Solubility of solids in supercritical fluids is core to applications of extraction, material formation and reactions. Phase equilibrium relationships of two forms are also core. For liquids, supercritical fluids dissolve into the liquid while at the same time the liquid compound dissolves into the supercritical fluid phase. Other forms of phase equilibria situations include settings in which a solute is absorbed or adsorbed into another phase (e.g. a solute dissolved into plastics or soil organic matter).
Extensive solid solubility data has been generated by the community over the last several decades. In 2007, Gupta and Shim published a compilation of solubility data for some 780 solutes. Many solutes have been measured more than once but few have been measured for a full range of temperature and pressure conditions. Many more solutes have published measurements after 2007. This data set is impressive but just scratches the surface.
The tunability of supercritical fluids is a key advantage but it means extensive data measurements are required. Solubility is a function of the supercritical fluid, temperature, pressure, cosolvents and the presence of other solutes. Empirical relationships have been introduced to help capture this data space. Empirical relations permit a degree of interpolation and potentially modest extrapolation with associated risk.
Chrastil (1982) offers one of the simplest and most widely used empirical relationships for the solid solubility as a function of temperature and density.
Chrastil’s relationship has a theoretical grounding in addition to reflecting the observed relationship between solubility and density on a log scale. Chrastil’s original form treated solubility in units of g/L; however, most published solubility data is in units of mol/mol. Chrastil’s relationship works equally well using with the solubility in mol/mol with the only difference being the magnitude of the coefficients.
Although Chrastil has been widely used and there are numerous publications with Chrastil coefficients, there are two challenges. The Chrastil coefficients are dependent on the source of the density values used. Published articles have used several different sources for density values (different PvT relationships) and some articles have not stated which PvT relationship has been used. Using Chrastil coefficients with a different PvT (than their original source) creates a discontinuity that can be severe.
The second challenge is associated with the fitting procedures employed. The original Chrastil article identifies solving for the coefficients by regressing a subset of the data (isotherm data) to determine the coefficient C (log-log slope) and then determining the coefficients A and B from the the isotherm intercepts. The resulting fitting performance is then reported as an Absolute Average Relative Deviation (AARD). Regression would typically be completed using a least squares method which is inconsistent with an AARD weighting and the isotherm regression on a log scale will add to the discontinuity as a log scale weights individual points different from a linear scale. Development of multiple variable regression techniques in tools such as Excel and Matlab would be typically executed on a linearization of the equation (log solubility and inverse T). These multiple variable regression techniques are also least squares based. The resulting Chrastil parameters are not consistent with a minimum AARD value. The resulting Chrastil parameters are offset from the true set of parameters providing a true minimum AARD. See illustrative examples of the magnitude of discontinuity created by these two challenges.