# Carbon dating example calculation

Almost everyone has heard on the news about archaeological findings claiming to have discovered a 12, years old bone, or a tomb with a skeleton of an individual who died 18, years ago. Do you remember when they found the famous tomb of Tutankhamun in Egypt? We have certainly been to a Natural History museum and saw exhibits of dinosaur or mammoth skeletons weather they are real or replicas. We might have seen displays of how our pre-historic ancestors looked like, and every display would indicate in the information box the date they were alive.

## Radiocarbon dating

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 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, k , we 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: Decay of radioactive isotopes Radioactive isotopes, such as 14 C, decay exponentially.

Modeling the decay of 14 C. Thus, we can write: Thus, our equation for modeling the decay of 14 C is given by,.

## In this section we will explore the use of carbon dating to determine the age of fossil remains. Problem 1- Calculate the amount ofC remaining in a sample. Radioactive dating can also use other radioactive For example, uranium ( which.

The ratio of carbon to carbon at the moment of death is the same as every other living thing, but the carbon decays and is not replaced. The carbon decays with its half-life of 5, years, while the amount of carbon remains constant in the sample. By looking at the ratio of carbon to carbon in the sample and comparing it to the ratio in a living organism, it is possible to determine the age of a formerly living thing fairly precisely. So, if you had a fossil that had 10 percent carbon compared to a living sample, then that fossil would be:.

Radiocarbon dating methods produce data that must then be further manipulated in order to calculate a resulting "radiocarbon age". The calculations to be performed on the measurements taken depend on the technology used, since beta counters measure the sample's radioactivity, whereas accelerator mass spectrometers AMS determine the ratio of the three different carbon isotopes in the sample.

During natural radioactive decay, not all atoms of an element are instantaneously changed to atoms of another element. The decay process takes time and there is value in being able to express the rate at which a process occurs.

## 5.7: Calculating Half-Life

Archaeologists use the exponential, radioactive decay of carbon 14 to estimate the death dates of organic material. The stable form of carbon is carbon 12 and the radioactive isotope carbon 14 decays over time into nitrogen 14 and other particles. Carbon is naturally in all living organisms and is replenished in the tissues by eating other organisms or by breathing air that contains carbon. At any particular time all living organisms have approximately the same ratio of carbon 12 to carbon 14 in their tissues. When an organism dies it ceases to replenish carbon in its tissues and the decay of carbon 14 to nitrogen 14 changes the ratio of carbon 12 to carbon

## Carbon Dating System

You can calculate half life if you know how much of the substance is left after a certain time, though typically it works the other way - the half life is known, and used to calculate age. Half life is defined as the time after which half of a sample of a radioactive material will have decayed. In other words, if you start with 1 kg of material with a half life of 1 year, then after 1 year you will have g. After another year you will have half of that, or g. After another year, you will have g, and so on. If, for example, we have the same 1kg sample of material with a half life of 1 year, how much do we have after 5. To do this, we need to use logarithms:. If, however, your goal is to determine the half life of carbon so you can use it to determine an age, then Google is your friend:

Radiocarbon dating also referred to as carbon dating or carbon dating is a method for determining the age of an object containing organic material by using the properties of radiocarbon , a radioactive isotope of carbon.

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.

## How do you calculate half life of carbon 14?

Petrology Tulane University Prof. Stephen A. Nelson Radiometric Dating Prior to the best and most accepted age of the Earth was that proposed by Lord Kelvin based on the amount of time necessary for the Earth to cool to its present temperature from a completely liquid state. Although we now recognize lots of problems with that calculation, the age of 25 my was accepted by most physicists, but considered too short by most geologists. Then, in , radioactivity was discovered. Recognition that radioactive decay of atoms occurs in the Earth was important in two respects: It provided another source of heat, not considered by Kelvin, which would mean that the cooling time would have to be much longer. It provided a means by which the age of the Earth could be determined independently. Principles of Radiometric Dating. Radioactive decay is described in terms of the probability that a constituent particle of the nucleus of an atom will escape through the potential Energy barrier which bonds them to the nucleus.

## Calculation of radiocarbon dates

Carbon is a radioactive isotope of carbon, containing 6 protons and 8 neutrons, that is present in the earth's atmosphere in extremely low concentrations. It is naturally produced in the atmosphere by cosmic rays and also artificially by nuclear weapons , and continually decays via nuclear processes into stable nitrogen atoms. Suppose we have a sample of a substance containing some carbon Suppose our sample initially contains nanograms of carbon Let's investigate what happens to the sample over time. First, we can solve the differential equation.

## Carbon Dating System

Unstable nuclei decay. However, some nuclides decay faster than others. For example, radium and polonium, discovered by Marie and Pierre Curie, decay faster than uranium. That means they have shorter lifetimes, producing a greater rate of decay. Here we will explore half-life and activity, the quantitative terms for lifetime and rate of decay. Why do we use the term like half-life rather than lifetime? The answer can be found by examining Figure

If you have a fossil, you can tell how old it is by the carbon 14 dating method. This is a formula which helps you to date a fossil by its carbon. The half life of carbon 14 is years. That means this is how long it takes for half the nuclei to decay. After years, if we start with a gram, we end up with half a gram. This rather complex formula shows you how to solve this puzzle using accepted scientific methods.

Radiometric dating is a means of determining the "age" of a mineral specimen by determining the relative amounts present of certain radioactive elements. By "age" we mean the elapsed time from when the mineral specimen was formed. Radioactive elements "decay" that is, change into other elements by "half lives. The formula for the fraction remaining is one-half raised to the power given by the number of years divided by the half-life in other words raised to a power equal to the number of half-lives. If we knew the fraction of a radioactive element still remaining in a mineral, it would be a simple matter to calculate its age by the formula. To determine the fraction still remaining, we must know both the amount now present and also the amount present when the mineral was formed. Contrary to creationist claims, it is possible to make that determination, as the following will explain:.

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Radioactive Carbon Dating Example