### A mathematical explanation of carbon dating and half-life

Radiocarbon Dating is the process of determining the age of a sample by examining the amount of 14 C remaining against the known half-life, 5, years. The reason this process works is because when organisms are alive they are constantly replenishing their 14 C supply through respiration, providing them with a constant amount of the isotope. However, when an organism ceases to exist, it no longer takes in carbon from its environment and the unstable 14 C isotope begins to decay.

From this science, we are able to approximate the date at which the organism were living on Earth.

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- 17.6: Radiocarbon Dating: Using Radioactivity to Measure the Age of Fossils and Other Artifacts?
- Carbon 14 dating 1.

Radiocarbon dating is used in many fields to learn information about the past conditions of organisms and the environments present on Earth. Radiocarbon dating usually referred to simply as carbon dating is a radiometric dating method. It uses the naturally occurring radioisotope carbon 14C to estimate the age of carbon-bearing materials up to about 58, to 62, years old. Carbon has two stable, nonradioactive isotopes: There are also trace amounts of the unstable radioisotope carbon 14 C on Earth.

Carbon has a relatively short half-life of 5, years, meaning that the fraction of carbon in a sample is halved over the course of 5, years due to radioactive decay to nitrogen The carbon isotope would vanish from Earth's atmosphere in less than a million years were it not for the constant influx of cosmic rays interacting with molecules of nitrogen N 2 and single nitrogen atoms N in the stratosphere.

Both processes of formation and decay of carbon are shown in Figure 1. Diagram of the formation of carbon forward , the decay of carbon reverse. Carbon is constantly be generated in the atmosphere and cycled through the carbon and nitrogen cycles.

### Radiometric dating

Once an organism is decoupled from these cycles i. When plants fix atmospheric carbon dioxide CO 2 into organic compounds during photosynthesis, the resulting fraction of the isotope 14 C in the plant tissue will match the fraction of the isotope in the atmosphere and biosphere since they are coupled.

After a plants die, the incorporation of all carbon isotopes, including 14 C, stops and the concentration of 14 C declines due to the radioactive decay of 14 C following. This follows first-order kinetics. The currently accepted value for the half-life of 14 C is 5, years. This means that after 5, years, only half of the initial 14 C will remain; a quarter will remain after 11, years; an eighth after 17, years; and so on.

The equation relating rate constant to half-life for first order kinetics is. In samples of the Dead Sea Scrolls were analyzed by carbon dating. From the measurement performed in the Dead Sea Scrolls were determined to be years old giving them a date of 53 BC, and confirming their authenticity.

## Radiocarbon Dating

Carbon dating has shown that the cloth was made between and AD. Thus, the Turin Shroud was made over a thousand years after the death of Jesus. Plutonium is used to make nuclear explosives. Plutonium has a half-life of 24, years, which means that it would take , years to decay to a safe amount.

## A mathematical explanation of carbon dating and half-life - Auto Transport

Plutonium decays exponentially into lead, but it causes concerns for humans because the tiny particles of plutonium react with oxygen and water and can be extremely flammable. Since the half-life of Plutonium is so high even in comparison to the carbon 14 half-life of 5, years humans must be very cautious of the way they dispose of plutonium. Scientists are looking for safe ways for disposing plutonium.

We can see exponential decay in other areas as well. I am just learning the recipe so it takes me more time to look back and forth and double check.

### IM Commentary

The more cookies I make, the more practice I have and the less time it takes me to bake the cookies. We can use exponential decay to represent a number of different things. Most importantly, exponential decay is not linear and the decrease is rapid at first, but not constant.

It is often used to describe population decreases or increases, which depicts exponential growth and can be seen using a graph of an exponential curve. Natasha Glydon Exponential decay is a particular form of a very rapid decrease in some quantity. If P o is the initial amount of pollutants in the kerosene, then the amount left, P , after n feet of pipe can be represented by the following equation: Carbon 14 Dating 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 Experts can compare the ratio of carbon 12 to carbon 14 in dead material to the ratio when the organism was alive to estimate the date of its death.