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Mechanistically this is because there are insufficient water molecules to effectively solvate the ions.
The oxygen atom is Lewis basic, so it is able to solvate many inorganic compounds.
The donor number is a measure of the ability of a solvent to solvate cations and Lewis acids.
This leads to less water being available to solvate the organic compound, reducing the solubility of the organic compound.
Protic solvents solvate anions (negatively charged solutes) strongly via hydrogen bonding.
The normal solvents of choice are both polar (to stabilize ionic intermediates in general) and protic (to solvate the leaving group in particular).
Polar solvents are necessary for this type of initiation both for stability of the anion-radical and to solvate the cation species formed.
Coordinating solvents such as ether or THF, are required to solvate (complex) the magnesium(II) center.
Also: Electrolytes dissociate in water because water molecules are dipoles and the dipoles orient in an energetically favorable manner to solvate the ions.
Hence the potassium cations are well "solvated" by the protein carbonyl groups, but these same carbonyl groups are too far apart to adequately solvate the sodium cation.
Hard solvents such as hydrogen fluoride, water and the protic solvents tend to solvate strong solute bases such as the fluorine anion and the oxygen anions.
The polar molecules of these solvents can solvate ions because they can orient the appropriate partially charged portion of the molecule towards the ion in response to electrostatic attraction.
The "best" way to solvate a system is to place explicit water molecules in the simulation box with the molecules of interest and treat the water molecules as interacting particles like those in the molecule.
Aprotic solvents such as acetone or dichloromethane tend to have large dipole moments (separation of partial positive and partial negative charges within the same molecule) and solvate positively charged species via their negative dipole.
The effect of the solvent is not only because of its acidity or basicity but also because of its dielectric constant and its ability to preferentially solvate and thus stabilize certain species in acid-base equilibria.