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Chapter 21: Problem 73

Several hundred pairs of sandals found in a cave in Oregon were found by carbon- 14 dating to be \(9.0 \times 10^{3}\) years old. What must have been the activity of the carbon- 14 in the sandals in disintegrations per minute per gram? Assume the original activity was \(15.3\) disintegrations per minute per gram.

Short Answer

Expert verified
The activity is approximately 5.14 disintegrations per minute per gram.

Step by step solution

01

Understand Carbon-14 Dating

Carbon-14 dating measures the decay of carbon-14, a radioactive isotope, which decreases over time. The disintegrations per minute per gram will decrease as the sandal ages.
02

Use the Exponential Decay Formula

The general formula for radioactive decay is \( N(t) = N_0 \times e^{-\lambda t} \), where \( N(t) \) is the remaining disintegrations per minute per gram, \( N_0 \) is the initial disintegrations per minute per gram, \( \lambda \) is the decay constant, and \( t \) is time in years.
03

Determine the Decay Constant

The decay constant \( \lambda \) for carbon-14 is given by \( \lambda = \frac{0.693}{t_{1/2}} \), where the half-life \( t_{1/2} \) for carbon-14 is approximately 5730 years. Thus, \( \lambda \approx \frac{0.693}{5730} \approx 1.21 \times 10^{-4} \text{ year}^{-1}.\)
04

Substitute Known Values into the Formula

Substituting the known values into the formula \( N(t) = N_0 \times e^{-\lambda t} \), we have \( N(t) = 15.3 \times e^{-1.21 \times 10^{-4} \times 9000} \).
05

Calculate the Remaining Activity

Calculating \( N(t) = 15.3 \times e^{-1.089} \), we find \( e^{-1.089} \approx 0.336 \). Therefore, \( N(t) \approx 15.3 \times 0.336 \approx 5.14 \) disintegrations per minute per gram.

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Key Concepts

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

Radioactive Decay
Radioactive decay is a natural process where unstable atomic nuclei lose energy by emitting radiation. This occurs in a spontaneous manner, involving the transformation from a parent isotope to a daughter isotope or other particles. Radioactive decay allows us to estimate the age of ancient objects, like the sandals found in the cave, by measuring the activity of isotopes such as carbon-14.
  • Radioactive isotopes like carbon-14 decay over time.
  • The rate of decay is constant and can be used to determine the age of objects.
  • Each isotope has a characteristic decay rate, often measured in terms of its half-life.
Carbon-14 is especially useful for dating materials that were once alive, owing to its predictable decay pattern. As organisms die, they no longer absorb carbon-14, causing the isotope to decay at a known rate, aiding in age determination.
Exponential Decay Formula
The exponential decay formula is central to understanding radioactive decay processes like those involved in carbon-14 dating. This mathematical expression describes how the number of decaying particles decreases over time. The general form of the formula is:\[ N(t) = N_0 \times e^{-\lambda t} \]Where:
  • \(N(t)\): Remaining disintegrations per minute per gram at time \(t\).
  • \(N_0\): Initial disintegrations per minute per gram (when the organism died).
  • \(e\): Base of the natural logarithm, approximately equal to 2.718.
  • \(\lambda\): Decay constant, specific to each isotope.
  • \(t\): Time elapsed since death, measured in years.
For the sandals, using this formula involved substituting known values to determine the current activity of carbon-14 in the sample.
Half-life Calculation
Half-life is the time required for half of the radioactive atoms in a sample to decay. This concept is essential in the field of radiometric dating, as it provides a timescale for observing radioactive decay.
For carbon-14, the approximated half-life is about 5730 years, which means every 5730 years, half of the original carbon-14 isotopes will have decayed.
  • The decay constant \(\lambda\) relates to the half-life through the formula:\[ \lambda = \frac{0.693}{t_{1/2}} \]
  • This relationship helps calculate \(\lambda\) when the half-life \(t_{1/2}\) is known.
This calculation plays a vital role in determining the decay constant, which then helps us use the exponential decay formula to ascertain the age or remaining activity of the sample.
Decay Constant
The decay constant \(\lambda\) is a key parameter in radioactivity. It defines the rate at which a specific isotope decays and is linked directly to the half-life. For carbon-14,\[ \lambda = \frac{0.693}{t_{1/2}} \approx 1.21 \times 10^{-4} \text{ year}^{-1} \]This small value reflects the slow rate of decay of carbon-14, making it suitable for dating archaeological samples that are thousands of years old. Understanding the decay constant allows scientists to predict how many disintegrations occur over time.
  • Provides the fraction of remaining nuclei that decays per unit time.
  • Specific to each radioactive isotope.
  • Essential for accurately applying the exponential decay formula.
Knowing the decay constant, as derived from the half-life of 5730 years, helps in computing the present-day activity of a sample, as seen in the sandal analysis.

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Most popular questions from this chapter

Neptunium- 237 was the first isotope of a transuranium element to be discovered. The decay constant is \(1.03 \times 10^{-14} / \mathrm{s}\). What is the half-life in years?

When technetium- \(99 m\) decays to technetium-99, a gamma photon corresponding to an energy of \(0.143 \mathrm{MeV}\) is emitted. What is the wavelength of this gamma emission? What is the difference in mass between Tc-99m and Tc-99?

In Chapter 7 (A Chemist Looks At: Zapping Hamburger with Gamma Rays) there is a discussion of how gamma radiation is used to kill bacteria in food. As indicated in the feature, there is concern on the part of some people that the irradiated food is radioactive. Why is this not the case? If you wanted to make the food radioactive, what would you have to do?

Describe how a Geiger counter works. How does a scintillation counter work?

What are the two types of nuclear reaction? Give an example of a nuclear equation for each type.
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