Question

A small sample of wood from an archaeological site in Clovis, New Mexico, was burned in \(\mathrm{O}_{2}\), and the \(\mathrm{CO}_{2}\) produced was bubbled through a solution of \(\mathrm{Ba}(\mathrm{OH})_{2}\) to produce a precipitate of \(\mathrm{BaCO}_{3} .\) When the \(\mathrm{BaCO}_{3}\) was collected by filtration, a \(1.000 \mathrm{~g}\) sample was found to have a radioactivity of \(4.0 \times 10^{-3}\) Bq. The half-life of \({ }^{14} \mathrm{C}\) is \(5715 \mathrm{y}\), and living organisms have a radioactivity due to \({ }^{14} \mathrm{C}\) of \(15.3\) disintegrations/min per gram of carbon. Assuming that the analysis was carried out in 1960, what is the age BP of the Clovis site?

Short answer

The Clovis site is approximately 26,615 years BP.

Step-by-step solution

Convert Units to Matching Basis

First, let's ensure both radioactivity measurements are in the same units. The current radioactivity of the sample is given as \(4.0 \times 10^{-3}\, \text{Bq/g}\), where Bq (Becquerel) is 1 disintegration per second. The radioactivity in living matter is \(15.3\, \text{disintegrations/min/g}\). Convert this to disintegrations per second: \[15.3\, \text{disintegrations/min} = \frac{15.3}{60}\, \text{disintegrations/s} \approx 0.255\, \text{Bq/g}.\]

Determine the Fraction of \(^{14}\mathrm{C}\) Remaining

Using the formula \(\text{Fraction Remaining} = \frac{N}{N_0}\), where \(N\) is the current activity, and \(N_0\) is the original activity:\[\frac{N}{N_0} = \frac{4.0 \times 10^{-3}}{0.255}.\] Calculate this fraction to find: \[\frac{N}{N_0} \approx 0.0157.\]

Apply the Radioactive Decay Formula

The formula for radioactive decay is:\[\left(\frac{N}{N_0}\right) = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}},\]where \(t_{1/2} = 5715 \; \mathrm{years}\). Solve for \(t\):\[t = \frac{\log{\left(\frac{N}{N_0}\right)}}{\log{0.5}} \times t_{1/2}.\]

Calculate the Age of the Clovis Site

Substituting the fraction and the known half-life:\[t = \frac{\log{(0.0157)}}{\log{(0.5)}} \times 5715 \approx 26625\, \text{years}.\]

Determine the Year Before Present (BP)

Since the analysis was done in 1960, the age of the site in BP (before present, with 'present' being 1950) is:\[26625 - (1960 - 1950) = 26615\, \text{years BP}.\]

Key concepts

half-life

The concept of half-life is fundamental in understanding radioactive decay and radiocarbon dating. It is defined as the time required for half of the radioactive isotopes in a sample to decay. This concept is essential in radiocarbon dating, which uses carbon-14, a naturally occurring radioactive isotope, to estimate the age of organic materials.

In radioactive decay processes, the half-life remains constant over time, allowing scientists to make accurate age determinations. For example, carbon-14 has a well-established half-life of 5715 years. This means that every 5715 years, half of the carbon-14 in a sample will have decayed into nitrogen-14.

Understanding half-life allows archaeologists to use the remaining concentration of carbon-14 to calculate the approximate time that has elapsed since the organism was alive. When they find only a small fraction of the original carbon-14 concentration, they can infer that a significant amount of time has passed since the organism's death.

radioactive decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. It is the foundational principle behind radiocarbon dating. In the context of carbon-14, the decay occurs gradually over thousands of years, transforming carbon-14 atoms into stable nitrogen-14 atoms.

The decay process follows a predictable exponential decay function, which can be described mathematically. The formula for radioactive decay is \[\left(\frac{N}{N_0}\right) = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}\]where

• \(N\) is the current amount of the substance that remains undecayed,
• \(N_0\) is the initial amount of the substance,
• \(t\) is the elapsed time,
• \(t_{1/2}\) is the half-life of the substance.

Using this formula, scientists can calculate the time that has passed since the organism containing the carbon-14 died, by comparing the current carbon-14 level with the original level in a living organism.

carbon-14

Carbon-14 is a radioactive isotope of carbon that plays a crucial role in archaeological and geological dating methods. It is continuously formed in the atmosphere through the interaction of cosmic rays with nitrogen atoms and is taken up by living organisms during their lifetime.

When an organism dies, it stops absorbing carbon-14, and the existing carbon-14 within it begins to decay. By measuring the remaining carbon-14 in a sample and using its known half-life of 5715 years, scientists can determine the time since death.

Radiocarbon dating using carbon-14 is particularly effective for dating artifacts up to about 50,000 years old, making it an invaluable tool in archaeology. It offers a means to estimate the age of fossils, ancient artifacts, and even geological samples, providing insights into the past climates, environments, and human activities. By understanding the distribution and decay of carbon-14, researchers can unlock details about the ancient world, with accuracy verified by cross-referencing other dating methods.