The species in the greatest concentration in a 0.500-molar solution of Ba(OH)2 in water is hydroxide ion (OH-).
When Ba(OH)2 is dissolved in water, it dissociates into Ba2+ and OH- ions. The molar concentration of Ba(OH)2 is 0.500 M, which means that there are 0.500 moles of Ba(OH)2 in every liter of solution. Since each mole of Ba(OH)2 produces 2 moles of OH- ions, the concentration of OH- ions in the solution is 2 * 0.500 M = 1.00 M.
Therefore, the species in the greatest concentration in a 0.500-molar solution of Ba(OH)2 in water is hydroxide ion (OH-).