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 Carbon 14 dating graph 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 14 C, a radioactive isotope of carbon with a relatively long half-life years. While 12 C is the most abundant carbon isotope, there is a close to constant ratio of 12 C to 14 C 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 14 C stops being replenished. At this point, the overall amount of 14 C in the organism begins to decay exponentially. Therefore, by knowing the amount of 14 C in fossil remains, you can determine how long ago an organism died by examining the departure of the observed 12 C to 14 C ratio from the expected ratio for a living organism.

Radioactive isotopes, such as 14 C, 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.

We can use our our general model for exponential decay to calculate the amount of carbon at any given time using the equation. Returning to our example of carbon, knowing that the half-life of 14 C is years, we can use this to find the constant, k.

Thus, we can write:. Simplifying this expression by canceling the N 0 on both sides of the equation gives. Solving for the unknown, kwe take the natural logarithm of both sides.

Other radioactive isotopes are also used to date fossils. The half-life for 14 C is approximately years, therefore the 14 C isotope is only useful for dating fossils up to about 50, years old. Fossils older than 50, years may have an undetectable amount of 14 C.

For older fossils, an isotope with a longer half-life should be used.

For example, the radioactive isotope potassium decays to argon with a half life of 1. Other isotopes commonly used for dating include uranium half-life of 4. Problem 1- Calculate the amount of 14 C remaining in a sample.

Problem 2- Calculate the age of a fossil. Problem 3- Calculate the initial amount of 14 C in a fossil. Problem 4 - Calculate the age of a fossil.

Problem 5- Calculate the amount of 14 C remaining after a given time has passed. Next Application: Allometry. Decay of radioactive isotopes Radioactive isotopes, such as 14 C, decay exponentially. Modeling the decay of 14 C. Thus, our equation for modeling the decay of 14 C is given by.