Ba(OH)2.8H2O ==>Ba^+2 + 2OH^- + 8H2O
How many mols of the compound did you start with initially. That will be 30.2 g Ba(OH)2.H2O/molar mass of that material.
Then you get 1 mol Ba^+, + 2 mols OH^- + 8 mols H2O. Add them to get total mols. The question may or may not be a trick question. Note that it asks for the total moles of IONS. If that is the intent of the problem, then the 8 mols H2O aren't included in the total since they are not ions.
How many total moles of ions are released when each of the following samples dissolves completely in water?
30.2g of Ba(OH)2 8H2O
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