SME
Inorganic Chemistry
Solid-state structures
Use this for “Iron is body-centered cubic with an atomic radius of 124 pm and M = 55.85 g/mol. Find its cell edge, density and packing fraction.”
Diagram
Atoms on corners count ⅛, on edges ¼, on faces ½ and inside 1: Z = 4 atoms per cell.
They
touch
along
a
face
diagonal:
4r
=
√
2
a.
This is an example. Type your own values to start a new problem.
ρ (g/cm³)
=
Z
×
M (g/mol)
÷
(
N
A
(
a (pm)
)
3
)
r
Atomic radius
example · pm
packing
Packing fraction
worked out (not typed) · %
Formulas
a
=
the
edge
per
radius
for
Z
atoms
per
cell
×
r
362 = the edge per radius for 4 atoms per cell × 128
ρ
=
Z
×
M
÷
(6.022
×
10
23
×
(
a
×
10
-10
)
3
)
8.895
=
4
×
63.55
÷
(6.022
×
10
23
×
(362 ×
10
-10
)
3
)
packing
=
Z
×
4
3
×
π
×
r
3
a
3
×
100
74.05
=
4
×
4
3
×
π
×
128
3
362
3
×
100
Assumptions
•
Type the structure as its atoms per cell Z: 1 simple cubic, 2 body-centered cubic, 4 face-centered cubic.
•
The atoms are hard spheres touching along an edge (a = 2r), the body diagonal (√3a = 4r) or a face diagonal (√2a = 4r).
•
The cell holds Z atoms of mass M ÷ NA each; a × 10⁻¹⁰ puts pm into cm, and NA = 6.022 × 10²³ mol⁻¹.
Step-by-step
WE KNOW
Z = 4, r = 128 pm, M = 63.55 g/mol
FIND
cell edge (a), density (ρ), packing fraction (packing)
Step 1 · Find cell edge (a)
Use a = the edge per radius for Z atoms per cell × r
Face-centered:
they
touch
along
a
face
diagonal,
√
2
a
=
4
r
,
so
a
=
2
√
2
r
.
a
=
2
×
√
2
×
r
a
=
2
×
√
2
×
128
a
=
2
×
1.414
×
128
a
=
2.828
×
128
a
=
362.04
pm
Step 2 · Find density (ρ)
Use ρ = Z × M ÷ (6.022 × 10²³ × (a × 10⁻¹⁰)³)
The
mass
of
the
cell’s
Z
atoms
over
its
volume,
with
the
edge
in
cm.
ρ
=
4
×
63.55
÷
(6.022
×
10
23
×
(362.04 ×
10
-10
)
3
)
ρ
=
4
×
63.55
(6.022 ×
10
23
) × (4.7454 ×
10
-23
)
ρ
=
4
×
63.55
28.58
ρ
=
254.2
28.58
ρ
=
8.895
g/cm³
Step 3 · Find packing fraction (packing)
Use packing = Z × 4 ÷ 3 × π × r³ ÷ a³ × 100
The
volume
of
the
Z
spheres
over
the
cell’s
volume,
as
a
percent.
packing
=
4
×
4
3
×
π
×
128
3
362.04
3
×
100
packing
=
4
×
4
3
×
π
×
2,097,152
4.7454 ×
10
7
×
100
packing
=
16
3
×
π
×
2,097,152
4.7454 ×
10
7
×
100
packing
=
5.3333
×
π
×
2,097,152
4.7454 ×
10
7
×
100
packing
=
16.755
×
2,097,152
4.7454 ×
10
7
×
100
packing
=
3.5138 ×
10
7
4.7454 ×
10
7
×
100
packing = 0.74046 × 100
packing = 74.05%
Check
Put the numbers back in. Both sides should match.
362.04 = the edge per radius for 4 atoms per cell × 128 ✓
8.895
=
4
×
63.55
÷
(6.022
×
10
23
×
(362.04 ×
10
-10
)
3
)
✓
74.05
=
4
×
4
3
×
π
×
128
3
362.04
3
×
100
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Radius ratio and the rock-salt cell
Use this for “Mg²⁺ is 72 pm and O²⁻ is 140 pm across. Predict the coordination number, then MgO’s rock-salt edge and density (40.30 g/mol).”
Plane spacing and Bragg’s law
Use this for “Tungsten is cubic with a = 316.5 pm. Find the spacing of its (110) planes and the angle 2θ of their peak with 154.2 pm X-rays.”
Lattice energy from the Born–Landé equation
Use this for “MgO has the rock-salt structure with r₀ = 210 pm and Born exponent 7. Estimate its lattice energy.”