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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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

Volume = a^3

= 4.426 x 10 ^ -23 cm^3

- 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

- 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

- 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

- What is the volume, cm3of the unit cell?

Volume = d^3

= 3.894x 10^-23 cm^3

- How many moles of atoms are within one unit cell?

atom per unit= 2

- 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

- Which element do you think that this is? C

- 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

- 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%

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

Coordination number = 8

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