A sample of wood from an archaeological excavation is dated...

A sample of wood from an archaeological excavation is dated by using a mass spectrometer to measure the fraction of ${ }^{14} \mathrm{C}$ atoms. Suppose 100 atoms of ${ }^{14} \mathrm{C}$ are found for every $1.0 \times 10^{15}$ atoms of ${ }^{12} \mathrm{C}$ in the sample. What is the wood's age?

Key Concepts

Isotope Ratio

The isotope ratio of carbon-14 to carbon-12 is a critical parameter in radiocarbon dating. In living organisms, this ratio is maintained at a nearly constant level due to continuous exchange with the environment. After death, the ratio decreases as carbon-14 decays, and the remaining ratio provides a quantitative measure to estimate the time elapsed since the death of the organism.

Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the masses and relative concentrations of atoms and molecules. In radiocarbon dating, it is used to precisely count the number of carbon-14 and carbon-12 atoms in a sample, which is essential for determining the ratio needed for calculating the age based on decay laws.

Exponential Decay

Exponential decay describes the process in which the amount of a radioactive substance decreases at a rate proportional to its current amount. In radiocarbon dating, the number of carbon-14 atoms decreases exponentially with time, following a mathematical model that relates the remaining fraction to the elapsed time.

Radiocarbon Dating

This method is used to determine the age of formerly living materials by measuring the decay of carbon-14. It is based on the fact that living organisms constantly exchange carbon with the atmosphere, establishing a fixed ratio of carbon isotopes. Once the organism dies, the carbon-14 begins to decay while the stable carbon-12 remains constant, so the decreasing ratio can be used to calculate the time elapsed.

Half-Life

The concept of half-life is central to radiometric dating. It is defined as the time required for half of the radioactive nuclei in a sample to decay. In the case of carbon-14, its half-life of about 5730 years provides the time scale needed to convert the measured isotope ratio into the age of the sample.

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