Worksheet - Face-Centered Cubic Problems - AP level

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Here are the problems:
Problem #1: Palladium crystallizes in a face-centered cubic unit cell. Its density is 12.023 g/cm3. Calculate the atomic radius of palladium.
Problem #2: Nickel crystallizes in a face-centered cubic lattice. If the density of the metal is 8.908 g/cm3, what is the unit cell edge length in pm?
Problem #3: Nickel has a face-centered cubic structure with an edge length of 352.4 picometers. What is the density?
Problem #4: Calcium has a cubic closest packed structure as a solid. Assuming that calcium has an atomic radius of 197 pm, calculate the density of solid calcium.
Problem #5: Krypton crystallizes with a face-centered cubic unit cell of edge 559 pm. (a) What is the density of solid krypton? (b) What is the atomic radius of krypton? (c) What is the volume of one krypton atom?
Problem #6: You are given a small bar of an unknown metal. You find the density of the metal to be 11.5 g/cm3. An X-ray diffraction experiment measures the edge of the face-centered cubic unit cell as 4.06 x 10-10 m. Find the gram-atomic weight of this metal and tentatively identify it.
Problem #7: A metal crystallizes in a face-centered cubic lattice. The radius of the atom is 0.197 nm. The density of the element is 1.54 g/cm3. What is this metal?
Problem #8: The density of an unknown metal is 2.64 g/cm3 and its atomic radius is 0.215 nm. It has a face-centered cubic lattice. Determine its atomic weight.
Problem #9: Metallic silver crystallizes in a face-centered cubic lattice with L as the length of one edge of the unit cube. What is the center-to-center distance between nearest silver atoms?
Problem #10: Iridium has a face centered cubic unit cell with an edge length of 383.3 pm. The density of iridium is 22.61 g/cm3. Use these data to calculate a value for Avogadro's Number.


Problem #1: Palladium crystallizes in a face-centered cubic unit cell. Its density is 12.023 g/cm3. Calculate the atomic radius of palladium.

Solution:

1) Calculate the average mass of one atom of Pd:

106.42 g mol¯1 ÷ 6.022 x 1023 atoms mol¯1 = 1.767187 x 10¯22 g/atom

2) Calculate the mass of the 4 palladium atoms in the face-centered cubic unit cell:

1.767187 x 10¯22 g/atom times 4 atoms/unit cell = 7.068748 x 10¯22 g/unit cell

3) Use density to get the volume of the unit cell:

7.068748 x 10¯22 g ÷ 12.023 g/cm3 = 5.8793545 x 10¯23 cm3

4) Determine the edge length of the unit cell:

[cube root of] 5.8793545 x 10¯23 cm3 = 3.88845 x 10¯8 cm

5) Determine the atomic radius:

Remember that a face-centered unit cell has an atom in the middle of each face of the cube. The square represents one face of a face-centered cube:

Here is the same view, with 'd' representing the side of the cube and '4r' representing the 4 atomic radii across the face diagonal.

Using the Pythagorean Theorem, we find:

d2 + d2 = (4r)2

2d2 = 16r2

r2 = d2 ÷ 8

r = d ÷ 2(√2)

r = 1.3748 x 10¯8 cm

You may wish to convert the cm value to picometers, the most common measurement used in reporting atomic radii. Try it before looking at the solution to the next problem.


Problem #2: Nickel crystallizes in a face-centered cubic lattice. If the density of the metal is 8.908 g/cm3, what is the unit cell edge length in pm?

Solution:

This problem is like the one above, it just stops short of determining the atomic radius.

1) Calculate the average mass of one atom of Ni:

58.6934 g mol¯1 ÷ 6.022 x 1023 atoms mol¯1 = 9.746496 x 10¯23 g/atom

2) Calculate the mass of the 4 nickel atoms in the face-centered cubic unit cell:

9.746496 x 10¯23 g/atom times 4 atoms/unit cell = 3.898598 x 10¯22 g/unit cell

3) Use density to get the volume of the unit cell:

3.898598 x 10¯22 g ÷ 8.908 g/cm3 = 4.376514 x 10¯23 cm3

4) Determine the edge length of the unit cell:

[cube root of] 4.376514 x 10¯23 cm3 = 3.524 x 10¯8 cm

5) Convert cm to pm:

cm = 10¯2 m; pm = 10¯12 m.

Consequently, there are 1010 pm/cm

(3.524 x 10¯8 cm) (1010 pm/cm) = 352.4 pm


Problem #3: Nickel has a face-centered cubic structure with an edge length of 352.4 picometers. What is the density?

This problem is the exact reverse of problem #2. (See problem 5a below for an example set of calculations.)

Solution:

1) Convert pm to cm
2) Calculate the volume of the unit cell
3) Calculate the average mass of one atom of Ni
4) Calculate the mass of the 4 nickel atoms in the face-centered cubic unit cell
5) Calculate the density (value from step 4 divided by value from step 2)

Problem #4: Calcium has a cubic closest packed structure as a solid. Assuming that calcium has an atomic radius of 197 pm, calculate the density of solid calcium.

Solution:

1) Convert pm to cm:

197 pm x (1 cm/1010 pm) = 1.97 x 10¯8 cm

2) Determine the edge length of the unit cell:

Use the Pythagorean Theorem (see problem #1 for a discussion):

r = d ÷ 2(√2)

1.97 x 10¯8 cm = d ÷ 2(√2)

d = 5.572 x 10¯8 cm

3) Determine the volume of the unit cell:

(5.572 x 10¯8 cm)3 = 1.730 x 10¯22 cm3

4) Determine mass of 4 atoms of Ca in a unit cell (cubic closest packed is the same as face-centered cubic):

40.08 g/mol divided by 6.022 x 1023 atoms/mol = 6.6556 x 10¯23 g/atom

6.6556 x 10¯23 g/atom times 4 atoms = 2.66224 x 10¯22 g

5) Determine density:

2.66224 x 10¯22 g divided by 1.730 x 10¯22 cm3 = 1.54 g/cm3

Problem #5: Krypton crystallizes with a face-centered cubic unit cell of edge 559 pm.

a) What is the density of solid krypton?
b) What is the atomic radius of krypton?
c) What is the volume of one krypton atom?
d) What percentage of the unit cell is empty space if each atom is treated as a hard sphere?

Solution to a:

1) Convert pm to cm:

559 pm x (1 cm/1010 pm) = 559 x 10¯10 cm = 5.59 x 10¯8 cm

2) Calculate the volume of the unit cell:

(5.59 x 10¯8 cm)3 = 1.7468 x 10¯22 cm3

3) Calculate the average mass of one atom of Kr:

83.798 g mol¯1 divided by 6.022 x 1023 atoms mol¯1 = 1.39153 x 10¯22 g

4) Calculate the mass of the 4 krypton atoms in the face-centered cubic unit cell:

1.39153 x 10¯22 g times 4 = 5.566 x 10¯22 g

5) Calculate the density (value from step 4 divided by value from step 2):

5.566 x 10¯22 g / 1.7468 x 10¯22 cm3 = 3.19 g/cm3

Solution to b:

Use the Pythagorean Theorem (see problem #1 for a discussion):

r = d ÷ 2(√2)

r = 5.59 x 10¯8 cm ÷ 2(√2)

r = 1.98 x 10¯8 cm

Solution to c:

V = (4/3) π r3

V = (4/3) (3.14159) (1.98 x 10¯8 cm)3

V = 3.23 x 10¯23 cm3

Solution to d:

1) Calculate the volume of the 4 atoms in the unit cell:

3.23 x 10¯23 cm3 times 4 = 1.29 x 10¯22 cm3

2) Calculate volume of cell not filled with Kr:

1.7468 x 10¯22 cm3 minus 1.29 x 10¯22 cm3 = 4.568 x 10¯23 cm3

3) Calculate % empty space:

4.568 x 10¯23 cm3 divided by 1.7468 x 10¯22 cm3 = 0.2615

26%


Problem #6: You are given a small bar of an unknown metal. You find the density of the metal to be 11.5 g/cm3. An X-ray diffraction experiment measures the edge of the face-centered cubic unit cell as 4.06 x 10-10 m. Find the gram-atomic weight of this metal and tentatively identify it.

Solution:

1) Convert meters to cm:

4.06 x 10-10 m = 4.06 x 10-8 cm

2) Determine the volume of the unit cube:

(4.06 x 10-8 cm)3 = 6.69234 x 10-23 cm3

3) Determine the mass of the metal in the unit cube:

11.5 g/cm3 times 6.69234 x 10-23 cm3 = 7.696193 x 10-22 g

4) Determine atomic weight (based on 4 atoms per unit cell):

7.696193 x 10-22 g is to 4 atoms as x grams is to 6.022 x 1023 atoms

x = 116 g/mol (to three sig figs)

This weight is close to that of indium.


Problem #7: A metal crystallizes in a face-centered cubic lattice. The radius of the atom is 0.197 nm. The density of the element is 1.54 g/cm3. What is this metal?

Solution:

1) Convert nm to cm:

0.197 nm x (1 cm/107 nm) = 1.97 x 10¯8 cm

2) Determine the edge length of the unit cell:

Use the Pythagorean Theorem (see problem #1 for a discussion):

r = d ÷ 2(√2)

1.97 x 10¯8 cm = d ÷ 2(√2)

d = 5.572 x 10¯8 cm

3) Determine the volume of the unit cell:

(5.572 x 10¯8 cm)3 = 1.72995 x 10¯22 cm3

4) Determine grams of metal in unit cell:

1.72995 x 10¯22 cm3 times 1.54 g/cm3 = 2.6641 x 10¯22 g

5) Determine atomic weight (based on 4 atoms per unit cell):

2.6641 x 10¯22 g is to 4 atoms as x grams is to 6.022 x 1023 atoms

x = 40.11 g/mol

The metal is calcium.


Problem #8: The density of an unknown metal is 2.64 g/cm3 and its atomic radius is 0.215 nm. It has a face-centered cubic lattice. Determine its atomic weight.

Solution:

1) Convert nm to cm:

0.215 nm x (1 cm/107 nm) = 2.15 x 10¯8 cm

2) Determine the edge length of the unit cell:

Use the Pythagorean Theorem (see problem #1 for a discussion):

r = d ÷ 2(√2)

2.15 x 10¯8 cm = d ÷ 2(√2)

d = 6.08112 x 10¯8 cm

3) Determine the volume of the unit cell:

(6.08112 x 10¯8 cm)3 = 2.2488 x 10¯22 cm3

4) Determine grams of metal in unit cell:

2.2488 x 10¯22 cm3 times 2.64 g/cm3 = 5.9368 x 10¯22 g

5) Determine atomic weight (based on 4 atoms per unit cell):

5.9368 x 10¯22 g is to 4 atoms as x grams is to 6.022 x 1023 atoms

x = 89.4 g/mol


Problem #9: Metallic silver crystallizes in a face-centered cubic lattice with L as the length of one edge of the unit cube. What is the center-to-center distance between nearest silver atoms?

a) L/2
b) 21/2 L
c) 2L
d) L/21/2
e) None of the above answers are valid.

Solution:

Call center-to-center distance = d. There are two of them on the face diagonal.

Therefore, by the Pythagorean Theorem:

L2 + L2 = (2d)2

2L2 = 4d2

(L2 ) / 2 = d2

L / 21/2 = d

The answer is d


Problem #10: Iridium has a face centered cubic unit cell with an edge length of 383.3 pm. The density of iridium is 22.61 g/cm3. Use these data to calculate a value for Avogadro's Number.

Solution:

1) Use the edge length to get the volume of the unit cell:

383.3 pm = 3.833 x 10¯8 cm

(3.833 x 10¯8 cm)3 = 5.6314 x 10¯23 cm3

2) Use the density to get the mass of Ir in the unit cell:

22.61 g/cm3 times 5.6314 x 10¯23 cm3 = 1.27326 x 10¯21 g

3) Use the atomic weight of Ir to determine how many moles of Ir are in the unit cell:

1.27326 x 10¯21 g divided by 192.217 g/mol = 6.624075 x 10¯24 mol

4) Use 4 atoms per face-centered unit cell to set up the following ratio and proportion:

4 atoms is to 6.624075 x 10¯24 mol as x is to 1.000 mol

x = 6.038 x 1023 atoms

For a different take on the solution to this problem, go here and take a look at the answer by m w.


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