life, although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction. Protozoa perform, in a single cell, the same life functions as those carried on by the complex tissues and organs of humans and other highly developed organisms. The attributes of life are inherent in such minute structures as viruses, bacteria, and genes, just as they are in the whale and the giant sequoia. In seeking an understanding of life, scientists have broken down many barriers that once separated the physical sciences from the biological sciences; a result of the growth of biochemistry, biophysics, and other interrelated fields of study has been a better understanding of the composition and functioning of living tissues of all kinds.

Characteristics of Life

Organization is found in the basic living unit, the cell, and in the organized groupings of cells into organs and organisms. Metabolism includes the conversion of nonliving material into cellular components (synthesis) and the decomposition of organic matter (catalysis), producing energy. Growth in living matter is an increase in size of all parts, as distinguished from simple addition of material; it results from a higher rate of synthesis than catalysis. Irritability, or response to stimuli, takes many forms, from the contraction of a unicellular organism when touched to complex reactions involving all the senses of higher animals; in plants response is usually much different than in animals but is nonetheless present. Adaptation, the accommodation of a living organism to its present or to a new environment, is fundamental to the process of evolution and is determined by the individual's heredity. The division of one cell to form two new cells is reproduction; usually the term is applied to the production of a new individual (either asexually, from a single parent organism, or sexually, from two differing parent organisms), although strictly speaking it also describes the production of new cells in the process of growth.

The Basis of Life

Much of the history of biology and of philosophy as related to biology has been marked by a division of thought between vitalistic (or animistic) and mechanistic (or materialistic) concepts. In the most antithetic interpretations of these concepts, the vitalistic school maintains that there is a vital force that distinguishes the living from the nonliving and the mechanistic school holds that there is no essential difference between the animate and inanimate and that all life can be explained by physical and chemical laws. Such diametrically opposed views have actually seldom been held by investigators of either school; elements of both are usually involved. The animistic school, largely predicated on the inexplicability of the basic phenomena of life, has been greatly overshadowed by the accumulating weight of scientific data. As more and more is learned of the minute details of the structure and composition of the substances that make up the cell (to the extent that some have been synthesized chemically), it has become increasingly apparent that living matter is made up of the same (and only those) elements found in inorganic material, except that they are differently organized.

The Origin of Life

Fundamental religious concepts center around special creation and belief in the infusion of life into inanimate substance by God or another superhuman entity. On the other hand, many scientists have hypothesized that during an early geological period there gradually formed in the atmosphere increasingly complex organic substances composed of available inorganic compounds and water, utilizing ultraviolet rays and electrical discharges as energy sources. At a certain stage they formed a diffuse solution of "nutrient broth." Then in some way they were drawn together and developed the capacity for self-renewal and self-reproduction. In 1953, S. L. Miller synthesized several of the most basic amino acids in a glass flask by introducing an electrical discharge into an atmosphere of water vapor and some simple compounds thought to have been present naturally at the time when life first developed on earth. A more recent theory now widely held is that life originated in a volcanic setting more than 3.5 billion years ago, perhaps in hot deep-sea vents, utilizing a biochemistry based largely on sulfur and iron. The theory that life on earth came in a simple form from another planet has had small currency, although the discovery by Melvin Calvin of molecules resembling genetic material in meteors has given it some force.


See M. Calvin, Chemical Evolution (1969); E. Borek, The Sculpture of Life (1973); N. D. Newell, Creation and Evolution (1985); S. W. Fox and K. Dose, Molecular Evolution and the Origins of Life (3d ed. 1990); R. Fortey, Life (1998).

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 describing how long it takes atoms to undergo radioactive decay, but also applies in a wide variety of other situations.

The term "half-life" dates to 1907. The original term was "half-life period", but that was shortened to "half-life" starting in the early 1950s.

Half-lives are very often used to describe quantities undergoing exponential decay—for example radioactive decay. However, a half-life can also be defined for non-exponential decay processes. For a general introduction and description of exponential decay, see the article exponential decay. For a general introduction and description of non-exponential decay, see the article rate law.

Number of
0 1/1 100
1 1/2 50
2 1/4 25
3 1/8 12 .5
4 1/16 6 .25
5 1/32 3 .125
6 1/64 1 .563
7 1/128 0 .781
... ... ...
n 1/2^n 100(1/2^n)
The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.

Probabilistic nature of half-life

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 "half-life is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom with a half-life of 1 second, there will not be "half of an atom" left after 1 second. There will be either zero atoms left or one atom left, depending on whether or not the atom happens to decay.

Instead, the half-life is defined in terms of probability. It is the time when the expected value of the number of entities that have decayed is equal to half the original number. For example, one can start with a single radioactive atom, wait its half-life, and measure whether or not it decays in that period of time. Perhaps it will and perhaps it will not. But if this experiment is repeated again and again, it will be seen that it decays within the half life 50% of the time.

In some experiments (such as the synthesis of a superheavy element), there is in fact only one radioactive atom produced at a time, with its lifetime individually measured. In this case, statistical analysis is required to infer the half-life. In other cases, a very large number of identical radioactive atoms decay in the time-range measured. In this case, the central limit theorem ensures that the number of atoms that actually decay is essentially equal to the number of atoms that are expected to decay. In other words, with a large enough number of decaying atoms, the probabilistic aspects of the process can be ignored.

There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a computer program. See the following websites: , ,

Formulae for half-life in exponential decay

An exponential decay process can be described by any of the following three equivalent formulae:

N_t = N_0 (1/2)^{t/t_{1/2}}
N_t = N_0 e^{-t/tau}
N_t = N_0 e^{-lambda t}
*N_0 is the initial quantity of the thing 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,
*t_{1/2} 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 t_{1/2}, τ, and λ are all directly related in the following way:
t_{1/2} = frac{ln (2)}{lambda} = tau ln(2)
where ln(2) is the natural logarithm of 2 (approximately 0.693).

By plugging in and manipulating these relationships, we get all of the following equivalent descriptions of exponential decay, in terms of the half-life:

N_t = N_0 (1/2)^{t/t_{1/2}} = N_0 2^{-t/t_{1/2}} = N_0 e^{-tln(2)/t_{1/2}}
t_{1/2} = t/log_2(N_0/N_t) = t/(log_2(N_0)-log_2(N_t)) = (log_{2^t}(N_0/N_t))^{-1} = tln(2)/ln(N_0/N_t)
Regardless of how it's written, we can plug into the formula to get

  • N_t=N_0 at t=0 (as expected—this is the definition of "initial quantity")
  • N_t=(1/2)N_0 at t=t_{1/2} (as expected—this is the definition of half-life)
  • N_t approaches zero when t approaches infinity (as expected—the longer we wait, the less remains).

Decay by two or more processes

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T1/2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:
frac{1}{T_{1/2}} = frac{1}{t_1} + frac{1}{t_2}
For three or more processes, the analogous formula is:
frac{1}{T_{1/2}} = frac{1}{t_1} + frac{1}{t_2} + frac{1}{t_3} + cdots
For a proof of these formulae, see Decay by two or more processes.


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

  • The current flowing through an RC circuit or RL circuit decays with a half-life of RCln(2) or ln(2)L/R, respectively.
  • In a first-order chemical reaction, the half-life of the reactant is ln(2)/lambda, where λ is the reaction rate constant.
  • 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.

Half-life in non-exponential decay

Many quantities decay in a way not described by exponential decay—for example, the evaporation of water from a puddle, or (often) the chemical reaction of a molecule. In this case, the half-life is defined the same way as before: The time elapsed before half of the original quantity has decayed. However, unlike in an exponential decay, the half-life depends on the initial quantity, and changes over time as the quantity decays.

As an example, the radioactive decay of carbon-14 is exponential with a half-life of 5730 years. If you have a quantity of carbon-14, half of it (on average) will have decayed after 5730 years, regardless of how big or small the original quantity was. If you wait another 5730 years, one-quarter of the original will remain. On the other hand, the time it will take a puddle to half-evaporate depends on how deep the puddle is. Perhaps a puddle of a certain size will evaporate down to half its original volume in one day. But if you wait a second day, there is no reason to expect that precisely one-quarter of the puddle will remain; in fact, it will probably be much less than that. This is an example where the half-life reduces as time goes on. (In other non-exponential decays, it can increase instead.)

For specific, quantitative examples of half-lives in non-exponential decays, see the article Rate equation.

A biological half-life is also a type of half-life associated with a non-exponential decay, namely the decay of the activity of a drug or other substance after it is introduced into the body.

See also


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