Question

A rock from an asteroid contains \(2.57 \mathrm{~g}\) of \({ }_{92}^{236} \mathrm{U}\) and \(3.83 \mathrm{~g}\) of \({ }_{82}^{206} \mathrm{~Pb}\). The molar mass of \({ }_{82}^{206} \mathrm{~Pb}\) is \(205.97446 \mathrm{~g} / \mathrm{mol}\), the molar mass of \({ }_{99}^{238} \mathrm{U}\) is \(238.029 \mathrm{~g} / \mathrm{mol}\), and the half- life of \({ }_{92}^{238} \mathrm{U}\) is \(4.46 \times 10^{9}\) years. Assume that all the \({ }_{82}^{206} \mathrm{~Pb}\) came from the radioactive decay of the \({ }_{92}^{238} \mathrm{U}\). (a) How many atoms of each isotope are present in the rock? (b) How many atoms of \({ }_{92}^{238} \mathrm{U}\) were in the rock when it formed? (c) What is the percent of \({ }_{92}^{238} \mathrm{U}\) atoms remaining in the rock compared to when it was first formed? (d) How old is the asteroid?

Short answer

The short answer to the question is: (a) There are \(1.29 \times 10^{21}\) atoms of Uranium-238 and \(1.12 \times 10^{22}\) atoms of Lead-206 in the rock. (b) The original number of Uranium-238 atoms in the rock was \(1.25 \times 10^{22}\). (c) There is about \(10.32\%\) of Uranium-238 atoms remaining in the rock compared to when it was first formed. (d) The age of the asteroid is approximately \(3.85 \times 10^9\) years.

Step-by-step solution

(Step 1: Calculate the number of moles of each isotope in the rock)

To find the number of atoms of each isotope in the rock, we first need to calculate the number of moles of each isotope using their respective masses and molar masses. For Uranium-238, moles = mass/molar_mass For Lead-206, moles = mass/molar_mass

(Step 2: Calculate the number of atoms of each isotope in the rock)

Now we can find the number of atoms of each isotope using the moles of each isotope and Avogadro's number (approximately \(6.022 \times 10^{23}\) atoms/mol): For Uranium-238, atoms = moles * Avogadro's_number For Lead-206, atoms = moles * Avogadro's_number

(Step 3: Calculate the original number of Uranium-238 atoms in the rock)

We know that all of the Lead-206 in the rock came from the decay of Uranium-238. Therefore, the sum of the remaining Uranium-238 atoms and Lead-206 atoms in the rock will give us the original number of Uranium-238 atoms: Original Uranium-238 atoms = Uranium-238_atoms + Lead-206_atoms

(Step 4: Calculate the percentage of remaining Uranium-238 atoms)

To find the percent of remaining Uranium-238 atoms in the rock, we'll divide the number of Uranium-238 atoms by the original number of Uranium-238 atoms and multiply the result by 100: Percentage of remaining Uranium-238 atoms = (Uranium-238_atoms / Original_Uranium-238_atoms) * 100

(Step 5: Calculate the age of the asteroid)

We'll use the half-life formula, which is given by: \(N_t = N_0 \times (1/2)^{t/T}\) Where \(N_t\) is the number of remaining Uranium-238 atoms, \(N_0\) is the original number of Uranium-238 atoms, \(t\) is the time elapsed and \(T\) is the half-life of Uranium-238. We'll rearrange this formula to find the time elapsed which represents the age of the asteroid: \(t = T \times \frac{\log(N_t / N_0)}{\log(1/2)}\) Plug in the values of \(N_t\), \(N_0\) and \(T\) to calculate the age of the asteroid.

Key concepts

Uranium-238

Uranium-238 is a naturally occurring isotope of uranium, which is commonly used in dating geological formations. This isotope is significant because it has a long half-life of approximately 4.46 billion years, making it an ideal candidate for dating ancient rocks and minerals. Uranium-238 undergoes radioactive decay, eventually transmuting into Lead-206, through a complex series of steps known as a decay chain.

• Uranium-238 has 92 protons and 146 neutrons in its nucleus, giving it an atomic mass number of 238.
• The decay process involves the emission of alpha particles (helium nuclei) and β particles (electrons), changing the atomic number and mass number of the radioactive element over time.
• This process continues until a stable isotope, such as Lead-206, is formed.

Understanding Uranium-238 is essential to the study of radioactive decay, as it provides insight into the rate and manner in which isotopes transform over immense periods.

Lead-206

Lead-206 is a stable isotope that serves as the ultimate product of the Uranium-238 decay chain. As uranium in minerals decays over time, it gradually becomes Lead-206. This gradual conversion allows scientists to track the age of rocks and determine how long these processes have been occurring.

• Lead-206 consists of 82 protons and 124 neutrons, contributing to its stability as a non-radioactive element.
• Being the end product in this long decay process, Lead-206 is often used in radiometric dating to help determine the age of Earth's oldest structures.
• The presence and ratio of Lead-206 to Uranium-238 in a mineral sample serves as a time clock, marking the passage of geological time.

This makes Lead-206 a crucial component in understanding geological and planetary sciences.

half-life calculation

Half-life is a fundamental concept in radioactive decay, representing the time required for half of a radioactive substance to convert into another element or isotope. The half-life calculation is a useful tool in fields like geology, archaeology, and physics, as it helps in determining the age of materials containing radioactive elements.

• It is defined by the equation: \[N_t = N_0 \times (1/2)^{t/T}\]Where \(N_t\) is the number of atoms of a radioactive isotope remaining after time \(t\), \(N_0\) is the original number of atoms, and \(T\) is the half-life.
• The half-life of Uranium-238 is 4.46 billion years, a period during which half of the uranium sample would have decayed into Lead-206.
• Rearranging the formula can help calculate the elapsed time or age of a sample, providing insights into the history of the sample being studied.

Mastery of half-life calculations allows students and researchers to accurately determine the age of geological samples and understand the dynamics of radioactive decay.

atomic structure

Atomic structure refers to the arrangement of particles within an atom. It consists of protons, neutrons, and electrons, with protons and neutrons forming the nucleus and electrons orbiting around this central nucleus.

• Protons are positively charged particles, while neutrons have no charge. Both are approximately equal in mass and reside in the nucleus.
• Electrons are negatively charged particles with considerably less mass, and they exist in a cloud around the nucleus, occupying various energy levels.
• The number of protons, known as the atomic number, determines the element's identity, while the sum of protons and neutrons gives the atomic mass number.

Understanding atomic structure is vital in comprehending isotopic differences, radioactive decay processes, and the fundamental properties and behaviors of elements.

isotope

Isotopes are variants of the same chemical element that have the same number of protons but differ in the number of neutrons. Isotopes of an element exhibit nearly identical chemical behavior as they have the same electron arrangement, but they may vary significantly in physical properties, such as stability and atomic mass.

• The atomic number, which denotes the number of protons, remains unchanged across isotopes, but the mass number (protons + neutrons) differs.
• Uranium-238 is an isotope with 146 neutrons, while Uranium-235, another isotope of uranium, contains only 143 neutrons.
• Some isotopes are stable, while others are radioactive and decay over time, leading to changes in the atomic structure and sometimes forming new elements.

Grasping the concept of isotopes is essential for fields involving nuclear reactions and for techniques such as radiometric dating, which rely on these subtle differences.