Carbon dating to determine the age of fossil remains

In this section we will explore the use of carbon dating to determine the age of fossil remains.

Carbon is a key element in biologically important molecules. During the lifetime of an organism, carbon is brought into the cell from the environment in the form of either carbon dioxide or carbon-based food molecules such as glucose; then used to build biologically important molecules such as sugars, proteins, fats, and nucleic acids. These molecules are subsequently incorporated into the cells and tissues that make up living things. Therefore, organisms from a single-celled bacteria to the largest of the dinosaurs leave behind carbon-based remains.

Carbon dating is based upon the decay of ¹⁴C, a radioactive isotope of carbon with a relatively long half-life (5700 years). While ¹²C is the most abundant carbon isotope, there is a close to constant ratio of ¹²C to ¹⁴C in the environment, and hence in the molecules, cells, and tissues of living organisms. This constant ratio is maintained until the death of an organism, when ¹⁴C stops being replenished. At this point, the overall amount of ¹⁴C in the organism begins to decay exponentially. Therefore, by knowing the amount of ¹⁴C in fossil remains, you can determine how long ago an organism died by examining the departure of the observed ¹²C to ¹⁴C ratio from the expected ratio for a living organism.

Decay of radioactive isotopes

Radioactive isotopes, such as ¹⁴C, decay exponentially. The half-life of an isotope is defined as the amount of time it takes for there to be half the initial amount of the radioactive isotope present.

For example, suppose you have N₀ grams of a radioactive isotope that has a half-life of t* years. Then we know that after one half-life (or t* years later), you will have

grams of that isotope.

t* years after that (i.e. 2t* years from the initial measurement), there will be

grams.

3t* years after the initial measurement there will be

grams,

and so on.

We can use our our general model for exponential decay to calculate the amount of carbon at any given time using the equation,

N (t) = N₀e ^kt .

Modeling the decay of ¹⁴C.

Returning to our example of carbon, knowing that the half-life of ¹⁴C is 5700 years, we can use this to find the constant, k. That is when t = 5700, there is half the initial amount of ¹⁴C. Of course the initial amount of ¹⁴C is the amount of ¹⁴C when t = 0, or N₀ (i.e. N(0) = N₀e ^k⋅0 = N₀e⁰ = N₀). Thus, we can write:

  .

Simplifying this expression by canceling the N₀ on both sides of the equation gives,


.

Solving for the unknown, k, we take the natural logarithm of both sides,

.

Thus, our equation for modeling the decay of ¹⁴C is given by,

.

Other radioactive isotopes are also used to date fossils.

The half-life for ¹⁴C is approximately 5700 years, therefore the ¹⁴C isotope is only useful for dating fossils up to about 50,000 years old. Fossils older than 50,000 years may have an undetectable amount of ¹⁴C. For older fossils, an isotope with a longer half-life should be used. For example, the radioactive isotope potassium-40 decays to argon-40 with a half life of 1.3 billion years. Other isotopes commonly used for dating include uranium-238 (half-life of 4.5 billion years) and thorium-232 (half-life 14.1 billion years).

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Test yourself

Problem 1- Calculate the amount of ¹⁴C remaining in a sample

Problem 2- Calculate the age of a fossil

Problem 3- Calculate the initial amount of ¹⁴C in a fossil

Problem 4 - Calculate the age of a fossil

Problem 5- Calculate the amount of ¹⁴C remaining after a given time has passed.

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