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Crystalline Structure

A.      Primitive Cubic Unit Cells

  • Polonium crystallizes in a primitive cubic unit cell. The side of a unit cell is measured through X-ray diffraction to be 3.80 Å. What is the radius of a polonium atom in Å?

we know that in primitive cubic unit cell side = 2r

so

3.8 = 2r

R = 1.9

so

the radius of a polonium atom is 1.9 Ao

  • What is the volume, cm3 of a single polonium atom?(Assume that atoms are spherical. Do you remember what the formula for the volume of a sphere is?)

Volume of sphere = (4/3) x pi x r3

so

Volume of single polonium atom = (4/3) x pi x (1.9 x 10-8)^3

Volume of single polonium atom = 2.871634 x 10-23 cm3

  1. What is the total volume, cm3 of polonium atoms inside the unit cell?

Only one atom in unit cell

so

Total volume of polonium atoms inside the unit cell

= 2.871634 x 10^-23cm3

  • What is the volume of a unit cell of polonium, cm3?(Do you remember the formula for the volume of a cube?)

Volume of unit cell = a3

Volume of unit cell = ( 3.8 x 10-8)^3

Volume of unit cell = 5.4872 x 10-23 cm3

  1. What is the percent of occupied volume in a unit cell of polonium?

% volume occupied = volume of polonium atom / volume of unit cell

% volume occupited = 2.871634 x 10-23 x 100 / 5.4872 x 10-23

% volume occupied = 52.33

  1. What is the mass, g of a unit cell of polonium?

Molar mass of polonium = 209 g /mol

so

Mass of one atom = molar mass / avagadro number

Mass of one atom = 209 / (6.022 x 10^23)

Mass of one atom = 3.4706 x 10-22 g

as

One unit cell contains one polonium atom

Mass of unit cell = 3.4706 x 10-22 g

  1. What is the density, g/cm3 of a sample of polonium?

Density = mass of unit cell / volume

Density = 3.4706 x 10-22 / (5.4872 x 10-23)

Density = 6.325 g/cm3

  1. What is the coordination number of an element that arranges in a primitive cubic unit cell?

The coordination number of simple cubic unit cell is 6

B.      Face-Centered Cubic Unit Cells

  • Aluminum crystallizes in a face-centered cubic unit cell. The density of aluminum is 4.05 g/cm3. What is the mass, g of aluminum in the unit cell?

gram sper atom Aluminum

= 26.98 g mol^-1/ 6.022x 10^23 atomsmol^-1

=4.480×10^23 g/atoms

  1. What is the volume, cm3 of the unit cell in cubic centimeters?

density= mass/ volum

Volum=(1.792×10^-22 g/unit cell)/ 4.05 cm^-3

= 4.42469 x 10^-23 cm^3 per unit cell

  1. What is the side of the unit cell in centimeters.

Density= Mass x no of atom / a^3 x Na

4.05= (26.99 x 4)/a^3 x 6.023 x 10^23

a^3= (26.99 x 4)/ 4.05 x 6.023 x 10^-23

a= 7.62 x 10^-7

  1. What is the radius of aluminum in angstroms?

a = 2π √2

7.62 x 10^-7 = 2x π x 1.414

π= 2.7 x 10 ^-7 cm

  1. What is the total volume, cm3 of aluminum atoms inside the unit cell?

Volume = a^3

= 4.426 x 10 ^ -23 cm^3

  1. What is the percent of occupied volume in a unit cell of aluminum?

No of atom = 4

pakage factor =(No. of atoms) (vol. of each atom) / (vol. of unit cell)

P.factor= 0.74

74% is occupied by atom and 26% is empty

  1. What is the coordination number of an element that arranges in a face-centered cubic unit cell?

The coordination number of face center cubic unit cell is 12

C.      Body-Centered Cubic Unit Cells

  1. An element crystallizes in a body-centered cubic unit cell. The radius of this element is found to be is 2.20Å. The density is 0.991 g/cm3. What is the side of the unit cell in centimeters?

Radius (r) = 2.20 A

4 r = 4.80 A

d^2= (√20)^2 = (4r)^2

d^2= (d √2)^2= (4.80)^2

3d^2 = 23.04

d^2= 23.04/3

d = (√23.09/3) = 3.39 A

3.39×10^-10m = 3.39x 10 ^-8cm

  1. What is the volume, cm3of the unit cell?

Volume = d^3

= 3.894x 10^-23 cm^3

  1. How many moles of atoms are within one unit cell?

atom per unit= 2

  1. What is the molar mass of this element?

= (0.991/1cm^3) x 3.899x 10 ^-23 cm^3 = 3.86×10^-23 g

2 atoms contain one unit cell

= 3.86 x 10 ^-23g/2 mol = 1.93x 10 ^-23g

Molar mass= 1.93x 10 ^-23 x 6.022x 10 ^-23

= 11.6

  1. Which element do you think that this is? C
  1. What is the total volume, cm3 of this element inside the unit cell?

R= 202x 10 ^-8cm

Total volum occupied by atom= 4 x 2 x 3.14 x (2.2 x 10^-8 cm)^3= 89.176 x 10 ^-24 cm^3

  1. What is the percent of occupied volume in a unit cell of this element?

edge length = 4.R/1.73= 5.08x 10^-8 cm

Volum of cube= (5.08 x 10 ^-8 cm)^3 = 131.1 x 10^-24 cm^3

Percentage occupied = 89.176 x 10^-24 cm ^-3 x 100/131.1x 10^-24 cm^3

= 68.02%

  1. What is the coordination number of an element that arranges in a body-centered cubic unit cell?

Coordination number = 8

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