SME
Inorganic Chemistry · Problem type
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.”
This is an example. Type your own values to start a new problem.
U (kJ/mol)
=
−
N
A
×
A
×
z₊
×
z₋
×
e²
1
−
1
n
÷
(4πε₀
r₀ (pm)
)
Formulas
U
=
−1.389 ×
10
5
×
A
×
z₊
×
z₋
×
1 − 1 ÷
n
r₀
−753.2
=
−1.389 ×
10
5
×
1.7476
×
1
×
1
×
1 − 1 ÷ 8
282
Assumptions
•
z₊ and z₋ are the sizes of the charges (2 for Mg²⁺ and O²⁻); U is given off as the gaseous ions form the solid, so it is negative.
•
Madelung constant A by structure: rock salt 1.7476, cesium chloride 1.7627, zinc blende 1.6381.
•
Born exponent n by the ions’ noble-gas shells: He 5, Ne 7, Ar 9, Kr 10, Xe 12; average two different ions (NaCl: 8).
•
Point charges, purely ionic: NA = 6.022 × 10²³ mol⁻¹, e = 1.602 × 10⁻¹⁹ C, ε₀ = 8.854 × 10⁻¹² F/m.
Step-by-step
WE KNOW
A = 1.7476, z₊ = 1, z₋ = 1, r₀ = 282 pm, n = 8
FIND
lattice energy (U)
Step 1 · Find lattice energy (U)
Use U = −1.389 × 10⁵ × A × z₊ × z₋ × (1 − 1 ÷ n) ÷ r₀
The
constants
first:
NA
e
2
4πε₀
=
1.389 ×
10
5
kJ·pm/mol,
so
r₀
stays
in
pm.
Coulomb
attraction
summed
over
the
whole
crystal
by
A
,
less
the
share
1
n
the
ions’
repulsion
takes
back.
U
=
−1.389 ×
10
5
×
1.7476
×
1
×
1
×
1 − 1 ÷ 8
282
U
=
(−1.389
×
10
5
)
×
1.7476
×
1 − 0.125
282
U
=
(−1.389
×
10
5
)
×
1.7476
×
0.875
282
U
=
−242,741.64
×
0.875
282
U
=
−
212,398.935
282
U
=
−753.2
kJ/mol
Check
Put the numbers back in. Both sides should match.
−753.2
=
−1.389 ×
10
5
×
1.7476
×
1
×
1
×
1 − 1 ÷ 8
282
✓
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
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.”