Problem 45 - Chemistry Structure and Dynamics

Question

A lump of beeswax was excavated in England near a collection of Bronze Age objects that are between 2500 and 3000 years old. Radiocarbon analysis of the beeswax suggests an activity roughly \(90.3 \%\) of that observed for living tissue. Was this beeswax part of the hoard of Bronze Age objects, or did it date from another period? \(\left({ }^{14} \mathrm{C}: t_{1 / 2}=5730 \mathrm{yr}\right)\)

Step-by-Step Solution

Answer

The beeswax is not from the Bronze Age; it is about 853 years old.

Step 1: Understanding the Decay Formula

The decay of carbon-14 is exponential and can be calculated using the formula:\[ A = A_0 \cdot (\frac{1}{2})^{t/t_{1/2}} \]where:- \(A\) is the current activity level,- \(A_0\) is the initial activity level for living tissue,- \(t\) is the time elapsed since the object was part of a living organism,- \(t_{1/2}\) is the half-life of carbon-14, which is given as 5730 years in this problem.

Step 2: Relate Current Activity to Initial Activity

We are given an activity level of 90.3% as compared to living tissue, meaning:\[ A = 0.903A_0 \]Substitute this into the decay formula:\[ 0.903A_0 = A_0 \cdot (\frac{1}{2})^{t/5730} \]

Step 3: Simplifying the Equation

Simplify the equation by dividing both sides by \(A_0\):\[ 0.903 = (\frac{1}{2})^{t/5730} \]This allows us to focus on solving for \(t\), the age of the beeswax.

Step 4: Converting to Logarithms

To solve for \(t\), we need to use logarithms. Take the natural logarithm (base \(e\)) of both sides:\[ \ln(0.903) = \ln((\frac{1}{2})^{t/5730}) \]

Step 5: Simplifying with Logarithms

Apply the power rule of logarithms, \(\ln(a^b) = b\ln(a)\) to simplify the right-hand side:\[ \ln(0.903) = \frac{t}{5730} \cdot \ln(\frac{1}{2}) \]

Step 6: Solving for t

Rearrange the equation to solve for \(t\):\[ t = \frac{5730 \cdot \ln(0.903)}{\ln(\frac{1}{2})} \]By calculating this, we find:\[ t \approx 853 \text{ years} \]

Step 7: Comparing with Bronze Age Timeline

Since the Bronze Age objects date from 2500 to 3000 years ago, and the beeswax dates to approximately 853 years ago, the beeswax is not part of the Bronze Age hoard.

Key Concepts

Carbon-14 Decay

Carbon-14 decay is a fascinating process that is central to understanding how radiocarbon dating works. Carbon-14 is a radioactive isotope of carbon that naturally occurs in all living organisms. It is unstable, meaning it decays over time. This decay happens at a predictable rate, which is why it's so useful for dating ancient biological materials.

When an organism is alive, it maintains a constant level of carbon-14 by absorbing carbon from its environment. However, once it dies, the intake stops, and carbon-14 begins to decay. This decay process is exponential, meaning that carbon-14 diminishes by half over a fixed period known as the half-life. The half-life of carbon-14 is 5730 years, which makes it particularly useful for dating materials that are thousands of years old. By measuring the remaining carbon-14 in a specimen, we can calculate how long it has been since the organism died, thus dating the object.

Half-Life Calculation

To understand radiocarbon dating, grasping the concept of half-life is crucial. The half-life is the time taken for half of the carbon-14 in a sample to decay. For carbon-14, this period is 5730 years. This consistent half-life is a critical aspect that allows scientists to perform accurate age estimations.

In practical terms, if you start with a certain amount of carbon-14, after 5730 years, only half of it will remain. After another 5730 years, only a quarter will be left, and so on. This progression helps in constructing a timeline for an object made of organic material, like beeswax. In our problem, this principle helps us determine whether the beeswax might be part of the Bronze Age hoard by calculating how many half-lives have passed since it last absorbed carbon-14.

Exponential Decay Formula

The exponential decay formula is key to solving problems related to radiocarbon dating. In essence, it is written as: \[ A = A_0 \cdot \left(\frac{1}{2}\right)^{t/t_{1/2}} \]where:

• \(A\) is the current amount of carbon-14.
• \(A_0\) is the original amount of carbon-14 when the organism was still alive.
• \(t\) represents the time since the organism died.
• \(t_{1/2}\) is the half-life of carbon-14, which is 5730 years.

This exponential relationship is because of the consistent and predictable decay pattern of carbon-14. For the beeswax, its current activity level relative to the living tissue is known, which allowed us to calculate the time that has elapsed since the beeswax was last part of a living system, by rearranging and solving this formula using logarithms.

Bronze Age

The Bronze Age is a historical period characterized by the use of bronze, a metal alloy primarily composed of copper combined with tin. It dates back to a period roughly between 3300 BCE and 1200 BCE, depending on the location. For regions like England, the Bronze Age is generally considered to be from about 2500 BCE to 800 BCE.

In archaeological studies, objects from this era, such as tools, ornaments, and hoards, provide profound insights into the early technological advancements and society's lifestyle at that time. The bronze hoard, like mentioned in our exercise, would be expected to be over 2500 years old due to the typical timeline of the Bronze Age. By determining the age of items such as the beeswax using radiocarbon dating and examining its association with known Bronze Age artifacts, historians and archaeologists can build a more complete picture of the past.