Problem 63 - Physics for Scientists and Engineers with Modern Physics

Question

An old wooden tool is found to contain only \(6.0 \%\) of the \({ }_{6}^{14} \mathrm{C}\) that an equal mass of fresh wood would. How old is the tool?

Step-by-Step Solution

Answer

The tool is approximately 24,763 years old.

Step 1: Understand Carbon Dating

Carbon-14 ( _{6}^{14}C ) is a radioactive isotope used in radiocarbon dating to estimate the age of carbonaceous materials. When a living organism dies, it stops taking in carbon-14, and its existing supply begins to decay. The half-life of carbon-14, the time it takes for half of a given amount of isotope to decay, is approximately 5730 years.

Step 2: Define the Given Values

We are given that the current amount of carbon-14 in the wood is 6.0% of the original amount when the wood was fresh. This percentage can be used to find the number of half-lives that have passed since the wood was cut.

Step 3: Calculate the Number of Half-Lives

The decay formula for the amount of a radioactive isotope is given by N(t) = N_0 imes (1/2)^{t/T}, where N(t) is the amount remaining after time t, N_0 is the initial amount, and T is the half-life. Rearranging to solve for the number of half-lives gives us: \[ \left(\frac{N(t)}{N_0}\right) = \left(\frac{1}{2}\right)^{t/T} \]Taking the natural logarithm: \[ t/T = \frac{\ln(N(t)/N_0)}{\ln(1/2)}\]Substitute the given percentage (6.0% or 0.06) into the equation: \[ \frac{\ln(0.06)}{\ln(0.5)} = \text{number of half-lives} \].This equals approximately 4.32 half-lives.

Step 4: Calculate the Age of the Tool

Since the number of half-lives (4.32) and the half-life of carbon-14 (5730 years) are known, we can find the age (age) by \[ \text{age} = \text{number of half-lives} \times \text{half-life of } _{6}^{14}C \] \[ \text{age} = 4.32 \times 5730 \text{ years} \approx 24763 \text{ years} \].Thus, the tool is approximately 24,763 years old.

Key Concepts

Radioactive Isotope

Radioactive isotopes are atoms that are unstable and decay over time, emitting radiation in the process. This transformation results in the atom changing into a different element or a different isotope of the same element. Understanding radioactive isotopes is crucial in many scientific fields, including carbon dating.

• They are characterized by their unstable nuclei, which strive to achieve stability by releasing energy.
• This release of energy often involves emitting particles or radiation, which can be measured.
• The rate at which a radioactive isotope decays is constant and can be described by its half-life.

Radioactive isotopes are remarkably significant in studying the age of ancient artifacts, such as using carbon-14 in radiocarbon dating.

Half-life

The concept of a half-life is fundamental in understanding radioactive decay. It represents the time required for half of a given amount of a radioactive isotope to decay.

• Every radioactive isotope has a unique half-life.
• It's a fixed property that doesn't get affected by external conditions like temperature or pressure.
• The concept is used to model how quickly an isotope changes and helps predict how much of it remains after a certain period.

In the context of carbon dating, carbon-14 has a half-life of approximately 5730 years. This steady decay rate enables scientists to estimate the age of objects containing remnants of ancient organic materials.

Radiocarbon Dating

Radiocarbon dating is a scientific method used to determine the age of organic materials.

• Employing the radioactive isotope carbon-14, this method measures the amount of carbon-14 remaining in a sample.
• As carbon-14 decays over time, the less carbon-14 found in an object, the older it likely is.
• The method is commonly used in archaeology, geology, and other fields to date historical artifacts and geological events.

Radiocarbon dating relies on the known half-life of carbon-14 to calculate how long it has been since the specimen stopped absorbing carbon. This technique is essential to our understanding of historical timelines.

Carbon-14

Carbon-14 is a specific radioactive isotope of carbon, denoted as _{6}^{14}C. It plays a crucial role in the radiocarbon dating method.

• Produced naturally in the atmosphere through the interaction of cosmic rays with nitrogen.
• Present in living organisms through the carbon cycle – while alive, organisms replenish carbon-14 by consuming carbon-containing substances.
• Once an organism dies, it ceases to absorb carbon-14, and the isotope begins to decay at a known rate.

Understanding carbon-14 is key to using radiocarbon dating effectively, as its decay rate over the half-life of 5730 years allows researchers to accurately determine the age of ancient biological materials.