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Chapter 6: Problem 47

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
a) Approximately 3.7 billion years old. b) Approximately 3 billion years old.

Step by step solution

01

Understanding the Problem

We are given Moon rocks with known percentages of remaining uranium-238 and corresponding lead formed. We need to calculate their ages using uranium-238's half-life of 4.5 billion years.
02

Use the Radioactive Decay Formula

The radioactive decay formula is \( N(t) = N_0 \cdot (\frac{1}{2})^{(t/T)} \), where \( N(t) \) is the remaining quantity of uranium-238, \( N_0 \) is the initial quantity, \( t \) is the age we want to find, and \( T \) is the half-life (4.5 billion years).
03

Solve for the Age in Part (a)

For part (a), \( N(t)/N_0 = 0.55 \). Substitute into the formula: \( 0.55 = (\frac{1}{2})^{(t/4.5)} \). Take logs and solve for \( t \): \( \log(0.55) = \frac{t}{4.5} \cdot \log(0.5) \). Thus, \( t = 4.5 \cdot (\frac{\log(0.55)}{\log(0.5)}) \). Calculation gives \( t \approx 3.7 \) billion years.
04

Solve for the Age in Part (b)

For part (b), \( N(t)/N_0 = 0.63 \). Substitute into the formula: \( 0.63 = (\frac{1}{2})^{(t/4.5)} \). Take logs and solve for \( t \): \( \log(0.63) = \frac{t}{4.5} \cdot \log(0.5) \). Thus, \( t = 4.5 \cdot (\frac{\log(0.63)}{\log(0.5)}) \). Calculation gives \( t \approx 3 \) billion years.

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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 in which unstable atomic nuclei lose energy by emitting radiation. This process transforms the original radioisotope into a more stable form. It occurs at a constant rate, which is instrumental for radiometric dating. Typically, in this process, the unstable nucleus changes by releasing particles such as alpha particles, beta particles, or gamma rays.

Here are some key points:
  • Spontaneous Process: Decay occurs without any external influence.
  • Predictable Rates: While individual events are random, large numbers of particles decay at predictable rates.
  • Exponential Nature: The rate is proportional to the number of undecayed nuclei.
This predictable set of events allows scientists to date materials, such as Moon rocks, accurately.
Uranium-238
Uranium-238 is a naturally occurring isotope of uranium and is key in the technique known as uranium-lead dating. This isotope undergoes radioactive decay over a long period of time, transitioning through a decay chain before finally stabilizing as lead-206.

Some important aspects include:
  • Long Half-Life: Uranium-238 has a half-life of about 4.5 billion years, making it suitable for dating geological timescales.
  • Decay Chain: It undergoes a series of decays, involving alpha and beta emissions, before settling as lead-206.
  • Abundance: It is the most abundant uranium isotope, ensuring its availability for scientific studies.
The use of uranium-238 in dating provides reliable age estimates for older geological materials, such as moon rocks.
Half-Life
Half-life is a term used to describe the time required for a quantity to reduce to half its initial value. In radiometric dating, this concept helps in determining the age of objects. Specifically, for uranium-238, one half-life is about 4.5 billion years. This means that after this period, half of any given sample of uranium-238 will have decayed into lead.

Key features of half-life:
  • Constant Rate: Each isotope has a characteristic half-life, unchanged by external conditions.
  • Mathematical Utility: Using the half-life formula, scientists can calculate the time elapsed since the formation of a rock.
  • Application: The relatively long half-life of uranium-238 makes it suitable for dating ancient rocks, including those from the Moon.
Understanding half-life is crucial for interpreting radioactive decay data in radiometric dating studies.
Moon Geology
Moon geology involves studying the composition, structure, and history of the Moon's surface and inner layers. Radiometric dating techniques, such as using uranium-238, have been critical in revealing Moon's age and geological history.

Insights and findings from Moon geology:
  • Age of the Moon: Radiometric dating indicates the Moon formed approximately 4.5 billion years ago.
  • Diverse Compositions: The Moon's rocks vary in composition, with some resembling volcanic rocks while others are more akin to rocks found on Earth.
  • Impact History: Moon surfaces feature craters and basins formed by impacts, offering clues to the solar system's history.
  • Absence of Atmosphere: Without an atmosphere, geological processes on the Moon are mostly static once formed.
Studying these characteristics advances our understanding of both the Moon and our own planet鈥檚 past.

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Most popular questions from this chapter

Which lists the major steps of solar system formation in the correct order? (a) collapse, accretion, condensation (b) collapse, condensation, accretion (c) accretion, condensation, collapse

An Early solar Wind. Suppose the solar wind had cleared away the solar nebula before the seeds of the jovian planets could gravitationally draw in hydrogen and helium gas. How would the planets of the outer solar system be different? Would they still have many moons? Explain your answer in a few sentences.

Dating the Past. The method of radiometric dating that tells us the age of our solar system is also used to determine when many other past events occurred. For example, it is used to determine ages of fossils that tell us when humans first evolved and ages of relics that teach us about the rise of civilization. Research one key aspect of human history in which radiometric dating helps us piece the story together. Write two or three paragraphs on how radiometric dating is used in this case (such as what materials are dated and what radioactive elements are used) and what these studies have concluded. Does your understanding of the method lead you to accept the results? Why or why not?

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.)

What do we mean by the solar nebula? What was it made of, and where did it come from?
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