Problem 36 - Chemistry The Molecular Science

Question

A sample of wood from a Thracian chariot found in an excavation in Bulgaria has a \({ }^{14} \mathrm{C}\) activity of 11.2 disintegrations per minute per gram. Estimate the age of the chariot and the year it was made. \(\left(t_{1 / 2}\right.\) for \({ }^{14} \mathrm{C}\) is \(5.73 \times 10^{3}\) years, and the activity of \({ }^{14} \mathrm{C}\) in living material is 15.3 disintegrations per minute per gram.)

Step-by-Step Solution

Answer

The chariot is approximately 3786 years old, made around 1763 BCE.

Step 1: Understanding the Problem

We have two key activity values: the current \(^{14} ext{C} \) activity of the wood from the Thracian chariot is 11.2 disintegrations/min/gram, and the normal \(^{14} ext{C} \) activity in living organisms is 15.3 disintegrations/min/gram. The half-life of \(^{14} ext{C} \) is given as 5730 years. We need to estimate the age of the chariot and determine the year it was made.

Step 2: Establish the Decay Formula

Carbon-14 decay can be modeled by the exponential decay formula: \[ N(t) = N_0 e^{-kt} \]where \( N(t) \) is the current activity, and \( N_0 \) is the initial activity. The rate constant \( k \) is given by \[ k = \frac{\ln(2)}{t_{1/2}} \].

Step 3: Calculate the Decay Constant

Calculate the decay constant \( k \) using the provided half-life \[ k = \frac{\ln(2)}{5730} \approx 0.000121 \text{ year}^{-1} \].

Step 4: Apply the Decay Formula

Use the exponential decay formula to find the age \( t \):\[ 11.2 = 15.3 \, e^{-0.000121 \, t} \]Solve for \( t \):\[ \frac{11.2}{15.3} = e^{-0.000121 \, t} \]Apply logarithms:\[ \ln\left(\frac{11.2}{15.3}\right) = -0.000121 \, t \]Calculate \( t \).

Step 5: Solve for Age

Calculate \( t \) by solving the equation:\[ t = \frac{\ln\left(\frac{11.2}{15.3}\right)}{-0.000121} \approx 3786 \text{ years} \].

Step 6: Estimate the Year the Chariot Was Made

Subtract the calculated age \( 3786 \) years from the current year (2023) to estimate the year the chariot was made:\[ 2023 - 3786 = -1763 \]. Since negative years correspond to BCE, the chariot was likely made around 1763 BCE.

Key Concepts

Carbon-14 Half-life

The concept of a half-life is crucial in understanding how radiocarbon dating works. Carbon-14, a radioactive isotope of carbon, decays over time. Its half-life represents the time needed for half of any given sample to transform into nitrogen-14 through radioactive decay. This process occurs naturally in the atmosphere and gets incorporated into living organisms. As a result, all living beings have a relatively constant level of Carbon-14.
Once the organism dies, it stops absorbing Carbon-14, and the isotope begins to decay. For Carbon-14, the half-life is approximately 5730 years. This means that in 5730 years, any sample of Carbon-14 will have half the amount it initially had.
Understanding the half-life is vital as it allows scientists to estimate the age of archaeological samples by comparing the current level of Carbon-14 to the initial level. This comparison is performed using the exponential decay formula. By knowing how much Carbon-14 has decayed, we can calculate how long it’s been since the organism's death.

Exponential Decay Formula

The exponential decay formula is key in radiocarbon dating. It provides a mathematical representation of how the quantity of Carbon-14 in a sample decreases over time. The formula is:

• \( N(t) = N_0 e^{-kt} \)

where \( N(t) \) is the remaining activity at time \( t \), \( N_0 \) is the initial activity when the organism was alive, and \( k \) is the decay constant.
The decay constant \( k \) is calculated using the formula \( k = \frac{\ln(2)}{t_{1/2}} \). For Carbon-14, with a half-life of 5730 years, \( k \approx 0.000121 \text{ year}^{-1} \).
This formula is essential because it allows us to link the current level of Carbon-14 to the time that has elapsed since the death of the organism. Thus, by measuring the current carbon activity, scientists can backtrack to find how many years have passed since the activity was at its peak.

Archaeological Age Estimation

Estimating the age of an archaeological find using radiocarbon dating involves comparing the current Carbon-14 activity to that of a living organism. For a given sample, scientists measure the number of disintegrations per minute per gram. Then, using the exponential decay formula, they determine the time elapsed since the organism's death.
For instance, in the Thracian chariot example, we know the current activity is 11.2 disintegrations/min/gram, while living organisms have 15.3 disintegrations/grams.
By plugging these values into the formula \( 11.2 = 15.3 \, e^{-0.000121 \, t} \), we can calculate \( t \), which tells us how many years have passed. In this case, it resulted in approximately 3786 years since the wood was last alive.

• Subtract this period from the current year to find out when the artifact was made.

Subtracting from 2023 leads us to an estimated origin year of 1763 BCE, illustrating how this method reliably dates ancient artifacts.