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Calculate the percent of volume that is actually occupied by spheres in a face-centered cubic lattice of identical spheres. You can do this by first relating the radius of a sphere, r , to the length of an edge of a unit cell, l . (Note that the spheres do not touch along an edge but do touch along the diagonal of a face.) Then calculate the volume of a unit cell in terms of r . The volume occupied by spheres equals the number of spheres per unit cell times the volume of a sphere (4 πr 3 /3).

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General Chemistry - Standalone boo...

11th Edition
Steven D. Gammon + 7 others
Publisher: Cengage Learning
ISBN: 9781305580343
BuyFind

General Chemistry - Standalone boo...

11th Edition
Steven D. Gammon + 7 others
Publisher: Cengage Learning
ISBN: 9781305580343

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Chapter
Section
Chapter 11, Problem 11.116QP
Textbook Problem
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Calculate the percent of volume that is actually occupied by spheres in a face-centered cubic lattice of identical spheres. You can do this by first relating the radius of a sphere, r, to the length of an edge of a unit cell, l. (Note that the spheres do not touch along an edge but do touch along the diagonal of a face.) Then calculate the volume of a unit cell in terms of r. The volume occupied by spheres equals the number of spheres per unit cell times the volume of a sphere (4πr3/3).

Expert Solution
Interpretation Introduction

Interpretation:

In a face centred cubic lattice of identical spheres the percent volume that is occupied by spheres has to be calculated.

Concept introduction:

  • Crystal structure: Crystal structure is arrangement of group of atoms or ions or molecule in the crystalline material.
  • Unit cell: A simplest repeating unit in the crystal structure. Every unit cell is described in terms of lattice point.  Example for unit cell: cubic, monoclinic, tetragonal, orthorhombic, rhombohedral, hexagonal and triclinic.
  • Face centred cubic cell: In a face centred cubic cell, all corners are occupied by an atom and each centre of the face contains one atom.  In face-centered cubic unit cell, each of the six corners is occupied by every single atom.  Each face of the cube is occupied by one atom.

Explanation of Solution

Given data

Radius of a sphere is r

Edge length of the unit cell is l

Volume occupied by spheres equals the number of spheres per unit cell

Volume of sphere is 4πr33

To determine: Percent volume that is occupied by spheres

The spheres touch along the diagonal of a face, d, the radius of the sphere is,

r = d/4 = l(2)/4 or

l =4r2

Because of unit cell have two spheres, the volume occupied by the sphere is,

Vcell= β =[4r2]3

For a face centred cubic structure, there are four spheres per cell

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Chapter 11 Solutions

General Chemistry - Standalone book (MindTap Course List)
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