SME
Inorganic Chemistry · Problem type
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).”
Diagram
Atoms on corners count ⅛, on edges ¼, on faces ½ and inside 1: Z = 4 formula units per cell.
They touch along an edge: 2(r₊ + r₋) = a.
This is an example. Type your own values to start a new problem.
r₊
Cation radius
example · pm
r₋
Anion radius
example · pm
r₊/r₋
Radius ratio
calculated
CN
Coordination number
worked out (not typed)
a
Cell edge
calculated · pm
M
Formula mass
example · g/mol
ρ
Density
calculated · g/cm³
Formulas
r₊/r₋ = r₊ ÷ r₋
0.5635
=
102
181
CN = the neighbors a radius ratio of r₊/r₋ predicts
6 = the neighbors a radius ratio of 0.5635 predicts
a
=
2
×
(r₊
+
r₋)
566 = 2 × (102 + 181)
ρ
=
4
×
M
÷
(6.022
×
10
23
×
(
a
×
10
-10
)
3
)
2.141
=
4
×
58.44
÷
(6.022
×
10
23
×
(566 ×
10
-10
)
3
)
Assumptions
•
Hard-sphere ions: each cation touches its anions, and the anions just fit round it.
•
Ratio r₊ ÷ r₋ below 0.225 → 3 neighbors, 0.225 to 0.414 → 4, 0.414 to 0.732 → 6, above 0.732 → 8.
•
Rock salt: ions touch along the edge, a = 2(r₊ + r₋), and the cell holds 4 formula units. The edge and density assume the rock-salt structure (CN 6) even when the ratio predicts 4 or 8.
•
The density uses NA = 6.022 × 10²³ mol⁻¹, with the edge in cm (1 pm = 10⁻¹⁰ cm).
Step-by-step
WE KNOW
r₊ = 102 pm, r₋ = 181 pm, M = 58.44 g/mol
FIND
radius ratio (r₊/r₋), coordination number (CN), cell edge (a), density (ρ)
Step 1 · Find radius ratio (r₊/r₋)
Use r₊/r₋ = r₊ ÷ r₋
Divide the cation’s radius by the anion’s: the bigger the cation, the more anions fit round it.
r₊/r₋
=
102
181
r₊/r₋ = 0.5635
Step 2 · Find coordination number (CN)
Use CN = the neighbors a radius ratio of r₊/r₋ predicts
Below 0.225 only three anions fit round the cation, below 0.414 four, up to 0.732 six, and above that eight.
CN = the neighbors a radius ratio of 0.5635 predicts
CN = 6
Step 3 · Find cell edge (a)
Use a = 2 × (r₊ + r₋)
Along an edge of rock salt an anion, a cation and an anion touch: two of each radius.
a
=
2
×
(102
+
181)
a
=
2
×
283
a
=
566
pm
Step 4 · Find density (ρ)
Use ρ = 4 × M ÷ (6.022 × 10²³ × (a × 10⁻¹⁰)³)
The mass of the cell’s 4 formula units over its volume, with the edge in cm.
ρ
=
4
×
58.44
÷
(6.022
×
10
23
×
(566 ×
10
-10
)
3
)
ρ
=
4
×
58.44
(6.022 ×
10
23
) × (1.8132 ×
10
-22
)
ρ
=
4
×
58.44
109.2
ρ
=
233.76
109.2
ρ
=
2.141
g/cm³
Check
Put the numbers back in. Both sides should match.
0.5635
=
102
181
✓
6 = the neighbors a radius ratio of 0.5635 predicts ✓
566 = 2 × (102 + 181) ✓
2.141
=
4
×
58.44
÷
(6.022
×
10
23
×
(566 ×
10
-10
)
3
)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Solid-state structures
Main lesson
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.”