Chemistry Ionic Bond Madelung Nacl Pos

The Madelung constant for salt

The Madelung constant is a pure number that captures how the geometry of a crystal lattice multiplies the electrostatic attraction between its ions. For sodium chloride, ordinary table salt, the constant is 1.748. The number appears in the formula for lattice energy, the energy released when gaseous ions condense into a solid crystal: energy equals the constant times a charge factor divided by the ion separation. A larger constant means a more tightly bound crystal.

The constant was introduced by Erwin Madelung in 1918 as a way to sum the infinite series of attractions and repulsions in an ionic crystal. In a salt crystal, each sodium ion is surrounded by six chlorine ions, which are surrounded by more sodium ions, and so on. The alternating signs make the sum converge slowly, and the value 1.748 is the result of that delicate cancellation. The same geometric idea extends to other structures: cesium chloride gives 1.763, and zinc blende gives 1.638.

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration named madelung_nacl_pos. The declaration proves, in the formal system, that the Madelung constant for sodium chloride is greater than zero. The proof is direct: the constant is defined as the decimal number 1.748, and the system verifies that this number is positive. The theorem is a small anchor in a larger derivation of ionic bonding, where the framework models electron transfer, shell closure, and lattice energy.

What the declaration does not claim is more interesting than what it proves. It does not derive the value 1.748 from first principles; the number is a definition, a model choice, not a theorem. It does not prove that salt actually forms ionic bonds, nor that the Madelung constant explains why sodium and chlorine react. Those claims live in other declarations, which connect the constant to electronegativity differences and lattice energy proxies. The positivity theorem is a sanity check: it confirms that the defined constant has the sign a physical lattice energy needs, nothing more.

The practical consequence is modest but real. A machine-checked library that builds chemistry from a few axioms must start somewhere, and a positive Madelung constant is a load-bearing floor. It tells a reader that the framework's model of salt is not accidentally inverted, that the electrostatic attraction is attractive. The deeper question, whether the framework's φ-derived stability values match measured lattice energies, remains an empirical check, not a proved theorem.

/-- The Madelung constant for NaCl is greater than 0. -/
theorem madelung_nacl_pos : madelungNaCl > 0 := by
  simp only [madelungNaCl]
  norm_num

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

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

What this page does not claim

The value 1.748 is not derived from first principles; it is a defined model constant. The positivity theorem does not prove that sodium chloride forms ionic bonds in nature. No claim is made that the framework's lattice energy model matches experimental measurements.

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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