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] [Al^{3+}] = K_{Bas.S.P.} × [HO^{−}]^{−3}. I For the sake of a certain simplicity in the result, we will, for the moment, consider aluminium hydroxide to ionize as an acid according to Al(HO)_{3} ⇄ AlO_{3}^{3−} + 3 H^{+}, which would resemble the basic ionization. Then we would have [AlO_{3}^{3−}] × [H^{+}]^3 = K′_{Ac.S.P.}, and, using the relation [H^{+}] = K_{HOH} / [HO^{−}], we have