Half-life

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]


Overview

The half-life of a quantity, subject to exponential decay, is the time required for the quantity to decay to half of its initial value. The concept originated in the study of radioactive decay, but applies to many other fields as well, including phenomena which are described by non-exponential decays.

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
0 1/1 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^0
1 1/2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^1
2 1/4 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^2
3 1/8 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^3
4 1/16 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^4
5 1/32 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^5
6 1/64 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^6
7 1/128 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^7
... ... ...
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^N Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/2^N

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t_{1/2} obeys this relation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t_{1/2} = \frac{\ln (2)}{\lambda}

where

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \ln (2) is the natural logarithm of 2 (approximately 0.693), and
  • λ is the decay constant, a positive constant used to describe the rate of exponential decay.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t_{1/2} = \ln (2) \cdot \tau

Examples

The constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda can represent many different specific physical quantities, depending on what process is being described.

  • In an RC circuit or RL circuit, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda is the reciprocal of the circuit's time constant. For simple RC and RL circuits, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda equals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 1/RC or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): R/L , respectively.
  • In first-order chemical reactions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda is the reaction rate constant.
  • In radioactive decay, it describes the probability of decay per unit time: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): dN = \lambda N dt , where dN is the number of nuclei decayed during the time dt, and N is the quantity of radioactive nuclei.
  • 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. Year introduced: 1974 (1971).

Decay by two or more processes

Some quantities decay by two processes simultaneously (see Decay by two or more processes). In a fashion similar to the previous section, we can calculate the new total half-life Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T_{1/2} and we'll find it to be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,

or, in terms of the two half-lives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t_1 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t_2

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,

i.e., half their harmonic mean.

Derivation

Quantities that are subject to exponential decay are commonly denoted by the symbol Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N , the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N at a time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t is given by the formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N(t) = N_0 e^{-\lambda t} \,

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N_0 is the initial value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N (at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t = 0 )

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t = 0 , the exponential is equal to 1, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N(t) is equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N_0 . As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t approaches infinity, the exponential approaches zero. In particular, there is a time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): t_{1/2} \, such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N(t_{1/2}) = N_0\cdot\frac{1}{2}.

Substituting into the formula above, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}}, \,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): e^{-\lambda t_{1/2}} = \frac{1}{2}, \,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): - \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2}, \,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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. See in [2] 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. [3], [4]. A similar experiment is performed with isotopes of a very short half-life, for example, see Fig 5 in [5]. See how to write a computer program that simulates radioactive decay including the required randomness in [6] and experience the behavior of the last atoms. 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.

See also

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References

  1. John Ayto "20th Century Words" (1999) Cambridge University Press.

External links

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