Problem 91 - Chemistry: The Central Science in SI Units, Global Edition

Question

Charcoal samples from Stonehenge in England were burned in \(\mathrm{O}_{2},\) and the resultant \(\mathrm{CO}_{2}\) gas bubbled into a solution of \(\mathrm{Ca}(\mathrm{OH})_{2}\) (limewater), resulting in the precipitation of \(\mathrm{CaCO}_{3}\). The \(\mathrm{CaCO}_{3}\) was removed by filtration and dried. A 788 -mg sample of the \(\mathrm{CaCO}_{3}\) had a radioactivity of \(1.5 \times 10^{-2}\) Bq due to carbon-14. By comparison, living organisms undergo 15.3 disintegrations per minute per gram of carbon. Using the half-life of carbon-14, 5700 yr, calculate the age of the charcoal sample.

Step-by-Step Solution

Answer

The age of the charcoal sample, calculated using its carbon-14 radioactivity and given half-life of 5700 years, is approximately 3450 years.

Step 1: Calculate the ratio of radioactivity

First, we need to find the current radioactivity of the sample in disintegrations per minute per gram (DPM/g). The sample has a radioactivity of \(1.5 \times 10^{-2}\) Bq, which is equivalent to \(1.5 \times 10^{-2}\) disintegrations per second. To convert this to disintegrations per minute, multiply by 60 seconds per minute: \[ \frac{1.5 \times 10^{-2}\,\text{disintegrations/s}}{1\,\text{min}} \times \frac{60\,\text{s}}{1\,\text{min}} = 0.9\,\text{disintegrations/min} \] Next, we need to find the mass of carbon in the sample, which can be found from the mass of the precipitated \(\mathrm{CaCO}_{3}\). The 788-mg sample of \(\mathrm{CaCO}_{3}\) contains carbon, and the mass ratio of carbon to \(\mathrm{CaCO}_{3}\) is: \[ \frac{\text{mass of carbon}}{\text{mass of }\mathrm{CaCO}_{3}} = \frac{\text{molar mass of carbon}}{\text{molar mass of }\mathrm{CaCO}_{3}} \] The molar mass of carbon is 12.01 g/mol, and the molar mass of \(\mathrm{CaCO}_{3}\) is 100.09 g/mol, so the mass ratio is: \[ \frac{\text{mass of carbon}}{788\,\text{mg}} = \frac{12.01\,\text{g/mol}}{100.09\,\text{g/mol}} \] Now we can find the mass of carbon in the sample: \[ \text{mass of carbon} = \frac{12.01\,\text{g/mol}}{100.09\,\text{g/mol}} \times 788\,\text{mg} = 94.2\,\text{mg} \] Now, we can find the radioactivity of the sample in disintegrations per minute per gram (DPM/g): \[ \frac{0.9\,\text{disintegrations/min}}{94.2\,\text{mg}} = 9.56 \times 10^{-3}\,\text{DPM/mg} \] Then, we will find the ratio of the current radioactivity to the initial radioactivity of living organisms: \[ \frac{9.56 \times 10^{-3}\,\text{DPM/mg}}{15.3\,\text{DPM/mg}} = 6.25 \times 10^{-4} \]

Step 2: Calculate the age of the sample using the half-life formula

Now that we have the ratio of the current radioactivity to the initial radioactivity, we can use the half-life formula to calculate the age of the charcoal sample. The formula is: \[ t = \frac{\ln(\frac{\text{Current radioactivity}}{\text{Initial radioactivity}})}{-\lambda} \] Where \(t\) is the age of the sample, and \(\lambda\) is the decay constant, which can be calculated from the half-life using the formula: \[ \lambda = \frac{\ln(2)}{\text{half-life}} \] Using the given half-life of carbon-14, 5700 years, we can find the decay constant \(\lambda\): \[ \lambda = \frac{\ln(2)}{5700\,\text{yr}} = 1.21 \times 10^{-4}\,\text{yr}^{-1} \] Now we can substitute the values of the ratio of radioactivity and decay constant into the age formula to find the age of the sample: \[ t = \frac{\ln(6.25 \times 10^{-4})}{-1.21 \times 10^{-4}\,\text{yr}^{-1}} \approx 3450\,\text{yr} \] The age of the charcoal sample is approximately 3450 years.

Key Concepts

Half-life

The concept of half-life is essential in understanding radioactive decay. It is the time required for half of the radioactive nuclei in a sample to decay. For carbon-14, a common isotope used in radiocarbon dating, the half-life is about 5700 years. This means that after 5700 years, only half of the original carbon-14 atoms remain in a sample.
Why is this important? Knowing the half-life allows scientists to date ancient objects like the charcoal from Stonehenge. By measuring how much carbon-14 remains, we can estimate how long it has been since the organism was alive.

• This calculation involves understanding exponential decay since the quantity decreases by a fraction (half) over each period.
• Half-life is crucial in archeology and geology for dating artifacts and fossils.

Carbon-14

Carbon-14 is a radioactive isotope of carbon. It is naturally occurring and forms when cosmic rays interact with nitrogen in the atmosphere. Living organisms absorb carbon-14 during their lifetime.
When an organism dies, it stops absorbing carbon-14, and the isotope begins to decay at a known rate (half-life of 5700 years). Radiocarbon dating uses this predictable decay to estimate the age of artifacts.

• Carbon-14 makes up a small fraction of the total carbon but is invaluable for dating ancient materials.
• Its presence in materials like charcoal can be used to track historical timelines accurately.

Disintegrations per minute

Disintegrations per minute (DPM) are a way to measure radioactivity. It refers to the number of decays or disintegrations that occur in one minute.
In this exercise, we calculated the difference in DPM between the present sample and living organisms. Charcoal had 0.9 disintegrations per minute compared to living organisms' 15.3 DPM.

• Converting from disintegrations per second (Becquerel) to DPM helps understand the level of radioactivity.
• This measure helps researchers determine how much of the radioactive isotope remains.

Decay constant

The decay constant (\( \lambda \)) is a crucial part of radiocarbon dating. It describes the probability of decay per unit time. It's derived from the half-life and is calculated using \( \lambda = \frac{\ln(2)}{\text{half-life}} \).
This constant is used in the formula to find the age of an archaeological sample, where \( t = \frac{\ln(\frac{\text{Current radioactivity}}{\text{Initial radioactivity}})}{-\lambda} \).

• The decay constant allows us to translate half-life into a usable metric for calculating age.
• It's an essential tool for converting observed radioactivity into a timeline.