# Solubility at any temperature

## Approximating the solubility at any temperature from a single (or more) solubility value(s)

### 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.

### References

[1] Solubility of Stearic Acid in Various Organic Solvents and Its Prediction using Non-ideal Solution Models. Heryanto, R. and Hasan, M. and Abdullah, E.C. and Kumoro, A.C. (2007) ScienceAsia, 33 . pp. 469-472. doi: 10.2306/scienceasia1513-1874.2007.33.469
[2] ONS Mendeley collection with tag temperature dependence