# Radiometric dating

**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. The most common form of radiometric dating is radiocarbon dating, which is based on the decay of carbon-14 to carbon-12. Other types of radiometric dating include rubidium-strontium, samarium-neodymium, potassium-argon, argon-argon, uranium-thorium, optically stimulated luminescence dating and uranium-lead. These methods differ in accuracy, cost, and range of applicability.

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*. Radiometric dating is based on forms of radioactive decay in which the atomic nucleus emits a neutron or alpha particle and becomes 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.

The accuracy of a method of dating depends in part on the half-life of the radioactive isotope involved. For instance, carbon-14 has a half-life of less than 6000 years. After an organism has been dead for 60,000 years, so little carbon-14 is left in it that accurate dating becomes impossible. On the other hand, the concentration of carbon-14 falls off so steeply that the age of relatively young remains can be determined accurately to within a few decades. The isotope used in uranium-thorium dating has a longer half-life, but other factors make it more accurate than radiocarbon dating.

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2 Example application 3 See also |

## 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 λ is the decay constant in the reciprocal of the units of the half-life.#### Derivation

Given a half-life*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

If*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)

2. Compute the number of carbon nuclei in a 25 gram sample (the gram molecular weight of carbon is 12.011 grams per mole).

- 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