In potash alum $[\text{K}_2\text{SO}_4 \cdot \text{Al}_2(\text{SO}_4)_3 \cdot 24\text{H}_2\text{O}]$, the ratio of $\text{K}^+$ ions to $\text{SO}_4^{2-}$ ions is:

Double salts are crystalline compounds formed by the simultaneous crystallisation of two different simple salts in definite molar proportions from a common solution. They are held together by physical forces of crystallisation rather than chemical bonding, and upon dissolution in water, they completely dissociate into the individual ions of the constituent simple salts — as if the two salts had simply been dissolved separately. Important examples: Potash alum KAl(SO4)2·12H2O (= K2SO4·Al2(SO4)3·24H2O): dissolves to give K+(aq), Al3+(aq), and SO4^2-(aq). Mohr's salt FeSO4·(NH4)2SO4·6H2O: dissolves to give Fe2+(aq), NH4+(aq), SO4^2-(aq). It is used in titrations as a source of Fe2+ because it is more stable against oxidation (to Fe3+) than simple FeSO4, due to the mildly acidic environment provided by ammonium sulfate. Ferric alum (NH4)Fe(SO4)2·12H2O: dissolves to give NH4+(aq), Fe3+(aq), SO4^2-(aq). Used as an indicator in permanganate titrations and as a mordant. Sodium potassium tartrate (Rochelle salt) NaKC4H4O6·4H2O: a double salt of sodium tartrate and potassium tartrate, used in the original Fehling's solution for reducing sugar detection. Isomorphism: different alums with the same general formula M+M3+(SO4)2·12H2O crystallise in the same crystal structure (octahedral crystals) regardless of the specific M+ and M3+ ions — classic example of isomorphism, reflecting the dominant influence of electrostatic interactions in crystal packing rather than the specific size and electronic properties of the ion (as long as they are within a certain range). This allows alum crystals to grow epitaxially on other alum crystals, a phenomenon exploited in classic crystal growing experiments.

Alums (M+M3+(SO4)2·12H2O) form a large family of compounds with consistent structural features and diverse practical applications. Preparation of potash alum: dissolve equimolar amounts of K2SO4 (174.26 g/mol) and Al2(SO4)3·18H2O (666.43 g/mol) in minimum hot water, mix the two solutions, filter if necessary, and concentrate by evaporation until crystallisation begins. Cooling the saturated solution gives beautiful octahedral crystals of KAl(SO4)2·12H2O. Alternatively: dissolve Al metal in KOH (aq) to give KAlO2, acidify with H2SO4, evaporate, and crystallise. Structure: in the alum crystal, each M+ ion is surrounded by 12 oxygen atoms from 6 water molecules (forming a regular cuboctahedral arrangement). Each M3+ ion is also surrounded by 12 oxygen atoms from 6 water molecules (separate from the M+ coordination sphere). The SO4^2- ions occupy the spaces in the structure, linking the two types of [M(H2O)6]^n+ cation units through hydrogen bonds and electrostatic interactions. This means the structure contains distinct [K(H2O)6]+, [Al(H2O)6]3+, and SO4^2- units (hence the formula can be written as [K(H2O)6][Al(H2O)6](SO4)2), with the second 6 water molecules from the formula unit (total 12) associated with each M+ and M3+ centre. Crystal system: cubic. Uses of potash alum: historically one of the most widely used industrial chemicals, traded across the ancient world, extensively used in medicine, textiles, water treatment, paper, tanning, and food. Water purification: Al3+ ions hydrolyse in neutral-to-slightly basic water to form colloidal Al(OH)3 floc: Al3+ + 3H2O → Al(OH)3(s) + 3H+; the gelatinous Al(OH)3 colloid is positively charged and adsorbs negatively charged suspended particles (clay, bacteria, algae) through electrostatic attraction and physical entrapment, coagulating them into larger aggregates (flocs) that settle rapidly. Styptic: alum contracts blood vessels and protein-coagulating action stops minor bleeding.

The distinction between double salts and coordination (complex) salts is of fundamental importance in inorganic chemistry and must be clearly understood. A double salt: (a) is formed by co-crystallisation of two or more simple salts from a common aqueous solution; (b) in solution, dissociates completely and independently into all constituent simple ions; (c) the component ions can be detected by ordinary chemical tests (e.g., Mohr's salt gives test for Fe2+, NH4+, and SO4^2- ions; potash alum gives test for K+, Al3+, and SO4^2- ions). A coordination (complex) salt: (a) contains at least one complex ion (metal surrounded by ligands) that remains intact (does not fully dissociate) in dilute solution; (b) in solution, gives the free complex ion plus any counter-ions outside the coordination sphere; (c) the metal ion within the complex cannot be directly detected by ordinary precipitation tests (because its concentration as free ion is very low, governed by the stability constant). Example: K4[Fe(CN)6] — in solution, gives 4K+ and [Fe(CN)6]4- (the hexacyanoferrate(II) complex ion remains intact). Does NOT give test for free Fe2+ (adding NaOH gives no Fe(OH)2 precipitate) or free CN- (adding AgNO3 does not give AgCN precipitate directly). This distinction was central to Alfred Werner's proof of his coordination theory — he showed that certain cobalt-ammonia-chloride complexes did NOT give tests for free Cl- with AgNO3 for the chlorides in the coordination sphere, while those outside gave immediate precipitates.

Colloidal systems — dispersions of particles with dimensions in the range 1 nm to 1 μm (10^-9 to 10^-6 m) — have unique properties that distinguish them from true solutions (particles < 1 nm) and suspensions (particles > 1 μm). Colloids show: Tyndall effect (scattering of light beam, making it visible — "milky" appearance), Brownian motion (random thermal motion of particles, stabilises them against settling by gravity), high surface area and adsorption properties, electrophoresis (migration in electric field, indicating surface charge), and coagulation (aggregation and settling upon addition of electrolytes). Classification of colloids by phase: aerosol (gas dispersion medium, liquid or solid dispersed: fog, smoke), foam (gas dispersed in liquid: shaving cream), emulsion (liquid dispersed in liquid: milk, mayonnaise), sol (solid dispersed in liquid: colloidal gold, Fe(OH)3 sol), gel (liquid dispersed in solid: jelly, silica gel). Coagulation of lyophobic sols (those not protected by a solvation layer): occurs when electrolytes are added that neutralise the surface charge of colloidal particles. Hardy-Schulze rule: the coagulating power of an ion increases sharply with its charge — trivalent ions are much more effective coagulants than divalent, which are more effective than monovalent. Al3+ is 103 times more effective than Na+ as a coagulant (hence use of alum in water treatment). Coagulation of negatively charged clay particles (the commonest type of suspended particles in turbid water) by Al3+ from added alum: the positive Al3+ and the gelatinous Al(OH)3 floc (from hydrolysis) both coagulate the negatively charged clay particles.

Aluminium (Al, Z=13, [Ne]3s²3p¹) is the most abundant metal in the Earth's crust (8.1% by mass, third most abundant element overall after O and Si), yet it was more valuable than gold until the development of the Hall-Heroult electrolytic process in 1886, because aluminium metal was so difficult to produce by chemical reduction methods. Properties: low density (2.70 g/cm³ vs iron at 7.87 g/cm³), good electrical and thermal conductivity, corrosion resistance (due to thin, adherent, transparent Al2O3 passivation layer that forms instantly on exposure to air), high strength-to-weight ratio. Extraction: bauxite (Al2O3·2H2O, gibbsite + boehmite + diaspore) is purified by the Bayer process (dissolution in hot NaOH: Al2O3 + 2NaOH + 3H2O → 2NaAl(OH)4; precipitation of pure Al(OH)3 by seeding and neutralisation; calcination to Al2O3). Pure Al2O3 (alumina) is then dissolved in molten cryolite (Na3AlF6, added to lower the melting point from 2050°C to ~980°C) and electrolysed (Hall-Héroult process): cathode: Al3+ + 3e- → Al (liquid Al at bottom of cell). Anode: O2- → O2 (graphite anodes slowly oxidised by O2 to CO2 and CO). Amphoteric behaviour of aluminium: Al reacts with both acids (Al + 3HCl → AlCl3 + 3/2 H2) and strong bases (2Al + 2NaOH + 2H2O → 2NaAlO2 + 3H2). Al2O3 is also amphoteric. This amphoteric behaviour reflects Al's position at the boundary between metals and non-metals (metalloid character in some respects, though Al is classified as a metal).

Group 13 (IIIA) elements — boron (B), aluminium (Al), gallium (Ga), indium (In), thallium (Tl) — show an interesting progression from non-metallic (B is a metalloid/semiconductor) through metallic (Al, Ga, In are silvery metals) to heavy metal with unusual properties (Tl shows both +1 and +3 states, with +1 more stable for the heavier element due to the inert pair effect). All have the valence electron configuration ns²np¹, with 3 valence electrons and the maximum common oxidation state of +3. The inert pair effect: down the group (especially at Tl), the ns² electrons become increasingly reluctant to participate in bonding because of the high penetration of s-orbitals to the nucleus (relativistic effect stabilises s-electrons at heavy elements, making them less available for bonding). Hence Tl+1 is very stable (like an alkali metal), while Tl+3 is strongly oxidising. This contrasts with Al+3 being very stable and Al+1 not observed under normal conditions. Boron vs aluminium differences: B forms exclusively covalent bonds (no B³⁺ ion exists in solution); Al forms ionic compounds (Al³⁺ in solution) as well as covalent. B2O3 is acidic oxide (gives boric acid with water); Al2O3 is amphoteric. BCl3 is monomeric (trigonal planar, sp²); AlCl3 dimerises to Al2Cl6 (bridged by two Cl atoms, each Al has 4-coordinate tetrahedral geometry). Boron maximum covalency = 4 (no d-orbitals); Al can have covalency 4 (AlCl4-) and higher (AlF6³-). Gallium melts at 30°C (melts in the hand) — used in thermometers. Indium and gallium are essential components of LCD displays (ITO = indium-tin oxide, the transparent electrode) and semiconductors (GaAs, InP, GaN for LEDs and lasers). Thallium compounds are highly toxic (act like potassium in biological systems, disrupting K⁺-dependent processes).