Problem 46
Question
Geologists who study volcanoes can develop historical profiles of previous eruptions by determining the \(^{14} \mathrm{C} /^{12} \mathrm{C}\) ratios of charred plant remains entrapped in old magma and ash flows. If the uncertainty in determining these ratios is \(0.1 \%,\) could radiocarbon dating distinguish between debris from the eruptions of Mt. Vesuvius that occurred in the years 472 and \(512 ?\) (Hint: Calculate the \(^{14} \mathrm{C} /^{12} \mathrm{C}\) ratios for samples from the two dates.)
Step-by-Step Solution
Verified Answer
Answer: Yes, radiocarbon dating can distinguish between the eruptions in 472 and 512, as the difference between the \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios (-0.48%) is greater than the uncertainty of 0.1%.
1Step 1: Calculate the decay constant for \(^{14}\mathrm{C}\)
The half-life (t) of \(^{14}\mathrm{C}\) is approximately 5,730 years. To calculate the decay constant (k), let's use the following formula:
\( k = \frac{\ln{2}}{t}\)
Now, substitute the half-life value and calculate k:
\( k = \frac{\ln{2}}{5730} \approx 1.2097 \times 10^{-4}\) (per year)
2Step 2: Calculate the \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios for samples from the eruptions in 472 and 512
We need to find the factor that relates the \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios at the two different times. We can use the equation for radioactive decay:
\( N(t) = N_0 e^{-kt}\)
Here, \(N(t)\) is the number of radioactive carbon-14 nuclei at time t, \(N_0\) is the initial number of nuclei, and k is the decay constant we calculated in step 1. To determine the \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios, we should divide the \(^{14}\mathrm{C}\) amount at both times by the constant \(^{12}\mathrm{C}\) amount:
\(\frac{N(t_1)}{N_{12}} \div \frac{N(t_2)}{N_{12}} = \frac{N_0 e^{-k t_1}}{N_0 e^{-k t_2}}\)
Here, \(N_{12}\) is the number of stable carbon-12 nuclei, which will remain constant and cancel out. \(t_1\) and \(t_2\) correspond to the eruption years 472 and 512, respectively.
For step 3, let's find the difference between the eruption years:
\( t_2 - t_1 = 512 - 472 = 40\) years
3Step 3: Calculate the ratio of the \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios at two different times
Now we can plug the value of the time difference into the equation:
\( \frac{N_0 e^{-k (t_1 + 40)}}{N_0 e^{-k t_1}} = e^{-k(40)}\)
Substitute the decay constant k we calculated earlier:
\( e^{-1.2097 \times 10^{-4}(40)} \approx 0.9952\)
4Step 4: Determine if radiocarbon dating can distinguish between the eruptions
The \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios from samples of the eruptions in 472 and 512 differ by a factor of 0.9952. The uncertainty in determining the ratios is 0.1%. To check if radiocarbon dating can distinguish between the eruptions, let's compare this factor with the uncertainty:
\(0.9952 - 1 = -0.0048\)
In percentage form:
\(-0.0048 \times 100 = -0.48\%\)
The difference between the \(^{14}\mathrm{C}/^{12}\mathrm{C}\) ratios (-0.48%) is greater than the uncertainty in determining these ratios (0.1%), which means radiocarbon dating can distinguish between debris from the eruptions of Mt. Vesuvius that occurred in the years 472 and 512.
Key Concepts
Carbon-14 DecayIsotopic RatiosGeological Dating Methods
Carbon-14 Decay
Carbon-14 decay is a cornerstone of radiocarbon dating, a technique used by scientists to determine the age of organic materials.
Carbon-14, or , is a radioactive isotope of carbon that is naturally found in the atmosphere. Living organisms continuously absorb carbon-14 through their interactions with the environment, maintaining a constant ratio of carbon-14 to the stable carbon-12 isotope within their cells. However, once an organism dies, it stops absorbing carbon, and the carbon-14 it contains begins to decay without being replaced.
The process of carbon-14 decay follows what is known as a half-life, which is the time it takes for half of the initial amount of carbon-14 to transform into nitrogen-14. For carbon-14, this half-life is about 5,730 years. As a result, by measuring the remaining amount of carbon-14 in a sample and comparing it to the expected initial amount (typically inferred from the stable carbon-12), scientists can calculate the time elapsed since the organism’s death. The equation \( N(t) = N_0 e^{-kt} \) represents this decay, where \( N(t) \) is the number of carbon-14 atoms at time \( t \), \( N_0 \) is the initial number of carbon-14 atoms, and \( k \) is the decay constant.
Carbon-14, or , is a radioactive isotope of carbon that is naturally found in the atmosphere. Living organisms continuously absorb carbon-14 through their interactions with the environment, maintaining a constant ratio of carbon-14 to the stable carbon-12 isotope within their cells. However, once an organism dies, it stops absorbing carbon, and the carbon-14 it contains begins to decay without being replaced.
The process of carbon-14 decay follows what is known as a half-life, which is the time it takes for half of the initial amount of carbon-14 to transform into nitrogen-14. For carbon-14, this half-life is about 5,730 years. As a result, by measuring the remaining amount of carbon-14 in a sample and comparing it to the expected initial amount (typically inferred from the stable carbon-12), scientists can calculate the time elapsed since the organism’s death. The equation \( N(t) = N_0 e^{-kt} \) represents this decay, where \( N(t) \) is the number of carbon-14 atoms at time \( t \), \( N_0 \) is the initial number of carbon-14 atoms, and \( k \) is the decay constant.
Isotopic Ratios
Isotopic ratios play an integral role in radiocarbon dating and other dating techniques. They provide a link between a sample's current chemical signature and its age.
In radiocarbon dating, the isotopic ratio of interest is typically the ratio of carbon-14 to carbon-12 (\(^{14}C/^{12}C\)). Since carbon-12 is stable and does not decay over time, its amount in a sample remains constant. On the contrary, carbon-14 decreases at a predictable rate, making the ratio between the two isotopes a key metric for determining age.
When comparing isotopic ratios from different samples, such as in the exercise with volcanic eruptions, the decayed carbon-14 amount in each sample is divided by the constant amount of carbon-12. This creates a ratio that can be compared across samples, even from different times. Small differences in these ratios can indicate significant differences in age, which is why precise measurements (like the 0.1% uncertainty mentioned in the problem) are crucial for accurate dating.
In radiocarbon dating, the isotopic ratio of interest is typically the ratio of carbon-14 to carbon-12 (\(^{14}C/^{12}C\)). Since carbon-12 is stable and does not decay over time, its amount in a sample remains constant. On the contrary, carbon-14 decreases at a predictable rate, making the ratio between the two isotopes a key metric for determining age.
When comparing isotopic ratios from different samples, such as in the exercise with volcanic eruptions, the decayed carbon-14 amount in each sample is divided by the constant amount of carbon-12. This creates a ratio that can be compared across samples, even from different times. Small differences in these ratios can indicate significant differences in age, which is why precise measurements (like the 0.1% uncertainty mentioned in the problem) are crucial for accurate dating.
Geological Dating Methods
Geological dating methods, including radiocarbon dating, are tools used by scientists to determine the age of Earth materials such as rocks, fossils, and sediment layers. These methods are based on the understanding of radioactive decay and the behavior of isotopes over time.
Apart from radiocarbon dating which targets organic remains, there are other dating methods such as uranium-lead dating for dating zircon crystals in rocks, potassium-argon dating for volcanic materials, and thermoluminescence for heated minerals. These methods use different elements and their decay products to calculate ages.
Precision, like the 0.1% uncertainty highlighted in the exercise, is pivotal in these methods. Achieving high precision allows scientists to distinguish between events that occurred close in time, such as the eruptions of Mt. Vesuvius in 472 and 512 AD. By understanding the decay rates and isotopic ratios, researchers can build historical profiles and timelines for geological events, which are essential for reconstructing Earth's history and the development of life.
Apart from radiocarbon dating which targets organic remains, there are other dating methods such as uranium-lead dating for dating zircon crystals in rocks, potassium-argon dating for volcanic materials, and thermoluminescence for heated minerals. These methods use different elements and their decay products to calculate ages.
Precision, like the 0.1% uncertainty highlighted in the exercise, is pivotal in these methods. Achieving high precision allows scientists to distinguish between events that occurred close in time, such as the eruptions of Mt. Vesuvius in 472 and 512 AD. By understanding the decay rates and isotopic ratios, researchers can build historical profiles and timelines for geological events, which are essential for reconstructing Earth's history and the development of life.
Other exercises in this chapter
Problem 41
Why is \(^{40} \mathrm{K}\) dating \((t_{1 / 2}=1.28 \times 10^{9}\) years) useful only for rocks older than 300,000 years?
View solution Problem 42
Where does the \(^{14} \mathrm{C}\) found in plants come from?
View solution Problem 49
What is the difference between a level of radioactivity and a dose of radioactivity?
View solution Problem 50
What are some of the molecular effects of exposure to radioactivity?
View solution