Chapter 12, Problem 50GQ

### Chemistry & Chemical Reactivity

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

Chapter
Section

### Chemistry & Chemical Reactivity

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

volumeâ€‰ofâ€‰theâ€‰sphereâ€‰=â€‰43Î r3VolumeÂ ofÂ theÂ cubeÂ =â€‰(2r)3OneÂ unitÂ cellÂ ofÂ theÂ simpleÂ cubicÂ containsÂ onlyÂ oneÂ atomÂ therefore,volumeâ€‰ofâ€‰fourâ€‰atomsâ€‰=â€‰1Ã—â€‰43Î r3TheÂ packingÂ efficiencyÂ canÂ beÂ calculatedÂ byÂ theÂ percentÂ ofÂ spaceÂ occupiedÂ byÂ spheresÂ presentÂ inÂ aÂ unitÂ cell.PackingÂ efficiencyâ€‰=â€‰volumeoccupiedâ€‰â€‰byâ€‰â€‰theâ€‰â€‰spheresâ€‰presentâ€‰inâ€‰unitâ€‰cellTotalâ€‰volumeâ€‰ofâ€‰theâ€‰unitâ€‰cellPackingÂ efficiencyâ€‰=â€‰1Ã—â€‰43Î r3â€‰(2r)3Ã—â€‰100PackingÂ efficiencyâ€‰ofâ€‰FCC=â€‰52.4â€‰%

Calculate the packing efficiency of FCC

One unit cell of the fcc contains four atoms therefore,

volumeâ€‰ofâ€‰theâ€‰sphereâ€‰=â€‰43Î r3VolumeÂ ofÂ theÂ cubeÂ =â€‰(2r2)3OneÂ unitÂ cellÂ ofÂ theÂ fccÂ containsÂ fourÂ atomsÂ therefore,volumeâ€‰ofâ€‰fourâ€‰atomsâ€‰=â€‰4Ã—â€‰43Î r3TheÂ packingÂ efficiencyÂ canÂ beÂ calculatedÂ byÂ theÂ percentÂ ofÂ spaceÂ occupiedÂ byÂ spheresÂ presentÂ inÂ aÂ unitÂ 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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