Solubility+at+any+temperature

The Buchowski Equation
The Buchowski equation can be used to approximate the mole fraction solubility, x, of a compound at any temperature. The equations is as follows: where lambda represents the non-ideality of the solution (for an ideal solution, lambda=1), H is the enthalpy of solution, T is the temperature of the solution in Kelvin, and T_m is the melting point temperature of the solute in Kelvin.[1]

Mole fraction solubility, x, can be converted to molarity C, using the following equation: where d and mw are the solute density and molecular weight respectively; and D and MW are the solvent density and molecular weight respectively.

If we then assume that lambda can be approximated as follows (perhaps a pretty big assumption): we see that the Buchowski equation can be re-written in terms of molar concentration (with a single unknown parameter H - which can be estimated from a single known concentration and the miscible concentration, assuming the melting point of the compound is known), allowing us to predict the molar concentration at any temperature using just a single measured solubility value. Examining the above equation, we see, as expected, a plot of ln(C) against 1/T should be linear and that, as T approaches T_m, C approaches the miscible value of d/mw.

Conclusion
While the approximation (3) used to obtain (4) is in many cases unrealistic, it suggests that may be a good way to approximate the molar concentration at any temperature using only a single measured solubility value.