A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC crystal structure. Given the density of the metal is 10 g/cm3 and the atomic weight/mass of the metal is 100 g/mol, estimate the atomic radius of the metal. Note: Avogadro's number NA = 6.022×1023 atoms/mol
A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC crystal structure. Given the density of the metal is 10 g/cm3 and the atomic weight/mass of the metal is 100 g/mol, estimate the atomic radius of the metal. Note: Avogadro's number NA = 6.022×1023 atoms/mol
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
Related questions
Question
![### Estimating the Atomic Radius of a Hypothetical Metal
**Problem Statement:**
A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC (Face-Centered Cubic) crystal structure. Given the density of the metal is 10 g/cm³ and the atomic weight/mass of the metal is 100 g/mol, estimate the atomic radius of the metal.
**Given Data:**
1. Density (\(\rho\)) of the metal: 10 g/cm³
2. Atomic weight/mass (M) of the metal: 100 g/mol
3. Avogadro's number (\(N_A\)): \(6.022 \times 10^{23}\) atoms/mol
**Solution:**
To estimate the atomic radius, we use the following steps:
1. **Determine the Number of Atoms per cm³:**
The density equation for atoms in a unit volume of the metal can be expressed as:
\[\rho = \frac{n \cdot M}{V\cdot N_A}\]
Where \(n\) is the number of atoms per unit cell and \(V\) is the volume of the unit cell. Rearranging for \(V\):
\[V = \frac{n \cdot M}{\rho \cdot N_A}\]
For an FCC structure, the number of atoms per unit cell \(n\) is 4.
2. **Calculate the Volume of the Unit Cell:**
\[V = \frac{4 \cdot 100 \text{ g/mol}}{10 \text{ g/cm}^3 \cdot 6.022 \times 10^{23} \text{ atoms/mol}}\]
3. **Convert the Volume to Appropriate Units:**
Calculate \(V\) to find the unit cell volume in cm³ and subsequently convert it to cubic angstroms (ų) if necessary for precision in determining the atomic radius.
4. **Determine the Atomic Radius:**
The volume of a cubic unit cell \(V\) can also be related to the lattice parameter \(a\) (where \(V = a^3\)). For an FCC lattice, the relationship between the lattice parameter \(a\) and the atomic radius \(R\) is:
\[a = \frac{4R}{\sqrt{2}}\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9f391b6-5f04-4d98-9bc5-d57c88046352%2F5463e54d-a26e-4217-9d1c-454d585110d9%2Fs8f9nzh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Estimating the Atomic Radius of a Hypothetical Metal
**Problem Statement:**
A hypothetical metal, which is a single crystal and made up of one chemical element, has an FCC (Face-Centered Cubic) crystal structure. Given the density of the metal is 10 g/cm³ and the atomic weight/mass of the metal is 100 g/mol, estimate the atomic radius of the metal.
**Given Data:**
1. Density (\(\rho\)) of the metal: 10 g/cm³
2. Atomic weight/mass (M) of the metal: 100 g/mol
3. Avogadro's number (\(N_A\)): \(6.022 \times 10^{23}\) atoms/mol
**Solution:**
To estimate the atomic radius, we use the following steps:
1. **Determine the Number of Atoms per cm³:**
The density equation for atoms in a unit volume of the metal can be expressed as:
\[\rho = \frac{n \cdot M}{V\cdot N_A}\]
Where \(n\) is the number of atoms per unit cell and \(V\) is the volume of the unit cell. Rearranging for \(V\):
\[V = \frac{n \cdot M}{\rho \cdot N_A}\]
For an FCC structure, the number of atoms per unit cell \(n\) is 4.
2. **Calculate the Volume of the Unit Cell:**
\[V = \frac{4 \cdot 100 \text{ g/mol}}{10 \text{ g/cm}^3 \cdot 6.022 \times 10^{23} \text{ atoms/mol}}\]
3. **Convert the Volume to Appropriate Units:**
Calculate \(V\) to find the unit cell volume in cm³ and subsequently convert it to cubic angstroms (ų) if necessary for precision in determining the atomic radius.
4. **Determine the Atomic Radius:**
The volume of a cubic unit cell \(V\) can also be related to the lattice parameter \(a\) (where \(V = a^3\)). For an FCC lattice, the relationship between the lattice parameter \(a\) and the atomic radius \(R\) is:
\[a = \frac{4R}{\sqrt{2}}\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, chemistry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781305957404
Author:
Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:
Cengage Learning
Chemistry
Chemistry
ISBN:
9781259911156
Author:
Raymond Chang Dr., Jason Overby Professor
Publisher:
McGraw-Hill Education
Principles of Instrumental Analysis
Chemistry
ISBN:
9781305577213
Author:
Douglas A. Skoog, F. James Holler, Stanley R. Crouch
Publisher:
Cengage Learning
Organic Chemistry
Chemistry
ISBN:
9780078021558
Author:
Janice Gorzynski Smith Dr.
Publisher:
McGraw-Hill Education
Chemistry: Principles and Reactions
Chemistry
ISBN:
9781305079373
Author:
William L. Masterton, Cecile N. Hurley
Publisher:
Cengage Learning
Elementary Principles of Chemical Processes, Bind…
Chemistry
ISBN:
9781118431221
Author:
Richard M. Felder, Ronald W. Rousseau, Lisa G. Bullard
Publisher:
WILEY