Question

Be sure to show all calculations clearly and state your final answers in complete sentences. You are analyzing Moon rocks that contain small amounts of uranium-238, which decays into lead with a half-life of about 4.5 billion years. a. In one rock from the lunar highlands, you determine that \(55 \%\) of the original uranium- 238 remains; the other \(45 \%\) decayed into lead. How old is the rock? b. In a rock from the lunar maria, you find that \(63 \%\) of the original uranium-238 remains; the other 37\% decayed into lead. Is this rock older or younger than the highlands rock? By how much?

Short answer

The highlands rock is 3.74 billion years old. It is 1.18 billion years older than the maria rock, which is 2.56 billion years old.

Step-by-step solution

Understanding Half-Life

The half-life of a radioactive substance is the time required for half of the substance to decay. For uranium-238, this half-life is 4.5 billion years.

Using Decay Formula

If a substance has decayed so that only \( N(t) \) percent of it remains, we can use the formula \( N(t) = N_0 \times (\frac{1}{2})^{\frac{t}{T}} \), where \( N_0 \) is the initial quantity, \( N(t) \) is the remaining quantity, \( t \) is the time, and \( T \) is the half-life.

Calculating Age of the Highlands Rock

For the highlands rock, we have \( N(t) = 0.55 \) and \( T = 4.5 \). We plug these values into the formula and solve for \( t \): \( 0.55 = (\frac{1}{2})^{\frac{t}{4.5}} \). Take the natural logarithm of both sides to solve for \( t \): \( \ln(0.55) = \frac{t}{4.5} \times \ln(0.5) \). Solving yields \( t \approx 3.74 \) billion years.

Calculating Age of the Maria Rock

For the maria rock, \( N(t) = 0.63 \) and using the same formula, we have \( 0.63 = (\frac{1}{2})^{\frac{t}{4.5}} \). Taking the natural logarithm of both sides: \( \ln(0.63) = \frac{t}{4.5} \times \ln(0.5) \). Solving gives \( t \approx 2.56 \) billion years.

Comparing Ages

The age of the highlands rock is approximately 3.74 billion years, while the maria rock is approximately 2.56 billion years. Therefore, the highlands rock is older than the maria rock.

Calculating the Age Difference

To find the age difference, subtract the age of the maria rock from the age of the highlands rock: \( 3.74 - 2.56 = 1.18 \) billion years. Therefore, the highlands rock is 1.18 billion years older than the maria rock.

Key concepts

Uranium-238 Decay

Uranium-238 decay is an essential process in geological dating, particularly when analyzing ancient rocks from the Earth and the moon. Uranium-238 (\( ^{238}U \)) is a radioactive isotope that transforms into lead-206 (\( ^{206}Pb \)) over time, releasing particles and energy. This decay is spontaneous and happens continuously over time.
The decay process is important for scientists because it acts as a natural clock. By measuring the amount of uranium-238 and lead-206 in a rock sample, they can determine the time that has passed since the rock formed—allowing for estimations of its age. This natural clock is precisely what characterizes radiometric dating.
This transformation from uranium to lead is extremely gradual, taking billions of years, which makes it perfect for dating geological processes that have occurred over long periods.

Half-life Calculation

The concept of half-life is essential in understanding radioactive decay and radiometric dating. Half-life refers to the time it takes for half of a given amount of a radioactive substance to decay into another element. For uranium-238, this half-life is approximately 4.5 billion years. Hence, every 4.5 billion years, half of any sample of uranium-238 will have decayed into lead-206.
To calculate the age of a rock based on the remaining uranium-238, scientists use the formula:\[N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}}\]where:

• \(N(t)\) is the remaining quantity of uranium-238.
• \(N_0\) is the initial quantity of uranium-238.
• \(t\) is the time the substance has been decaying.
• \(T\) is the half-life of uranium-238, 4.5 billion years in this case.

By using this formula, one can solve for \(t\), determining how many years have passed since the rock formed. Understanding this calculation facilitates determining the age of geological samples accurately.

Geological Dating

Geological dating refers to the process by which the age of geological materials is determined. Radiometric dating, which involves measuring the decay of radioactive isotopes, is a primary method for dating rocks and minerals.
Radioisotopes, like uranium-238, are compared to their decay products (such as lead-206) to deduce the age of rocks. This process is essential because it lets scientists piece together the history of the Earth and other planetary bodies, such as the Moon. For example, by measuring the decay of uranium and comparing it to the amount of lead in a sample, scientists can calculate how long the decay process has been occurring, providing an age for the rock.
Additionally, this process unveils information about Earth's formation, the evolution of its landscape, and significant events in its history. The ability to date rocks accurately is crucial for constructing the timeline of natural processes and planetary evolution.

Moon Rocks Analysis

Moon rocks provide invaluable information about the history of the Moon, and indirectly, the Earth. They were brought back by astronauts during NASA's Apollo missions, allowing scientists to study rocks that are billions of years old. They are often analyzed using radiometric dating techniques, especially uranium-lead dating.
The analysis of moon rocks enables scientists to determine the age of these rocks and, therefore, events in the Moon's past, such as volcanic activity and the formation of its surface. Differences in the amounts of uranium and lead isotopes between samples can reveal not only the age of the rocks but also offer insights into the Moon's geological history.
For instance, in the exercise provided, rocks from the lunar highlands and lunar maria have been dated utilizing uranium-238 decay analysis. The conclusion drawn from comparing their ages showcases the ancient volcanic activity and surface formation processes that shaped the Moon. Understanding these timelines aids in elucidating the broader history of our solar system.