frequencies above a threshold called the Frenkel frequency. Since most energy is contained in these high-frequency modes, a simple modification of the Debye model is sufficient to yield a good approximation to experimental heat capacities of simple liquids. Because of high crystal binding energies, the effects of vibrational mode freezing are observed in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3R per mole of atoms of the Dulong–Petit theoretical maximum. Amorphous materials can be considered a type of liquid at temperatures above the glass transition temperature. Below the glass transition temperature amorphous materials are in the solid (glassy) state form. The specific heat has characteristic discontinuities at the glass transition temperature which are caused by the absence in the glassy state of percolating clusters made of broken bonds (configurons) that are present only in the liquid phase. Above the glass transition temperature percolating clusters formed by broken bonds enable a more floppy structure and hence a larger degree of freedom for atomic motion which results in a higher heat capacity of liquids. Below the glass transition temperature there are no extended clusters of broken bonds and the heat capacity is smaller because the solid-state (glassy) structure of amorphous material is more rigid. The discontinuities in the heat capacity are typically used to detect the glass transition temperature where a supercooled liquid transforms to a glass. Effect of hydrogen bonds Hydrogen-containing polar molecules like ethanol, ammonia, and water have powerful, intermolecular hydrogen bonds when in their liquid phase. These bonds provide another place where heat may be stored as potential energy of vibration, even at comparatively low temperatures. Hydrogen bonds account for the fact that liquid water stores nearly the theoretical limit of 3R per mole of atoms, even at relatively low temperatures (i.e. near the freezing point of water). See also Quantum statistical mechanics Heat capacity ratio Statistical mechanics Thermodynamic equations Thermodynamic databases for pure substances Heat equation Heat transfer coefficient Heat of mixing Latent heat Material properties (thermodynamics) Joback method (Estimation of heat capacities) Specific heat of melting (Enthalpy of fusion) Specific heat of vaporization (Enthalpy of vaporization) Volumetric heat capacity Thermal mass R-value (insulation) Storage heater Frenkel line References Physical quantities Thermodynamic properties