[AlO_{3}^{3−}] = [HO^{−}]^3 × K′_{Ac.S.P.} × K_{HOH}^{−3}. II Adding equations I and II we find [Al^{3+}] + [AlO_{3}^{3−}] = K_{Bas.S.P.} × [HO^{−}]^{−3} + [HO^{−}]^3 × K′_{Ac.S.P.} × K_{HOH}^{−3} III Aluminium hydroxide will be most completely precipitated when [Al^{3+}] + [AlO_{3}^{3−}] is a ‹minimum›, the values [Al^{3+}] and [AlO_{3}^{3−}] measuring the solubility of aluminium as aluminium-ion and as aluminate-ion. If we put [Al^{3+}] + [AlO_{3}^{3−}] = ‹y› and [HO^{−}] = ‹x›, we can find the value ‹x› (the concentration of the hydroxide-ion) for which ‹y› is a minimum. We have ‹y› = K_{Bas.S.P.} × ‹x›^{−3} + ‹x›^3 × K′_{Ac.S.P.} × K_{HOH}^{−3}, and find, by means of the calculus,[393] that ‹y› is a minimum, when ‹x› = +(K_{HOH}^3 × K_{Bas.S.P.} / K′_{Ac.S.P.})^{1/6}. If aluminium hydroxide were as strong an acid as it is a base, ‹i.e.› if K_{Bas.S.P.} = K′_{Ac.S.P.}, we would have, simply, ‹x› = [HO^{−}] = ((1.2E−14)^3)^{1/6} = √(1.2E−14) (at 25°), which is the concentration of the hydroxide-ion in pure water at 25° (p. 176). In other words, a perfectly neutral solution would then give us the conditions for as complete a precipitation as possible. But aluminium hydroxide is a stronger base than acid, K_{Bas.S.P.} > K′_{Ac.S.P.}, and consequently we find for ‹x› = [HO^{−}] = (K_{HOH}^3 × K_{Bas.S.P.} / K′_{Ac.S.P.})^{1/6}, a value somewhat ‹greater› than the concentration of the hydroxide-ion in pure water, ‹i.e.› we must use a slightly ‹alkaline› medium—which agrees with common practice. In other words, there is less danger of losing aluminium hydroxide in the form of aluminate, owing to the ‹weaker acid› character of the hydroxide, than there is of losing it in the form of aluminium-ion. The most favorable degree of alkalinity for the precipitation would depend on the relation of K_{Bas.S.P.}. and K′_{Ac.S.P.}. The exact values for K_{Bas.S.P.} and K′_{Ac.S.P.}, the two solubility-product constants, and for the corresponding ionization constants, which would show the same ‹ratio›, are still not known. But, if, for the sake of an illustration, we take recourse to assumed values for these constants, we find that the solubility of aluminium, as aluminium-ion and as aluminate-ion, is, by calculation, as anticipated, a ‹minimum› for a solution, which contains the concentration of HO^{−} calculated (for ‹x›) in the manner indicated above. And the further interesting conclusion is reached that this minimum loss of aluminium [p198] hydroxide would occur when [Al^{3+}] = [AlO_{3}^{3−}]—which would correspond to a saturated solution of aluminium aluminate, Al(AlO_{3}).