   Chapter 12, Problem 49GQ

Chapter
Section
Textbook Problem

You can get some idea of how efficiently spherical atoms or ions are packed in a three-dimensional solid by seeing how well circular atoms pack in two dimensions. Using the drawings shown here, prove that B is a more efficient way to pack circular atoms than A. The unit cell of A contains portions of four circles and one hole. In B, packing coverage can be calculated by looking at a triangle that contains portions of three circles and one hole. Show that A fills about 80% of the available space, whereas B fills closer to 90% of the available space. Interpretation Introduction

Interpretation:

From the picture, picture B is more efficiency than picture A and diagram B is leads to the surface coverage of 78.5% has to be proved.

Concept introduction:

The area of the triangle is calculated by using following formula,

areaoftriangle=1/2×base×height

Explanation
• Diagram A is leads to the surface coverage of 78.5% which is shown below,

=areacoveredbythecircletotalarea=areaofonecircle(2rcircle)2=Πr24r2=0.785=78.5%

• Diagram B is leads to the surface coverage of 78.5% which is shown below,

areaoftriangle=1/2×base×heightareaoftriangle=12×2r×3rareacovered

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