Problem 97 - Chemistry The Science in Context

Question

The discovery of six skeletons in an Italian cave at the beginning of the 20 th century was considered a significant find in Stone Age archaeology. The age of these bones has been debated. The first attempt at radiocarbon dating indicated an age of 15,000 years. Redetermination of the age in 2004 indicated an older age for two bones, 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 is approximately 4.40.

Step 1: Determine the number of half-lives for each sample

First, we need to determine the number of half-lives that have passed for each given time period. To do this, we will divide the given time period by the half-life of \(^{14}\mathrm{C}\), which is 5,730 years. For 15,000 years: Number of half-lives = \(\frac{15,000}{5,730} = 2.62\) half-lives For 25,000 years: Number of half-lives = \(\frac{25,000}{5,730} = 4.36\) half-lives

Step 2: Calculate the remaining \(^{14}\mathrm{C}\) ratio in each sample

Now, we need to find how much \(^{14}\mathrm{C}\) remains in each sample after the given time period. We will use the formula: Remaining ratio = \(0.5^n\) Where n is the number of half-lives. For the 15,000-year-old sample: Remaining ratio = \(0.5^{2.62} = 0.229\) For the 25,000-year-old sample: Remaining ratio = \(0.5^{4.36} = 0.052\)

Step 3: Calculate the ratio of \(^{14}\mathrm{C}\) in the 15,000-year-old sample to the 25,000-year-old sample

Finally, we divide the remaining ratio of the 15,000-year-old sample by the remaining ratio of the 25,000-year-old sample to find the ratio of \(^{14}\mathrm{C}\) between the two samples. Ratio = \(\frac{0.229}{0.052} = 4.40\) The ratio of \(^{14}\mathrm{C}\) in a sample 15,000 years old to one 25,000 years old is approximately 4.40.

Key Concepts

Half-life

The half-life of a substance is the time it takes for half of it to decay. It remains a crucial concept in understanding radioactive decay, including radiocarbon dating.
For carbon-14 ( {}^{14}C eq so, the half-life is roughly 5,730 years.
This means that after 5,730 years, only half of the original carbon-14 remains in a material.
In the context of archaeology, this constant decay rate helps determine the age of organic materials, such as bones or wood, that contain carbon.
To figure out the number of half-lives that have passed, you divide the age of the sample by the half-life.

• For a 15,000-year-old sample, about 2.62 half-lives have elapsed.
• For a 25,000-year-old sample, about 4.36 half-lives have elapsed.

Understanding half-lives allows us to accurately calculate the remaining amount of carbon-14 in ancient samples.

Carbon-14

Carbon-14 ( {}^{14}C eq is an isotope found in small amounts within every living organism.
It is useful for determining the ages of archaeological finds since it decays at a consistent rate after the organism dies, ceasing the intake of carbon.
Through radiocarbon dating, scientists can measure the remaining carbon-14 in a specimen and calculate its age.
Unlike its stable counterparts (carbon-12 and carbon-13), carbon-14 is radioactive, meaning it slowly transforms into nitrogen-14 over time.

• This transformation occurs at a predictable rate, with a known half-life, making it ideal for dating ancient organic materials.
• The method is typically accurate for materials up to around 50,000 years old.

This scientific tool helps archaeologists build timelines and understand historical timelines based on organic sample rates.

Archaeology

Archaeology is the study of past human activity through artifacts, monuments, and other physical remains.
This field allows us to understand human history and prehistory, providing insights into how ancient civilizations lived.
One vital practice in archaeology is dating these artifacts accurately, and radiocarbon dating is a significant method made possible by the half-life of carbon-14.
By calculating the amount of carbon-14 remaining in organic materials, archaeologists can infer the age of a sample.

• Radiocarbon dating enables researchers to assign dates to archeological finds, improving our understanding of historical developments and cultural progressions.
• This method supports piecing together past events and timelines, crucial for establishing connections between different sites and societies.

Overall, archaeology uncovers our shared past, with radiocarbon dating playing a key role in this mission.

Stone Age

The Stone Age marks a period in human history where early humans mainly made tools and artifacts from stone.
Spanning millions of years, it is divided into three periods: Paleolithic, Mesolithic, and Neolithic.
Archaeologists study the Stone Age to understand the evolution of human societies and technological advancements in ancient times.
The use of radiocarbon dating allows researchers to pinpoint when specific Stone Age events or developments occurred.

• This period is characterized by profound changes in human behavior, from foraging in the Paleolithic times to farming in the Neolithic.
• Understanding the chronological sequence of events during the Stone Age also assists in grasping the broader picture of human evolution.

The accuracy of dating techniques like radiocarbon dating is crucial for charting the development of early human societies through the ages.