What does the solubility of KNO3 depend on?

This is to do with the fact that #KNO_3# is an ionic compound.

Ionic compounds dissolve in water and covalent compounds do not. The best example of this is NaCl (Sodium Chloride: table salt) - this is an ionic salt and dissolves readily in water. A covalent compound such as sand (Silicon Dioxide: #SiO_2#) does not dissolve in water.

This happens because the dipole water molecules attract the positive and negative ions and split them apart - in covalent compounds such as #SiO_2# there is no electrical charge on the atoms, so they are therefore harder to break down.

On a side note: a dipole is a molecule or atom where there is a greater concentration of electrons in one area than another - this causes one side to be slightly positive and the other side to be slightly negative. For example, if you imagine a Helium atom (which has two electrons) - and both of the electrons are at the left side of the atom, the right side will have none. This causes the side with more electrons to have a slightly negative charge, and the side with less to have a slightly positive charge.

In the case of ionic breakdown, then, the positive side of the water molecules attracts the negative #NO_3^(-)# part of the ion, while the negative #K^(+)# ion is attracted to the negative side of the water dipole.