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Moreover, does the amount of solvent affect solubility?
No (as long as there is solvent… if there is no solvent, the rate is zero). The amount of solvent can affect only how much of the solute dissolves, assuming that you have more solute than would saturate the solvent.
Beside above, what are factors that affect solubility?
Solubility is the maximum amount of a substance that will dissolve in a given amount of solvent at a specific temperature. There are two direct factors that affect solubility: temperature and pressure. Temperature affects the solubility of both solids and gases, but pressure only affects the solubility of gases.
In addition, molecular polarity affects solubility in that polar molecules are best solvated by polar solvent molecules and nonpolar molecules are best solvated by nonpolar solvent molecules; i.e., "like dissolves like". If, however, the sum of the bond polarities is not zero, the molecule is polar.