We have, in this instance, the case of a very weak, difficultly soluble acid, aluminium hydroxide, forming a salt with a weak, soluble base, ammonium hydroxide. The conditions determining the ‹solubility› of aluminium hydroxide in ammonium hydroxide, as an aluminate NH_{ 4}AlO_{ 2}, may be shown as follows: for the acid ionization of aluminium hydroxide, Al(OH)_{ 3} ⇄ AlO_{ 2}^{ −} + H^{ +} + H_{ 2}O (p. 172); the solubility-product for a saturated solution is [AlO_{ 2}^{ −}] × [H^{ +}] = K_{ Ac.S.P.}. Further, from [H^{ +}] × [HO^{ −}] = K_{ HOH}, we find [H^{ +}] = K_{ HOH} / [HO^{ −}]. Then [AlO_{ 2}^{ −}] = [HO^{ −}] × K_{ Ac.S.P.} / K_{ HOH}, which shows that the solubility of aluminium hydroxide, as aluminate, is proportional to the concentration [HO^{ −}] of the hydroxide-ion in the solution. For NH_{ 4}OH we have [NH_{ 4}^{ +}] × [HO^{ −}] / ([NH_{ 3}] + [NH_{ 4}OH]) = 0.000,018 (p. 161), and consequently, [HO^{ −}] = 0.000,018 × ([NH_{ 3}] + [NH_{ 4}OH]) / [NH_{ 4}^{ +}]. Then [HO^{ −}] is the smaller, the smaller the excess of ammonium hydroxide used (which is approximately equal to ([NH_{ 3}] + [NH_{ 4}OH])) and the greater the concentration [NH_{ 4}^{ +}] of the ammonium-ion, ‹i.e.› of the added ammonium salt. The solubility of Al(OH)_{ 3}, as aluminate, in ammonium hydroxide and ammonium chloride is, therefore, directly proportional to the excess of ammonium hydroxide, and indirectly proportional to the concentration of the ammonium salt present.[392]
«The Favorable Conditions for a Maximum Precipitation of an Amphoteric Hydroxide.»—The precipitation of aluminium hydroxide depends also on the solubility-product of aluminium hydroxide, ionized as a base. For Al(HO)_{ 3} ⇄ Al^{ 3+} + 3 HO^{ −}, in a saturated solution, [Al^{ 3+}] × [HO^{ −}]^3 = K_{ Bas.S.P.}. It is evident, that an excess of the precipitating hydroxide-ion would be favorable to the precipitation in this form, and that the reduction of the concentration of the hydroxide-ion, ‹while acting favorably›, as just shown, ‹in preventing the solution of the hydroxide as an aluminate, must be, to some extent, detrimental to a maximum precipitation of the hydroxide as a base›. One may ask, therefore, ‹what the most favorable concentration of the hydroxide-ion› must be for a quantitative precipitation of aluminium hydroxide. The problem may be treated as follows: According to the solubility-product relation for the basic ionization, we have, in a solution saturated with aluminium hydroxide, [p197]