Problem 119 - Chemistry: An Atoms-Focused Approach

Question

The first attempt at radiocarbon dating six skeletons discovered in an Italian cave at the beginning of the 20 th century indicated an age of 15,000 years. Redetermination of the age in 2004 indicated an older age for two bones of between 23,300 and 26,400 years. What is the ratio of \(^{14} \mathrm{C}\) in a sample 15,000 years old to one 25,000 years old?

Step-by-Step Solution

Answer

Answer: The ratio of carbon-14 in a 15,000-year-old sample to a 25,000-year-old sample is approximately 1.677.

Step 1: Calculate decay constant (k) using the half-life formula

The half-life formula is given by: \(t_{1/2} = \frac{ln(2)}{k}\). We are given the half-life of C-14, \(t_{1/2}\) = 5,730 years, so we can find the decay constant (k) by rearranging the formula: \(k = \frac{ln(2)}{t_{1/2}}\). Calculating the decay constant (k): \(k = \frac{ln(2)}{5730} \approx 1.21 \times 10^{-4} \text{years}^{-1}\)

Step 2: Calculate the percentage of remaining C-14 in both samples

We can use the decay constant to find the remaining C-14 after a certain time using the formula: \(percentage = e^{-kt}\), where t is the age of the sample in years. Calculating the percentage of remaining C-14 in the 15,000-year-old sample: \(percentage_{15000} = e^{-1.21 \times 10^{-4} \times 15,000} \approx 0.593\) Calculating the percentage of remaining C-14 in the 25,000-year-old sample: \(percentage_{25000} = e^{-1.21 \times 10^{-4} \times 25,000} \approx 0.354\)

Step 3: Calculate the ratio of remaining C-14 in both samples

We can find the ratio by dividing the percentage of C-14 in the 15,000-year-old sample by the percentage of C-14 in the 25,000-year-old sample. Calculating the ratio of C-14: \(ratio = \frac{percentage_{15000}}{percentage_{25000}} = \frac{0.593}{0.354} \approx 1.677\) The ratio of C-14 in a 15,000-year-old sample to one 25,000 years old is approximately 1.677.

Key Concepts

Decay Constant

The decay constant is a crucial factor in understanding the radioactive decay process. It refers to the probability per unit time that a given radioactive atom will decay. In radiocarbon dating, it helps us determine how fast a radioactive isotope like carbon-14 (\(^{14}\mathrm{C}\)) decreases over time. Using the decay constant, we can calculate how much \(^{14}\mathrm{C}\) is left in a sample from a long time ago. To find the decay constant (\(k\)), we use the formula where the half-life (\(t_{1/2}\)) is the time it takes for the substance to reduce to half its original amount. The formula used is:\[k = \frac{\ln(2)}{t_{1/2}}.\]In the context of carbon-14 dating, we know its half-life is 5730 years, and thus:\[k = \frac{\ln(2)}{5730} \approx 1.21 \times 10^{-4} \text{ years}^{-1}.\]This decay constant helps in calculating how long it takes for a specific amount of \(^{14}\mathrm{C}\) to decay, which is key in dating archaeological artifacts.

Half-life of C-14

The half-life of a radioactive substance is an essential concept in understanding decay processes. It is defined as the time required for half of the radioactive isotopes in a sample to decay. For carbon-14 (\(^{14}\mathrm{C}\)), the half-life is about 5730 years.This relatively long half-life makes carbon-14 very useful in dating objects ranging from a few hundred to tens of thousands of years old. During this time, the quantity of \(^{14}\mathrm{C}\) decreases as it converts into \(^{14}\mathrm{N}\) (nitrogen-14) through a process called beta decay. Here's what happens:

• When an organism dies, it stops taking in carbon-14, allowing scientists to measure the remaining \(^{14}\mathrm{C}\).
• Over time, the \(^{14}\mathrm{C}\) remains decay at a predictable rate, allowing for accurate age estimations.

By understanding the half-life of \(^{14}\mathrm{C}\), scientists can calculate the age of archaeological finds by comparing the remaining \(^{14}\mathrm{C}\) in the sample to what it initially was.

Exponential Decay Formula

Exponential decay describes the process by which quantities decrease at a rate proportional to their current value. In the case of radiocarbon dating, we use an exponential decay formula to model how carbon-14 (\(^{14}\mathrm{C}\)) decreases over time. This model helps determine the age of archaeological samples.The exponential decay formula is given by:\[N(t) = N_0 e^{-kt}\]where:

• \(N(t)\) is the quantity of \(^{14}\mathrm{C}\) remaining at time \(t\).
• \(N_0\) is the initial quantity of \(^{14}\mathrm{C}\).
• \(k\) is the decay constant.
• \(t\) is the time elapsed since the organism's death.

By using this formula, scientists can calculate the percentage of carbon-14 left in the sample and, subsequently, estimate the time that has passed since the death of an organism. For example, given a 15,000-year-old sample, the calculation within exponential decay shows how much \(^{14}\mathrm{C}\) remains compared to a 25,000-year-old sample. These findings are essential for accurately dating historical artifacts and understanding our past.