## 5.7: Calculating Half-Life

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,.

## Radioactive Half-Life Formula

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. To calculate the age of a substance using isotopic dating, use the equation below:. How long will it take for Ra has a half-life of years. Radioactive dating can also use other radioactive nuclides with longer half-lives to date older events.

For example, uranium which decays in a series of steps into lead can be used for establishing the age of rocks and the approximate age of the oldest rocks on earth. Since U has a half-life of 4. In a sample of rock that does not contain appreciable amounts of Pb, the most abundant isotope of lead, we can assume that lead was not present when the rock was formed.

Therefore, by measuring and analyzing the ratio of U Pb, we can determine the age of the rock. This assumes that all of the lead present came from the decay of uranium If there is additional lead present, which is indicated by the presence of other lead isotopes in the sample, it is necessary to make an adjustment. Potassium-argon dating uses a similar method.

K decays by positron emission and electron capture to form Ar with a half-life of 1. If a rock sample is crushed and the amount of Ar gas that escapes is measured, determination of the Ar K ratio yields the age of the rock.

Other methods, such as rubidium-strontium dating Rb decays into Sr with a half-life of As of , the oldest known rocks on earth are the Jack Hills zircons from Australia, found by uranium-lead dating to be almost 4. An ingenious application of half-life studies established a new science of determining ages of materials by half-life calculations.

After one half-life, a 1. Present day estimates for the age of the Earth's crust from this method is at 4 billion years. Isotopes with shorter half-lives are used to date more recent samples.

Chemists and geologists use tritium dating to determine the age of water ocean and fresh. In addition, tritium dating can be useful in determining the age of wines and brandies. The half-life of an isotope is used to describe the rate at which the isotope will decay and give off radiation. Using the half-life, it is possible to predict the amount of radioactive material that will remain after a given amount of time.

Its half-life is approximately years. Skills to Develop Describe what is meant by the term half-life and what factors affect half-life. Calculate the amount of radioactive material that will remain after an integral number of half-lives. Calculate the age of a material based upon its half-life. Describe how carbon is used to determine the age of carbon containing objects. Give examples of other isotopes used in radioactive dating. Appreciate the half-life of isotopes involved in nuclear weapons and reactors.

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Rate of Radioactive Decay During natural radioactive decay, not all atoms of an element are instantaneously changed to atoms of another element. Solution To determine the number of half-lives n , both time units must be the same.

To calculate the age of a substance using isotopic dating, use the equation below: Summary and Vocabulary The half-life of an isotope is used to describe the rate at which the isotope will decay and give off radiation. Radiation that comes from environment sources including the earth's crust, the atmosphere, cosmic rays, and radioisotopes. These natural sources of radiation account for the largest amount of radiation received by most people. The half-life of a radioactive substance is the time interval required for a quantity of material to decay to half its original value.