Chemistry Ionic Bond Lattice Energy Increases With Charge

The charge rule

Ionic bonding is the electrostatic attraction between oppositely charged ions. When a metal atom loses an electron to a non-metal atom, the resulting positive and negative ions arrange into a crystal lattice. The lattice energy is the energy released when those ions come together from infinity to form the solid; it is what makes ionic compounds stable. The classical picture, taught in every introductory chemistry course, is that this energy grows with the charges on the ions and shrinks as the ions get farther apart.

The standard formula, the Born-Landé equation, expresses this as a balance between an attractive Coulomb term and a short-range repulsive term. The attraction scales directly with the product of the ion charges divided by the distance between them. So a 2+ ion paired with a 1- ion should bind more strongly than a 1+ ion paired with a 1- ion, at the same separation. This is why magnesium oxide melts at 2852 °C while sodium chloride melts at 801 °C: the doubled charge on the cation roughly quadruples the dominant Coulomb contribution.

In Recognition Science, the framework models this relationship explicitly. Its machine-checked library of formal theorems defines a proxy for lattice energy as the product of the two ion charges divided by a distance proxy, and then proves a result named lattice_energy_increases_with_charge: for any positive distance, the proxy for a 1+ / 1- pair is strictly less than the proxy for a 2+ / 1- pair. The proof is a direct consequence of the definition; the kernel checks that multiplying the charge by two doubles the numerator while the denominator stays fixed.

What the result does not do is predict a numerical lattice energy for any real compound. The proxy omits the Madelung constant, which encodes the geometric arrangement of ions, and it omits the repulsive Born exponent that prevents the lattice from collapsing. The framework's library does define those quantities for specific structures, such as sodium chloride with a Madelung constant of 1.748, but the charge-increase result itself does not invoke them. It establishes a monotonicity fact, not a quantitative prediction.

The consequence for a reader is simple: within the framework's model, the textbook rule that higher charge means stronger ionic bonding is not an assumption but a proved statement. The framework derives it from the same electrostatic definition that classical chemistry uses, and the machine-checked proof guarantees the inference is valid. What remains outside the result is the full physical picture, where geometry and repulsion modify the simple inverse-distance scaling.

/-- Lattice energy proxy increases with ionic charge product.
    U(1,1) < U(2,1) since 1/d < 2/d for d > 0. -/
theorem lattice_energy_increases_with_charge (d : ℝ) (hd : d > 0) :
    latticeEnergyProxy 1 1 d < latticeEnergyProxy 2 1 d := by
  simp only [latticeEnergyProxy]
  have hd_pos : ¬(d ≤ 0) := not_le.mpr hd
  simp only [hd_pos, ite_false]
  -- 1/d < 2/d when d > 0
  have h1 : (1 : ℤ).toNat = 1 := rfl
  have h2 : (2 : ℤ).toNat = 2 := rfl
  simp only [h1, h2]
  -- 1 * 1 / d < 2 * 1 / d when d > 0
  have : (1 : ℝ) * 1 / d < 2 * 1 / d := by
    apply div_lt_div_of_pos_right _ hd
    norm_num
  simpa using this

/-- The Born-Landé equation for lattice energy (dimensionless proxy).
    U ∝ M * z+ * z- / r₀ * (1 - 1/n)
    where M is the Madelung constant, z are charges, r₀ is interionic distance,
    and n is the Born exponent.
    We use a simplified proxy: U ∝ charge_product / distance_proxy
-/
def latticeEnergyProxy (charge1 charge2 : ℤ) (distanceProxy : ℝ) : ℝ :=
  if distanceProxy ≤ 0 then 0
  else
    (charge1.toNat * charge2.toNat : ℝ) / distanceProxy

/-- Madelung constant for NaCl structure (rock salt). -/
def madelungNaCl : ℝ := 1.748

What this page does not claim

The theorem does not predict a numerical lattice energy for any real compound. The theorem does not incorporate the Madelung constant or the Born repulsive exponent. The framework does not claim that the 2+ / 1- ratio of lattice energies is exactly two.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/IonicBond.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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