The element along with Barium that can be found in hard water is Calcium. Calcium contributes to the total hardness in water and does not cause any ill health effects. On the other hand, Barium can cause serious toxic effects to the heart and blood vessels in large concentrations.
Barium sulfate is insoluble in water, while barium carbonate will dissolve. As barium carbonate dissolves in water, it dissociates, and the barium ions are freed. The barium ions are toxic, and that is the crux of the issue.
Since both barium chloride and barium sulfate contain one mole of barium atoms pert mole of compound, the moles of barium sulfate will be the same, 0.100, when barium has the limiting concentration in the production of the sulfate.
Barium is not poisonous in the form of its sulfate. That's because barium sulfate is insoluble in water. Unless a barium compound dissolves with the subsequent release of barium ions, it is not going to be highly toxic.
Barium is not a mineral, but is a chemical element. Barium, an alkaline metal, is quite reactive, and it is never found free in nature. Rather, it forms compounds that make up some minerals like barite, which is barium sulfate, and witherite, which is barium carbonate.