**Radiometric dating**is the determination of the date at which materials were formed by analyzing the decay of radioactive isotopes that were incorporated into the material when it was created and which presumably have not diffused out. Probably the best known form of radiometric dating is radiocarbon dating, which uses carbon-14. Rubidum-Strontium dating is also popular.

Most materials decay radioactively to some extent, but the decay rates of most are so long that, for all practical purposes, they can be considered inert. The remainder are said to be *radioactive*. Radioactive materials can decay in any of several ways, emitting either a particle or radiation and changing to a different element or isotope. The decay rate of radioactive materials does not depend on temperature, chemical environment, or similar factors. For dating purposes, the important parameter is the half-life of the reaction - the time it takes for half the material to decay. Half lives of various isotopes vary from microseconds to billions of years. Materials useful for radiometric dating have half lives from a few thousand to a few billion years.

Some types of radiometric dating assume that the initial proportions of a radioactive substance and its decay product are known. The decay product should not be a small-molecule gas that can leak out, and must itself have a long enough half life that it will be present in significant amounts. In addition, the initial element and the decay product should not be produced or depleted in significant amounts by other reactions. The procedures used to isolate and analyze the reaction products must be straightforward and reliable.

In contrast to most systems, isochron dating using rubidium-strontium does not require knowledge of the initial proportions.

Several systems are known that satisfy these constraints including carbon-14-carbon-12, Rb-Sr, Sm-Nd, K-Ar, Ar-Ar, and U-Pb. Carbon-14 has a fairly short half life and is used for dating recent organic remains. It is useful for periods up to perhaps 60,000 years and is thus very important to historians and archeologists as a method of determining the age of human artifacts. The other isotopes have half lives of hundreds of millions of years and are used for dating igneous rock formations.

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2 Example application 3 Related Articles |

## Formulae related to radioactive decay

### Nuclei N remaining at time t

The formula for the nuclei *N* remaining at time *t* is

_{0}sample after time t has elapsed, and

*λ*is the decay constant. (See exponential decay).

### Decay constant

- where
*h*is the half-life.

#### Derivation

Starting with

*N*=*N*_{0}*e*^{-λt}

*h*, after said half-life there will be half of the original amount remaining, so the formula can be changed to:

- (1/2)
*N*_{0}=*N*_{0}*e*^{-λh}

- 1/2 =
*e*^{-λh} - ln (1/2) = -
*λh* - ln 2 =
*λh* *λ*= (ln 2)/*h*

### Decay rate at time t

_{0}is the initial radioactivity.

#### Derivation

Starting with

*N*=*N*_{0}*e*^{-λt}*R*= d*N*/d*t*= -*λN*_{0}*e*^{-λt}

*t*= 0

*R*_{0}= -*λN*_{0}

*R*= -*λN*_{0}*e*^{-λt}*R*=*R*_{0}*e*^{-λt}

## Example application

*How old is a 25 gram charcoal sample that has an activity of 5 Bq?*

^{-12}(see Radiocarbon_dating).

### Solution

1. Compute the decay constant for carbon-14 (in seconds for simplification)

*λ*= ln(2) / Half-life = 0.693/(5730 yr * 31558464 sec per yr) = 3.83 × 10^{-12}Bq

- Number of Carbon nuclei = 25 g (6.02 × 10
^{23}mol^{-1}) / (12.011 g/mol) = 1.26x10^{24}

*N*_{0}= (1.3x10^{-12})(1.26 × 10^{24}) = 1.6 × 10^{12}initial number of C-14 nuclei*R*_{0}=*λN*_{0}= (3.83 × 10^{-12}Bq)(1.6 × 10^{12}) = 6.1 Bq

- 5 Bq = 6.1 Bq
*e*^{-λt} *t*= ln(6.1 Bq/5 Bq) / (3.83 × 10^{-12}Bq)- t = 5.1 × 10
^{10}s = 1.6 × 10^{3}yr