When the ionization of aluminium hydroxide, as an acid, is considered to take place according to Al(OH)_{3} ⇄ AlO_{2}^{−} + H^{+} + H_{2}O, which agrees best with its real behavior (p. 172), we can find, similarly, that [Al^{3+}] + [AlO_{2}^{−}] is a ‹minimum›, when aluminium hydroxide is precipitated in such a way, that an excess ‹x› of the hydroxide-ion is used, and ‹x› = [HO^{−}] = (3 K_{HOH} × K_{Bas.S.P.} / K_{Ac.S.P.})^{0.25},—where K_{Ac.S.P.} represents the solubility-product constant for [AlO_{2}^{−}] × [H^{+}]. That a minimum loss of aluminium hydroxide would be suffered when the favorable excess of the hydroxide-ion (‹x›) is calculated on the basis of the equation as given, may readily be seen by again assuming definite values for K_{Bas.S.P.} and K_{Ac.S.P.}. It also appears that this minimum loss[394] of aluminium includes one-third as many Al^{3+} ions, as AlO_{2}^{−} ions—a relation corresponding, again, to a saturated solution of aluminium aluminate, Al(AlO_{2})_{3}. FOOTNOTES: [354] When all three of the hydrogen atoms in the hydroxide are ionized, an aluminate ion, AlO_{3}^{3−} is formed: Al(OH)_{3} ⇄ AlO_{3}^{3−} + 3 H^{+}. But, as in the case of other weak polybasic acids, a single hydrogen atom is far more readily ionized than are the remaining two (p. 102), and the ion Al(OH)_{2}O^{−}, which is formed by the ‹primary› ionization, readily loses water and forms the anhydride ion AlO_{2}^{−}. The most important aluminates are derivatives of this ion. [355] See the table at the back of Smith's ‹Inorganic Chemistry›, or p. 149 of Remsen's ‹Inorganic Chemistry›. [356] The displacement of hydrogen by a metal, like sodium, is the result of the displacement of the ‹hydrogen-ion› (see Chapters XIV and XV). The hydrogen-ion in fused sodium hydroxide is probably formed chiefly by the secondary ionization of the hydroxide-ion (HO^{−} ⇄ H^{+} + O^{2−}) (see Chap. XIII). We cannot have positive ions, Na^{+}, with negative ions, O^{2−}, without having some ions NaO^{−}. (O^{2−} + Na^{+} ⥂ NaO^{−}), NaOH, undoubtedly, is much ‹too› ‹weak› an ‹acid› to form salts with bases in the presence of water. Such salts would be decomposed by water (see below, p. 180), as sodium oxide, indeed, is decomposed; we have Na—O—Na + HOH ⇄ 2 NaOH (see Chapter XIII for a detailed discussion of this action). These relations sufficiently account for the fact that salts of sodium hydroxide, in which it has the functions of an acid, are not commonly formed. (‹Cf.› Abegg, ‹Anorganische Chemie›, II, (1) p. 247.) [357] See J. J. Thomson, ‹Corpuscular Theory of Matter›, pp. 103–141. [358] See Mendeléeff, ‹Principles of Chemistry›, I, 22 (1891), in regard to the rôle of "even" and "uneven" series in the system. [359] In regard to the indications of the amphoteric character of stronger acids, see Chapter XV.