# Radioactive half-life

**the radioactive half-life** is the time, in which an exponentially with the time removing value halved itself. With exponential growth one speaks of a duplication time.

An exponential behavior is present for example, if the temporal change of a quantity proportionallyto the quantity is, as with the decay of radioactive isotopes.

## Table of contents |

## for opinion

the quantity of a radioactive substance remained after a radioactive half-life halves itself in the course of the next radioactive half-life, i.e. it remains ^{to 1}/_{2} *^{ 1}/_{2} =^{ 1}/_{4}; after3 radioactive half-lives^{ 1}/_{8}, then ^{1}/_{16}, ^{1}/_{32}, ^{1}/_{64} and so on, until in the long run only an individual core is remaining. The decay of this one -- like also every different one --Core cannot be predicted however, since a probability for its decay can be only indicated within a given time. The probability that a regarded core disintegrates within the first radioactive half-life, amounts to 50% that it disintegrates within 2 radioactive half-lives50% + 25% = 75%, with 3 radioactive half-lives the value amounts to 50% + 25% + 12.5% = 87.5%, etc.

## Zerfallsgesetz

(see also Zerfallsgesetz)

It is a radioactive preparation with N_{ 0} cores; at the time t=0still none of the cores disintegrated. With the activity the differential equation applies

- < math> for N \ cdot \ lambda = \ frac {\ mathrm D N} {\ mathrm D t}< /math>

- <math> - \ lambda \ \ mathrm D cdot t= \ frac {\ mathrm D N} {N}< /math>

- <math> \ - \ lambda \ cdot \ mathrm D t= \ int \ frac {\ mathrm D N} {N} \ Leftrightarrow - \ lambda t int + C_1= \ LN (N) +C_2< /math>

For t=0still N of 0 cores is _{present} after a condition. Thus applies to <math> C_1< /math>

- <math> C_1= \ LN \ left (N_0 \ right) +C_2< /math>

- <math> - \ lambda t + \ LN \ left (N_0 \ right) + C_2= \ LN (N) + C_2< /math>

- <math> - \ lambda t + \ LN \ left (N_0 \ right) = \ LN (N)< /math>

- <math> - \ lambda t = \ LN (N) - \ LN \ left (N_0 \ right) =\ LN \ left (\ frac {N} {N_0} \ right)< /math>

- <math> e^ {- \ lambda t} = \ frac {N} {N_0}< /math>

- <math> N (t) = N_0 \ cdot e^ {- \ lambda t}< /math>

Here the speed of the acceptance is certain by *the disintegration constant* λ. It is the reciprocal of the life span <math> \ rope = 1 \ lambda< /math>.
With the radioactive decay are thus after the time *t* of N_{ 0} output cores still N remaining.

## mathematical definition of the radioactive half-life

is <math> T_ {\ frac {1} {n}}< /math> the time, after that the output quantity <math> N_0< /math> on the 1/n-fache dropped (for the radioactive half-life applies n=2):

- <math> N (T_ {\ frac {1} {n}}) = \ frac {N_0} {n} = N_0 \ cdot e^ {- \ lambda T_ {\ frac {1} {n}}}< /math>

Afterwards N_0 /math becomes on both <sides> through< math> divided and logarithmiert.

- <math> \ LN \ left (\ frac {1} {n} \ right) = - \ lambda \ T_ cdot {\ frac {1} {n}} </math>

From this follows then considering the logarithm laws:

- <math> T_ {\ frac {1} {n}} = \ frac {\ LN \ left (n \ right)}{\ lambda}< /math>

Particularly to the radioactive half-life applies (n=2):

- <math> T_ {\ frac {1} {2}} = \ frac {\ LN \ left (2 \ right)}{\ lambda}< /math>

From it resultsfor the Zerfallsgesetz:

- <math> N (t) = N_0 \ cdot 2^ {- \ frac {t} {T {\ frac {1} {2}}}}< /math>

## examples

### radioactive radioactive half-life

**the physical radioactive half-life** is in nuclear physics that time interval, which elapses statistically seen, until the quantity of a certain radioactive nuclide on thoseHalf sank, i.e. was converted into other atoms. For each nuclide the radioactive half-life is a constant.

The number of remaining cores at a certain time is given by the Zerfallsgesetz.

Radioactive half-lives of some radioactive nuclides:

Element symbol radioactive half-life Bismuth ^{209}Biapprox. 1,9·10 ^{ 19}yearsuranium ^{238}U4.5 billion Years plutonium ^{239}Pu24000 years carbon ^{14}C5730 years tritium ^{3}H12.36 years cesium ^{137}Cs30 years radium ^{236}RA1622 years radon ^{222}Rn3.8 days francium ^{223}Fr22 minutes thorium ^{223}Th0.9 seconds polonium ^{212}Po0.3 µs

pure mathematically regarded disappears the radioactive substance is thus never, physically naturally with the transformation of the last atom a border set. Oftenone uses as estimation for the length of time up to the Bedeutungslosigkeit of a radioactive contamination, which 10 times radioactive half-life, which a decrease to 2^{,-10} - corresponds to subject (= 1/1024).

See also: Life span (physics)

### radio carbon method

(see also radio carbon method)

The radioactive carbon nuclide^{ 14} C is contained in a firm relationship in the carbon dioxide of our atmosphere. By the proportionate installation of the nuclide with photosynthesis into the biomass of the plants and over the food chain it continues to come also in the body of allOrganism to a firm relationship between normal^{ 12} C and radioactive ^{14} C. If an organism dies, then it hears on with photosynthesis and/or. with food intake. That has the consequence that the portion of ^{14} C off exactlyfrom 5730 years decreases to this time according to the radioactive decay exponentially with a radioactive half-life. On the basis the ionizing residual radiation, which proceeds from a dead organism, one can determine original 14 C of the portion by this radio carbon method, like^{ much} per cent still availableare and in the consequence the time of the death of the organism and thus the age of the find determine.

#### example of use

of the bars of a historical building has still 90% of the original equilibrium portion of ^{14} C in fresh plant mass. Thenapplies to the applied decay time:

- <math> \ mathrm {t} = \ mathrm {t} _ {1/2} \ cdot \ mathrm {log} _2 (0 {,} 9) = 5730a \ cdot \ mathrm {logs} _2 \ left (0 {,} 9 \ right) = -870,98a< /math>

That means that the tree, from which the bar was made it was struck before approximately 871 years.

Dating is not on thatYear exactly. The possible accuracy depends on the quantity of available sample material and the spent counting duration. For the measurement by means of accelerator mass spectrometry substantially smaller quantities of the sample material are sufficient.

### biological radioactive half-life

**the biological radioactive half-life** designates in the special time intervalt_{ 1/2}, in which in a biological organism (humans, animal, plant, single-celled organism) the content of a inkorporierten radioactive, toxic or pharmaceutical substance dropped by biological or physical processes (metabolism, elimination, radioactive decay, etc.) to half.

In that Pharmakokinetik calls one radioactive half-life the time, in which half of the taken up medicament separated amplifier-off-changed and/or. Since the biological radioactive half-life consists of different processes, which possess partial different concentration dependence, them are not always independent of the initial concentrationthe examined material.

### Bibliometri radioactive half-lives

in the Bibliometrie can be determined with the investigation of publications different radioactive half-lives. *Brooks* examined as one of the first radioactive half-lives in this area.

**The radioactive half-life of literature** amounts to about 5 years. Thisapplies both to the reading and the number of quotation ions. That is, the fact that a work *on the average* each year around approximately 14% less often from a library entliehen or is quoted as in the preceding (apart from classical authors and thatnewest works).

**The radioactive half-life of hyper+on the left of** amounts to about 51 months. That is, that after one year about 15% all are no longer valid hyper+on the left of.

### radioactive half-lives of passenger car

the resale value of a passenger car sinks approximately exponentially with his age,with a radioactive half-life of approx. 4-5 years. For example the price of a new vehicle of the model Volkswagen-Beetle is today (2006) with approx. 20 ' 000 EUR. The model from the year 2002 costs today approx. 10 ' 000 EUR, that of 1998 with approx. 5 ' 000 EUR.