Problem 83 - Chemistry: The Central Science

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 \(788-\mathrm{mg}\) 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, 5715 yr, calculate the age of the charcoal sample.

Step-by-Step Solution

Answer

The age of the charcoal sample is approximately 2893 years.

Step 1: Calculate the number of moles of CaCO3

To calculate the age of the sample, we first need to find the number of moles of CaCO3. This can be done by using the formula: moles of CaCO3 = (mass of CaCO3) / (molecular weight of CaCO3) The molecular weight of CaCO3 is 40.08 (Ca) + 12.01 (C) + 3 * 16.00 (O) = 100.09 g/mol. Thus, we can find the number of moles of CaCO3 as: moles of CaCO3 = (788 mg) / (100.09 g/mol) = 0.00788 mol

Step 2: Calculate the number of moles of C in CaCO3

Since there is a 1:1 molar ratio between calcium carbonate (CaCO3) and carbon (C), the number of moles of C will be equal to the moles of CaCO3: moles of C = 0.00788 mol

Step 3: Calculate the mass of C in the sample

We can now calculate the mass of carbon in the sample as: mass of C = (moles of C) * (molecular weight of C) mass of C = (0.00788 mol) * (12.01 g/mol) = 0.09456 g

Step 4: Calculate the initial radioactivity

We are given that living organisms undergo 15.3 disintegrations per minute per gram of carbon. We can calculate the initial radioactivity (R0) of the charcoal sample (in disintegrations per minute) using this information as follows: R0 = (15.3 dis/min/g) * (0.09456 g) = 1.44808 dis/min

Step 5: Finding the final radioactivity

We are given that the final radioactivity of the sample is 1.5 x 10^(-2) Bq. We need to convert this to dis/min to accurately compare it to the initial radioactivity: 1 Bq = 1 disintegration/second, thus, 1.5 x 10^(-2) Bq = 1.5 x 10^(-2) disintegrations/second Now, we can convert it to dis/min: final radioactivity (Rf) = 1.5 x 10^(-2) dis/sec * 60 sec/min = 0.9 dis/min

Step 6: Use decay formula to calculate the age of the sample

Now we can use the decay formula to find the age of the sample: \(Rf = R0 * (1/2)^{t/T_{1/2}}\) Where Rf is the final radioactivity, R0 is the initial radioactivity, t is the age of the sample, and \(T_{1/2}\) is the half-life of carbon-14. Rearranging the formula to solve for t: \(t = \frac{T_{1/2} * \log{(\frac{R0}{Rf})}}{\log{2}}\) Plugging the values: \(t = \frac{5715 * \log{(\frac{1.44808}{0.9})}}{\log{2}}\) Calculating the age of the sample: t ≈ 2892.99 years The age of the charcoal sample is approximately 2893 years.

Key Concepts

Half-life of Carbon-14

Radiocarbon dating uses the properties of carbon-14 ( C^{14} ), a radioactive isotope of carbon. One important feature of C^{14} is its half-life, which is the time it takes for half of a given amount of the isotope to decay. The half-life of carbon-14 is approximately 5715 years. This time span allows scientists to date ancient organic materials.

Unlike other forms of dating that focus on non-organic materials, radiocarbon dating provides a timeline for historical events and developments. This process relies on the predictable decay rate of carbon-14. After determining how much carbon-14 remains in a sample compared to a living organism, researchers can calculate its approximate age. Using the half-life, the dating is done by applying the formula involving exponential decay. This formula calculates the remaining percentage of carbon-14 in the sample, which corresponds to the time that has elapsed since the death of the organism.
Understanding the half-life allows us to interpret radiocarbon measurements meaningfully. For instance, if a sample shows a radioactive decay indicating half of the original amount of carbon-14 remains, it suggests the sample is approximately 5715 years old.

Disintegrations per minute

In radiocarbon dating, the decay of carbon-14 is measured in terms of disintegrations per minute (dpm), which denotes the number of times carbon-14 atoms decay in one minute. It reflects the radioactivity level and helps infer the sample's age.
Living organisms maintain a relatively constant level of carbon-14 activity, registering about 15.3 disintegrations per minute per gram of carbon. This baseline helps researchers determine the original radioactivity level of a sample (known as R_0 ), which is essential for age calculations.

When dealing with ancient samples, we're interested in comparing this original activity against the current activity ( R_f ). The exponential decay principle is used to determine what fraction of the original dpm remains. By comparing these two measurements, it is possible to estimate how much time has passed since the organism's death. More rapid declines in dpm imply more significant time has elapsed since the sample was last alive.

Molar calculations

To analyze and date samples, molar calculations play a crucial role. They help convert the mass of a chemical component, like calcium carbonate ( CaCO_3 ), into moles, which then aids in determining the number of carbon atoms present. This step is necessary to understand the composition and to carry out radiocarbon dating practices.

The first calculation involves determining the number of moles of calcium carbonate using its molecular weight. Given the molecular weight of CaCO_3 as 100.09 g/mol, the sample's mass is converted from milligrams to grams, which facilitates the molar conversion. In our example, a 788 mg sample is equal to approximately 0.00788 mol of CaCO_3 . Once we've calculated the moles of CaCO_3 , we can directly find the moles of carbon (C) because there's a one-to-one molar relationship between CaCO_3 and carbon.

By multiplying the number of moles of carbon by the molecular weight of carbon, which is 12.01 g/mol, the actual mass of carbon in the sample can be deduced. This mass is essential for calculating the initial radioactivity and eventually determining the sample's age.

Charcoal analysis

Charcoal analysis in archaeological studies often involves assessing the carbon content to determine its age. By burning charcoal in oxygen, the resultant carbon dioxide ( CO_2 ) can be trapped in a solution like limewater, resulting in calcium carbonate ( CaCO_3 ) formation. This CaCO_3 can be analyzed for carbon-14 content.

The precise understanding of how charcoal ages allow historians and archaeologists to attribute chronological dates to ancient human activities. Stonehenge's charcoal samples, for instance, are invaluable for time stamping its construction and usage by analyzing the carbon-14 present in the CaCO_3 extracted from burnt charcoal.

The dried CaCO_3 precipitate provides a measurable quantity of carbon-14, which reveals the disintegration rate per time unit. Through this methodical process, scientists derive the historical timeline of an artifact based on the known half-life of carbon-14 and other disintegration metrics, offering insights into past climatic and environmental conditions.