Asolution of hydrochloric acid is mixed with an equal volume solution of barium hydroxide. when ph paper is inserted into the resulting solution, the ph was determined to be 7.0. if the initial concentration of the hydrochloric acid solution was 0.5 m, what is the concentration of the barium hydroxide solution? explain your steps and balanced equation.

The balanced equation for the reaction between Ba(OH)₂ and HCl is ;
Ba(OH)₂ + 2HCl > BaCl₂ + 2H₂O
When equal volumes of both acid and base are added, the pH is 7 which means its neutral. 
This means that the H⁺ ions and OH⁻ are equal and they have neutralised each other. All the ions have reacted to form water molecules.
Dissociation of base and acid are as follows;
Ba(OH)₂ ---> Ba²⁺ + 2OH⁻
1 mol of Ba(OH)₂ dissociates to 2 mol H⁺
HCl ---> H⁺ + Cl⁻
n moles of HCl give out n mol of H⁺ ions
H⁺ = OH⁻ which is why solution is neutral
then OH⁻ moles present are n  mol 
then Ba(OH)₂ moles present should be n/2 mol
molarity of HCl = 0.5 mol/L
since volumes are equal number of moles are directly proportional to molarity 
molarity of Ba(OH)₂ = 0.5/2 = 0.25 M