   Chapter 12, Problem 50GQ

Chapter
Section
Textbook Problem

Consider the three types of cubic units cells. (a) Assuming that the spherical atoms or ions in a primitive cubic unit cell just touch along the cube’s edges, calculate the percentage of occupied space within the unit cell. (Recall that the volume of a sphere is (4/3)πr3, where r is the radius of the sphere.) (b) Compare the percentage of occupied space in the primitive cell (pc) with the bcc and fcc unit cells. Based on this, will a metal in these three forms have the same or different densities? If different, in which is it most dense? In which is it least dense?

(a)

Interpretation Introduction

Interpretation:

Percentage of occupied space within the unit cell has to be calculated.

Concept introduction:

The packing efficiency can be calculated by the percent of space occupied by spheres present in a unit cell.packing efficiency=volumeoccupiedbythespherespresentinunitcellTotalvolumeoftheunitcell×100

Volume of the cube = a3

Explanation

Calculate the packing efficiency of primitive cell

volumeofthesphere=43Πr3Volume of the cube =(2r)3One unit cell of the simple cubic contains only one atom therefore,volumeoffouratoms=1×43Πr3The packing efficiency can be calculated by the percent of space occupied by spheres present in a unit cell.Packing efficiency=volumeoccupiedbythespherespresentinunitcellTotalvolumeoftheunitcellPacking efficiency=43Πr3(2r)3×100Packing efficiencyofFCC=52.4%

Calculate the packing efficiency of FCC

One unit cell of the fcc contains four atoms therefore,

volumeofthesphere=43Πr3Volume of the cube =(2r2)3One unit cell of the fcc contains four atoms therefore,volumeoffouratoms=43Πr3The packing efficiency can be calculated by the percent of space occupied by spheres present in a unit cell

(b)

Interpretation Introduction

Interpretation:

On comparing the percentage of occupied space in PC with BCC and FCC, most dense and least dense has to be identified.

Concept introduction:

The packing efficiency can be calculated by the percent of space occupied by spheres present in a unit cell.packing efficiency=volumeoccupiedbythespherespresentinunitcellTotalvolumeoftheunitcell×100

Volume of the cube = a3

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