Problem 44 - Chemistry Structure and Dynamics

Question

Charcoal samples from Stonehenge in England emit \(62.3 \%\) of the disintegrations per gram of carbon per minute expected for living tissue. What is the age of this charcoal? \(\left({ }^{14} \mathrm{C}: t_{1 / 2}=5730 \mathrm{yr}\right)\)

Step-by-Step Solution

Answer

The charcoal is about 4151 years old.

Step 1: Understand the Problem

We need to find the age of the charcoal sample. The problem states that the charcoal emits 62.3% of the disintegrations per gram of carbon per minute as living tissue does. We are given the half-life of carbon-14, which is 5730 years.

Step 2: Apply the Radioactive Decay Formula

The decay of carbon-14 follows an exponential form: \[ N = N_0 e^{-kt} \], where \( N \) is the current quantity, \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is time. But we can use the formula for half-life: \( N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}} \) because we are given the half-life and the percentage of the current activity.

Step 3: Calculate the Decay Constant

The decay constant \( k \) is related to the half-life by the formula: \[ k = \frac{\ln(2)}{t_{1/2}} \]. For this problem, \( k = \frac{\ln(2)}{5730} \approx 1.2097 \times 10^{-4} \space \text{per year} \).

Step 4: Calculate the Age of the Charcoal

We rearrange the half-life formula to solve for \( t \), using \( N/N_0 = 0.623 \): \[ 0.623 = \left(\frac{1}{2}\right)^{t/5730} \]. Taking the natural logarithm of both sides, we have \[ \ln(0.623) = \ln\left(\left(\frac{1}{2}\right)^{t/5730}\right) = \frac{-t}{5730} \ln(2) \]. So, \[ t = \frac{-5730 \cdot \ln(0.623)}{\ln(2)} \approximately 4151 \text{ years}. \]

Step 5: Conclusion

The age of the charcoal sample is approximately 4151 years.

Key Concepts

Carbon-14 Dating

Carbon-14 dating, also known as radiocarbon dating, is an essential tool in archaeology and geology for determining the age of ancient organic materials. This method relies on the fact that carbon-14 - found naturally in the atmosphere - is absorbed by living organisms. Once they die, they stop absorbing it, and the carbon-14 they contain starts to decay. Carbon-14 is radioactive and decays over time into a stable isotope, nitrogen-14. This radioactive decay follows a predictable pattern, allowing scientists to measure the remaining levels of carbon-14 in a sample and compare it with the initial levels found in living organisms. By doing so, they can calculate how long it has been since the organism died. This is particularly useful for dating materials up to about 50,000 years old, offering insights into historic events like the age of archaeological sites and remnants, such as the charcoal found at Stonehenge.

Half-Life Calculation

A crucial concept in radioactive decay is half-life. The half-life of a radioactive isotope is the time it takes for half of the isotope to decay. For carbon-14, this period is 5730 years. This means that, every 5730 years, half of the carbon-14 in a sample will have decayed into nitrogen-14.In the exercise, you are given that the charcoal sample exhibits 62.3% of the original level of disintegrations expected from living tissue. This means it has not yet reached a point where half of the carbon-14 has decayed, but rather a bit more than half remains.Knowing the half-life allows us to apply the formula: \[ N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}} \]where - \(N\) is the remaining quantity,- \(N_0\) is the initial amount, - \(t\) is the time elapsed, and - \(t_{1/2}\) is the half-life. By rearranging this formula, we can calculate the age of the sample based on how much carbon-14 remains.

Exponential Decay Formula

The concept of exponential decay is central to understanding carbon-14 dating. Radioactive decay processes are exponential in nature, meaning they proceed at a rate proportional to the amount of substance remaining. In mathematical terms, this can be expressed with the formula:\[ N = N_0 e^{-kt} \]where - \(N\) is the remaining quantity of the substance,- \(N_0\) is the initial quantity, - \(k\) is the decay constant, and - \(t\) is the time that has passed.For half-life calculations, an alternative formula is often more directly useful:\[ N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}} \]These formulas reflect that the rate at which a substance decays is dependent not on time, but on the changing quantity of the substance itself. Utilizing the decay constant \(k\), calculated as \( \frac{\ln(2)}{5730} \) for carbon-14, and the remaining proportion of the sample's original activity, allows us to solve effectively for the time, \(t\). This uses the natural logarithm to solve exponential equations, hence converting them into linear forms that are easier to manipulate.