 # 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) πr 3 , 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? ### Chemistry & Chemical Reactivity

10th Edition
John C. Kotz + 3 others
Publisher: Cengage Learning
ISBN: 9781337399074 ### Chemistry & Chemical Reactivity

10th Edition
John C. Kotz + 3 others
Publisher: Cengage Learning
ISBN: 9781337399074

#### Solutions

Chapter
Section
Chapter 12, Problem 50GQ
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?

Expert Solution

(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 of Solution

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

Expert Solution

(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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