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Chapter 8: Problem 46

Be sure to show all calculations clearly and state your final answers in complete sentences. Lunar Rocks. You are dating Moon rocks based on their proportions of uranium-238 (half-life of about 4.5 billion years) and its ultimate decay product, lead. Find the age for each of the following. a. A rock for which you determine that \(55 \%\) of the original uranium-238 remains, while the other \(45 \%\) has decayed into lead b. A rock for which you determine that \(63 \%\) of the original uranium-238 remains, while the other 37\% has decayed into lead

Short Answer

Expert verified
Part (a): The rock is about 3.87 billion years old. Part (b): The rock is about 2.88 billion years old.

Step by step solution

01

Understand the Half-Life Formula

The half-life formula gives us the remaining quantity of a substance. It is written as \( N(t) = N_0 \times (1/2)^{t/T} \), where \( N(t) \) is the remaining quantity, \( N_0 \) is the initial quantity, \( t \) is the time that has passed, and \( T \) is the half-life.
02

Set Up the Equation for Part (a)

In the problem, 55% of the original uranium-238 remains. Thus, \( N(t) = 0.55N_0 \). Substitute this into the half-life formula \( 0.55N_0 = N_0 \times (1/2)^{t/4.5} \). The \( N_0 \) terms cancel out, leaving us with \( 0.55 = (1/2)^{t/4.5} \).
03

Solve for Time in Equation for Part (a)

To solve \( 0.55 = (1/2)^{t/4.5} \), take the logarithm of both sides:\[ \log(0.55) = \frac{t}{4.5} \cdot \log(0.5). \] Solve for \( t \):\[ t = \frac{4.5 \cdot \log(0.55)}{\log(0.5)} \approx 3.87 \text{ billion years}. \]
04

Set Up the Equation for Part (b)

In this part, 63% of the original uranium-238 remains, or \( N(t) = 0.63N_0 \). Substitute into the half-life formula: \( 0.63N_0 = N_0 \times (1/2)^{t/4.5} \). Cancelling the \( N_0 \) terms, we get \( 0.63 = (1/2)^{t/4.5} \).
05

Solve for Time in Equation for Part (b)

To solve \( 0.63 = (1/2)^{t/4.5} \), take the logarithm of both sides:\[ \log(0.63) = \frac{t}{4.5} \cdot \log(0.5). \] Solve for \( t \):\[ t = \frac{4.5 \cdot \log(0.63)}{\log(0.5)} \approx 2.88 \text{ billion years}. \]
06

Conclusion

For part (a), the lunar rock is approximately 3.87 billion years old, and for part (b), the lunar rock is approximately 2.88 billion years old.

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Key Concepts

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

Uranium-Lead Dating
Uranium-lead dating is a powerful technique utilized to determine the age of rocks and minerals. It leverages the properties of uranium isotopes that decay into stable lead isotopes over time. This dating method is widely used in geochronology, especially for dating lunar rocks and other ancient geological samples.
  • This process involves the decay of uranium-238 into lead-206 and uranium-235 into lead-207.
  • The decay occurs at a known rate, which makes it possible to calculate the age of a rock by measuring the ratio of uranium to lead.

This method is important for understanding the history of our planet and the Moon, providing insights into their formation and evolution. The strength of this method lies in its ability to provide precise ages for samples that are billions of years old.
Half-Life Calculations
Half-life calculations are central to understanding radioactive decay processes. The half-life is the time it takes for half of a given amount of a radioactive substance to decay into a stable form. This predictable nature of decay allows scientists to use mathematical equations to estimate the age of a sample.
  • The half-life of uranium-238 is about 4.5 billion years, making it suitable for dating extremely old rocks.
  • The formula used in half-life calculations is \( N(t) = N_0 \times \left(\frac{1}{2}\right)^{t/T} \), where \( N(t) \) is the amount remaining, \( N_0 \) is the initial amount, \( t \) is time, and \( T \) is the half-life.

By applying this formula to the remaining percentage of a substance, scientists can derive the age of geological samples such as lunar rocks.
Radioactive Decay
Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This is a natural process involving the transformation of a radioactive element into a stable element over time. In the context of geological dating, it is crucial to understanding how elements like uranium turn into lead.
  • There are different types of decay, such as alpha, beta, and gamma decay, each with distinct characteristics and effects.
  • In lunar rock dating, uranium-238 undergoes alpha decay, where it emits an alpha particle and transforms into a series of other elements before finally becoming lead-206.

The known, consistent rate of decay helps scientists determine the time elapsed since a rock or mineral was last in its molten state, capturing a snapshot of its age.
Geochronology
Geochronology is the science of determining the age of rocks, fossils, and sediments. It utilizes various methods, including radioactive dating, to quantify geological time scales and reconstruct the timing of events in Earth's history. This field allows scientists to piece together the chronology of processes that have shaped the Earth and other planetary bodies like the Moon.
  • Accurate dating is achieved by studying isotopic decay, such as uranium-lead dating, which has revolutionized our understanding of geological time.
  • Geochronology has applications beyond simple dating: it helps in understanding plate tectonics, the history of life, and the evolution of our planetary system.

This science helps to systematically organize the history of the Earth based on solid evidence from the very rocks of which it is composed, providing a comprehensive timeline of geological events.

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Most popular questions from this chapter

Two Kinds of Planets. The jovian planets differ from the terrestrial planets in a variety of ways. Using sentences that members of your family would understand, explain why the jovian planets differ from the terrestrial planets in each of the following: composition, size, density, distance from the Sun, and number of satellites.

Be sure to show all calculations clearly and state your final answers in complete sentences. What Are the Odds? The fact that all the planets orbit the Sun in the same direction is cited as support for the nebular hypothesis. Imagine that there's a different hypothesis in which planets can be created orbiting the Sun in either direction. Under this hypothesis, what is the probability that eight planets would end up traveling in the same direction? (Hint: It's the same probability as that of flipping a coin eight times and getting all heads.)

Suppose we found a solar system with the property described. (These are not real discoveries.) In light of what you've learned about the formation of our own solar system, decide whether the discovery should be considered reasonable or surprising. Explain your reasoning. A solar system has five terrestrial planets in its inner solar system and three jovian planets in its outer solar system.

Describe the technique of radiometric dating. What is a half-life?

The Stardust Puzzle. Search the Internet for recent progress in explaining the Stardust mission's discovery of minerals from a comet nucleus that must have formed very close to the Sun. Write a one- to two-page report, describing how the result has affected our ideas about the processes in the early solar nebula.
See all solutions

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