Coulomb's Law and Lattice Energy

Key Ideas

• Coulomb's law: F = k·q₁q₂/r² — opposite charges attract, and the force grows with charge and falls with distance.
• Lattice energy (U): the energy released when 1 mole of an ionic solid forms from its gaseous ions (or equivalently, the energy required to separate 1 mole of the solid into gaseous ions). It is always reported as a positive quantity.
• Trend 1 — charge: lattice energy increases as the ion charges increase (Mg²⁺O²⁻ ≫ Na⁺Cl⁻).
• Trend 2 — size: lattice energy increases as the ions get smaller (closer charges attract more strongly).
• Qualitative form: U ∝ (Q₁·Q₂)/r, where Q are the ion charges and r is the distance between ion centers (sum of the two ionic radii).
• Born–Haber cycle: an energy cycle (Hess's law) that combines ionization energy, electron affinity, sublimation, and bond dissociation energies to compute lattice energy indirectly.

Equations and Variables

• Coulomb's law: F = k·(q₁·q₂)/r²
  • F = electrostatic force (N)
  • q₁, q₂ = charges (C)
  • r = distance between charges (m)
  • k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant)
• Lattice energy (qualitative): U ∝ (Q₁·Q₂)/r
  • Q₁, Q₂ = magnitudes of the ion charges
  • r = interionic distance (sum of the ionic radii)

How It Works

1. In an ionic solid, every cation is surrounded by anions and vice versa; each pair interacts by Coulomb's law.
2. Doubling the charges (e.g., Na⁺Cl⁻ vs. Mg²⁺O²⁻) quadruples each pairwise attraction (2 × 2 = 4), so the lattice energy rises sharply.
3. Shrinking the ions moves the opposite charges closer together, decreasing r and increasing the attraction.
4. Lattice energy is therefore large for small, highly charged ions (MgO, Al₂O₃) and small for large, singly charged ions (CsI, KBr).
5. Because lattice energy is difficult to measure directly, it is computed with a Born–Haber cycle that sums the energies of the steps (sublimation, ionization, bond dissociation, electron affinity) that convert elements into gaseous ions, then uses Hess's law with the measured enthalpy of formation.

Worked Example

Predict which has the larger lattice energy — NaCl or MgO — and explain.

• Charges: Na⁺/Cl⁻ are 1+/1−, while Mg²⁺/O²⁻ are 2+/2−. The charge product is 1 vs. 4, so the Coulombic attraction is about four times stronger in MgO.
• Sizes: Mg²⁺ (72 pm) and O²⁻ (140 pm) are also smaller than Na⁺ (102 pm) and Cl⁻ (181 pm), so r is smaller in MgO.

Both factors favor MgO, so MgO has the much larger lattice energy — measured values are about 769 kJ/mol for NaCl versus ~3795 kJ/mol for MgO.

Second check — rank lattice energy: LiF > NaF > KBr > CsI. LiF has the smallest ions (short r) with 1+/1− charges; as the ions grow down the group, r increases and the lattice energy falls.

Common Confusions

• "Lattice energy is small for highly charged ions." — Wrong: it increases with charge; Mg²⁺O²⁻ has a much larger lattice energy than Na⁺Cl⁻.
• "Bigger ions make stronger lattices." — Wrong: larger ions are farther apart (bigger r), so the lattice energy is smaller.
• "Lattice energy is the energy to break one bond." — It is the energy for a whole mole of the crystal, not a single ion pair.
• "Coulomb's law applies only to a single pair of ions." — It describes each pair, but lattice energy sums all the pairwise interactions across the entire crystal.