Abstract
The Kauzmann paradox is spoken of as a property of supercooled liquids. This paper shows that crossing zero follows from holding the heat-capacity difference constant—held constant, however small, it always crosses. No new theorem or law is claimed.
First, the entropy difference is $+4.530353$ at $T=250$ and $-2.163953$ at $T=200$ (Section 2).
Second, it crosses zero at $\mathbf{214.959393}$, or $\mathbf{0.716531}$ of the melting point (Section 2).
Third, that matches the closed form $\exp(-\Delta S_{m}/\Delta C_{p})$ at $\mathbf{0.716531310574}$ (Section 2).
Fourth, this is the core of the paper. Lowering the difference to $1$ still crosses, at $T_{K}=\mathbf{1.36\times10^{-2}}$—only the crossing temperature falls (Section 3).
Fifth, let the difference vanish with temperature and it crosses only when the total exceeds $\Delta S_{m}$ (Section 3).
Sixth, the separator is whether the heat-capacity difference was held constant or made to vanish (Section 4).
Keywords: Kauzmann paradox, glass transition, supercooled liquid, entropy, heat capacity.
Scope (what is not claimed)
No new theorem or law is claimed—the Kauzmann paradox and the candidate resolutions are standard. No theory of glasses—whether an ideal glass transition exists, or dynamic arrest suffices, is not the subject. No real substance—a melting point of $300$, melting entropy of $10$ and heat-capacity difference of $30$ are model values, not measurements of any material. The temperature dependence was not measured—only two cases are compared: held constant, and vanishing in proportion to $T$. The real dependence differs by substance. The third law is not proved—that heat capacities vanish as $T\to0$ is used as a premise, not derived. Relation to earlier papers: Paper 352 showed that the glass transition temperature moves with the speed of measurement—that is the kinetic side; this is the thermodynamic one. Paper 498 showed continuity in composition and discontinuity in identity—both are cases where entropy shows a different face depending on which variable is moved. Paper 481 showed that a refutation can be blamed on any assumption—“the entropy goes negative” can be blamed on the extrapolation. Paper 300 showed that a shared root can be decided—holding constant and letting vanish are distinct roots. What is added is solving $T_{K}=214.959393$ by bisection, confirming the closed form to twelve digits, showing that lowering the difference to $30$, $5$ and $1$ still crosses, showing that with a vanishing difference the total decides, and placing the separator in the extrapolation.
1. Notation and the Question
Setting: A liquid that gained a melting entropy $\Delta S_{m}$ at $T_{m}$ is cooled without crystallising. With $\Delta C_{p}$ for the heat-capacity difference between liquid and crystal, the entropy difference at $T$ is
$$\Delta S(T)=\Delta S_{m}-\int_{T}^{T_{m}}\frac{\Delta C_{p}}{T'}\,dT'$$and holding $\Delta C_{p}$ constant gives $\Delta S(T)=\Delta S_{m}-\Delta C_{p}\ln(T_{m}/T)$. Below, $T_{m}=300$, $\Delta S_{m}=10$ and $\Delta C_{p}=30$, all of the order of J/(mol K).
Observation 1.1 The Kauzmann paradox is spoken of as a property of supercooled liquids. Cool far enough and the liquid’s entropy falls below the crystal’s—a picture of matter behaving that way.
Observation 1.2 But the crossing lies outside the measured range. Measurement stops at the glass transition—beyond it is extrapolation, and extrapolation needs assumptions.
Question: Does the crossing come from the substance, or from the assumption.
2. The Extrapolation Crosses Zero
Computation 2.1: Tabulate $\Delta S(T)$.
| Temperature $T$ | $\Delta S(T)$ |
|---|---|
| $300.0$ | $+10.000000$ |
| $250.0$ | $+4.530353$ |
| $214.0$ | $\mathbf{-0.134194}$ |
| $200.0$ | $-2.163953$ |
| $150.0$ | $-10.794415$ |
Observation 2.2 The sign changes between $214$ and $250$. From $+4.530353$ to $-0.134194$—the boundary is bracketed.
Computation 2.3: Bisection gives $T_{K}=\mathbf{214.959393}$, or $\mathbf{0.716531}$ of the melting point. Recomputing $\Delta S$ there returns $0.000000000000$.
Observation 2.4 It matches the closed form. $0.716531310574$ against $\exp(-\Delta S_{m}/\Delta C_{p})=0.716531310574$—to twelve digits: an independent check.
Observation 2.5 It matches what is reported. For many glass formers $T_{K}/T_{m}$ is about $0.7$—the model’s $0.7165$ lies in that range.
Observation 2.6 Below $T_{K}$ it is negative. $-3.160815$—the liquid’s entropy falls below the crystal’s, colliding with the third law.
Observation 2.7 Above $T_{K}$ it is positive. $+2.859305$—the other side of the boundary, so $T_{K}$ really is where the sign turns.
3. It Crosses Because the Difference Was Held Constant
Computation 3.1: Lower $\Delta C_{p}$ and see where $T_{K}$ goes.
| $\Delta C_{p}$ | $T_{K}$ | As a fraction of $T_{m}$ |
|---|---|---|
| $30.0$ | $\mathbf{2.149594\times10^{2}}$ | $7.165\times10^{-1}$ |
| $5.0$ | $4.060058\times10^{1}$ | $1.353\times10^{-1}$ |
| $1.0$ | $\mathbf{1.361998\times10^{-2}}$ | $4.540\times10^{-5}$ |
Observation 3.2 A smaller difference still crosses. At $\Delta C_{p}=1$, $T_{K}=1.36\times10^{-2}$—only the crossing temperature falls.
Observation 3.3 The reason is that the logarithm diverges. $\Delta C_{p}\ln(T_{m}/T)$—it grows without bound as $T\to0$, so any finite $\Delta S_{m}$ is eventually exceeded.
Observation 3.4 So “it must cross” follows from holding the difference constant. Not from the substance—the assumption contains the conclusion.
Computation 3.5: Let $\Delta C_{p}(T)=\Delta C_{p,0}(T/T_{m})$, as the third law requires, and the total lost down to $T=0$ saturates at $\Delta C_{p,0}$.
| $\Delta C_{p,0}$ | Total lost | Against $\Delta S_{m}=10$ |
|---|---|---|
| $30.0$ | $30.000000$ | crosses |
| $5.0$ | $\mathbf{5.000000}$ | does not cross |
Observation 3.6 A vanishing difference need not cross. $5.000000<10$—it crosses only when the total exceeds $\Delta S_{m}$.
Observation 3.7 And the third law requires heat capacities to vanish as $T\to0$. They cannot stay constant—the extrapolation’s assumption is incompatible with the third law itself.
4. The Separator
The separator: was the heat-capacity difference held constant, or made to vanish as the third law requires.
Observation 4.1 Held constant, it must cross. The logarithm diverges—the paradox is built into the extrapolation.
Observation 4.2 Made to vanish, the total decides. $30$ crosses, $5$ does not—the answer may differ by substance.
Observation 4.3 The two are distinct roots by the criterion of Paper 300. Both may be called “the Kauzmann temperature”—one a number the assumption always produces, one a number the substance decides.
Observation 4.4 Paper 352 did the same thing on the kinetic side. The glass transition temperature moves with the speed of measurement—there, “written like a material constant but not constant”; here, “written like a property of matter but a consequence of an assumption”.
Observation 4.5 It neighbours Paper 498. Entropy showing a different face depending on which variable is moved—there composition against identity, here temperature against the assumption.
Observation 4.6 This is one of the separators of Paper 481. When “the entropy goes negative” arrives as a refutation—whether the third law failed or the extrapolation was poor sends the repair elsewhere.
Observation 4.7 To be honest—the real temperature dependence was not measured. Only constant and linearly vanishing were compared—the real dependence differs by substance, and knowing its form would settle whether $T_{K}$ exists. This paper does not have that form.
5. Roll Call and Closing
- $+4.530353$ at $T=250$ (Computation 2.1): $-2.163953$ at $T=200$.
- $T_{K}=214.959393$ (Computation 2.3): $0.716531$ of $T_{m}$.
- Matching the closed form to twelve digits (Observation 2.4): $\exp(-\Delta S_{m}/\Delta C_{p})$.
- Matching the reported $0.7$ (Observation 2.5): the model gives $0.7165$.
- $\Delta C_{p}=1$ still crosses (Computation 3.1): the core.
- The logarithm diverges (Observation 3.3): any $\Delta S_{m}$ is exceeded.
- A vanishing difference lets the total decide (Computation 3.5): $5<10$ does not cross.
- The third law requires vanishing (Observation 3.7): incompatible with the assumption.
- The separator is constant or vanishing (Observation 4.3): distinct roots by Paper 300.
- The real dependence is unmeasured (Observation 4.7): that form is not held here.
Closing: The Kauzmann paradox is spoken of as a property of supercooled liquids—cool far enough and the liquid’s entropy falls below the crystal’s. But the crossing lies outside the measured range: measurement stops at the glass transition, and beyond it is extrapolation, which needs assumptions. The extrapolation crosses zero—with $T_{m}=300$, $\Delta S_{m}=10$ and $\Delta C_{p}=30$, $\Delta S$ runs $+10.000000$, $+4.530353$, $\mathbf{-0.134194}$ and $-10.794415$ at $T=300$, $250$, $214$ and $150$, changing sign between $214$ and $250$. Bisection gives $T_{K}=\mathbf{214.959393}$, or $\mathbf{0.716531}$ of the melting point, and recomputing there returns $0.000000000000$. The ratio matches the closed form $\exp(-\Delta S_{m}/\Delta C_{p})=0.716531310574$ to twelve digits. Reported values of $T_{K}/T_{m}$ for glass formers are about $0.7$, and the model’s $0.7165$ lies in that range. Below $T_{K}$ it is $-3.160815$, colliding with the third law; above it, $+2.859305$. It crosses because the difference was held constant—lowering $\Delta C_{p}$ to $30$, $5$ and $1$ gives $T_{K}=2.149594\times10^{2}$, $4.060058\times10^{1}$ and $\mathbf{1.361998\times10^{-2}}$. A smaller difference still crosses; only the crossing temperature falls. The reason is that $\Delta C_{p}\ln(T_{m}/T)$ grows without bound as $T\to0$, so any finite $\Delta S_{m}$ is eventually exceeded. So “it must cross” follows from holding the difference constant, not from the substance: the assumption contains the conclusion. Letting $\Delta C_{p}(T)=\Delta C_{p,0}(T/T_{m})$ as the third law requires, the total lost saturates at $\Delta C_{p,0}$, so $30$ crosses while $\mathbf{5}$ does not. And since the third law requires heat capacities to vanish as $T\to0$, the extrapolation’s assumption is incompatible with the third law itself. One thing separates them—was the heat-capacity difference held constant, or made to vanish. Held constant, the logarithm diverges and it must cross: the paradox is built into the extrapolation. Made to vanish, the total decides—$30$ crosses, $5$ does not—and the answer may differ by substance. The two are distinct roots by the criterion of Paper 300: both may be called “the Kauzmann temperature”, but one is a number the assumption always produces and one a number the substance decides. Paper 352 did the same on the kinetic side: there “written like a material constant but not constant”, here “written like a property of matter but a consequence of an assumption”. It neighbours Paper 498, where entropy showed a different face depending on which variable was moved—there composition against identity, here temperature against the assumption. This is one of the separators of Paper 481: when “the entropy goes negative” arrives as a refutation, whether the third law failed or the extrapolation was poor sends the repair elsewhere. To be honest—the real temperature dependence was not measured. Only constant and linearly vanishing were compared, while the real dependence differs by substance; knowing its form would settle whether $T_{K}$ exists, and this paper does not have that form.
References
- Angell, C. A. (1995). Formation of Glasses from Liquids and Biopolymers. Science.
- Kauzmann, W. (1948). The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures. Chemical Reviews.
- Stillinger, F. H., Debenedetti, P. G., & Truskett, T. M. (2001). The Kauzmann Paradox Revisited. Journal of Physical Chemistry B.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Glass Transition Temperature Moves with the Speed of Measurement [Paper 352]. Zenodo. 10.5281/zenodo.22695723
- Yamagishi, Y. (2026). A Broken Law Can Be Blamed on Any of the Assumptions [Paper 481]. Zenodo. 10.5281/zenodo.22733927
- Yamagishi, Y. (2026). Continuous in Composition, Discontinuous in Whether the Species Are the Same [Paper 498]. Zenodo. 10.5281/zenodo.22760218
Reference
Yamagishi, Y. (2026). The Extrapolation Crosses Zero Because the Heat-Capacity Difference Was Held Constant [Paper 503]. YOI: https://www.yamagishiyuuki.com/yoi/503
DOI: 10.5281/zenodo.22861694
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 503. This paper is a preprint and has not been peer reviewed.