   Chapter 12.6, Problem 3.4ACP

Chapter
Section
Textbook Problem

Determine the percentage of space occupied by tin atoms in both the tetragonal and cubic crystal lattices. The atomic radius of tin is 141 pm.

Interpretation Introduction

Interpretation:

The percentage of the space occupied by the tin atoms in tetragonal and cubic crystal lattice has to be calculated.

Concept introduction:

The volume of the atom is given below,

V=(a)3

The density of the unit cell is calculated as follows,

Massvolume = density

Mass = number of moles × molar massNumber of moles = Number of atoms per unit cellAvogadro number

The Volume of the one atom is given below

V = 43Πr3

Explanation

Number of atoms per unit is given below,

It has 8 atoms in the corner, 6 atom in the face, 4 atoms in the body.

Number of atoms per unit=(8corneratoms×1/8+6faceatom×1/2+4bodyatoms×1)Number of atoms per unit=8density =5.769g/cm3molarmassoftin=118.710 g/mol

Z = number of atoms per unit.

The Avogadro number NA=6.02×1023

Massoftheunitcell=8 × 118.7106.023×1023Massoftheunitcell=1.5767×10-21g

Volume=MassDensityVolume=1.5767×10-21g5.769g/cm3Volume=2.733×10-22cm3Volume=2.733×108pm3

Volume of the one atom is given below

V = 43Πr3

Therefore, Volume of the eight atoms is given below,

V = 8×43Πr3where,r=141pmV = 8×43Π(141 pm)3V = 9.397×107 pm3

The percentage of space occupied is

The percentage of space occupied is= 9.397×107 pm32.733×108pm3×100=0

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