§ C-3 Ionic Crystal Lattice & Lattice Energy (NaCl) (Grade 8–11 · Chemistry)
Rock-salt unit cell, Madelung constant, lattice energy & thermal phonons. In an ionic crystal, every cation is octahedrally coordinated by six anions, forming a continuous three-dimensional electrostatic lattice.
Theory Summary & Scientific Mechanism
Crystalline solids possess long-range periodic order across a 3D lattice. In ionic crystals (such as NaCl, face-centered cubic rocksalt lattice with 6:6 coordination), cations and anions are bound by omnidirectional electrostatic attraction. In covalent network crystals (diamond, quartz SiO₂), atoms form a continuous 3D scaffold of rigid directional bonds (sp³ tetrahedra in diamond). In metallic crystals, positive ion cores are immersed in a mobile, delocalized Fermi sea of conduction electrons.
The cohesive energy of an ionic crystal is given by the Born–Landé equation: U₀ = -(N_A M |z₊ z₋| e²)/(4πε₀ r₀)≤ft(1 - (1)/(n_B) ), where M is the Madelung constant capturing the geometric lattice sum of electrostatic interactions (M = 1.7476 for NaCl), r₀ is the equilibrium interionic distance, and n_B ≈ 8–12 characterizes short-range Born repulsion between closed electron shells.
Increasing absolute temperature T amplifies the root-mean-square thermal vibration amplitude of lattice sites langle u²rangle ∝ (k_B T)/(k_bond). By Lindemann’s melting criterion, when the thermal vibration amplitude reaches ≈ 10–15% of the interatomic spacing r₀, long-range order breaks down and the crystal melts. Hence crystals with high ionic charges and short bond lengths (MgO, Al₂O₃, diamond) exhibit exceptionally high melting points.
Everyday Analogy
Table salt NaCl forms tiny cubes because Na⁺ and Cl⁻ ions alternate in a 3D checkerboard! In diamond, every carbon atom locks onto four neighbors in a rigid 3D tetrahedral network, making diamond the hardest natural material.
Interactive Question & Derivation
How do the crystal lattice energy |Uₗat| and melting point of an ionic solid change when ionic charges (z⁺, z⁻) increase or interionic distance r₀ decreases?
- A: Lattice energy |Uₗat| ∝ (z⁺ z⁻)/(r₀) increases dramatically, raising crystal hardness and melting temperature (e.g., MgO vs. NaCl)
- B: Lattice energy decreases and the crystal sublimes at room temperature
- C: Lattice energy is independent of ionic charges and radii
Study Hint: Look at the Born–Landé lattice energy relation U₀ ∝ (|z₊ z₋|)/(r₀): what happens to electrostatic attraction when ionic charges double (e.g. from Na⁺Cl⁻ to Mg²⁺O²⁻) and interionic distance r₀ decreases?
Proof & Explanation: By the Born–Landé equation, ionic lattice energy is directly proportional to the charge product z⁺ z⁻ and inversely proportional to the interionic distance r₀ = r₊ + r₋. Moving from monovalent Na⁺Cl⁻ (Uₗat ≈ 787 kJ/mol, Tₘ = 801 ^°C) to divalent Mg²⁺O²⁻ nearly quadruples lattice energy (3790 kJ/mol, Tₘ = 2852 ^°C).
Key Equations & 3D Simulation Parameters
|Uₗat| = (1389 · M · z⁺ z⁻)/(r₀(pm))≤ft(1 - (1)/(n) ) kJ/mol
- Interionic distance r₀ (r₊ + r₋): 2.1 … 3.5 Å (default: 2.82 Å)
- Ion charge magnitude |z±| (1=NaCl, 2=MgO): 1 … 2 e (default: 1 e)
- Crystal temperature T: 100 … 1000 K (default: 300 K)
- Born repulsion exponent n: 6 … 12 (default: 9)