Chemistry Van Der Waals Noble Gas Bp Increases Kr Xe

The krypton-xenon step

Noble gases are the right-hand column of the periodic table: helium, neon, argon, krypton, xenon, radon. They rarely form chemical bonds, yet their boiling points climb steadily down the column, from helium at about 4 kelvin to radon at about 211 kelvin. The standard explanation is that heavier atoms have more electrons spread over a larger volume, so temporary imbalances in their electron clouds are easier to induce. Those fleeting dipoles attract neighboring atoms, and the attraction grows with atomic size. The step from krypton to xenon, from about 120 kelvin to about 165 kelvin, is one link in that chain.

The Recognition Science library contains a formal, machine-checked proof of exactly this one comparison. The declaration noble_gas_bp_increases_kr_xe states that the boiling point assigned to krypton, 119.93 kelvin, is less than the boiling point assigned to xenon, 165.05 kelvin. The proof is a direct calculation over the library's own table of values, not a measurement and not a derivation from first principles. It establishes the ordering of two numbers already present in the code. The same library proves the full chain of five inequalities, from helium through radon, as a single theorem.

In Recognition Science, the framework models the ledger, a discrete record of recognition events, as the source of temporary asymmetries in electron distribution. The library defines a polarizability proxy that grows with the atom's period, and a London dispersion proxy that scales with the product of two such proxies divided by the sixth power of distance. The boiling point table is a definitional choice, a list of measured values entered into the formal system, not a consequence of the ledger. The proof that krypton boils before xenon therefore confirms the internal consistency of the table, not the physical mechanism behind it.

What the declaration does not claim is more interesting. It does not prove that polarizability causes the boiling point trend, because the table is an input, not an output. It does not establish the 1/r⁶ London dispersion law as a theorem about real atoms; that law is modeled by a proxy function. It does not even claim that the boiling point values are physically accurate, only that the numbers stored in the library satisfy the stated inequalities. The physical bridge from the ledger to electron clouds remains open, and this declaration does not cross it.

theorem noble_gas_bp_increases_kr_xe : nobleGasBoilingPoint 36 < nobleGasBoilingPoint 54 := by
  simp only [nobleGasBoilingPoint]
  norm_num

/-- Complete ordering of noble gas boiling points. -/
theorem noble_gas_bp_full_ordering :
    nobleGasBoilingPoint 2 < nobleGasBoilingPoint 10 ∧
    nobleGasBoilingPoint 10 < nobleGasBoilingPoint 18 ∧
    nobleGasBoilingPoint 18 < nobleGasBoilingPoint 36 ∧
    nobleGasBoilingPoint 36 < nobleGasBoilingPoint 54 ∧
    nobleGasBoilingPoint 54 < nobleGasBoilingPoint 86 := by
  refine ⟨?_, ?_, ?_, ?_, ?_⟩
  · exact noble_gas_bp_increases_he_ne
  · exact noble_gas_bp_increases_ne_ar
  · exact noble_gas_bp_increases_ar_kr
  · exact noble_gas_bp_increases_kr_xe
  · exact noble_gas_bp_increases_xe_rn

/-- Boiling points of noble gases (Kelvin). -/
def nobleGasBoilingPoint : ℕ → ℝ
| 2  => 4.22    -- He
| 10 => 27.07   -- Ne
| 18 => 87.30   -- Ar
| 36 => 119.93  -- Kr
| 54 => 165.05  -- Xe
| 86 => 211.4   -- Rn
| _  => 0

What this page does not claim

The declaration does not prove that polarizability causes the boiling point trend. The declaration does not establish the London dispersion law as a theorem about real atoms. The declaration does not assert the physical accuracy of the boiling point values.

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

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