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Chapter 4: Problem 17

All living things contain carbon 12, which is stable, and carbon 14 , which is radioactive. While a plant or animal is alive, the ratio of these two isotopes of carbon remains unchanged, since the carbon 14 is constantly renewed; after death, no more carbon 14 is absorbed. The half-life of carbon 14 is 5730 years. If charred logs of an old fort show only \(70 \%\) of the carbon 14 expected in living matter, when did the fort burn down? Assume that the fort burned soon after it was built of freshly cut logs.

Short Answer

Expert verified
The fort burned down approximately 3000 years ago.

Step by step solution

01

Understanding the Problem

When an organism dies, it stops absorbing carbon-14, which begins to decay according to its half-life. The half-life of carbon-14 is 5730 years, meaning every 5730 years, half of the existing carbon-14 decays. We are told that only 70% of the carbon-14 remains, so we need to find out how much time it took for 30% of the carbon-14 to decay.
02

Use the Exponential Decay Formula

The formula for radioactive decay is given by \[ N(t) = N_0 \times (0.5)^{t/T_{1/2}} \]where \( N(t) \) is the remaining quantity of carbon-14 at time \( t \), \( N_0 \) is the initial quantity, \( T_{1/2} \) is the half-life, and \( t \) is the time elapsed. We have \( N(t)/N_0 = 0.7 \) and \( T_{1/2} = 5730 \) years.
03

Solve for Time Elapsed (\( t \))

Rearrange the equation to solve for \( t \):\[ 0.7 = (0.5)^{t/5730} \]Take the natural logarithm of both sides:\[ \ln(0.7) = \ln((0.5)^{t/5730}) \]This simplifies to:\[ \ln(0.7) = \frac{t}{5730} \times \ln(0.5) \]
04

Calculate \( t \)

Now solve for \( t \):\[ t = \frac{\ln(0.7)}{\ln(0.5)} \times 5730 \]Calculate:\[ t \approx \frac{-0.3567}{-0.6931} \times 5730 \approx 3000 \text{ years} \]
05

Interpret the Result

The solution indicates that it has been approximately 3000 years since the carbon-14 in the logs began to decay, which corresponds to the time since the fort burned down.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Carbon-14 Dating
Carbon-14 dating is a technique used by scientists to determine the age of ancient objects made from organic materials. This method relies on the radioactive decay of carbon-14, a naturally occurring isotope of carbon. Living organisms continually take in carbon-14 from their environment, keeping a steady proportion of carbon-14 relative to carbon-12.
However, when the organism dies, the carbon-14 intake stops, and the isotope starts to decay at a predictable rate. By measuring the remaining carbon-14 in a sample and comparing it to the initial level expected in a living organism, scientists can calculate how long it has been since the organism died.
  • This process is incredibly useful for dating ancient artifacts, fossils, and archaeological sites.
  • Carbon-14 dating is most effective for samples up to about 50,000 years old, beyond which there's insufficient carbon-14 left to measure accurately.
Exponential Decay
Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. This concept is critical in understanding radioactive decay, such as the decay of carbon-14. In this process, the amount of carbon-14 reduces over time, following an exponential decay model.
The fundamental nature of exponential decay means that the more of a substance there is, the faster it decays. The decay formula for radioactive substances is \[ N(t) = N_0 \times (0.5)^{t/T_{1/2}} \] where:
  • \( N(t) \) is the remaining quantity of the substance at time \( t \).
  • \( N_0 \) is the initial quantity of the substance.
  • \( T_{1/2} \) is the half-life, or the time it takes for half of the substance to decay.
This formula is central for calculating the age of specimens in carbon-14 dating.
Half-life Calculation
The half-life is a key concept in radioactive decay, representing the time required for half of the radioactive atoms in a sample to decay. For carbon-14, this time is 5730 years. Understanding half-life helps with calculating how much time has passed since an organism died, which is essential for dating purposes.
Using the concept of half-life, we can set up equations to determine the elapsed time since a sample stopped absorbing carbon-14. In our exercise, with only 70% of the carbon-14 remaining, this indicates that about 30% has decayed.
  • By re-working the decay formula, scientists can estimate the time since decay began, using the changes in carbon-14 levels.
  • Half-life creates a predictable and reliable way to measure time passed based solely on the radioactive properties of carbon isotopes.
Quantitative Reasoning
Quantitative reasoning involves the mathematical interpretation of data and the strategic use of numbers to understand complex problems, like those involving radioactive decay. When dealing with carbon-14 dating, being able to work through equations and logarithms is essential to derive meaningful conclusions.
For example, determining the time since the fort's logs stopped receiving carbon-14 requires forming equations based on decay properties and solving them for the variable \( t \) (time).
This involves:
  • Understanding the decay process and being comfortable with the properties of logarithms.
  • Manipulating formulas to isolate the desired variables, enhancing problem-solving skills.
  • Evaluating your mathematical results to ensure they make logical sense within the context of the problem.
Quantitative reasoning thus serves as a bridge between theoretical knowledge and practical application, making it indispensable in scientific analysis.

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Most popular questions from this chapter

An object thrown from the edge of a 42 -foot cliff follows the path given by \(y=-\frac{2 x^{2}}{25}+x+42\) (Figure 10 ). An observer stands 3 feet from the bottom of the cliff. (a) Find the position of the object when it is closest to the observer. (b) Find the position of the object when it is farthest from the observer.

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