Problem 38 - Chemistry and Chemical Reactivity

Question

A piece of charred bone found in the ruins of a Native American village has a \(^{14} \mathrm{C}:^{12} \mathrm{C}\) ratio that is \(72 \%\) of the radio found in living organisms. Calculate the age of the bone fragment.

Step-by-Step Solution

Answer

The bone fragment is approximately 2,840 years old.

Step 1: Understand the Problem

We need to calculate the age of a bone fragment based on its \( ^{14}\text{C}:^{12}\text{C} \) ratio compared to that of living organisms. The given ratio is 72% of that found in living organisms.

Step 2: Recall the Half-life of Carbon-14

The half-life of \( ^{14}\text{C} \) is approximately 5730 years. This is the time it takes for half of the \( ^{14}\text{C} \) in a sample to decay.

Step 3: Use the Exponential Decay Formula

The decay of \( ^{14}\text{C} \) can be modeled by the formula \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the remaining amount of \( ^{14}\text{C} \) at time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant. First, we need to find \( k \) using the half-life.

Step 4: Calculate the Decay Constant \( k \)

The decay constant \( k \) is related to the half-life by \( k = \frac{\ln(2)}{5730} \). Calculate \( k \).

Step 5: Solve for the Age \( t \)

Set up the equation using the information given: \( 0.72 = e^{-kt} \). Solve for \( t \) to find the age of the bone.

Key Concepts

Half-life

The concept of "half-life" plays a crucial role in understanding radiocarbon dating. Half-life refers to the amount of time it takes for half of a given amount of a radioactive isotope, such as carbon-14, to decay.
For carbon-14, the half-life is approximately 5730 years. This means if you start with a sample containing a certain number of carbon-14 atoms, after 5730 years, only half of those atoms will remain; the rest will have decayed into a different element.
Understanding half-life is essential because it allows scientists to use the predictable reduction of carbon-14 to estimate how long it's been since the organism containing it perished. This is particularly useful for dating ancient archaeological finds like bone fragments.

Exponential Decay Formula

Radiocarbon dating relies on what is known as the exponential decay formula, which models how the amount of a radioactive substance decreases over time. The formula is:

• \( N(t) = N_0 e^{-kt} \)

Here:

• \( N(t) \) is the quantity of carbon-14 remaining at time \( t \).
• \( N_0 \) is the initial quantity of carbon-14.
• \( k \) is the decay constant, which you'll calculate separately.
• \( t \) is the time that has passed since the organism died.

This formula helps determine the age of archaeological samples by comparing the current value of carbon-14 in the sample with the expected value if it were still living.
By rearranging the formula, we can solve for \( t \), thus dating the sample based on how much carbon-14 remains.

Decay Constant Calculation

To use the exponential decay formula effectively, you must calculate the decay constant \( k \), which depends on the half-life of the radioactive isotope. The relationship between \( k \) and the half-life \( T_{1/2} \) is given by:

• \( k = \frac{\ln(2)}{T_{1/2}} \)

For carbon-14, its known half-life is approximately 5730 years.
Logarithmically, \( \ln(2) \) approximates to 0.693. Substituting these values into the formula helps us derive the decay constant \( k \).
This constant is a pivotal component in exponential decay calculations, allowing us to translate the percentage of remaining carbon-14 into an actual age.

Age Calculation

With the decay constant in hand, calculating the age of an artifact becomes a straightforward task. Here is a step-by-step approach:

• Start by acknowledging the artifact's current \( ^{14}C:^{12}C \) ratio. For the charred bone discussed, this is 72%.
• Apply the exponential decay equation: \( 0.72 = e^{-kt} \).
• Take the natural logarithm of both sides to isolate \( t \).
• Solve for \( t \) using \( t = \frac{-\ln(0.72)}{k} \).

Upon substituting the calculated decay constant \( k \) and resolving this equation, you'll discover the approximate age of the sample.
The beauty of this method is its precision, granting archaeologists and researchers a robust tool for mapping out the history of ancient biological materials.