   Chapter 11, Problem 11.116QP ### General Chemistry - Standalone boo...

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

#### Solutions

Chapter
Section ### General Chemistry - Standalone boo...

11th Edition
Steven D. Gammon + 7 others
ISBN: 9781305580343
Textbook Problem

# 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).

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

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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