Chemistry Glass Transition Kauzmann Pos

The Kauzmann ratio

The Kauzmann ratio is the quotient of a liquid's glass transition temperature Tg and its melting temperature Tm, both measured in kelvin. For a wide range of glass-forming materials, this ratio clusters near 2/3. Silica and germanium dioxide sit close to the mark, while molecular liquids like o-terphenyl and many polymers fall below it. The ratio matters because it describes how far a liquid can be cooled below its melting point before it solidifies into a glass rather than a crystal.

The French physicist Walter Kauzmann drew attention to this region in 1948, when he pointed out the thermodynamic puzzle that a supercooled liquid's entropy would fall below that of its crystal if the liquid kept cooling slowly enough. The 2/3 ratio is a rough empirical observation about where many liquids actually give up and vitrify, not a law of thermodynamics. It is a convenient benchmark, not a precise constant: real materials spread around it, and the spread is part of what glass scientists study.

In the Recognition Science framework, the ratio is encoded as a definition and two small theorems. The definition kauzmannRatio sets the value to the rational number 2/3. The first theorem, kauzmann_pos, proves that this number is greater than zero. The second, kauzmann_lt_one, proves that it is less than one. Both proofs are immediate arithmetic: 2/3 is plainly between 0 and 1. The declarations do not derive the ratio from deeper principles; they assert it as a chosen model input and then check the two obvious bounds.

What the declarations do not claim is more interesting than what they prove. They do not show that any real material has a Kauzmann ratio of exactly 2/3, nor that the ratio is universal. They do not explain why the ratio sits near 2/3, and they do not connect it to the entropy crisis Kauzmann described. The framework's own docstring lists the 2/3 value as a prediction and names a falsifier: if measured Tg/Tm deviates significantly from 2/3, the prediction fails. The theorem kauzmann_pos is a statement about the number 2/3, not about any glass.

The value of the theorem is therefore modest but real. It records, in machine-checked form, that the framework's chosen benchmark is a positive number less than one, which is what a ratio of two positive temperatures must be. That is all. The reader who wants to know whether silica is a strong glass or whether o-terphenyl is fragile will find the fragility index bounds in the same file, defined as ranges from 16 to 30 for strong glasses and 100 to 200 for fragile ones. The positivity theorem itself settles nothing about any actual substance.

/-- Kauzmann ratio is positive. -/
theorem kauzmann_pos : kauzmannRatio > 0 := by
  simp only [kauzmannRatio]
  norm_num

/-- Kauzmann ratio is less than 1. -/
theorem kauzmann_lt_one : kauzmannRatio < 1 := by
  simp only [kauzmannRatio]
  norm_num

/-- Kauzmann ratio Tg/Tm ≈ 2/3. -/
def kauzmannRatio : ℝ := 2 / 3

What this page does not claim

The theorem does not prove that any real material has a Kauzmann ratio of exactly 2/3. It does not derive the 2/3 value from the framework's forcing chain or from thermodynamics. It does not address the entropy crisis or the thermodynamic puzzle Kauzmann raised.

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

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Derived articles

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