Talk:Kauzmann paradox
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[edit] Possible Error?
This article seems like it could be a slight misinterpretation of Kauzmann's paradox.
RE: In thermodynamics, the Kauzmann Paradox is the apparent result that it is possible to obtain a supercooled liquid with an entropy lower than that of its corresponding crystal.
The latter scenario, although rare, is not impossible in nature and therefore cannot be defined as being paradoxical. For example, consider systems of hard spheres, which are known to crystallize at high pressures/low temperatures. This phenomenon implies that the free energy of the crystal is lower and hence the entropy of the crystal is higher than that of the liquid since there is no internal energy contribution for hard spheres. The crystal has more entropy because the additional free volume per particle gained by crystallizing increases the total number of states accessible to the system.
The author was correct to point that extrapolating the liquid entropy to zero temperature poses a problem, since the entropy becomes negative, which is impossible by definition. Kauzmann theorized that to avoid this scenario the rate of change of entropy with respect to temperature must suddenly change, which is precisely what happens at the glass transition, hence the paradox.
68.41.6.53 08:38, 2 September 2006 (UTC)A
[edit] Kauzmanns own explanation
If I remember correctly Kauzmann speculated that the free energy barrier of crystallisation could vanish at low temperatures. This means that the supercooled liquid would become unstable. The original article also mentions a Kauzmann paradox for volumes.
[edit] Where is the paradox
It is also regarded paradoxial that the liquid should have a lower configurational entropy than the crystal. At sufficiently low temperature a finite system will always be vibrating around a minimum of the potential energy function. We can introduce a socalled configurational entropy to describe the number of populated energy minima. This entropy is related to the heat capacities.
A perfect crystal has a configurational entropy of zero. This means that the ensemble only populates a single potential energy minimum.
Kauzmann predicted a temperature at which the configurational entropy liquid vanishes as well. This means that the supercooled liquid is trapped in the global energy minium (of the luquid subset of phase space). Many people regard this prediction as paradoxial or even wrong. See also
http://arxiv.org/abs/cond-mat/0212487
[edit] Please make a redirect
Please make a redirect of ideal glass transition temperature to this page. Then we can make a link to here from Glass (disambiguation)
