Unit Cell

Crystalline solids represent structures with well-organized patterns and shapes. 

Crystalline Solids vs. Amorphous Solids

Concept: Crystalline Solids

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Unlike crystalline solidsamorphous solids represent structures that lack an organized patterns or shapes. 

When examining the particles within a crystal you may observe them tightly packed in an organized pattern. The smallest portion of which is termed the unit cell

Simple Cubic (SC) Unit Cell

There are 7 crystal systems and 14 types of unit cells that naturally occur. The simple cubic unit cell represents one of the simplest types. 

Concept: Simple Cubic Unit Cell

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The Simple Cubic Unit Cell is composed of a cube with an atom at each corner. In the Lattice Diagram each of the unit cells are combined to form a crystal lattice. In the Space Filling Unit Cell Diagram we are shown that the unit cell uses only 1/8 of each of the 8 corner atoms. 

Body-Centered Cubic (BCC) Unit Cell 

The body-centered cubic unit cell is composed of a cube with one atom at each of its corners and one atom at the center of the cube. 

Concept: Body Centered Cubic Unit Cell

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Face-Centered Cubic (FCC) Unit Cell 

The face-centered cubic unit cell is composed of a cube with one atom at each of its corners and one atom in the center of each cube face. 

Concept: Face Centered Cubic Unit Cell

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Unit Cell Calculations

Example: Unit Cell Calculations 1

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Example: Unit Cell Calculations 1

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Example: Unit Cell Calculations 1

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Example: Unit Cell Calculations

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Example: Unit Cell Calculations 2

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Example: Unit Cell Calculations 2

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Example: Unit Cell Calculations 2

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Example: Unit Cell Calculations 3

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Example: Unit Cell Calculations 3

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Example: Unit Cell Calculations 3

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Unit Cell Additional Practice Problems

When gaseous Na+ and Cl− ions form gaseous NaCl ion pairs, 548 kJ/mol of energy is released. Why, then, does NaCl occur as a solid under ordinary conditions?

Watch Solution

The density of solid W is 19.3 g/cm3

a) How many atoms are present per cubic centimeter of W? 

b) As a solid, W adopts a body-centered cubic unit cell. How many unit cells are present per cubic centimeter of W? 

c) What is the volume of a unit cell of this metal? 

d) What is the edge length of a unit cell of W?

Watch Solution

Rank the crystal lattice structures in order of decreasing efficiency of space in the structure.

Simple cubic, Body centered cubic, Face centered cubic, Hexegonal close packed

Watch Solution

Consider a body-centered cubic unit cell as shown here.

a. How many corner atoms (orange) are shown in this image?

b. What fraction of each corner atom is inside the boundaries of the cube? 

c. How many body atoms shown in this image? 

d. What fraction of each body atom is inside the boundaries of the cube? 

e. If you sum all the fractions of atoms, how many atoms are actually inside a body-centered cubic unit cell?

Watch Solution

Consider a face-centered cubic unit cell as shown here.

a. How many corner atoms are shown in this image?

b. What fraction of each corner atom is inside the boundaries of the cube?

c. How many face atoms are shown in this image?

d. What fraction of each face atom is inside the boundaries of the cube?

e. If you sum all the fractions of atoms, how many atoms are actually inside a face-centered cubic unit cell?

Watch Solution

Consider a simple cubic (a.k.a, primitive cubic) unit cell as shown here.

(i) How many atoms are shown in this image?

(ii) What fraction of each atom is inside the boundaries of the cube?

(iii) If you sum all (the fractions of atoms, how many atoms are actually inside a simple cubic unit cell?

Watch Solution

What is the number of nearest neighbors in a body-centered-cubic lattice?

(A) 12

(B) 8

(C) 6

(D) 4

Watch Solution

The edge of a body-centered-cubic unit cell (which contains two atoms per unit cell) of an element Y was found to be 3.16 x 10 –8 cm. The density of the metal is 19.35 g•cm –3. What is the approximate molar mass of Y?

a) 65.4 g•mol –1 

b) 92.0 g•mol –1

c) 184 g•mol –1

d) 238 g•mol –1

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Watch Solution

Sodium chloride, NaCl, crystallizes in a face-centered cubic lattice of chloride ions, with the smaller sodium ions occupying holes between the chloride ions. How many Cl  ions are in contact with any single Na+ ions?

a) 4

b) 6

c) 8

d) 12

Watch Solution

How many atoms are in one body-centered cubic unit cell of a metal?

A) 1

B) 2

C) 3

D) 4

Watch Solution

Which term describes the number of atoms surrounding an atom in a crystal lattice 

a) Unit cell 

b) Coordination number

c) Crystalline solid

d) Packing efficiency

Watch Solution

Which of the following statements about crystalline and amorphous solids is TRUE?

A) A crystalline solid is composed of atoms or molecules arranged with long-range repeating order.

B) An example of a crystalline solid is glass.

C) An example of an amorphous solid is table salt (NaCl).

D) An amorphous solid is composed of atoms or molecules with a majority of its volume empty.

E) All of the above statements are TRUE.

Watch Solution

Gold crystallizes in a face-centered cubic structure. What is the edge length of the unit cell if the atomic radius of gold is 144 pm?

a) 204 pm

b) 288 pm

c) 333 pm

d) 407 pm

Watch Solution

Determine the radius of an Al atom (in pm) if the density of aluminum is 2.71 g/cm 3. Aluminum crystallizes in a face centered cubic structure with an edge length of 2√2 r. 

a) 143 pm

b) 227 pm

c) 96 pm

d) 172 pm

e) 193 pm

Watch Solution

Rank the three cubic crystalline structures in order of increasing space between the particles of the crystal.

Note: FCC = face-centered cubic, BCC = Body-centered cubic, and SC = simple cubic.

a) SC < BCC = FCC

b) FCC < BCC < SC

c) BCC < FCC < SC

d) FCC < SC < BCC

e) SC = BCC < FCC

Watch Solution

Potassium metal crystallizes in a body-centered cubic structure with a unit cell edge length of 5.31 angstroms. The radius of a potassium atom is __________ angstroms.

A. 2.30       

B. 2.66        

C. 5.31        

D. 1.33         

E. 1.88

Watch Solution

Which one of the following cannot form a solid with a lattice based on the sodium chloride structure?

A. CuO       

B. NaBr       

C. LiF       

D. CuCl2       

E. RbI

Watch Solution

Which of the following statements a through d is incorrect?

a) Ionic solids are insulators

b) The binding forces in a molecular solid include London dispersion forces.

c) Ionic solids have high melting points.

d) Molecular solids have high melting points.

e) All of the statements a through d are correct.

Watch Solution

Aluminum has a face-centered cubic unit structure and a density of 2.716 g/cm3. Calculate the edge length of the unit cell.

a) 4.041 x 10 -8 cm

b) 3.992 x 10 -8 cm

c) 3.615 x 10 -8 cm

d) 3.247 x 10 -8 cm

e) 2.836 x 10 -8 cm

Watch Solution

Vanadium crystallizes in a body centered cubic structure and has an atomic radius of 131 pm. Determine the density of vanadium, if the edge length of a bcc structure is 4r/√3.

A) 3.06 g/cm3

B) 12.2 g/cm3

C) 6.11 g/cm3

D) 2.77 g/cm3

E) 8.46 g/cm3

 

Watch Solution

What is the edge length of a face-centered cubic unit cell made up of atoms having a radius of 128 pm?

 

A) 181 pm

B) 362 pm

C) 512 pm

D) 1020 pm

E) 81 pm

Watch Solution

Which one of the following elements is considered an insulator?

a. Fe

b. Ga

c. N

d. Ge

e. Si

Watch Solution