Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Carbon-14 Dating. The half-life of carbon-14 is about 5700 years. a. You find a piece of cloth painted with organic dyes. By analyzing the dye in the cloth, you find that only \(77 \%\) of the carbon-14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has \(6.2 \%\) of the carbon- 14 that it must have had when it was living. Estimate when the wood was cut. c. Suppose a fossil is 570,000 years old, which is 100 half-lives of carbon-14. What fraction of its original carbon-14 would remain? Use your answer to explain why carbon-14 generally is not useful for dating fossils of this age.

Short answer

a. Cloth painted ~2082 years ago. b. Wood cut ~21643 years ago. c. Trace amount remains; C-14 dating ineffective for fossils 570,000 years old.

Step-by-step solution

Understanding the half-life formula

The formula to find the amount of a substance remaining after a certain period is given by: \[ N = N_0 \left( \frac{1}{2} \right)^\frac{t}{T} \]where:- \(N\) is the remaining quantity of the substance.- \(N_0\) is the original quantity of the substance.- \(T\) is the half-life of the substance.- \(t\) is the time passed.The goal is to find \(t\).

Solve Part a - Finding the age of the cloth

Given that \(77\%\) of the carbon-14 remains, we substitute into the formula:\[ 0.77 = \left( \frac{1}{2} \right)^\frac{t}{5700} \]Taking the natural logarithm on both sides:\[ \ln(0.77) = \frac{t}{5700} \cdot \ln\left(\frac{1}{2}\right) \]Solving for \(t\), we get:\[ t = 5700 \cdot \frac{\ln(0.77)}{\ln(0.5)} \approx 2082 \text{ years} \]Thus, the cloth was painted approximately 2082 years ago.

Solve Part b - Estimating the age of the wood

The remaining \(6.2\%\) of carbon-14 indicates:\[ 0.062 = \left( \frac{1}{2} \right)^\frac{t}{5700} \]Taking logarithms:\[ \ln(0.062) = \frac{t}{5700} \cdot \ln\left(\frac{1}{2}\right) \]Solving for \(t\):\[ t = 5700 \cdot \frac{\ln(0.062)}{\ln(0.5)} \approx 21643 \text{ years} \]Therefore, the wood was cut approximately 21643 years ago.

Solve Part c - Carbon-14 in a 570,000-year-old fossil

Given the fossil is 570,000 years old, which is 100 half-lives, the fraction remaining is:\[ \left(\frac{1}{2}\right)^{100} \]Calculating this gives an extremely small value, essentially approximating to zero.Thus, only a trace amount of carbon-14 remains, making it ineffective for dating fossils this old given measurement limitations.

Key concepts

Half-life

The concept of half-life is fundamental in understanding radioactive decay processes such as Carbon-14 dating. It is defined as the time required for half of the atoms in a radioactive sample to decay.
A substance’s half-life is a constant that allows us to predict how much of the substance will remain after a given amount of time.
For Carbon-14, this half-life is approximately 5700 years. This means that if you start with a certain amount of Carbon-14, after 5700 years, only half of it will remain.
After another 5700 years, half of the remaining Carbon-14 will have decayed, leaving only a quarter of the original amount.
The formula used to describe this process in calculations is:

• \( N = N_0 \left( \frac{1}{2} \right)^\frac{t}{T} \) where:
• \(N\) is the quantity remaining after time \(t\),
• \(N_0\) is the initial quantity, and
• \(T\) is the half-life.

This mathematical relationship allows archaeologists and scientists to estimate the age of ancient artifacts by measuring the remaining Carbon-14 and comparing it to its expected initial value.

Radioactive decay

Radioactive decay is a process through which unstable nuclei lose energy by emitting radiation.
This decay happens spontaneously, transforming the original atom into a different element or a different isotope of the same element.
In the context of Carbon-14, it is an isotope of carbon that decays over time.
Key points about radioactive decay:

• It is a random process at the level of a single atom: we cannot predict when a particular atom will decay, but we can predict the decay rate on a larger scale.
• The decay results in the production of a different element or isotope, such as nitrogen in the case of Carbon-14.
• The rate of decay is described by the half-life, which helps calculate how old an object is based on the remaining amount of a radioactive isotope.

Understanding radioactive decay is essential for fields like archaeology and geology, where it is used to date ancient objects and formations by tracing how much of a radioactive material remains.

Archaeological dating

Archaeological dating using Carbon-14 is one of the primary methods used to date ancient organic materials.
This technique takes advantage of the predictable radioactive decay of Carbon-14 within organic materials over time.
Carbon-14 dating allows us to determine how long ago a living organism died, as illustrated in scenarios such as dating a cloth or wood that archaeologists may find.
Here's how it works:

• Living organisms take in Carbon-14 from the atmosphere during their lifetime, maintaining a balance with the atmosphere's levels.
• When the organism dies, it stops absorbing Carbon-14, and the isotope starts to decay at a known rate (half-life of 5700 years). The remaining amount of Carbon-14 thus acts as a timestamp for the time since the organism's death.
• By measuring how much Carbon-14 remains in a sample compared to how much it originally had, scientists can estimate the time that has elapsed since the organism died.

However, Carbon-14 becomes ineffective for dating materials beyond a few tens of thousands of years. This is because, after around 50,000 years, the remaining activity of Carbon-14 is so minute that it becomes difficult to measure accurately.
For older fossils, other dating techniques, such as potassium-argon dating, are used to overcome these limitations.