# How does the composition of a mixed solvent affect solubility?

I am trying to find out how varying the proportion of each solvent in a binary (water/ethanol) solvent solution affects the solubility of an organic acid. I know the solubilities of the acid in each solution. Is it as simple as a linear relationship between solvent composition, or is there something else going on?

This question deals with the same topic, and the conclusion there is that it is immensely difficult to predict solubility in mixed-solvent systems. My question is different for two reasons - I only want a qualitative answer, and I know the solubilities for the compound in both pure solvents.

In general, solubility is determined by the intermolecular interaction between a solvent and a solute. Electrostatic forces, hydrogen bonding, electron-pair donor/electron-pair acceptor interactions and solvophobic interactions are examples. The sum of the interaction forces can be related to the polarity of the molecule.

The dissolution of a substance requires that the interaction energy of the solute molecules and the interaction energy of the solvent molecules be overcome. In other words, for the solvation of A in a solvent B, we expect a stronger interaction A-B than A-A or B-B. Considering this, you will have the trend of solubility of polar and non-polar solutes in solvents: the old "like dissolves like" rule.

Now, if you have a solvent mixture, the interaction between unlike solvent molecules is important and leads to deviations from the ideal behaviour from Raoult's law, for example the water/alcohol mixture. Solvent mixtures can exhibit different physical properties compared with their pure components. Coming back to intermolecular forces, it has been found that the ratio of the solvent in the solvent shell of solvation can be different from that in the bulk solution, leading to a preferential solvation. This is caused by the variation of free energy of solvation of the solute in different solvents.

The acid-base equilibria is affected by a change in the 1) acidity or basicity of the solvent, 2) the relative permittivity of the solvent, $\epsilon_{r}$, and 3) its ability to solvate the species in the equilibria. There exists some formulas that relates the equilibrium constant and the relative permittivity (Born, Onsager, etc). In the case of the acetic acid, for example, the relative permittivity will have a strong influence in the solvation of the anionic and cationic species of the equilibria $$\text{CH}_{3}\text{CO}_{2}\text{H}\rightleftarrows\text{CH}_{3}\text{CO}_{2}^{-}+\text{H}^{+}$$ since the two charges attract each other. The good solvation of the ions in a high relative permittivity solvent shift to the right the ionization. Consider that for water ($\epsilon_{r}\approx80$) and ethanol ($\epsilon_{r}\approx33$) the miscibility is similar, but also have distinct pKa's. Therefore, taking into account only the electrical interaction, between solute and solvent, is a poor approximation. The other problem is that it is practically impossible to find solvents that differ only in their relative permittivities and not in their acidity or basicity.

So, for instance, data on solubility in mixed solvents may be very difficult to obtain, but qualitatively the whole thing works in the form of intermolecular interactions.

The conclusions here are essentially the same is in the question that you linked; even a qualitative answer is not at all straight forward, and a simple interpolation of solvent concentration given the known solubilities of a compound in two different solvents may well give you vastly wrong solubilities in the mixed solvent.

Also, it looks to me as if you have the necessary reagents and information to do a quick, qualitative test this theory. Try measuring the solubilities of a couple compounds over a range of mixtures of water / ethanol and see how that compares to values predicted by interpolation.