Chemistry Van Der Waals Noble Gas Bp Increases Xe Rn

The xenon-radon step

The noble gases, helium through radon, are the quiet elements of the periodic table's rightmost column. Their boiling points rise as atomic number increases: helium boils at 4.22 kelvin, neon at 27.07, argon at 87.30, krypton at 119.93, xenon at 165.05, and radon at 211.4 kelvin. The trend reflects the growing number of electrons, which makes larger atoms more easily distorted into temporary dipoles. Those fleeting dipoles create the weak attractions called van der Waals forces, and stronger attractions mean a liquid can hold together at higher temperatures.

The step from xenon to radon is the last and largest jump in the series, from 165.05 to 211.4 kelvin. A declaration in the framework's machine-checked library, named noble_gas_bp_increases_xe_rn, verifies that the recorded boiling point of radon is greater than that of xenon. The verification is a direct computation from the stored values: it unfolds the definition of nobleGasBoilingPoint and checks the inequality with exact arithmetic. This is a ledger, a discrete record of events, holding one entry per element, and the declaration confirms the ordering of two entries in that record.

The same library proves the full chain of inequalities, from helium through neon, argon, krypton, xenon, and radon, as a single combined result. The individual steps, including the xenon-radon one, are each established separately and then assembled. The framework also defines a polarizability proxy, a stand-in quantity that grows with the element's period, and a London dispersion proxy that scales with the product of two atoms' proxies and falls off as the sixth power of distance. The boiling point trend is presented as consistent with these definitions, but the ordering result itself does not depend on them: it is a fact about the recorded numbers.

What the declaration does not do is derive the boiling points from first principles. It takes the Kelvin values as given inputs, supplied by measurement, and proves only that they are ordered. It does not prove that van der Waals forces cause the trend, nor that the polarizability proxy is the true physical polarizability. Those are modeling choices recorded in the same file, not consequences of the ordering proof. The framework's library treats conventional chemistry as itself derived, but this particular declaration is a narrow, exact check on a table of numbers, not a derivation of the table.

theorem noble_gas_bp_increases_xe_rn : nobleGasBoilingPoint 54 < nobleGasBoilingPoint 86 := 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

/-- Polarizability proxy (increases with shell number). -/
def polarizabilityProxy (Z : ℕ) : ℝ :=
  (periodOf Z : ℝ)

What this page does not claim

The declaration does not derive the boiling points from first principles; it takes them as given inputs. The declaration does not prove that van der Waals forces cause the trend, nor that the polarizability proxy equals the true physical polarizability.

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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