First, we are going to use H-H equation :
when PH = Pka + ㏒ [acetate / acetic acid]
when we have Ka value = 1.8 x 10^-5 so, we can get the Pka value:
Pka = - ㏒Ka
= - ㏒ 1.8 x 10^-5
= 4.74
by substitution:
∴ 4.58 = 4.74 + ㏒[acetate / acetic acid]
∴acetate/acetic acid = 0.692
so we need moles of acetate = volume *molarity
= 0.5 L * 0.25 M
= 0.125 moles
when the acetate/ acetic acid ratio = 0.692 and moles acetate = 0.125 moles
∴ moles acetic acid = 0.125 moles / 0.692 moles = 0.182 moles
∴ volume use for 0.182 moles = 0.182 / 0.5 = 0.364 L
∴volume of buffer solution = 0.5 L + 0.364L
= 0.864 L