Textbook Question
Molybdenum crystallizes with the body-centered unit cell. The radius of a molybdenum atom is 136 pm. Calculate the edge length of the unit cell and the density of molybdenum
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Ch.13 - Solids & Modern Materials
Tro6th EditionChemistry: A Molecular ApproachISBN: 9780137832217
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Table of contents
- Ch.1 - Matter, Measurement & Problem Solving90
- Ch.2 - Atoms & Elements94
- Ch.3 - Molecules and Compounds103
- Ch.4 - Chemical Reactions and Chemical Quantities46
- Ch.5 - Introduction to Solutions and Aqueous Solutions65
- Ch.6 - Gases100
- Ch.7 - Thermochemistry85
- Ch.8 - The Quantum-Mechanical Model of the Atom51
- Ch.9 - Periodic Properties of the Elements79
- Ch.10 - Chemical Bonding I: The Lewis Model70
- Ch.11 - Chemical Bonding II: Molecular Shapes, VSEPR & MO Theory68
- Ch.12 - Liquids, Solids & Intermolecular Forces44
- Ch.13 - Solids & Modern Materials42
- Ch.14 - Solutions77
- Ch.15 - Chemical Kinetics88
- Ch.16 - Chemical Equilibrium57
- Ch.17 - Acids and Bases121
- Ch.18 - Aqueous Ionic Equilibrium126
- Ch.19 - Free Energy & Thermodynamics80
- Ch.20 - Electrochemistry87
- Ch.21 - Radioactivity & Nuclear Chemistry61
- Ch.22 - Organic Chemistry114
Chapter 13, Problem 39
Rhodium has a density of 12.41 g/cm3 and crystallizes with the face-centered cubic unit cell. Calculate the radius of a rhodium atom.
Verified step by step guidance1
Identify the type of unit cell: Rhodium crystallizes in a face-centered cubic (FCC) unit cell.
Recall the relationship between the edge length (a) of the FCC unit cell and the atomic radius (r): In an FCC unit cell, the face diagonal is equal to 4 times the atomic radius, and the face diagonal can also be expressed as \( \sqrt{2}a \).
Use the formula for density: \( \text{Density} = \frac{\text{mass of unit cell}}{\text{volume of unit cell}} \). The mass of the unit cell can be calculated using the number of atoms per unit cell (4 for FCC) and the molar mass of rhodium.
Calculate the edge length (a) of the unit cell using the density formula and the given density of rhodium.
Solve for the atomic radius (r) using the relationship \( r = \frac{\sqrt{2}a}{4} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Density and Unit Cell
Density is defined as mass per unit volume, and in the context of crystalline solids, it can be related to the unit cell structure. A face-centered cubic (FCC) unit cell contains four atoms per unit cell, and the density can be calculated using the formula: density = (mass of atoms in unit cell) / (volume of unit cell). Understanding this relationship is crucial for determining the radius of the atom.
Face-Centered Cubic Structure
In a face-centered cubic (FCC) structure, atoms are located at each corner of the cube and at the center of each face. The edge length of the cube can be related to the atomic radius, as the diagonal of the face of the cube contains four atomic radii. This geometric relationship is essential for calculating the radius of the rhodium atom from the unit cell dimensions.
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Face Centered Cubic Example
Atomic Radius Calculation
The atomic radius can be calculated from the edge length of the FCC unit cell using the formula: radius = (√2 * edge length) / 4. To find the edge length, one can rearrange the density formula to solve for volume and subsequently for edge length. This calculation is key to determining the size of the rhodium atom based on its density and crystal structure.
Related Practice
Textbook Question
Identify each solid as molecular, ionic, or atomic. a. Ru(s)
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Textbook Question
Platinum crystallizes with the face-centered cubic unit cell. The radius of a platinum atom is 139 pm. Calculate the edge length of the unit cell and the density of platinum in g/cm3.
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Textbook Question
Plonium crystallizes with a simple cubic structure. It has a density of 9.3 g/cm3, a radius of 167 pm, and a molar mass of 209 g/mol. Use these data to calculate Avogadro's number (the number of atoms in one mole).
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