# Why is the half-life a constant quantity

## Half-life

Lexicon> letter H> half-life

Acronym: HWZ

Definition: the time within which an exponentially decreasing quantity reaches half of its original value

English: half-life, half-value time

Categories: Basic concepts, nuclear energy, physical principles, ecology and environmental technology

Formula symbol: T1/2

Unit: seconds, hours, days, years

Author: Dr. Rüdiger Paschotta

How to quote; suggest additional literature

Original creation: December 14, 2014; last change: 05/28/2020

URL: https://www.energie-lexikon.info/halbwertszeit.html

The Half-life is a term that is often used in connection with radioactivity, among other things. The half-life of a certain radioactive isotope is the period of time after which half of the original amount of the isotope has been broken down by the radioactive decay.

Since the probability of a radioactive (unstable) atomic nucleus to decay in the next moment is always the same (regardless of the time for which the atomic nucleus already existed), an exponential law of decay results in the following form:

in which m(t) that currently t is the existing mass of the material and α is a constant that depends on the material. The larger the value of α, the faster the decay.

The exponential factor reaches the value 1/2 if αt = ln 2. The half-life is T1/2 = ln 2 / α. After ten half-lives, the amount of material has increased by a factor of 210 = 1024 decreased.

Since radioactive processes are random (stochastic), the equations given relate to so-called expected values ​​(mean values). However, since one usually has to do with a very large number of atomic nuclei, the actual values ​​follow the expected values ​​with a very high degree of accuracy; random deviations are very small.

The half-lives of different radioactive substances are extremely different: They can be tiny fractions of a second, but also many billions of years or anywhere in between. Even for one and the same decay mechanism (e.g. alpha decay) the half-life can vary enormously. The following table lists the half-lives of some technically important isotopes.

isotopeHalf-lifeRemarks
Cobalt 605.27 yearsUse for cancer radiation
Iodine 1318 daysFission product from nuclear reactors
Cesium 13730.17 yearsFission product from nuclear reactors
Radon 2223.82 daysPart of the decay series of uranium and thorium
Uranium 235703.8 million years0.7% contained in natural uranium
Uranium 2384.47 billion yearsMain component of natural uranium
Plutonium 23924 110 yearsarises through “breeding processes” (with neutron irradiation) in nuclear reactors

Short-lived isotopes (i.e. those with a short half-life) emit more radioactive radiation per gram of the material than long-lived isotopes. However, the degree of danger also depends on various other factors.

It would not make sense to specify a half-life for a mixture of different radioactive substances, since the total amount of radioactive material and radiation does not decrease exponentially. (For example, the total amount could decrease quickly at first, then much more slowly.) The half-life must therefore always relate to a specific isotope.

The intensity of the radiation decreases proportionally to the amount of substance - but only in simple cases!

With radioactive decay, a certain isotope is converted into another, whereby in some cases several different isotopes can arise from one original isotope (if different decay mechanisms occur).

If the decay only results in isotopes that are themselves stable, the intensity of the radioactive radiation decreases proportionally to the amount of the radioactive isotope.

If, on the other hand, unstable isotopes themselves are at least partially formed, these then decay with their own half-life. In the latter case, the intensity of the radioactivity does not necessarily decrease exponentially. If, for example, a short-lived isotope (with a short half-life) is converted into a much longer-lived one, the total radiation initially decreases quickly to a certain level, but from then on much more slowly. Conversely, if a longer-lived isotope creates a much shorter-lived one, the overall radiation can decrease exponentially - see the following example.

### Example: the decay of radioactive cesium 137

As an example, the following diagram shows the decay of radioactive cesium 137, which occurs as a fission product in nuclear fission in nuclear reactors. It has a half-life of 30.17 years and emits beta radiation (fast electrons) when it decays. The decay product is barium 137, which mostly occurs in a so-called metastable (excited) state, which changes to the stable ground state with emission of gamma radiation with a half-life of 2.55 minutes. Since the half-life of metastable barium 137 is so much shorter than that of cesium 137, the time course of the intensity of the total radioactive radiation is also exponential with high accuracy, with the half-life corresponding to that of cesium 137 (i.e. 30.17 years).

### Biological and effective half-life

If a radioactive substance gets into the human body, its amount can be reduced not only by radioactive decay, but also by excretion e.g. B. be reduced via the urine or stool. The so-called biological half-life describes the decrease in the amount of substance solely due to biological processes. Furthermore, one speaks of the effective half-life, which describes the overall resulting decrease in the amount of substance. It is below the biological and physical half-life. This is especially true for long-lived radioactive substances as long as they are not permanently built into the bones, for example.

How can a radioactive isotope occur naturally when its half-life is far shorter than the age of the earth?

Most naturally occurring radioactive isotopes have a very long half-life - simply because otherwise they would have decayed practically completely since the Earth was formed. However, it can happen that a long-lived radioactive isotope decays into a short-lived one, so that this can also occur naturally on earth.

Interestingly, radon would be less dangerous if its half-life were much longer. Its concentration in the air of a building would then hardly be higher, since the removal of the radon takes place anyway mainly through ventilation and not through radioactive decay. The amount of radioactive decay products formed would then be much lower.

### Half-life in other contexts

The term half-life can also be used outside the range of radioactivity, provided that a quantity decreases exponentially. For example, this applies at least approximately to the temperature difference between the interior of a hot water tank and its surroundings when neither hot water is withdrawn nor heat is supplied. Depending on the quality of the thermal insulation and the size of the storage tank, the half-life can be between a few hours and many weeks or even several months. A half-life of at least a few months is of course necessary for seasonal storage.

An exponential decrease in the amount of substance also occurs when substances are broken down in the earth's atmosphere. For example, methane is broken down into carbon dioxide and water vapor with a half-life of around 15 years. It is true that the methane concentration does not decrease exponentially, since new methane is constantly being released into the atmosphere. However, it would drop exponentially if there was suddenly no more supply.