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Radioactive dating

Radioactive dating is a technique that uses naturally occurring radioactivity to determine the age of a material, such as a rock or an ancient artifact. The basic approach is to estimate the original number of nuclei in a material and the present number of nuclei in the material (after decay), and then use the known value of the decay constant λ and [link] to calculate the total time of the decay, t .

An important method of radioactive dating    is carbon-14 dating    . Carbon-14 nuclei are produced when high-energy solar radiation strikes 14 N nuclei in the upper atmosphere and subsequently decay with a half-life of 5730 years. Radioactive carbon has the same chemistry as stable carbon, so it combines with the ecosphere and eventually becomes part of every living organism. Carbon-14 has an abundance of 1.3 parts per trillion of normal carbon. Therefore, if you know the number of carbon nuclei in an object, you multiply that number by 1.3 × 10 −12 to find the number of 14 C nuclei in that object. When an organism dies, carbon exchange with the environment ceases, and 14 C is not replenished as it decays.

By comparing the abundance of 14 C in an artifact, such as mummy wrappings, with the normal abundance in living tissue, it is possible to determine the mummy’s age (or the time since the person’s death). Carbon-14 dating can be used for biological tissues as old as 50,000 years, but is generally most accurate for younger samples, since the abundance of 14 C nuclei in them is greater. Very old biological materials contain no 14 C at all. The validity of carbon dating can be checked by other means, such as by historical knowledge or by tree-ring counting.

An ancient burial cave

In an ancient burial cave, your team of archaeologists discovers ancient wood furniture. Only 80 % of the original 14 C remains in the wood. How old is the furniture?

Strategy

The problem statement implies that N / N 0 = 0.80 . Therefore, the equation N = N 0 e λ t can be used to find the product, λ t . We know the half-life of 14 C is 5730 y, so we also know the decay constant, and therefore the total decay time t .

Solution

Solving the equation N = N 0 e λ t for N / N 0 gives us

N N 0 = e λ t .

Thus,

0.80 = e λ t .

Taking the natural logarithm of both sides of the equation yields

ln 0.80 = λ t ,

so that

−0.223 = λ t .

Rearranging the equation to isolate t gives us

t = 0.223 λ ,

where

λ = 0.693 t 1 / 2 = 0.693 5730 y .

Combining this information yields

t = 0.223 ( 0.693 5730 y ) = 1844 y .

Significance

The furniture is almost 2000 years old—an impressive discovery. The typical uncertainty on carbon-14 dating is about 5 % , so the furniture is anywhere between 1750 and 1950 years old. This date range must be confirmed by other evidence, such as historical records.

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Check Your Understanding A radioactive nuclide has a high decay rate. What does this mean for its half-life and activity?

Half-life is inversely related to decay rate, so the half-life is short. Activity depends on both the number of decaying particles and the decay rate, so the activity can be great or small.

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Visit the Radioactive Dating Game to learn about the types of radiometric dating and try your hand at dating some ancient objects.

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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