Phase equilibria from the exactly integrated Clapeyron equation

Abstract

In this work, the exactly integrated form of the Clapeyron equation found by Mosselman et al. has been used in a systematic manner to derive a comprehensive set of equations describing the first-order transition curves of pure substances. The application of each of these equations requires the knowledge of only one (reference) point on the particular equilibrium line, of the corresponding enthalpy of transition, and some ancillary data (molar volumes and heat capacities of the phases at equilibrium). No fitting to ( p, T ) experimental data is needed. In this respect the equations developed here can be regarded as a source for calculating a priori the phase equilibrium curves. The results have been tested for a number of selected pure substances of variable molecular complexity, and the uncertainties attached to the calculations have been assessed. Empirical equations currently used for first-order transitions are compared with those obtained from the exact integration. As far as we are aware, no equation was previously proposed for solid + solid equilibrium lines.

The fusion curves of xenon, krypton, and argon

“…Not long ago, we showed [18] that by exactly integrating the Clapeyron equation the fusion curves of pure substances, whose volumes expand on melting as is the case for the low-pressure solid phases of the heavier rare gases, are of the form,…”

Effect of pressure on melting temperature of carbon dioxide

“…When Dp = p À p 0 and DT = T À T 0 are small, DH m (T, p) % DH m (T 0 , p 0 ) and DV m (T, p) % DV m (T 0 , p 0 ) have a minor error where p 0 denotes zero pressure and T 0 is the temperature under p 0 [13]. However, as Dp and DT increase, exact functions of DH m (T, p) and DV m (T, p) must be known [13].…”

“…Since both DH m (T, p) and DV m (T, p) are functions of temperature and pressure, and the necessary separation of variables cannot be accomplished in any direct and known manner, the integration of equation (1) is complicated and has been carried out through approximate methods ever since the equation was first established in the 19th century. Recently, an exactly integrated form of the Clapeyron equation has been used in a systematic manner to derive a comprehensive set of equations describing the first-order transition curves of pure substances that consider only fundamental thermodynamic properties [13]. When Dp = p À p 0 and DT = T À T 0 are small, DH m (T, p) % DH m (T 0 , p 0 ) and DV m (T, p) % DV m (T 0 , p 0 ) have a minor error where p 0 denotes zero pressure and T 0 is the temperature under p 0 [13].…”

“…Recently, an exactly integrated form of the Clapeyron equation has been used in a systematic manner to derive a comprehensive set of equations describing the first-order transition curves of pure substances that consider only fundamental thermodynamic properties [13]. When Dp = p À p 0 and DT = T À T 0 are small, DH m (T, p) % DH m (T 0 , p 0 ) and DV m (T, p) % DV m (T 0 , p 0 ) have a minor error where p 0 denotes zero pressure and T 0 is the temperature under p 0 [13]. However, as Dp and DT increase, exact functions of DH m (T, p) and DV m (T, p) must be known [13].…”

“…The classic Clapeyron equation governing all first-order phase transitions of pure substances may be useful to determine the temperature-pressure (T-p) curve theoretically in the following form [13]:…”