Chemistry Van Der Waals Noble Gas Bp Increases He Ne

The noble gas boiling point theorem

Noble gases are the elements in group 18 of the periodic table: helium, neon, argon, krypton, xenon, and radon. Their boiling points rise steadily down the group, from helium at 4.22 kelvin to radon at 211.4 kelvin. The cause is the van der Waals force, a weak attraction between atoms that grows with atomic size. Larger atoms have more electrons, which can shift into temporary dipoles, creating stronger dispersion forces that hold the liquid together at higher temperatures. This trend is a standard result of physical chemistry, taught in every introductory course.

The Recognition Science library, a machine-checked collection of formal theorems, now contains a proof of the first step in this trend. The declaration noble_gas_bp_increases_he_ne proves that the boiling point of helium is less than the boiling point of neon, using the values 4.22 and 27.07 kelvin respectively. The proof is a direct computation: it expands the definition of the boiling point function and verifies the inequality with numerical arithmetic. Similar theorems prove each subsequent step, and a combined theorem assembles them into a full ordering: He < Ne < Ar < Kr < Xe < Rn.

What this theorem does not do is explain why the boiling points have these values. It takes the boiling points as given data, encoded in a function that maps atomic numbers to kelvin values. The theorem only checks that the list is strictly increasing. It does not derive the numbers from more fundamental principles, and it does not prove that the trend holds for all elements or under all conditions. The mechanism, the increasing polarizability of larger atoms, is described in the library's notes as a model, not as a theorem.

The framework also includes a separate, approximate connection between the Lennard-Jones potential minimum and the golden ratio. The minimum distance ratio 2^(1/6) ≈ 1.122 is within 0.01 of phi − 0.5 ≈ 1.118. This is a numerical observation, not a proved equality, and it does not affect the boiling point theorem. The boiling point proof stands on its own as a simple, verified fact about the listed values.

/-- Noble gas boiling points increase down the group (vdW strength increases). -/
theorem noble_gas_bp_increases_he_ne : nobleGasBoilingPoint 2 < nobleGasBoilingPoint 10 := 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

What this page does not claim

The theorem does not derive the boiling point values from more fundamental principles. The theorem does not prove that the trend holds for all elements or under all conditions. The phi connection is an approximate numerical observation, not a proved equality.

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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