Problem 50 - Thomas Calculus in SI Units

Question

The frozen remains of a young Incan woman were discovered by archeologist Johan Reinhard on Mt. Ampato in Peru during an expedition in 1995 . a. How much of the original carbon-14 was present if the estimated age of the "Ice Maiden" was 500 years? b. If a \(1 \%\) error can occur in the carbon-14 measurement, what is the oldest possible age for the Ice Maiden?

Step-by-Step Solution

Answer

a. 93.9% of carbon-14 remains; b. Oldest possible age is 577 years.

Step 1: Understanding Carbon-14 Decay

Carbon-14 ( ^{14}C) has a half-life of about 5730 years. This means that every 5730 years, half of the carbon-14 in a sample will decay. We need to use this decay rate to determine how much of the original carbon-14 remains after 500 years.

Step 2: Calculate Remaining Carbon-14

We use the decay formula: \[ N(t) = N_0 e^{-kt} \]where \(N(t)\) is the amount of carbon-14 remaining, \(N_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time. The decay constant \(k\) is given by \(k = \frac{\ln(2)}{5730}\). To find the fraction remaining after 500 years, evaluate \(\frac{N(t)}{N_0} = e^{-\frac{500 \times \ln(2)}{5730}} \approx 0.939\).

Step 3: Calculate Maximum Age with Measurement Error

If a 1% error occurs, the carbon-14 measurement could appear 1% less than the actual value, which means the Ice Maiden could appear younger than she actually is. To find the oldest possible age, the actual carbon-14 remaining would be \(\frac{N(t)}{N_0} = 0.939 \times 0.99 \approx 0.92961\). Using the decay formula backwards: \[ t = \frac{-5730 \times \ln(0.92961)}{\ln(2)} \approx 577 \] years.

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. This gradual transformation results in a change of the radioactive element into a different element over time. In the context of carbon-14 dating, this transformation is essential for understanding how archaeologists estimate the age of ancient remains. Carbon-14 ( ^{14}C) is a radioactive isotope of carbon, which is unstable and decays over time. During decay, carbon-14 atoms in antique samples are emitted, gradually reducing their number, which can be used to determine how long the organism has been dead. By measuring the amount of carbon-14 left in a sample, and knowing the rate at which it decays, scientists can calculate the age of the specimen. This process is invaluable for archaeology, as it allows researchers to date artifacts and remains with remarkable precision. It's important to understand that radioactive decay is a predictable process on average, even though the exact time of decay for a specific atom is random.

Decay Constant

The decay constant is a crucial parameter in the formula for radioactive decay. It provides the rate at which a substance will undergo radioactive decay. In the carbon-14 decay formula, the decay constant ( k) helps to calculate how fast the carbon-14 decreases in the sample over time. The decay constant is related to the half-life of the substance. Mathematically, it is found using the equation:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

where \( T_{1/2} \) is the half-life of the isotope. For carbon-14, \( T_{1/2} \) is approximately 5730 years. This means that the decay constant helps us predict what fraction of the substance remains after a certain period. Knowing about the decay constant allows us to derive the decay formula, which is used to calculate the remaining amount of carbon-14 at any given time.

Half-Life

The half-life of a radioactive substance is the time it takes for half of the radioactive elements in a sample to decay. For carbon-14, the half-life is about 5730 years. This means that every 5730 years, half of the carbon-14 turns into nitrogen, reducing the number of carbon-14 atoms left in a specimen. Understanding half-life is crucial for carbon-14 dating, as it provides a time frame for the decay process. When we know the half-life, we can calculate how many half-lives have passed since the death of the organism, providing an estimate of its age. To calculate the remaining amount of carbon-14 in a sample, we often use the formula:

• \( N(t) = N_0 e^{-kt} \)

This formula uses the decay constant and initial quantity of carbon-14, demonstrating the power of the half-life concept in estimating the age of ancient artifacts.

Measurement Error

Measurement error in carbon-14 dating can impact the estimated age of a sample. As an example, suppose there is a 1% error in the carbon-14 measurement. This could mean that the calculated age of a specimen is either slightly older or younger than the actual age. Errors in measurement can arise from various sources like equipment inaccuracies or external environmental factors. It's crucial to account for these possible discrepancies to provide an accurate estimate of age. In calculating the maximum possible age of a historical artifact, taking into account a possible measurement error is vital. For instance, if the carbon-14 detected is 1% less than expected, the actual age could be older than measured. Adjusting for a 1% error during calculation modifies the equation, providing an alternative estimation for the specimen's age so that researchers can be confident in their results. Measurement error, therefore, needs careful consideration in dating techniques to ensure precise archaeological assessments.