First of all, let's consider the solubility of Group II hydroxides.
Solubility of Group II hydroxides increase down a group, because solution = lattice enthalpy (defined endothermically) + solvation enthalpy (exothermic).
Comparing Mg2+ and Ba2+, lattice enthalpy between Mg2+ and OH- is much greater than solvation enthalpy resulting for ion-dipole forces between polar water molecules and the cation; as compared to the lesser difference between lattice enthalpy and solvation enthalpy for Ba2+.
In other words, it's not worth it, energetically speaking, to break apart the solid ionic compound (for Mg2+) to solvate it. Thus solution does not occur. But it's worth if for Ba2+, and solution occurs. Relatively speaking, that is. Ksp for Ba(OH)2 > Ksp for Mg(OH)2
Solubility of Group II hydroxides INCREASE down the group.
So why is the trend opposite for Group II sulfates? Why is magnesium sulfate soluble and barium sulfate insoluble? Why does solubility of Group II sulfate DRECREASE down the group?
Thanks for reading and thanks in advance for any replies!