Carbon from a cypress beam obtained from the tomb of an ancient Egyptian king gave \(9.2\) disintegrations/minute of \(C-14\) per gram of carbon. Carbon from living material gives \(15.3\) disintegrations/min of C-14 per gram of carbon. Carbon-14 has a half-life of 5730 years. How old is the beam?

Understanding the half-life of Carbon-14 (C-14) is crucial when studying radioactive carbon dating. The half-life is the amount of time it takes for half of the radioactive isotopes in a sample to decay. For Carbon-14, this period is approximately 5730 years.

This knowledge allows scientists to calculate the age of ancient organic materials by comparing the remaining amount of C-14 to the expected amount in a current, live sample. If we know the half-life, we can use this as a 'clock' to measure how much time has passed since the death of the organic material. It is important to remember that the half-life remains constant while the proportion of the isotope decreases over time due to its exponential decay nature.

Radioactive isotopes such as Carbon-14 follow an exponential decay model, described by the formula \(N(t) = N_0 \cdot e^{-kt}\) where \(N(t)\) is the number of remaining radioactive atoms at time \(t\), \(N_0\) is the initial number of radioactive atoms, \(e\) is the base of the natural logarithm, \(k\) is the decay constant, and \(t\) is the time elapsed.

In context, this model shows us that the decay of radioactive isotopes is not linear but decreases faster at the beginning and slows down as time passes. The decay constant \(k\) is unique to each radioactive isotope and strongly connected to its half-life, providing a clear link between the amount of isotope remaining, the time elapsed, and the half-life, all essential factors in radiocarbon dating.

To perform radiocarbon dating calculations, we first establish the decay constant from the known half-life of Carbon-14 using the equation \(k = \dfrac{ln(2)}{T_{1/2}}\).

Then, we compare the disintegration rate of the sample with the disintegration rate of a modern, or living, sample to calculate the fraction representing the amount of C-14 that has decayed. Finally, we use the exponential decay equation to solve for the elapsed time since the death of the sample, which represents its age.

Example Calculation

Given the disintegration rate of an ancient beam and a modern sample, we use these rates in conjunction with the exponential decay formula to determine the age of the beam. By rearranging and solving the equation, the calculated time gives us the approximate age of the ancient material. This process shines light on the history of our planet by allowing us to date artifacts, fossils, and even geologic events.