An equilibrium supercooled liquid’s entropy and enthalpy in the Kauzmann and the third law extrapolations, and a proposed experimental resolution

Abstract

In our current discussion of the thermodynamics and molecular kinetics of glass-forming liquids, the entropy is extrapolated below a liquid’s vitrification temperature Tg along a curve of progressively increasing slope until a temperature Tk is reached. Here the entropy and heat capacity, Cp, of the equilibrium liquid become equal to those of its crystal. Several observations have indicated fundamental difficulties with this extrapolation, thus suggesting the need for an alternative. We propose one alternative, in which Cp of an equilibrium liquid decreases along a sigmoid-shape path stretched over a broad temperature range from above Tg to 0 K. Its entropy and Cp become equal to those of its crystal at 0 K, as required by the third law of thermodynamics, and the enthalpy and volume remain higher. To elaborate, the available Cp data of 12 supercooled liquids have been interpolated between T>Tg and 0 K, and the enthalpy of their equilibrium state at 0 K, as well as the Gibbs free energy and enthalpy at T<Tg, determined. The enthalpy of the equilibrium liquid state at 0 K is 17%–37% of the enthalpy of melting, and for eight out of 12 liquids the Kauzmann extrapolation and our interpolation yield values within 5% of the average. Relative merits of the two resolutions of the entropy situation may be tested by the heat of solution, enthalpy loss and vapor pressure measurements of aged nonionic glasses and emf measurement of ionic glasses forming a half-cell of an electrochemical equilibrium. The anticipated enthalpy, Gibbs energy and vapor pressure change for the Kauzmann extrapolation of Cp and our interpolation are given at T<Tg for triphenylethene. As the equipment time needed for such measurements is only a few hours, such experiments may allow a study of the time-dependent thermodynamics of a glass more conveniently than other experiments.