You are asked to predict the ion with the greatest enthalpy of hydration, the one with the greatest hydration number, and the one that is most favored, in terms of entropy, to dissolve in water.
You are given the formulas of some ions.
Hydration enthalpy increases in magnitude as the ion charge increases and as radius decreases. Of these ions, Al^{3+} has both the highest charge and the smallest radius, so it has the largest magnitude hydration enthalpy. Hydration number also follows the same trends, and Al^{3+} will have the largest hydration number.
In terms of entropy of dissolution, the smaller the number of water molecules bonding to the dissolved ions, the more favored the dissolution, and therefore the ion with the smallest hydration number—that is, the ion with the lowest charge and the largest radius—will be most favored to dissolve in water. Of these, Ba^{2+} will have the smallest hydration number and will be the most favored to dissolve in terms of entropy.