Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Carbon 14 ( \(^{14} \mathrm{C}\) ) dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of \(^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \(^{14} \mathrm{C}\) absorbed by a tree growing today. \(\mathrm{A}\) piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal, given that the half-life of \(^{14} \mathrm{C}\) is about 5700 years?

Step-by-Step Solution

Answer

After solving the equation, we find that the time elapsed since the tree was burned to make the ancient charcoal is approximately 18900 years.

Step 1: Identify the necessary parameters

Carbon-14 decays according to the formula \(N = N_{0} \cdot 0.5^{t/T}\), where \(T\) is the half-life of the element, \(N\) is the current amount, and \(N_{0}\) is the initial amount. The current amount \(N\) is 15% of the initial amount \(N_{0}\).

Step 2: Substitute the known values

By substituting the values into the equation, the expression becomes \(0.15 = 0.5^{t/5700}\). Since both sides of the equation are not zero, it is possible to take the natural log (ln) of both sides.

Step 3: Solve for time \(t\)

By taking the natural log of both sides, the equation turns into \(ln(0.15) = \frac{t}{5700} ln(0.5)\). Solving for \(t\) by isolating it on one side gives us \(t = \frac{5700 \cdot ln(0.15)}{ln(0.5)}\). This equation can now be solved to get the time elapsed.

Key Concepts

Radioactive Decay

The concept of radioactive decay is crucial in understanding carbon-14 dating. Radioactive decay occurs when the nucleus of an unstable atom loses energy by emitting radiation. This decay continues at a consistent pace, specific to each type of isotope. In the case of carbon-14, it decays into nitrogen-14 over time.
This natural process is predictable, making it possible to use it as a "clock" to determine the age of organic materials, such as ancient charcoal. By measuring the remaining amount of carbon-14 in a specimen and comparing it to current levels, scientists can deduce how long the decay has been occurring. This method assumes that the isotope's rate of decay remains constant over time, a principle called half-life, which is pivotal in carbon-14 dating.

Half-Life Calculation

At the heart of calculating the age of an artifact using radioactive decay lies the concept of half-life. The half-life of a radioactive isotope is the time taken for half of the isotope in a sample to decay. For carbon-14, this period is approximately 5700 years.
Understanding half-life allows scientists to estimate the time elapsed since the formation of the sample by measuring how much of the radioactive isotope remains. In the given exercise, the ancient charcoal has only 15% of the carbon-14 compared to a new piece. This tells us that several half-life periods have passed.

• The precise calculation involves relating the remaining percentage of carbon-14 to half-life periods.
• The formula used here helps pinpoint how many half-lives have occurred, which is essential to calculate the age of the artifact.

Your task is to utilize this formula to find out exactly how many years have passed since the charcoal was burned.

Exponential Decay Formula

The exponential decay formula is key to solving carbon-14 dating problems. This formula quantifies the amount of a substance over time as it undergoes radioactive decay. The exercise provides us with the formula: \[ N = N_{0} \cdot 0.5^{t/T} \]where:

• \(N\) is the current amount of carbon-14.
• \(N_{0}\) is the initial amount of carbon-14.
• \(t\) represents the time that has passed.
• \(T\) is the half-life of the isotope, which for carbon-14 is 5700 years.

This formula captures the heart of radioactive decay by showing the decline of the isotope over time. Here, we know \(N\) is 15% of \(N_{0}\). Simply substitute \(0.15N_{0}\) for \(N\), and solve for \(t\) using logarithms, as shown in the given solution. This calculation forms the basis for determining the artifact's age, transforming data from decay into a historical timeline.