   Chapter 21, Problem 21.12E

Chapter
Section
Textbook Problem

What is the maximum percentage volume that can be taken up by the atoms in a simple cubic unit cell? How much less is it than close packing?

Interpretation Introduction

Interpretation:

The maximum percentage volume that can be taken up by the atoms in a simple cubic unit cell is to be stated. The difference between the corresponding answer and the maximum percentage volume that can be taken up by the atoms in close packing is to be stated.

Concept introduction:

A unit cell of the crystal is the three-dimensional arrangement of the atoms present in the crystal. The unit cell is the smallest and simplest unit of the crystal which on repetition forms an entire crystal. Unit cell can be a cubic unit cell or hexagonal unit cell. The classification of a unit cell depends on the lattice site occupied by the atoms.

Explanation

The structure of a simple cubic unit cell is represented as,

Figure 1

The number of the constituent particles present in one cube is 1.

The length of the unit cell and the radius of the constituent particles are related as shown below.

Figure 2

The mathematical relation between the length of the unit cell and the radius of the constituent particles is represented as,

r=a2 …(1)

Where,

a represents the length of the unit cell.

r represents the radius of the constituent particles.

The total volume of cube is calculated with the formula shown below as,

Vc=a3

The total volume of sphere is calculated with the formula shown below as,

Vs=43πr3

The percentage packing efficiency of a unit cell is represented as,

%P=zVsVc×100 …(2)

Where,

z represents the number of spheres in a unit cell.

Substitute the value of z, Vs and Vc in the equation (2).

%P=(1)(43πr3a3)(100)

Substitute the value of r in the above expression

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