Half-life

This article describes a scientific and mathematical term. For other meanings see half-life (disambiguation).

Uranium ore. The most abundant uranium isotope is ²³⁸U, with a half-life of 4.5 × 10⁹ years.

If a sample of material decays at a certain rate over time, its half-life is defined as the time it takes for the sample to decay to half its initial amount. This concept originated when studying the exponential decay of radioactive isotopes, but it is applied to other phenomena as well, including those described by non-exponential decay.

In the case of radioactive decay, each radioactive isotope has a particular half-life that is unaffected by changes in the physical or chemical conditions of the surroundings. This property is the basis for radiometric dating of rocks and fossils. In pharmacology, the half-life of a drug (in a biological system) is the time it takes for the drug to lose half its pharmacologic activity.

Mathematical calculation of half-life

Number of half-lives elapsed	Fraction remaining	As power of 2
0	1/1	$1/2^{0}$
1	1/2	$1/2^{1}$
2	1/4	$1/2^{2}$
3	1/8	$1/2^{3}$
4	1/16	$1/2^{4}$
5	1/32	$1/2^{5}$
6	1/64	$1/2^{6}$
7	1/128	$1/2^{7}$
...	...
$N$	$1/2^{N}$	$1/2^{N}$

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 $t_{1/2}$ obeys this relation:

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

where

• $\ln(2)$ is the natural logarithm of 2, and
• $\lambda$ is called 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:

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

Example of radioactive decay

Carbon-14 (¹⁴C) is a radioactive isotope that decays to produce the isotope nitrogen-14 (¹⁴N). The half-life of ¹⁴C is about 5,730 years. This means that if one starts with 10 grams of ¹⁴C, then 5 grams of the isotope will remain after 5,730 years, 2.5 grams will remain after another 5,730 years, and so forth.

The generalized constant $\lambda$ can represent many different specific physical quantities, depending on what process is being described.

• In first-order chemical reactions, $\lambda$ is the reaction rate constant.
• In pharmacology (specifically pharmacokinetics), the half-life of a drug is defined as the time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiologic activity.
• For electronic filters such as an RC circuit (resistor-capacitor circuit) or an RL circuit (resistor-inductor circuit), $\lambda$ is the reciprocal of the circuit's time constant $\tau$. (The symbol $\tau$ is the same as the mean lifetime mentioned above.) For simple RC or RL circuits, $\lambda$ equals $RC$ or $L/R$, respectively.

Decay by two or more processes

Some quantities decay by two processes simultaneously (see Exponential decay#Decay by two or more processes). In a fashion similar to that mentioned in the previous section, one can calculate the new total half-life $T_{1/2}$, and it is given as:

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

or, in terms of the two half-lives $t_{1}$ and $t_{2}$

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

i.e., half their harmonic mean.

Derivation

Quantities that are subject to exponential decay are commonly denoted by the symbol $N$. (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 $N_{0}$ is the initial value of $N$ (at $t=0$)

When $t=0$, the exponential is equal to 1, and $N(t)$ is equal to $N_{0}$. As $t$ approaches infinity, the exponential approaches zero. 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 readily determined by experiment. 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 [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. [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. 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

• Exponential decay
• Mean lifetime
• Radioactive decay
• Rate equation

References

ISBN links support NWE through referral fees

• Emsley, John. 2001. Nature's Building Blocks: An A to Z Guide to the Elements. Oxford: Oxford University Press. ISBN 0-19-850340-7.

• Magill, Joseph, and Jean Galy. 2004. Radioactivity Radionuclides Radiation. Berlin: Springer. ISBN 3540211160 and ISBN 978-3540211167.

• Brown, G.I. 2002. Invisible Rays: A History of Radioactivity. Sutton Publishing. ISBN 0750926678 and ISBN 978-0750926676.

External links

• Time constant

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