(II) An ancient wooden club is found that contains 73 g of...

(II) An ancient wooden club is found that contains 73 g of carbon and has an activity of 7.0 decays per second. Determine its age assuming that in living trees the ratio of $^{14}$C/$^{12}$C atoms is about 1.3 $\times$ 10$^{-12}$.

Key Concepts

Radiocarbon Dating

This is a method used to determine the age of carbon-containing materials by measuring the decay of carbon-14, a radioactive isotope. It is based on the principle that new organic matter incorporates carbon-14 at a known ratio, which begins to decay at a fixed rate once the organism dies. This technique is widely used in archaeology, geology, and other fields to date ancient biological samples.

Exponential Decay

Exponential decay describes the process by which the quantity of a radioactive isotope decreases over time at a rate proportional to its current value. The decay follows a specific mathematical formula, allowing the calculation of how much of the isotope remains after a given period, which is fundamental in radiometric dating methods.

Half-Life

The half-life is the time required for half of the radioactive atoms in a sample to decay. It is a critical parameter in radiocarbon dating as it governs the rate at which carbon-14 decays. Knowing the half-life allows scientists to relate the remaining amount of carbon-14 in a sample to the time elapsed since the death of the organism.

Activity Measurement

Activity refers to the rate at which a radioactive sample undergoes decay, typically measured in decays per second. By comparing the current activity of a sample with the known initial activity of a living organism, one can determine the time elapsed since the organism's death. This measurement is a key input in calculating the age of archaeological samples.

Isotope Ratio

The isotope ratio, such as the ratio of carbon-14 to carbon-12, is crucial in radiocarbon dating. Living organisms maintain a constant ratio of these isotopes, but after death, the carbon-14 decays while carbon-12 remains stable. By measuring the current isotope ratio in a sample and comparing it to the original ratio, one can estimate the time that has passed since the organism died.

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