A small sample of wood from an archaeological site in Clo-vis, New Mexico, was burned in O2 and the CO2 produced was bubbled through a solution of Ba(OH)2 to produce a precipitation of BaCO3. When the BaCO3 was collected by filtration, a 1.000 g sample was found to have a radioactivity of 4.0 x 10^-5 Bq. The half-life of 14C is 5715 y, and living organisms have a radioactivity due to 14C if 15.3 disintegrations/min per gram of carbon. If the analysis was carried out in 1960, what is the date of the Clovis site?

Verified step by step guidance

1. Calculate the decay constant (\(\lambda\)) using the formula \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(t_{1/2}\) is the half-life of 14C.
2. Convert the given radioactivity from disintegrations per minute to becquerels (Bq) for the living organisms' radioactivity, knowing that 1 disintegration per minute equals 1/60 Bq.
3. Use the decay law \(A = A_0 e^{-\lambda t}\) to find the ratio \(\frac{A}{A_0}\), where \(A\) is the current activity and \(A_0\) is the original activity when the organism was alive.
4. Solve for \(t\) in the decay equation to find the time elapsed since the wood was part of a living organism. Rearrange the equation to \(t = \frac{\ln(\frac{A_0}{A})}{\lambda}\).
5. Calculate the approximate year of the Clovis site by subtracting the time elapsed (from step 4) from the year of analysis (1960).

Key Concepts

Radiocarbon Dating

Radiocarbon dating is a method used to determine the age of organic materials by measuring the amount of carbon-14 (14C) they contain. Living organisms continuously absorb 14C from the atmosphere, but once they die, the 14C begins to decay at a known rate, characterized by its half-life of 5715 years. By comparing the remaining 14C in a sample to the expected levels in a living organism, scientists can estimate the time since the organism's death.

Radioactivity and Disintegration Rates

Radioactivity refers to the process by which unstable atomic nuclei lose energy by emitting radiation. The rate of disintegration, measured in becquerels (Bq), indicates how many nuclei decay per second. In this context, the radioactivity of the BaCO3 sample is used to infer the amount of 14C present, which is crucial for determining the age of the archaeological wood sample.

Precipitation Reactions

Precipitation reactions occur when two soluble substances react to form an insoluble product, known as a precipitate. In this case, the CO2 produced from burning the wood reacts with Ba(OH)2 to form BaCO3, which can be collected and analyzed. Understanding this reaction is essential for isolating the carbon from the wood sample, allowing for accurate measurement of its radioactivity.