Chemistry Atomic Radii Na Larger Than Cl

The sodium-chlorine radius comparison

Atomic radius is the distance from an atom's nucleus to its outermost electron shell. In the periodic table, a clear trend emerges: within a single period, radii decrease as atomic number increases, because the added protons pull the electron cloud inward. Moving down a group, radii increase, as each new shell sits farther from the nucleus. Sodium (Na, atomic number 11) sits at the start of period 3, while chlorine (Cl, atomic number 17) sits near its end, so sodium's radius is the larger of the two.

The Recognition Science framework models this with a ledger, a discrete record of events, where each atomic shell is a rung on a ladder. Its library, a machine-checked collection of formal theorems, defines a normalized radius for each element: a value between 0 and 1 that measures how far an element sits within its period, from the start of one shell closure to the next. The library then proves, as a formal theorem, that this normalized radius for sodium is greater than that for chlorine. The proof is a direct calculation on the definitions, not an empirical measurement.

This theorem is a statement about the framework's own model. It confirms that the model reproduces the qualitative periodic trend for these two elements. It does not claim that sodium's actual measured atomic radius exceeds chlorine's in every experimental context, nor does it assign a physical size in nanometers. The framework's radius is a dimensionless proxy, built from shell number and a screening factor, not a direct measurement of electron cloud extent.

The framework's library also proves similar comparisons, such as lithium being larger than fluorine, and verifies that shell numbers increase down a group, as with potassium and lithium. These results serve as consistency checks: they show the model's definitions behave like the periodic table's known patterns. The sodium-chlorine theorem is one such check, a small but concrete confirmation that the framework's structure aligns with textbook chemistry.

/-- Na (Z=11) has larger radius than Cl (Z=17) in Period 3.
    Na: (1 - (11-10)/(18-10)) = 7/8
    Cl: (1 - (17-10)/(18-10)) = 1/8 -/
theorem na_larger_than_cl : normalizedRadius 11 > normalizedRadius 17 := by
  simp only [normalizedRadius, prevClosure, nextClosure]
  norm_num

/-- Atomic radius proxy: combines shell and screening effects.
    radius = shellRadius * screeningFactor -/
def radiusProxy (Z : ℕ) : ℝ :=
  shellRadiusProxy Z * screeningFactor Z

What this page does not claim

The theorem does not state that sodium's measured atomic radius exceeds chlorine's in every experimental context. The theorem does not assign a physical size in nanometers to any atom. The framework's radius model is not a direct measurement of electron cloud extent.

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

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