactive decay, the exponential model does not apply for a small number of atoms (or a small number of atoms is not within the domain of validity of the formula or equation or table). The DIY experiments use pennies or M&M's candies. M&M's Chocolate Candies are Candy -coated pieces of Chocolate with the letter "m" inscribed on them produced by Mars Incorporated. [2], [3]. A similar experiment is performed with isotopes of a very short half-life, for example, see Fig 5 in [4]. See how to write a computer program that simulates radioactive decay including the required randomness in [5] and experience the behavior of the last atoms. Randomness is a lack of order Purpose, cause, or predictability Of particular note, atoms undergo radioactive decay in whole units, and so after enough half-lives the remaining original quantity becomes an actual zero rather than asymptotically approaching zero as with continuous systems. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output