Some applications of thermodynamics in crystal chemistry

Abstract

The integrated study of polymorphic phase changes in crystals by combining thermodynamic and crystallographic data is illustrated for four one-component systems — tin, carbon, adamantane and fullerene C 60— I h — of increasing complexity. A brief review of the basic thermodynamics of the solid state (laws of thermodynamics, polymorphism, order of transitions) reminds the reader that enthalphy ( H) and entropy ( S) are experimental quantities derivable from the values of the heat capacity at constant pressure ( C p ) measured as a function of temperature ( T). Combination of H and S through the Gibbs function gives the free energy as a function of T (G = H − TS). For a one-component crystal with two polymorphic forms, the separate roles of the enthalpy and entropy differences between the polymorphs can be distinguished. This is illustrated in quantitative fashion for the two polymorphs of tin at atmospheric pressure, where there is a first-order phase transformation from grey (diamond) to white (metallic) at 286 K. Application of the Clausius-Clapeyron equation (d P/d T = Δ S/Δ V) shows that metallic tin is the stable phase above ≈ 5 kbar at 0 K. This is entirely analogous to the treatment of the pressure dependence of the melting point of ice. In the next stage, pressure is added as a variable in order to treat the graphite-diamond polymorphism of carbon. Adamantane, for which calorimetric, phase-diagram and crystallographic data are available over a considerable range of temperature and pressure, is next reviewed in detail. The polymorphic change from tetragonal (stable below 208.6 K at atmospheric pressure) to cubic is almost, but not quite, first-order in character. A somewhat similar analysis is applied to available calorimetric and crystallographic data for fullerene C 60— I h, where the transition shows appreciable deviations from ideal first-order character. The next stage, not considered here, would be to proceed from the methods of thermodynamics to those of statistical mechanics, involving consideration of the detailed vibrational spectra of the solids.