buffer

Mg(OH)2 = Mg2+(aq) + 2OH-(aq)
Ksp = [Mg2+][OH-]2 = 1.8 x 10-11 mol3 dm-9

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In water:
Solubility of Mg(OH)2 = z mol dm-3

At eqm:
[Mg2+] = z mol dm-3
[OH] = 2z mol dm-3
Ksp = (z)(2z)2 = 1.8 x 10-11
4z3 = 1.8 x 10-11
z = 1.65 x 10-4

Solubility of Mg(OH)2 in water = 1.65 x 10-4 mol dm-3

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In the buffer of pH 9.3:
Solubility of Mg(OH)2 = y mol dm-3

At eqm:
pH = 9.3
pOH = 14 - 9.3 = 4.7
-log[OH-] = 4.7
Hence, [OH-] = 10-4.7 = 2 x 10-5 mol dm-3
Ksp = y(2 x 10-5)2 = 1.8 x 10-11
y = 0.045

Solubility of Mg(OH)2 in the buffer = 0.045 mol dm-3