Born–Landé Equation: Calculating Lattice Energy
Lattice energy is a fundamental concept in ionic bonding and solid-state chemistry. In competitive examinations such as IIT-JAM, GATE, and CSIR NET, questions often require calculating theoretical lattice energy using the Born–Landé equation. Unlike simple point-charge models, the Born–Landé equation balances long-range Coulombic attraction against short-range electron repulsion.
Definition and Sign Conventions
Lattice energy is expressed under two conventions across chemistry textbooks:
- Lattice formation energy: Energy released when one mole of an ionic crystal forms from gaseous ions at absolute zero (ΔU < 0 or ΔHlatt < 0). This is exothermic and negative.
- Lattice dissociation enthalpy: Enthalpy required to break one mole of an ionic crystal into gaseous ions at infinite separation (ΔHlatt > 0). Under IUPAC recommendations, this process is endothermic and positive.
Both conventions describe the same physical magnitude. In examinations, clarify whether your answer denotes binding energy (negative) or dissociation enthalpy (positive).
The Born–Landé Formula
Max Born and Alfred Landé expressed crystal potential energy by summing Coulombic attractions and adding an inverse-power repulsive term. Setting dU/dr = 0 at equilibrium separation r0 yields the Born–Landé formula:
Variables and SI Units:
- U0 = Molar lattice energy at 0 K (J·mol-1 or kJ·mol-1)
- NA = Avogadro constant = 6.022 × 1023 mol-1
- M = Madelung constant (dimensionless geometry factor)
- |z+|, |z-| = Formal ionic charges (dimensionless)
- e = Elementary charge = 1.602 × 10-19 C
- 1 / (4πε0) = Coulomb constant = 8.988 × 109 N·m2·C-2
- r0 = Equilibrium interionic separation = r+ + r- (m)
- n = Born exponent (dimensionless, typically 5 to 12)
- (1 - 1/n) = Born repulsion factor (dimensionless)
The bracketed term represents electrostatic attraction. The factor (1 - 1/n) accounts for short-range electron repulsion. For typical values of n (7 to 10), (1 - 1/n) equals 0.85 to 0.90, reducing electrostatic binding by 10% to 15%.
Madelung Constants and Born Exponents
The Madelung constant (M) reflects crystal packing geometry: Rock salt (NaCl) is 1.748; Caesium chloride (CsCl) is 1.763; Zinc blende (ZnS) is 1.638; Fluorite (CaF2) is 2.519.
Pauling values for Born exponent n depend on noble-gas configurations: [He] = 5, [Ne] = 7, [Ar] = 9, [Kr] = 10, [Xe] = 12. For compounds with different ions, use the arithmetic mean: n = (ncation + nanion) / 2.
Worked Numerical Examples
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: IIT-JAM, GATE, CSIR NET, CUET-PG
- Topic: Solid State Inorganic Chemistry — Lattice Energy and Born–Landé Equation
- Difficulty: Advanced
Question: Calculate the lattice energy of sodium chloride (NaCl) in kJ·mol-1 using the Born–Landé equation. Given: r0 = 2.820 × 10-10 m, M = 1.748, charges z+ = +1, z- = -1, and average Born exponent n = 8.00. Constants: NA = 6.022 × 1023 mol-1, e = 1.602 × 10-19 C, and 1 / (4πε0) = 8.988 × 109 N·m2·C-2.
Step-by-step solution:
Step 1: Compute e2 / (4πε0) = (1.602 × 10-19)2 × 8.988 × 109 = 2.3067 × 10-28 J·m.
Step 2: Calculate Coulombic attraction per mole.
Numerator = 6.022 × 1023 × 1.748 × 1 × 2.3067 × 10-28 = 2.4281 × 10-4 J·m·mol-1
Divide by r0 = 2.820 × 10-10 m:
Ucoulomb = (2.4281 × 10-4) / (2.820 × 10-10) = 8.610 × 105 J·mol-1 = 861.0 kJ·mol-1
Step 3: Apply Born repulsion factor.
(1 - 1 / n) = 1 - 1 / 8.00 = 0.875
U0 = - 861.0 × 0.875 = - 753.4 kJ·mol-1
Step 4: Check by second route.
Using molar factor C0 = (NA · e2) / (4πε0) = 138.91 kJ·nm·mol-1:
With r0 = 0.2820 nm, Ucoulomb = (138.91 × 1.748) / 0.2820 = 861.0 kJ·mol-1.
U0 = - 861.0 × 0.875 = - 753.4 kJ·mol-1. Both routes match.
Answer: -753.4 kJ·mol⁻¹ (or +753.4 kJ·mol⁻¹ for lattice dissociation enthalpy)
Question: Magnesium oxide (MgO) crystallizes in the rock salt lattice (M = 1.748) with r0 = 2.120 × 10-10 m. Both Mg2+ and O2- have neon electron configurations (n = 7.00). Calculate the lattice energy of MgO in kJ·mol-1.
Step-by-step solution:
Step 1: Calculate charge product.
|z+ · z-| = |(+2) · (-2)| = 4.
Step 2: Calculate Coulombic energy.
Numerator = 1.3891 × 10-4 × 1.748 × 4 = 9.7126 × 10-4 J·m·mol-1
Ucoulomb = (9.7126 × 10-4) / (2.120 × 10-10) = 4.5814 × 106 J·mol-1 = 4581.4 kJ·mol-1
Step 3: Apply Born repulsion factor.
(1 - 1 / n) = 1 - 1 / 7.00 = 6 / 7 ≈ 0.8571
U0 = - 4581.4 × (6 / 7) = - 3926.9 kJ·mol-1 ≈ - 3927 kJ·mol-1
Step 4: Check by second route (ratio comparison with NaCl).
Charge ratio = 4 / 1 = 4. Distance ratio = 2.820 / 2.120 = 1.330. Repulsion ratio = (6/7) / (7/8) = 0.980.
Predicted U0 = - 753.4 × (4 × 1.330 × 0.980) = - 3927 kJ·mol-1. Both routes match.
Answer: -3927 kJ·mol⁻¹ (or +3927 kJ·mol⁻¹ for lattice dissociation enthalpy)
Common Mistakes Students Make
- Forgetting unit conversions: Radii are often given in picometers (pm). 1 pm = 10-12 m. Leaving values in pm produces an answer off by twelve orders of magnitude.
- Confusing edge length (a) with interionic distance (r0): For rock salt, a = 2 · r0. For CsCl, a√3 = 2 · r0. Substituting cell edge a instead of r0 produces severe errors.
- Ignoring higher charges: For divalent salts like MgO, |z+ · z-| = 4. Omitting this factor underestimates lattice energy by four times.
- Miswriting the repulsion term: Writing (1 - n) instead of (1 - 1/n) gives a negative or huge non-physical factor.
Exam Relevance and Question Patterns
| Examination | Question Format | Key Concepts Tested |
|---|---|---|
| IIT-JAM Chemistry | Multiple Choice & Numerical Answer Type | Calculating lattice energy from ionic radii; lattice energy ratios; assigning Born exponents. |
| GATE Chemistry | Numerical Answer Type | Computation of lattice enthalpy in kJ/mol; extracting Born exponent n. |
| CSIR NET Chemical Sciences | Multiple Choice | Derivation of equilibrium condition dU/dr = 0; Born–Landé vs Born–Mayer models. |
| CUET-PG Chemistry | Multiple Choice | Identifying Madelung constants; periodic trends in lattice energy and solubility. |
Explore real crystal structures in the Crystal Lab, then test yourself on previous-year questions.
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