Problem 41 - Chemistry: The Central Science in SI Units, Global Edition

Question

A 10.00 -g plant fossil from an archaeological site is found to have a \({ }^{14} \mathrm{C}\) activity of 3094 disintegrations over a period of ten hours. A living plant is found to have a \({ }^{14} \mathrm{C}\) activity of 9207 disintegrations over the same period of time for an equivalent amount of sample with respect to the total contents of carbon. Given that the half-life of \({ }^{14} \mathrm{C}\) is 5715 years, how old is the plant fossil?

Step-by-Step Solution

Answer

The plant fossil is approximately 4489 years old, calculated by determining the activity ratio of \({ }^{14} \mathrm{C}\) between the fossil and a living plant, and then using the half-life formula for \({ }^{14} \mathrm{C}\) which has a half-life of 5715 years.

Step 1: Calculate the activity ratio

First, calculate the activity ratio of the fossil's \({ }^{14} \mathrm{C}\) activity to the living plant's \({ }^{14} \mathrm{C}\) activity: Activity ratio = \(\dfrac{\text{Activity in fossil}}{\text{Activity in living plant}}\) Activity ratio = \(\dfrac{3094}{9207}\) Now, calculate the exact value.

Step 2: Find the exact value of the activity ratio

Activity ratio = \(\dfrac{3094}{9207} \approx\) 0.3361

Step 3: Use the half-life formula

Now, we can use the half-life formula to find the age of the fossil. The formula for half-life is: \(N_t = N_0 \times (\dfrac{1}{2})^\dfrac{t}{t_{1/2}}\) Where: - \(N_t\) is the amount of radioactive substance at time \(t\) - \(N_0\) is the initial amount of radioactive substance - \(t_{1/2}\) is the half-life - \(t\) is the time that has passed In this case, we have the activity ratio (\(\dfrac{N_t}{N_0}\)), which is approximately 0.3361. The half-life of \({ }^{14} \mathrm{C}\) is 5715 years. We can rearrange the formula to solve for \(t\): \(t = t_{1/2} \times \dfrac{\log(\dfrac{N_t}{N_0})}{\log(\dfrac{1}{2})}\) Now, plug in the values.

Step 4: Calculate the age of the plant fossil

By plugging in the values, we get: \(t = 5715 \times \dfrac{\log(0.3361)}{\log(0.5)}\) Now, calculate the exact age.

Step 5: Find the exact age of the plant fossil

By finding the exact age, we get: \(t \approx 4489.3 \, \text{years}\) Thus, the plant fossil is approximately 4489 years old.

Key Concepts

Half-Life Calculation

Half-life is the time it takes for half of a sample of a radioactive substance to decay. It is a vital concept in the study of radioactive materials, including Carbon-14. Half-life calculations are crucial for determining the age of objects using radioactive decay. To calculate the age of a sample, we need to know:

• The half-life of the radioactive substance, which is 5715 years for Carbon-14.
• The ratio of the substance's activity in a sample to that in a living organism.

This information helps us estimate how many half-lives have occurred, thus giving insight into the age of the sample.

Radioactive Decay

Radioactive decay is a process where unstable nucleus emits radiation to become more stable. In Carbon-14 dating, we examine how this decay alters the amount of Carbon-14 in an object over time. For example, when a plant is alive, it continuously absorbs Carbon-14. But once it dies, no more Carbon-14 is absorbed, and the Carbon-14 already present starts decaying. This decay can be measured by counting the disintegrations over a given period. Understanding the decay rate of Carbon-14 helps us to find out the age of archaeological samples, such as fossils. The decreased radioactivity in a fossil compared to a living organism indicates how long it has stopped absorbing Carbon-14.

Archaeological Dating

Archaeological dating using Carbon-14, or radiocarbon dating, is a method to determine the age of an object containing organic material. It measures the remaining Carbon-14 levels compared to living organisms. When archaeologists find a fossil, they measure the Carbon-14 activity and compare it with a living equivalent to find its age.

• The more the Carbon-14 has decayed, the older the sample.
• This method helps date objects up to 50,000 years with reasonable accuracy.

Radiocarbon dating has revolutionized archaeology by providing a more precise timeline of human historical events and ancient life.