Inorganic Chemistry — Periodic Trends, Coordination, Solid State

Inorganic chemistry is the chemistry of every element — all 118 — and of every compound that is not built around a carbon–hydrogen backbone. It is the discipline that explains why gold is yellow, why mercury is liquid, why hemoglobin binds O2 reversibly while CO poisons it, why a zeolite cracks oil, why NdFeB lifts forklift loads, and why a 90 kg shell of uranium hexafluoride centrifuges its way into power reactors and warheads. Where organic-chemistry-foundations focuses on one carbon-centered niche, inorganic chemistry surveys the rest of the periodic landscape — main-group reactivity, transition-metal coordination, lanthanide and actinide f-block behavior, ionic and covalent solid-state architecture, bioinorganic active sites, and nuclear transformations.

This note is a Tier 1 reference: it states the laws, the equations, the numbers, the named reactions, and the laureates who pinned them down. Cross-references to neighboring disciplines — _index, _index, _index for the group-theoretic underpinnings, and _index for the computational-chemistry stack — are made liberally so the curious reader can follow the trail.

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1. Periodic table architecture

The modern periodic table arranges 118 elements (as of 2026, with oganesson Og-118 the heaviest confirmed in 2002 and named in 2016) into 7 periods (rows, corresponding to principal quantum number n = 1 through 7) and 18 groups (columns). Dmitri Mendeleev’s 1869 layout famously left gaps for ekasilicon (Ge, found 1886), ekaaluminium (Ga, 1875), and ekaboron (Sc, 1879) — predictions that vindicated the periodic law on the basis of chemical analogy before any quantum-mechanical justification existed.

1.1 Blocks: s, p, d, f

Electrons fill orbitals labeled by their azimuthal quantum number ℓ:

• s-block (ℓ = 0): groups 1–2 plus He. Two columns, valence ns. 14 elements.
• p-block (ℓ = 1): groups 13–18. Six columns, valence ns² np¹⁻⁶. 36 elements (including the noble gases).
• d-block (ℓ = 2): groups 3–12. Ten columns. The transition metals, valence (n−1)d¹⁻¹⁰ ns⁰⁻². 40 elements.
• f-block (ℓ = 3): the lanthanides (4f, Ce–Lu) and actinides (5f, Th–Lr). Fourteen columns each, drawn below the main table for typographic mercy. 28 elements.

Hydrogen sits awkwardly in group 1 (1s¹, like the alkali metals) but is chemically more p-like in many respects — it forms H⁻ (hydride), is a covalent diatomic in its elemental form, and has electronegativity 2.20 (Pauling), placing it near carbon rather than near cesium.

1.2 The Madelung n+ℓ rule (Aufbau order)

Orbitals fill in order of increasing n + ℓ, and for equal n + ℓ in order of increasing n. This gives the canonical sequence:

1s (1) → 2s (2) → 2p (3) → 3s (3) → 3p (4) → 4s (4) → 3d (5) → 4p (5) → 5s (5) → 4d (6) → 5p (6) → 6s (6) → 4f (7) → 5d (7) → 6p (7) → 7s (7) → 5f (8) → 6d (8) → 7p (8) …

The bolded crossings — 4s before 3d, 5s before 4d, 6s before 4f before 5d — are the source of the “long” 6th and 7th periods (32 elements each, including the f-blocks).

1.3 Effective nuclear charge Zeff and Slater’s rules

The effective nuclear charge Zeff = Z − S, where Z is the atomic number and S is the shielding constant. John C. Slater (1930) gave an empirical recipe:

• Group orbitals: [1s][2s,2p][3s,3p][3d][4s,4p][4d][4f][5s,5p]…
• For electrons in the same group (except [1s]): contribute 0.35 each (0.30 for 1s).
• For electrons in the n−1 shell, if the electron of interest is s/p: 0.85 each.
• For electrons in deeper shells (n−2 or lower): 1.00 each.
• For d or f electrons of interest: all electrons in lower groups contribute 1.00; same-group contribute 0.35.

Example: Cl (Z = 17). Configuration [Ne]3s²3p⁵. The 3p electron sees:

• Same-group (3s, 3p) electrons: 6 others × 0.35 = 2.10
• 2s/2p shell: 8 × 0.85 = 6.80
• 1s shell: 2 × 1.00 = 2.00
• S = 10.90, Zeff ≈ 17 − 10.90 = 6.10

A modern alternative — Clementi–Raimondi (1963) values — comes from Hartree–Fock self-consistent-field calculations and gives Zeff(3p,Cl) ≈ 6.12, in striking agreement.

1.4 Atomic radius

Atomic radius decreases across a period (Zeff rises, valence shell unchanged) and increases down a group (new shells added). Representative covalent radii (pm):

• Li 128, Be 96, B 84, C 76, N 71, O 66, F 57, Ne 58
• Na 166, K 203, Rb 220, Cs 244, Fr ≈ 260

The lanthanide contraction — 4f electrons shield poorly, so across Ce–Lu the 5d/6s radius shrinks by ~17 pm — makes the 5d row (Hf, Ta, W, Re, Os, Ir, Pt, Au) nearly the same size as the 4d row above (Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag). Hf and Zr are notoriously hard to separate (problem for nuclear-grade Zr cladding).

1.5 Ionization energy

First ionization energy IE₁ (kJ/mol, gas-phase, ground state → +1):

• H 1312, He 2372 (highest of all elements)
• Li 520, Be 899, B 800, C 1086, N 1402, O 1314, F 1681, Ne 2081
• Na 496, Mg 738, Al 577, Si 786, P 1011, S 1000, Cl 1251, Ar 1521
• K 419, Rb 403, Cs 376, Fr ≈ 380

Note the kinks: B < Be (because Be’s 2s² is more tightly held than B’s lone 2p), O < N (because N’s 2p³ half-filled subshell is extra-stable, and O’s fourth 2p electron pairs and feels exchange-loss).

Second, third, fourth IEs rise steeply. For Mg: IE₁ = 738, IE₂ = 1450, IE₃ = 7733 kJ/mol — the jump at IE₃ marks core-electron removal and explains why Mg never forms Mg³⁺ chemically.

1.6 Electron affinity

Energy released when a gas-phase atom gains an electron, X(g) + e⁻ → X⁻(g). By convention, more negative = more exothermic; many tables give the magnitude. Representative values (kJ/mol, sign convention “energy released”):

• Cl 349, F 328, Br 325, I 295 (halogens favor anion formation)
• O 141 (first EA); −744 (second EA, endothermic) — O²⁻ is unstable as a free gas-phase ion and exists only in lattices that compensate via lattice energy
• N ≈ 0 (half-filled 2p³ is too stable to want another electron)
• Noble gases: all positive (energy must be supplied)

1.7 Electronegativity scales

Three popular formulations:

Pauling (1932, Nobel 1954) — defined from bond-dissociation energies: χ_A − χ_B = √(D_AB − ½(D_AA + D_BB)) / 96.5

The “Pauling unit” is dimensionless. F = 4.0 (anchor), O = 3.5, N = 3.0, Cl = 3.0, C = 2.5, H = 2.2, Si = 1.9, Li = 1.0, Na = 0.93, Cs = 0.79 (least electronegative, with Fr also ~0.7).

Mulliken (1934, Nobel 1966) — average of IE and EA: χ_M = ½(IE + EA), in eV. Maps to Pauling via χ_P ≈ 0.336·(χ_M − 0.615).

Allred–Rochow (1958) — derived from electrostatic force at the covalent radius: χ_AR = 3590·Zeff/r² + 0.744, with r in pm.

All three rank elements similarly. F is always the highest; Cs/Fr the lowest.

1.8 Relativistic effects in heavy elements

For elements with Z > ~50, inner s-electrons move at a substantial fraction of c. The Lorentz factor γ = 1/√(1 − v²/c²) contracts the 1s orbital (and by self-consistency the 6s and 7s outer s-orbitals), which in turn changes chemistry:

• Gold (Au) is yellow — the relativistically contracted 6s lowers the 5d → 6s transition into the blue (~520 nm), so reflected light is yellow. Without relativity, Au would look like silver.
• Mercury (Hg) is liquid at 298 K — relativistic 6s² is so strongly bound that Hg–Hg metallic bonding is weak (the 6s “lone pair” doesn’t want to share). Melting point: −38.83 °C.
• Lead 6s inert pair — Pb favors +2 (6s²) over +4 because the 6s pair is relativistically stabilized and reluctant to be ionized. Same effect makes Tl⁺ > Tl³⁺ and Bi³⁺ > Bi⁵⁺ in stability.
• Cesium 6s relativistic contraction — partially explains why Cs is not quite as reactive as a naïve extrapolation from Li → Cs would predict.

See relativistic-quantum-chemistry for the four-component Dirac–Hartree–Fock treatment.

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2. Atomic structure

2.1 Hydrogen-like solutions

For a one-electron atom (H, He⁺, Li²⁺, …), the Schrödinger equation separates in spherical coordinates and yields ψ(n, ℓ, m_ℓ) = R(n,ℓ)(r)·Y(ℓ,m_ℓ)(θ, φ). The energies depend only on n:

E_n = −13.606 eV · Z² / n² (Rydberg formula)

For real multi-electron atoms, ℓ-degeneracy is lifted (penetration: s > p > d > f) and Aufbau order applies.

2.2 Quantum numbers and orbital shapes

• n (principal): 1, 2, 3, … sets shell and energy.
• ℓ (azimuthal): 0 to n−1, sets subshell shape. ℓ = 0 (s, spherical), 1 (p, dumbbell), 2 (d, cloverleaf + donut for d_z²), 3 (f, complex multilobed).
• m_ℓ: −ℓ … +ℓ, sets orientation. p has three (p_x, p_y, p_z), d has five, f has seven.
• m_s: ±½ (electron spin).

Pauli exclusion forbids two electrons sharing all four numbers — hence at most two electrons per orbital, with opposite spins.

2.3 Aufbau exceptions

The Madelung rule predicts ground-state electron configurations for most elements, but a few transition metals and lanthanides break it because half-filled and fully-filled subshells gain extra exchange stabilization:

• Cr (Z = 24): predicted [Ar]4s²3d⁴, actual [Ar]4s¹3d⁵ (half-filled d).
• Cu (Z = 29): predicted [Ar]4s²3d⁹, actual [Ar]4s¹3d¹⁰ (filled d).
• Mo (Z = 42): [Kr]5s¹4d⁵ (same trick as Cr).
• Ag (Z = 47): [Kr]5s¹4d¹⁰ (same as Cu).
• Au (Z = 79): [Xe]4f¹⁴5d¹⁰6s¹.
• Pd (Z = 46) is uniquely [Kr]4d¹⁰ — no 5s electron at all in the ground state.
• La (Z = 57) and Lu (Z = 71) and Ac/Lr complicate where f-block “starts.”

These exceptions matter for predicting oxidation-state chemistry: Cu⁺ (3d¹⁰) is diamagnetic and tetrahedral-soft; Cu²⁺ (3d⁹) is paramagnetic, blue/green, and Jahn–Teller distorted.

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3. Ionic bonding

3.1 Madelung constant

For an infinite ionic crystal, the electrostatic energy per formula unit involves an alternating lattice sum:

U_electrostatic = − N_A · M · z⁺z⁻ · e² / (4π ε₀ r₀)

where M is the Madelung constant (geometry-only). Values:

• NaCl (rock salt): M = 1.7476
• CsCl: M = 1.7627
• ZnS (zinc blende): M = 1.6381
• ZnS (wurtzite): M = 1.6413
• CaF₂ (fluorite): M = 2.5194 (per formula unit)
• TiO₂ (rutile): M = 2.408

3.2 Born–Landé equation

Adding the short-range Born repulsion (Pauli) term:

U_lattice = − (N_A · M · z⁺z⁻ · e²) / (4π ε₀ r₀) · (1 − 1/n)

where n is the Born exponent (5 for He-config ions, 7 for Ne, 9 for Ar, 10 for Kr, 12 for Xe). For NaCl: r₀ = 282 pm, M = 1.7476, n = 8 (average of Ne and Ar) → U_calc ≈ −755 kJ/mol. Experimental: −787 kJ/mol. Within 4 %.

A more refined version is Born–Mayer, where 1/n is replaced by ρ/r₀ (with ρ ≈ 30 pm, a softness parameter).

3.3 Born–Haber cycle (NaCl)

Hess’s law decomposes ΔH_f(NaCl, s) = −411 kJ/mol into:

1. Na(s) → Na(g): ΔH_sub = +108 kJ/mol
2. ½ Cl₂(g) → Cl(g): ½ × +242 = +121 kJ/mol
3. Na(g) → Na⁺(g) + e⁻: IE₁ = +496 kJ/mol
4. Cl(g) + e⁻ → Cl⁻(g): EA = −349 kJ/mol
5. Na⁺(g) + Cl⁻(g) → NaCl(s): U_lattice = ?

Closing the cycle: U_lattice = −411 − 108 − 121 − 496 + 349 = −787 kJ/mol. This is the gold-standard experimental lattice energy and the benchmark for Born–Landé.

3.4 Lattice-energy trends

• Higher charges → larger U: MgO (2370 kJ/mol) ≫ NaF (923).
• Smaller ions → larger U: LiF (1036) > CsF (740).
• Inverse square-ish on r₀, linear on z⁺z⁻.

Lattice energy drives most of solubility, melting point, hardness, and high-temperature stability of ionic solids. See ionic-crystals.

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4. Covalent bonding

4.1 Lewis structures (1916)

Gilbert N. Lewis introduced the dot diagram and the octet rule: main-group atoms tend toward 8 valence electrons (2 for H, He). Lewis’s electron-pair model preceded quantum mechanics by a decade but survives because it captures connectivity and lone pairs correctly for most molecules.

4.2 VSEPR (Gillespie–Nyholm, 1957)

Valence-shell electron-pair repulsion: electron pairs (bonding and lone) around a central atom arrange to minimize repulsion. Ronald Gillespie (McMaster) and Ronald Nyholm (UCL) systematized it. Geometry by total pair count (steric number SN):

SN	Shape (lone pairs in parens)	Example
2	Linear	BeCl₂, CO₂
3	Trigonal planar	BF₃, NO₃⁻
3	Bent (1 lp)	SO₂, O₃
4	Tetrahedral	CH₄, NH₄⁺
4	Trigonal pyramidal (1 lp)	NH₃
4	Bent (2 lp)	H₂O
5	Trigonal bipyramidal	PCl₅
5	Seesaw (1 lp)	SF₄
5	T-shape (2 lp)	ClF₃
5	Linear (3 lp)	XeF₂
6	Octahedral	SF₆, PF₆⁻
6	Square pyramidal (1 lp)	BrF₅
6	Square planar (2 lp)	XeF₄

Repulsion ordering: lone-pair–lone-pair > lone-pair–bond > bond–bond. This explains why H₂O has a 104.5° angle (compressed from tetrahedral 109.5°) and NH₃ has 107°.

4.3 Formal charge

FC = (valence e⁻) − (lone-pair e⁻) − ½(bonding e⁻). Used to choose between resonance structures: the best Lewis structure minimizes |FC| and places negative FC on the most electronegative atom.

4.4 Resonance

For molecules with delocalized π electrons (benzene, carbonate CO₃²⁻, nitrate NO₃⁻, ozone O₃), no single Lewis structure suffices — the true structure is a weighted average. Resonance energy of benzene ≈ 150 kJ/mol below the hypothetical “cyclohexatriene.”

4.5 Hypervalency debate

Classical Lewis/VSEPR pictures SF₆ as sp³d² hybridized at S, using 3d orbitals to accommodate 12 valence electrons. Computational evidence (Reed, Schleyer, others, 1990s) shows that S 3d orbital populations in SF₆ are <0.3 e⁻ — too small to support six full bonds. Modern picture: three-center four-electron (3c-4e) hyperbonds in which a lone pair on one F donates into an S–F σ* antibond, creating delocalized bonding without true d participation.

• SF₆: six 3c-4e bonds, formal charge on S is +2 (not 0).
• PCl₅: two axial bonds form a 3c-4e set; three equatorial bonds are conventional 2c-2e.
• XeF₂, XeF₄, I₃⁻: classic 3c-4e textbook cases.

Hybridization terminology (sp³d, sp³d²) is still taught but is understood now as bookkeeping rather than literal orbital description for elements beyond period 2.

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5. Hybridization

Linus Pauling (1931) introduced hybridization to reconcile Lewis structures with quantum mechanics. Standard schemes:

Hybrid	Shape	Example	Bond angle
sp	Linear	BeCl₂, CO₂, HCN	180°
sp²	Trigonal planar	BF₃, C in ethene	120°
sp³	Tetrahedral	CH₄, NH₃, H₂O	109.5° (ideal)
sp³d	Trigonal bipyramidal	PCl₅ (cf. caveat above)	90/120°
sp³d²	Octahedral	SF₆	90°

Caveat for heavier elements: the energy gap between (n−1)d and ns/np orbitals is too large for clean hybridization in periods 3+, and the 3d population evidence above shows it doesn’t really happen. Hybridization is robust for B, C, N, O (period 2), useful as bookkeeping for period 3, and increasingly unphysical beyond.

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6. Molecular orbital (MO) theory

Friedrich Hund and Robert Mulliken (Nobel 1966) developed MO theory. Atomic orbitals on bonded atoms combine linearly (LCAO) into molecular orbitals delocalized over the whole molecule.

6.1 Homonuclear diatomics

For H₂ through O₂, two AOs of each atom combine to form bonding (σ, π) and antibonding (σ*, π*) MOs. Bond order = ½(n_bonding − n_antibonding).

Molecule	Configuration	BO	Magnetism
H₂	(σ1s)²	1	dia
He₂	(σ1s)²(σ*1s)²	0	(not stable)
Li₂	(σ2s)²	1	dia
Be₂	(σ2s)²(σ*2s)²	0	(not stable in ground state)
B₂	(π2p)¹(π2p)¹	1	para (two unpaired)
C₂	(π2p)²(π2p)²	2	dia
N₂	(π2p)²(π2p)²(σ2p)²	3	dia
O₂	(σ2p)²(π2p)⁴(π2p)¹(π2p)¹	2	para
F₂	(π*2p)⁴ filled	1	dia
Ne₂	(σ*2p)² added	0	(not stable)

O₂ paramagnetism: two unpaired electrons in degenerate π*2p orbitals. Liquid O₂ sticks to a magnet — a classic demo. Lewis-structure dot diagrams cannot explain this; MO theory does it elegantly.

6.2 σ/π crossover

For Li₂ through N₂, σ2p sits above π2p because of s–p mixing (the 2s orbital overlaps with 2p_z of the partner, pushing σ2p upward). For O₂ and F₂, where the 2s–2p gap is wider, σ2p sits below π2p, and the diagram reverts to the “expected” order. This is the σ/π crossover and is the key to predicting B₂ (paramagnetic) versus a naïve forecast of diamagnetic.

6.3 Heteronuclear diatomics

Energy levels are unequal — the more electronegative atom has lower-energy AOs and contributes more to bonding MOs. HF: σ MO is dominantly F 2p_z (90 %) plus a little H 1s. Polarity follows naturally.

CO has a famously odd MO ordering and a lone pair on C (HOMO is σ on the carbon end), which is why CO binds metals through carbon, not oxygen.

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7. Crystal field theory (CFT) and ligand field theory (LFT)

Hans Bethe (1929) introduced CFT for spectroscopy of ions in crystals; John H. Van Vleck (Nobel 1977) generalized it to coordination compounds. Ligand field theory adds covalent (MO) corrections.

7.1 Octahedral splitting Δ_o

In an octahedral field, the five d orbitals split into:

• t₂g (d_xy, d_xz, d_yz) — lower by 0.4·Δ_o
• e_g (d_z², d_x²−y²) — higher by 0.6·Δ_o

The total separation Δ_o (or 10 Dq) typically falls in 100–400 kJ/mol for 3d transition metals. For [Ti(H₂O)₆]³⁺, Δ_o = 243 kJ/mol (20 300 cm⁻¹), giving an absorption at ~493 nm — purple solution.

7.2 Tetrahedral splitting Δ_t

In tetrahedral geometry, the splitting inverts (e set lower, t₂ higher) and is smaller:

Δ_t = (4/9)·Δ_o ≈ 0.44·Δ_o

So tetrahedral complexes are almost always high-spin for d⁴–d⁷ configurations (Δ_t < pairing energy P).

7.3 Square-planar splitting

A strong tetragonal distortion (elongate the z axis until z-ligands leave) gives a d_x²−y² very high, d_z² much lower, and a four-orbital ground state for d⁸: dxy < dxz, dyz < dz² < dx²−y². Famous d⁸ square-planar complexes: [Ni(CN)₄]²⁻, [PtCl₄]²⁻, [Pd(PPh₃)₂Cl₂].

7.4 High-spin vs low-spin

For d⁴ to d⁷ in O_h symmetry, electrons either fill all t₂g first (low-spin, requires Δ_o > P) or follow Hund (high-spin, Δ_o < P). Examples:

• [Fe(H₂O)₆]³⁺ d⁵: weak field, high-spin, μ = √35 ≈ 5.92 μ_B
• [Fe(CN)₆]³⁻ d⁵: strong field, low-spin, μ = √3 ≈ 1.73 μ_B
• [Co(H₂O)₆]²⁺ d⁷: high-spin, μ ≈ 3.87 μ_B (pink)
• [Co(NH₃)₆]³⁺ d⁶: low-spin, diamagnetic (yellow-orange)

7.5 Spectrochemical series

Ligands ordered by field strength (Δ_o they produce):

I⁻ < Br⁻ < S²⁻ < SCN⁻ < Cl⁻ < NO₃⁻ < F⁻ < OH⁻ < ox²⁻ < H₂O < NCS⁻ < py < NH₃ < en < bpy < phen < CN⁻ < CO

CO is the strongest common π-acceptor (largest Δ_o); I⁻ is the weakest. Mnemonic: from “iodide is feeble” to “carbon monoxide is fierce.”

7.6 Crystal-field stabilization energy (CFSE)

CFSE = (energy of d-electron distribution relative to a spherical field). For O_h:

CFSE = (−0.4·n_t2g + 0.6·n_eg)·Δ_o + (P × extra-pairings)

Example: d⁶ low-spin (t₂g⁶ e_g⁰): CFSE = −2.4·Δ_o + 2P_extra.

CFSE drives the Irving–Williams series stability (next section) and influences hydration enthalpies, ionic radii, and Jahn–Teller distortions.

7.7 Jahn–Teller distortion

Hermann Jahn and Edward Teller (1937) proved that any non-linear molecule in a degenerate electronic state distorts to lift the degeneracy. The textbook case: d⁹ Cu²⁺ in octahedral geometry. The lone electron occupies one of the degenerate e_g pair (d_z² or d_x²−y²); the molecule elongates along z, splitting e_g into d_z² (down) and d_x²−y² (up). The result: [Cu(H₂O)₆]²⁺ has four short equatorial Cu–O (~195 pm) and two long axial Cu–O (~238 pm). Many Cu(II) salts are blue-green for this reason. Also seen weakly in d⁴ high-spin (Cr²⁺, Mn³⁺) and in low-spin d⁷ (Ni³⁺).

7.8 The 18-electron rule (organometallics)

For low-oxidation-state d-block carbonyls, phosphines, and CpML_n complexes, the metal aims for 18 valence electrons (filled n s, n p, (n−1)d): the noble-gas configuration. Counting: metal d-electrons + 2 from each 2-electron σ-donor + extra from π-systems (Cp = 6, ethylene = 2, benzene = 6, NO linear = 3, NO bent = 1).

• Ni(CO)₄: Ni(0) d¹⁰ + 4×2 = 18. ✓
• Fe(CO)₅: Fe(0) d⁸ + 5×2 = 18. ✓
• Cr(CO)₆: Cr(0) d⁶ + 6×2 = 18. ✓
• Ferrocene Fe(Cp)₂: Fe(II) d⁶ + 2×6 = 18. ✓
• W(CO)₆: W(0) d⁶ + 6×2 = 18. ✓

Exceptions: 16-electron d⁸ square planars (Pt(II), Pd(II), Rh(I), Ir(I)) are stable because the 18-electron-occupied orbital lies high.

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8. Coordination chemistry

8.1 Werner’s complexes (1893, Nobel 1913)

Alfred Werner in Zurich proposed that transition metals have a “primary” (ionizable) and “secondary” (coordination) valence. He resolved the optical isomers of [Co(en)₃]Cl₃ in 1911 — proving for the first time that octahedral complexes can be chiral. Werner’s six-coordinate octahedral and four-coordinate tetrahedral/square-planar geometries underlie all of modern coordination chemistry.

8.2 IUPAC nomenclature

Rules (abridged):

1. Cation first, then anion (in formula and name).
2. Ligands alphabetical, then metal, then oxidation state in Roman numerals.
3. Multiplicative prefixes: di, tri, tetra (for simple ligands); bis, tris, tetrakis (for complex ligands or to avoid ambiguity).
4. Anionic ligands end in -o (chloro, hydroxo, cyano); neutrals named (H₂O = aqua, NH₃ = ammine, CO = carbonyl).
5. Anionic complexes end in -ate (ferrate, cuprate, aurate, plumbate).

Examples:

• [Co(NH₃)₆]Cl₃: hexaamminecobalt(III) chloride
• K₃[Fe(CN)₆]: potassium hexacyanoferrate(III)
• [Pt(NH₃)₂Cl₂]: cis- or trans-diamminedichloroplatinum(II) (cisplatin or transplatin)

8.3 Denticity

How many donor atoms a ligand binds through:

• Monodentate: H₂O, NH₃, Cl⁻, CN⁻, CO, py
• Bidentate: ethylenediamine (en, H₂NCH₂CH₂NH₂), oxalate (ox²⁻), acac⁻, bipy, phen, dppe
• Tridentate: dien (diethylenetriamine), terpy
• Tetradentate: salen, porphyrin (planar), trien, cyclam
• Pentadentate: EDTA (sometimes binds only 5 of its 6 donors, leaving one COO⁻ free)
• Hexadentate: EDTA (full coordination)
• Polydentate / macrocyclic: porphyrin, corrin (in B₁₂), cyclam, crown ethers, cryptands

8.4 Chelate effect

A chelating ligand (one that wraps around a metal with multiple donors) forms more stable complexes than the equivalent number of monodentates. Why? Entropy. Replacing six H₂O with three en releases 3 net molecules into solution, so ΔS > 0; ΔG = ΔH − TΔS becomes more negative.

K(Ni(en)₃²⁺) / K(Ni(NH₃)₆²⁺) ≈ 10¹⁰ — ten orders of magnitude favoring the chelate.

The macrocyclic effect adds another factor of 10²–10⁴ for ring-locked ligands (porphyrin, cyclam) because the metal-free macrocycle is already preorganized.

8.5 Hard and Soft Acids and Bases (HSAB)

Ralph Pearson (1963) classified Lewis acids and bases by polarizability:

Hard acids (small, highly charged, low polarizability): H⁺, Li⁺, Na⁺, K⁺, Mg²⁺, Ca²⁺, Al³⁺, Cr³⁺, Fe³⁺, Ti⁴⁺, Si⁴⁺.

Hard bases: F⁻, OH⁻, O²⁻, RO⁻ (alkoxides), NH₃, RNH₂, NO₃⁻, ClO₄⁻, RCOO⁻.

Soft acids (large, low charge, high polarizability): Cu⁺, Ag⁺, Au⁺, Hg²⁺, Pt²⁺, Pd²⁺, Tl⁺, Cd²⁺, Pt⁴⁺.

Soft bases: I⁻, S²⁻, RS⁻, R₂S, R₃P, CN⁻ (C end), CO, H⁻, R⁻, alkenes/arenes.

Borderline acids: Fe²⁺, Co²⁺, Ni²⁺, Cu²⁺, Zn²⁺, Pb²⁺, Sn²⁺, Bi³⁺. Borderline bases: Br⁻, NO₂⁻, SO₃²⁻, N₃⁻, pyridine, aniline.

HSAB rule: hard prefers hard, soft prefers soft. Examples:

• Au⁺ binds CN⁻ (cyanide leaching of gold ore).
• Hg²⁺ binds RS⁻ (mercury poisoning targets thiol enzymes).
• Mg²⁺/Ca²⁺ stick to phosphates and carboxylates (biology).
• Why is Fe³⁺ insoluble at pH 7 but Fe²⁺ less so? Fe³⁺ is harder; it binds OH⁻ tightly and precipitates Fe(OH)₃.

8.6 Common ligands by name

Ligand	Formula	Denticity	Notes
en	H₂NCH₂CH₂NH₂	2	Ethylenediamine
dien	HN(CH₂CH₂NH₂)₂	3	Diethylenetriamine
trien	H₂N(CH₂CH₂NH)₂CH₂CH₂NH₂	4	Triethylenetetraamine
EDTA⁴⁻	(OOCCH₂)₂NCH₂CH₂N(CH₂COO)₂⁴⁻	6	Hexadentate; water hardness titration
acac⁻	CH₃-C(O)-CH=C(O⁻)-CH₃	2	Acetylacetonate
Cp⁻	C₅H₅⁻	η⁵	Cyclopentadienyl, 6-e donor
Cp*	C₅(CH₃)₅⁻	η⁵	Pentamethylcyclopentadienyl
CO	C≡O	1	Carbonyl, σ-donor + π-acceptor
py	C₅H₅N	1	Pyridine
bpy	2,2’-bipyridine	2	π-acid; Ru(bpy)₃²⁺ is a famous photocatalyst
phen	1,10-phenanthroline	2	Like bpy but rigid
dppe	Ph₂PCH₂CH₂PPh₂	2	Bidentate phosphine
dppf	Fe(C₅H₄PPh₂)₂	2	Ferrocene-tethered phosphine
salen²⁻	Schiff base of salicylaldehyde + en	4	Used in Jacobsen epoxidation
porphyrin	C₂₀H₁₄N₄	4 (N’s)	Heme, chlorophyll, B₁₂ corrin (related)

8.7 Stability constants

For ML_n complexes, stepwise and overall constants:

• K_n = [ML_n]/([ML_(n−1)][L])
• β_n = ∏K_i = [ML_n]/([M][L]^n)

For [Cu(NH₃)₄]²⁺: log K₁ = 4.04, log K₂ = 3.43, log K₃ = 2.80, log K₄ = 1.48, log β₄ = 11.75.

Typically K_n decreases with n (statistical + electrostatic + chelate-ring effects).

8.8 Irving–Williams series

For divalent first-row transition-metal complexes, stability follows:

Mn²⁺ < Fe²⁺ < Co²⁺ < Ni²⁺ < Cu²⁺ > Zn²⁺

Why? CFSE rises through Ni²⁺, then Cu²⁺ benefits additionally from Jahn–Teller stabilization; Zn²⁺ has d¹⁰ (zero CFSE) and drops back. This series predicts biological binding preferences — Cu enzymes versus Zn enzymes — and ion-exchange behavior. Harry Irving and Robert Williams (1953) at Oxford.

8.9 Coordination geometries and isomerism

Common geometries by coordination number:

CN	Geometry	Point group	Example
2	Linear	D_∞h	[Ag(NH₃)₂]⁺, [CuCl₂]⁻
3	Trigonal planar	D₃h	[HgI₃]⁻
4	Tetrahedral	T_d	[CoCl₄]²⁻, [Zn(NH₃)₄]²⁺
4	Square planar	D₄h	[Pt(NH₃)₂Cl₂], [Ni(CN)₄]²⁻
5	Trigonal bipyramidal	D₃h	Fe(CO)₅
5	Square pyramidal	C₄v	[VO(acac)₂]
6	Octahedral	O_h	[Co(NH₃)₆]³⁺, [Fe(CN)₆]³⁻
7	Pentagonal bipyramidal	D₅h	[V(CN)₇]⁴⁻
8	Square antiprism	D₄d	[Mo(CN)₈]⁴⁻
9	Tricapped trigonal prism	D₃h	[ReH₉]²⁻

Isomerism in coordination compounds:

• Geometric (cis/trans): square-planar [Pt(NH₃)₂Cl₂] has cis (cisplatin, anticancer) and trans (inactive); octahedral [MA₄B₂] same.
• Facial / meridional (fac/mer): octahedral [MA₃B₃] — three same on a face (fac) or around the meridian (mer).
• Optical (Δ/Λ): chiral [M(en)₃] octahedrals exist as right-handed (Δ) and left-handed (Λ) enantiomers. Werner resolved these 1911–1914.
• Ionization isomers: [Co(NH₃)₅Br]SO₄ vs [Co(NH₃)₅SO₄]Br — same formula, different ion in vs out of coordination sphere.
• Linkage isomers: SCN⁻ binds through S (thiocyanato-S) or N (isothiocyanato-N); NO₂⁻ binds through N (nitro) or O (nitrito); CN⁻ usually C but sometimes N.
• Hydrate / solvate: CrCl₃·6H₂O exists as [Cr(H₂O)₆]Cl₃ (violet), [Cr(H₂O)₅Cl]Cl₂·H₂O (green), [Cr(H₂O)₄Cl₂]Cl·2H₂O (dark green).

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9. Organometallic chemistry

9.1 Ferrocene and the sandwich revolution

Ferrocene Fe(η⁵-C₅H₅)₂ was discovered serendipitously by Kealy and Pauson in 1951 (Nature 168, 1039). Its sandwich structure was solved by Geoffrey Wilkinson (Imperial College) and E. O. Fischer (Munich) independently in 1952. They shared the 1973 Nobel Prize in Chemistry for opening up organometallic π-complex chemistry. Ferrocene is air-stable, sublimes at 100 °C, undergoes Friedel-Crafts acylation, and forms hundreds of derivatives (ferrocenes are now in fuel additives, ferrocenyl drugs like ferroquine for malaria, and chiral phosphine ligands).

9.2 Olefin metathesis

Robert Grubbs (Caltech), Richard Schrock (MIT), and Yves Chauvin (IFP, France) won the 2005 Nobel Prize for olefin metathesis. The Chauvin mechanism (1971) invokes a metal carbene + alkene → metallacyclobutane → new carbene + new alkene. Grubbs catalysts (Ru-based, generations I, II, Hoveyda–Grubbs, etc.) tolerate air, moisture, and many functional groups; they are now standard in polymer chemistry (ROMP, ring-opening metathesis polymerization) and total synthesis.

9.3 Cross-coupling

The 2010 Nobel Prize in Chemistry went to Richard Heck, Ei-ichi Negishi, and Akira Suzuki for palladium-catalyzed C–C bond formation:

• Heck (1972): ArX + CH₂=CHR → ArCH=CHR + HX (with Pd(0) and base)
• Negishi (1977): ArX + R-ZnX → Ar-R (organozinc transmetalation)
• Suzuki (1979): ArX + R-B(OH)₂ → Ar-R (organoboron, with base)
• Also: Stille (R-SnR’₃), Sonogashira (terminal alkyne + Pd/Cu), Kumada (R-MgX), Buchwald-Hartwig (C-N coupling).

Cross-coupling is now the workhorse of pharmaceutical synthesis — pretty much every drug with a biaryl scaffold (losartan, valsartan, tamoxifen analogues, etc.) traces to a Pd-catalyzed step.

9.4 Tolman cone angle

Chadwick Tolman (DuPont, 1970s) introduced the cone angle θ — the apex angle of a cone centered 2.28 Å from the P, encompassing the van der Waals surface of the phosphine R groups. It quantifies steric bulk:

• PH₃: 87°
• PMe₃: 118°
• PPh₃: 145°
• P(o-tol)₃: 194°
• P(tBu)₃: 182°
• PCy₃: 170°
• dppe (chelating): 125° (per P)

Cone angle correlates with reactivity: bulky phosphines favor reductive elimination (Buchwald-Hartwig amination uses XPhos, SPhos, etc.).

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10. Main-group chemistry highlights

10.1 Borane chemistry

William N. Lipscomb (Harvard, Nobel 1976) worked out the structure and bonding of boron hydrides. Key insights:

• Boranes feature three-center two-electron (3c-2e) bonds, where two boron atoms and one hydrogen share a single pair of electrons (a “banana” bond).
• Diborane B₂H₆ has two terminal H per B (2c-2e) and two bridging H (3c-2e).
• Wade’s rules (Kenneth Wade, 1971) classify polyhedral boranes:
  • closo BnHn²⁻ (n vertices, n+1 skeletal pairs)
  • nido BnHn+4 (n+2 pairs, one vertex missing)
  • arachno BnHn+6 (n+3 pairs, two vertices missing)
• B₁₂H₁₂²⁻ is icosahedral, surprisingly stable, used in neutron-capture therapy and as anion for ionic liquids.
• Carboranes (C₂B₁₀H₁₂) replace two B–H vertices with C–H; ortho, meta, para isomers exist; o-carborane is extraordinarily robust (thermally stable to 500 °C+).

10.2 Silicates

Silicate anions classify by polymerization of SiO₄ tetrahedra. “Q” notation refers to the number of bridging oxygens per Si:

• Q⁰ (orthosilicate, no sharing): olivine (Mg,Fe)₂SiO₄, garnets, zircon ZrSiO₄.
• Q¹ (sorosilicate, sharing 1 O): hemimorphite Zn₄Si₂O₇(OH)₂·H₂O.
• Q² (cyclosilicate, ring): beryl Be₃Al₂Si₆O₁₈ (the structural mineral for emeralds and aquamarines), tourmaline.
• Q² (chain inosilicate): pyroxenes (e.g., enstatite MgSiO₃), and Q³ chains (amphiboles like tremolite).
• Q³ (sheet phyllosilicate): micas (muscovite, biotite), clays (kaolinite, montmorillonite), talc, chlorite.
• Q⁴ (framework tectosilicate): quartz SiO₂, feldspars (orthoclase KAlSi₃O₈, plagioclase series), zeolites.

10.3 Aluminosilicates and zeolites

Replacing some Si⁴⁺ with Al³⁺ in a tetrahedral framework creates a charge deficit, balanced by exchangeable cations (Na⁺, K⁺, Ca²⁺, NH₄⁺, H⁺). The pores form regular cages and channels. Zeolites with industrial impact:

• Linde A (LTA): Na/Ca exchange (water softeners).
• Y zeolite (FAU): fluid catalytic cracking (FCC) in petroleum refining — cracks heavy gas oil to gasoline/diesel.
• ZSM-5 (MFI): methanol-to-gasoline (Mobil 1976), xylene isomerization.
• SAPO-34: methanol-to-olefins (MTO, Dalian / DICP).

Zeolite synthesis and structure are themselves a subdiscipline; the IZA database lists 250+ framework types as of 2025. See zeolites-and-porous-solids.

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11. Transition-metal chemistry

11.1 The 3d row (Sc–Zn)

Z	Element	Common oxidation states	Notable chemistry
21	Sc	+3	Sc₂O₃ in light alloys; aerospace
22	Ti	+2, +3, +4	TiO₂ pigment; Ti–Al alloys
23	V	+2, +3, +4, +5	V₂O₅ contact process; redox-flow batteries
24	Cr	+2, +3, +6	Cr₂O₇²⁻; stainless steel
25	Mn	+2, +3, +4, +6, +7	MnO₄⁻ (purple); brain MRI artifact
26	Fe	+2, +3	Hemoglobin, magnetite, steel
27	Co	+2, +3	Vitamin B₁₂; lithium battery cathodes
28	Ni	+2	Ni alloys; hydrogenation catalyst
29	Cu	+1, +2	Wiring; Cu enzymes (cytochrome c oxidase)
30	Zn	+2	Galvanizing; Zn enzymes (carbonic anhydrase)

11.2 4d row (Y–Cd) and 5d row (La/Hf–Hg)

The 4d row is the home of noble catalysts: Mo (industrial HDS — Mo–S sulfide), Ru (metathesis, Haber-Bosch alt), Rh (Wilkinson hydrogenation, automotive 3-way), Pd (cross-coupling, hydrogenation), Ag (silver halide photography legacy, antimicrobial). Cd is toxic, +2, used historically in pigments and batteries.

5d: Hf (nuclear-reactor control rods), Ta (capacitors and supercapacitors), W (lightbulb filaments historically, WC tooling), Re (jet-engine superalloys), Os (osmium tetroxide oxidation), Ir (the densest stable element; OLED phosphorescent dyes), Pt (catalytic converters, anticancer), Au (jewelry, electronics; gold colloid catalysis discovered ~2000 — surprising, since bulk gold is inert), Hg (the only liquid metal at 298 K).

Lanthanide contraction makes Hf nearly identical in radius to Zr (only ~1 pm difference), which is why Hf and Zr are extremely difficult to separate by classical means.

11.3 Notable oxidation-state couples and redox

• Fe²⁺ / Fe³⁺: E° = +0.77 V (acid); rust chemistry; ferritin storage.
• Cu⁺ / Cu²⁺: E° = +0.16 V; disproportionation in water (Cu⁺ unstable as aquo).
• Mn²⁺ → MnO₄⁻: traverses Mn(II, III, IV, VI, VII); MnO₄⁻ in acid: E° = +1.51 V (strong oxidant, deep purple).
• V(II)→V(V): colors are violet [V(H₂O)₆]²⁺ → green V³⁺ → blue VO²⁺ → yellow VO₂⁺.

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12. Lanthanides (4f, Ce–Lu)

The 4f electrons are core-like — heavily shielded by 5s and 5p — so lanthanides behave very similarly chemically. +3 is overwhelmingly dominant; Ce can reach +4, Eu and Yb can reach +2.

12.1 Lanthanide contraction

The 4f electrons shield poorly, so Zeff rises across the row. Ionic radius of Ln³⁺ drops from La³⁺ (103 pm) to Lu³⁺ (86 pm). This contraction is what makes Hf ~ Zr (above), and it also forms the basis for separation by ion exchange (the smaller, later Ln³⁺ ions bind more tightly to chelates like α-hydroxyisobutyrate).

12.2 Industrial uses

• Nd₂Fe₁₄B magnets: the strongest permanent magnets known. Coercivity > 1 T. Used in EV motors, wind turbines, hard drives, headphones.
• SmCo₅ and Sm₂Co₁₇ magnets: lower energy product than NdFeB but higher Curie temperature (~700 °C) — used where heat tolerance matters.
• Y₃Al₅O₁₂:Ce (YAG): yellow phosphor pumped by blue InGaN LEDs → white LED lighting. The single biggest consumer of cerium worldwide.
• Eu²⁺ (red phosphor) and Tb³⁺ (green phosphor) in fluorescent and CRT tubes (legacy).
• Gd-DTPA, gadobutrol (Gadovist), gadoteridol (ProHance): MRI contrast agents. Gd³⁺ has 7 unpaired f-electrons (S = 7/2), shortening T₁; the DTPA chelate (or DOTA macrocycle) keeps free Gd³⁺ (toxic, can mimic Ca²⁺) sequestered.
• Er, Tm, Yb: solid-state laser dopants (Er:YAG at 2.94 μm for surgical/cosmetic lasers).
• Ce: pyrophoric — used in lighter flints (mischmetal alloy).
• Lanthanum nickel hydride LaNi₅H₆: classic hydrogen-storage material.

See rare-earth-magnets and permanent-magnet-motors.

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13. Actinides (5f, Th–Lr)

The actinides span +3 to +7 oxidation states. The 5f electrons are less core-like than 4f, so actinides behave more like d-block elements (greater variety in oxidation state and covalency, especially in the early actinides Pa, U, Np, Pu).

13.1 Uranium

Naturally occurring as U-238 (99.27 %), U-235 (0.72 %), and trace U-234. UF₆ gaseous diffusion / centrifuge enrichment raises U-235 to:

• ~3–5 % for LWR fuel
• ~20 % HALEU (limit for research reactors and some advanced reactors)
• ≥ 90 % HEU (weapons-grade)

Common compounds: UO₂ (fuel pellets), UF₄ (intermediate), UO₃ (yellowcake), uranyl ion UO₂²⁺ (linear O=U=O, 5f⁰ d⁰, yellow-green, fluorescent under UV — historical use in Vaseline glass). MOX (mixed-oxide) fuel is UO₂ + PuO₂ for reactor recycle of plutonium.

13.2 Plutonium

Pu was first synthesized by Seaborg, McMillan, Kennedy, Wahl at UC Berkeley in 1940–41. Glenn Seaborg (Nobel 1951) elucidated the transuranic chemistry.

Pu has six stable oxidation states (+3, +4, +5, +6, +7 transient, all coexisting in some solutions — a chemist’s nightmare). Aqueous Pu(IV) disproportionates partially into Pu(III) and Pu(V) / Pu(VI). The PUREX process (plutonium uranium reduction extraction) uses tributyl phosphate in kerosene to separate U and Pu from fission products in spent fuel.

The Manhattan Project (1943–45) at Los Alamos and Hanford produced ~6 kg of Pu-239 for the Trinity test and the Fat Man device. Pu-239: half-life 24 100 years.

13.3 Americium and lighter applications

Am-241 (half-life 432 years) is an α-emitter used in ionization-chamber smoke detectors (the gold standard for ionization detectors, ~0.3 μg per detector).

13.4 Transuranics

Elements Z = 93 (Np) through Z = 118 (Og) are all synthetic. Heavy-element chemistry is conducted on single atoms (“one-atom-at-a-time chemistry”) at GSI Darmstadt, JINR Dubna, RIKEN, LBNL. The heaviest with reasonably characterized chemistry is Sg (seaborgium, Z = 106), shown to behave like Mo/W in its +6 state.

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14. Solid-state chemistry

14.1 Close-packing and lattices

Most metals adopt one of three close-packed lattices:

• HCP (hexagonal close-packed): ABABAB… layers. c/a ideal = 1.633. Mg, Zn, Cd, Be, Ti (α), Co (low-T), Ru, Os.
• FCC (face-centered cubic, equivalent to cubic close-packed CCP): ABCABC… stacking. Al, Cu, Ag, Au, Ni, Pd, Pt, Pb, γ-Fe.
• BCC (body-centered cubic): not close-packed, but common. Na, K, Cs, V, Cr, Fe (α), Mo, W, Ta, Nb.

FCC and HCP both have packing efficiency 74 %; BCC, 68 %; simple cubic, 52 %.

14.2 Interstitial sites

In close-packed lattices:

• Octahedral holes: r/R = 0.414 (maximum sphere radius / host sphere radius). One per atom.
• Tetrahedral holes: r/R = 0.225. Two per atom.

These limits determine which ionic structures form: when r⁺/r⁻ is between 0.414 and 0.732, octahedral coordination (NaCl-type); 0.225–0.414 favors tetrahedral (ZnS-type); above 0.732, eightfold (CsCl-type) or 12-fold (perovskite A-site).

14.3 Canonical ionic structures

Structure	Formula type	Examples	Coordination
Rock salt	AB	NaCl, KCl, MgO, FeO, LiF, AgCl	6:6
CsCl	AB	CsCl, CsBr, CsI, NH₄Cl	8:8
Zinc blende	AB	ZnS, GaAs, CuCl, β-SiC, InP	4:4
Wurtzite	AB	ZnO, GaN, AlN, β-ZnS, CdS	4:4
Fluorite	AB₂	CaF₂, UO₂, ZrO₂ (cubic), CeO₂	8:4
Antifluorite	A₂B	Li₂O, Na₂O, K₂S	4:8
Rutile	AB₂	TiO₂ (rutile), SnO₂, MnO₂, RuO₂	6:3
Corundum	A₂B₃	Al₂O₃ (sapphire), Cr₂O₃, Fe₂O₃ (α-hematite)	6
Perovskite	ABO₃	CaTiO₃, BaTiO₃, SrTiO₃, LaMnO₃, CH₃NH₃PbI₃	12 (A) / 6 (B)
Spinel	AB₂O₄	MgAl₂O₄, Fe₃O₄ (magnetite, inverse), MgFe₂O₄	4 (A) / 6 (B)

14.4 Perovskite and the Goldschmidt tolerance factor

Victor Goldschmidt (1926) proposed a tolerance factor:

t = (r_A + r_O) / [√2 · (r_B + r_O)]

• t ≈ 1.0: ideal cubic perovskite (SrTiO₃, BaTiO₃ above T_c).
• 0.9 < t < 1.0: slight distortions (BaTiO₃ tetragonal at 298 K, ferroelectric).
• 0.7 < t < 0.9: orthorhombic distortion (CaTiO₃, GdFeO₃ type).
• t > 1.0: hexagonal polytypes.
• t < 0.7: ilmenite or other non-perovskite.

Perovskites underlie a huge fraction of functional ceramics: BaTiO₃ (capacitors, piezo), Pb(Zr,Ti)O₃ “PZT” (piezo), LaMnO₃ (CMR, colossal magnetoresistance), LaFeO₃ / LaCoO₃ (mixed conductors for SOFC cathodes), and the methylammonium-lead-iodide perovskites CH₃NH₃PbI₃ (PSCs, perovskite solar cells, >25 % efficiency by 2024).

14.5 Defects in solids

Schottky defect: cation + anion vacancy pair, charge-neutral. Common in alkali halides; raises ionic conductivity.

Frenkel defect: cation moves to an interstitial site, leaving a cation vacancy. Common in AgX (Ag⁺ small).

F-center: anion vacancy occupied by an electron. Colors otherwise-clear crystals (KCl irradiated → magenta, NaCl irradiated → yellow-brown).

Substitutional / interstitial dopants: P or As in Si (n-type, donor); B or Al in Si (p-type, acceptor). Concentrations of 10¹⁵–10²⁰ cm⁻³ engineer carrier density. See semiconductor-doping.

Kröger–Vink notation (Ferdinand Kröger and H. J. Vink, 1956): site_charge as superscript, defect species as subscript. Examples:

• V_O^•• (oxygen vacancy with effective +2 charge in oxide)
• Y_Zr^′ (Y³⁺ on Zr⁴⁺ site, effective −1 charge — the basis for YSZ)
• Li_Li^× (Li on its own site, neutral — ”×” superscript)
• e^′ (free electron, −1)
• h^• (hole, +1)

Charge balance: sum of effective charges over all defects = 0.

14.6 Ionic conductors

Selected fast-ion conductors:

• β-alumina (Na-β″-Al₂O₃): Na⁺ conduction through ~0.5 nm spaced layers. Used in Na-S batteries (NGK; grid storage).
• YSZ (yttria-stabilized zirconia): Zr₁₋ₓY_xO₂₋_x/2, oxide-ion conductor. The electrolyte for solid-oxide fuel cells (SOFC) at 700–1000 °C. Y³⁺ doping creates O vacancies (charge balance: 2Y_Zr^′ + V_O^••).
• GDC / SDC (Gd- or Sm-doped ceria): similar role at slightly lower temperatures.
• LISICON / NASICON family: Li₃PO₄ or Na₃Zr₂Si₂PO₁₂ frameworks for Li⁺ / Na⁺ batteries.
• LLZO (Li₇La₃Zr₂O₁₂) garnet: solid Li⁺ conductor, ~10⁻³ S/cm at 298 K. Candidate solid electrolyte for next-gen Li batteries.
• Sulfide superionic: Li₁₀GeP₂S₁₂ (LGPS) — 1.2 × 10⁻² S/cm, matches liquid-electrolyte conductivity. Kanno et al. 2011.

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15. Acid–base theories

Five overlapping definitions:

1. Arrhenius (1884): acid releases H⁺ in water, base releases OH⁻. Works for HCl + NaOH but not for NH₃ + HCl(g).
2. Brønsted–Lowry (1923): acid is a proton donor, base a proton acceptor. Conjugate acid/base pairs. Works in any solvent.
3. Lewis (1923): acid is an electron-pair acceptor, base is a donor. Encompasses BF₃ + NH₃ → F₃B–NH₃; explains coordination chemistry.
4. Solvent system: acid increases the “solvent cation” (H₃O⁺ in water, NH₄⁺ in liquid ammonia). Useful in non-aqueous solvents.
5. Pearson HSAB (1963) (above): refines Lewis theory by adding hard/soft polarizability classification.

In aqueous solution, pK_a values of representative inorganic acids:

• HF 3.17, HCl ≈ −7, HBr ≈ −9, HI ≈ −10.
• H₃PO₄: pK_a 2.15, 7.20, 12.35.
• H₂SO₄: pK_a1 ≈ −3, pK_a2 1.99.
• H₂CO₃ (CO₂ + H₂O): pK_a1 6.35, pK_a2 10.33.
• HNO₃: pK_a ≈ −1.4.

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16. Inorganic catalysis

16.1 Haber–Bosch ammonia synthesis

Fritz Haber (Nobel 1918) at Karlsruhe developed laboratory NH₃ synthesis; Carl Bosch (Nobel 1931) at BASF scaled it. Modern process:

• N₂ + 3H₂ ⇌ 2NH₃ (ΔH = −92 kJ/mol)
• Catalyst: K-promoted Fe (magnetite-derived, with Al₂O₃, CaO, K₂O promoters). Newer: Ru/C-K (Kellogg Advanced Ammonia Process, KAAP).
• Conditions: 150–300 bar, 400–500 °C. Typical industrial: 200 bar, 450 °C.
• Single-pass conversion ~15 %; ammonia is condensed out (boils −33 °C) and unreacted gas recycled.
• World production: ~175 Mt NH₃/year (2024). Half goes to fertilizer (urea, ammonium nitrate, ammonium phosphates).
• Energy footprint: ~1.8 % of global energy use; ~1.4 % of global CO₂ emissions (steam-methane-reforming for H₂). The “green ammonia” / electrolytic-H₂ route is the active decarbonization target.

16.2 Contact process (H₂SO₄)

• SO₂ + ½O₂ → SO₃, ΔH = −98 kJ/mol.
• Catalyst: V₂O₅ on silica, 420–620 °C. Mechanism involves V⁵⁺/V⁴⁺ redox cycling with bound SO₂/SO₃.
• SO₃ is absorbed into 98 % H₂SO₄ (not water — to avoid mist) → oleum → diluted to product.
• World production: ~280 Mt/yr (largest-volume industrial chemical).

16.3 Ostwald process (HNO₃)

• 4 NH₃ + 5 O₂ → 4 NO + 6 H₂O (catalyst: Pt–Rh gauze, 800–900 °C, 1–10 bar, contact time ~1 ms)
• 2 NO + O₂ → 2 NO₂
• 3 NO₂ + H₂O → 2 HNO₃ + NO

The Pt-Rh gauze is recovered and recycled; losses of Pt are a major operating cost. Wilhelm Ostwald (Nobel 1909) developed the process around 1902.

16.4 Fischer–Tropsch synthesis

CO + 2H₂ → −(CH₂)− + H₂O (then chain growth)

• Catalysts: Fe (cheaper, can use coal-derived syngas with low H₂/CO; Sasol in South Africa), Co (more selective for longer chains; Shell GTL plants in Bintulu and Pearl-GTL Qatar — 140 000 bpd, the world’s largest GTL).
• Products: gasoline, diesel, waxes (the Anderson-Schulz-Flory distribution).
• Developed in 1925 by Franz Fischer and Hans Tropsch at the Kaiser Wilhelm Institute Mülheim.

16.5 Polyolefin catalysis

• Phillips catalyst (J.P. Hogan and R.L. Banks, 1951): Cr/SiO₂ → HDPE. Still used today; >40 % of world HDPE.
• Ziegler–Natta (Karl Ziegler at Mülheim, Giulio Natta at Milan, Nobel 1963): TiCl₃ or TiCl₄ with AlR₃ cocatalyst. Produces stereoregular (isotactic) polypropylene. MgCl₂-supported versions (Montedison, Mitsui) gave 100× productivity gains in the 1970s.
• Metallocenes / Kaminsky catalysts (Walter Kaminsky, Hamburg, 1980s): Cp₂ZrCl₂ + MAO (methylaluminoxane). Single-site, very narrow MW distribution, tunable tacticity.
• Post-metallocenes: phenoxyimine “FI” catalysts, nickel-diimine (Brookhart), Ni/Pd α-diimine for branched polyethylene from ethylene alone.

16.6 Other notable inorganic catalysis

• Wacker process (PdCl₂/CuCl₂, 1959): ethylene + ½O₂ → acetaldehyde. Pd(II) → Pd(0), re-oxidized by Cu(II).
• SCR (selective catalytic reduction) for NOx in diesel exhaust: V₂O₅-WO₃/TiO₂ or Cu-/Fe-zeolites. NH₃ (from urea) reduces NO to N₂.
• Three-way catalytic converter: Pt + Pd + Rh on cordierite honeycomb with washcoat (CeO₂-ZrO₂). CO + NO + O₂ → CO₂ + N₂.
• Hydrodesulfurization (HDS): Co-Mo-S or Ni-Mo-S on Al₂O₃, removes S from petroleum streams as H₂S.
• Methanol synthesis: Cu/ZnO/Al₂O₃, 50–100 bar, 230–270 °C, CO + 2H₂ → CH₃OH.

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17. Bioinorganic chemistry

The intersection of inorganic chemistry and biology — essential elements as cofactors in enzymes and structural proteins.

17.1 Heme proteins (Fe-porphyrin)

• Hemoglobin (tetrameric, α₂β₂): four heme groups bind O₂ reversibly via Fe(II); cooperative binding gives sigmoidal saturation curve. Bohr effect: lower pH (active tissue) releases O₂.
• Myoglobin: monomeric Fe-heme O₂ storage in muscle. Sequenced by John Kendrew and Max Perutz, X-ray structures published in 1959–60 (Nobel 1962).
• Cytochromes a, b, c: electron-transport chain in mitochondria, all Fe-hemes; the protein environment tunes E° from −0.4 to +0.4 V.
• Cytochrome c oxidase: Cu_A binuclear, Cu_B + Fe-a₃ heme reduce O₂ to H₂O.
• Cytochrome P450: O₂ activation, hydroxylation of substrates (drug metabolism, steroid biosynthesis).

17.2 Chlorophyll (Mg-porphyrin)

Photosynthetic light absorption: chlorophyll a (Q_y absorbance at 662 nm), chlorophyll b (642 nm). Mg²⁺ centers a 20-carbon porphyrin macrocycle with a fifth ring; the long phytyl tail anchors the molecule in the thylakoid membrane. PSII (oxygen-evolving complex with Mn₄CaO₅ cluster) splits water; PSI reduces ferredoxin.

17.3 Vitamin B₁₂ (Co-corrin)

Cobalt centered in a 19-carbon corrin (porphyrin-like but with one less methene bridge). Co–C bond to 5’-deoxyadenosyl or methyl group. The first organometallic biomolecule discovered. Dorothy Crowfoot Hodgkin (Nobel 1964) solved its X-ray structure 1955–56. Robert Burns Woodward and Albert Eschenmoser synthesized it in 1972 — a 100-step total synthesis that defined an era of organic chemistry.

17.4 Nitrogenase

The only enzyme that fixes atmospheric N₂. FeMoco: Mo–7Fe–9S–C–homocitrate cluster discovered in 1992 (Kim and Rees X-ray) with a central carbide revealed in 2011 (Spatzal, Einsle, et al.). Operates at 298 K and 1 bar — chemistry that Haber–Bosch achieves only at 200 bar and 450 °C. The mechanism is still partially debated.

17.5 Cytochromes, ferredoxins, zinc fingers, blue copper proteins

• Ferredoxins: [2Fe-2S], [3Fe-4S], [4Fe-4S] iron-sulfur clusters; electron transport at very negative potentials.
• Zinc fingers: Zn²⁺ stabilizes small DNA-binding protein motifs (Cys₂His₂); each Zn organizes ~30 amino acids. Thousands of zinc-finger transcription factors in mammalian genomes.
• Plastocyanin (a “blue copper protein”): Cu(I/II) with distorted tetrahedral N₂S₂ coordination (2 His, 1 Cys, 1 Met). Tuned E° for plant electron transport. The Cu–S(Cys) charge transfer gives the deep blue color.
• Carbonic anhydrase: Zn²⁺ tetrahedrally coordinated (3 His + OH⁻), catalyzes CO₂ + H₂O ⇌ HCO₃⁻ + H⁺ at ~10⁶ s⁻¹.

See biochemistry-foundations for the cellular context.

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18. Nuclear chemistry

18.1 Decay modes

• α decay: emission of He-4 nucleus. ⁴²⁾ ²³⁸U → ²³⁴Th + α. Mostly for Z > 82.
• β⁻ decay: n → p + e⁻ + ν̄_e. Examples: ¹⁴C → ¹⁴N, T = 5730 yr.
• β⁺ decay / electron capture: p → n + e⁺ + ν_e. Examples: ¹⁸F → ¹⁸O, T = 110 min (PET tracer).
• γ decay: nuclear de-excitation, photon emission, no change in Z or A.
• Spontaneous fission: heavy nuclei split (e.g., ²⁵²Cf neutron source).
• Neutron emission, proton emission, cluster decay: rarer.

18.2 Half-life and decay law

N(t) = N₀ · exp(−λt); T_½ = ln 2 / λ.

Activity A = λN (in becquerels, Bq = 1 decay/s; or curies Ci = 3.7 × 10¹⁰ Bq).

18.3 Decay chains

• Uranium series: ²³⁸U → ²³⁴Th → ²³⁴Pa → ²³⁴U → … → ²¹⁰Pb → ²¹⁰Bi → ²¹⁰Po → ²⁰⁶Pb (stable). 14 steps, dominated by α and β⁻.
• Actinium series: ²³⁵U → ²⁰⁷Pb.
• Thorium series: ²³²Th → ²⁰⁸Pb.
• Neptunium series: ²³⁷Np → ²⁰⁹Bi (now-extinct since formation of Earth — half-life only 2 Myr).

18.4 Fission and fusion

• Fission: ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + ~200 MeV. Other fissile nuclei: ²³⁹Pu, ²³³U (Th cycle). Critical mass for bare-sphere ²³⁵U ~52 kg; for ²³⁹Pu ~10 kg.
• Fusion: D + T → ⁴He + n + 17.6 MeV. Cross-section peaks at ~64 keV ion energy. ITER under construction in France targets Q = 10 ratio of fusion power to heating.

18.5 Medical radioisotopes

• ⁹⁹ᵐTc (γ, 140 keV, T_½ = 6.0 h): the workhorse of nuclear medicine — bone scans, cardiac perfusion, thyroid, brain. Produced by ⁹⁹Mo (from ²³⁵U fission) decay; eluted from “Mo cows” in radiopharmacies daily.
• ¹³¹I (β⁻ + γ, T_½ = 8.0 d): thyroid imaging and ablation; thyroid cancer treatment.
• ¹⁸F (β⁺, T_½ = 110 min): FDG-PET (fluorodeoxyglucose) for oncology imaging. Produced on-site in medical cyclotrons.
• ⁶⁸Ga (β⁺, T_½ = 68 min): from ⁶⁸Ge/⁶⁸Ga generator; DOTA-conjugated peptides (DOTATATE for neuroendocrine tumors).
• ¹⁷⁷Lu (β⁻ + γ, T_½ = 6.7 d): theranostic — Lutathera (DOTATATE-¹⁷⁷Lu) approved 2018 for NETs; Pluvicto (PSMA-617-¹⁷⁷Lu) approved 2022 for prostate cancer. Growing market.
• ²²⁵Ac (α, T_½ = 10 d): targeted α therapy under clinical investigation.

See nuclear-medicine-instrumentation and radiopharmaceuticals.

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19. Industrial inorganic chemicals — top 10 by volume (global, ~2024)

Rank	Chemical	Annual production (Mt)	Primary use
1	H₂SO₄	~280	Fertilizer (phosphates), batteries, refining
2	N₂	~150 (from air sep)	Cryogen, inerting, fertilizer feedstock
3	O₂	~120	Steelmaking BOF, medical, oxidation processes
4	Ethylene	~210 (organic — listed)	Polyethylene, ethylene oxide, EDC
5	Lime CaO	~430 (incl. construction)	Cement, steel slag flux, FGD
6	NH₃	~185	Urea, ammonium nitrate / phosphates, HNO₃
7	NaOH	~80	Pulp & paper, alumina (Bayer), soap, water treatment
8	Cl₂	~70	PVC (via EDC/VCM), bleaches, disinfection
9	Propylene	~140 (organic)	Polypropylene, propylene oxide
10	H₃PO₄	~75 (as P₂O₅ basis ~50)	Fertilizer, food acid, detergents

Other notables: H₂ (~95 Mt, ~96 % from SMR), HNO₃ (~80 Mt), urea (~190 Mt — but classified as fertilizer end-product), HCl (~22 Mt), TiO₂ (~10 Mt pigment), Cl₂-derived chlorinated organics, soda ash Na₂CO₃ (~60 Mt). Cross-link to chemical-process-economics and industrial-feedstocks.

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20. Worked numerical example: lattice energy of MgO via Born–Haber

1. Mg(s) → Mg(g): ΔH_sub = +148 kJ/mol
2. Mg(g) → Mg⁺(g): IE₁ = +738
3. Mg⁺(g) → Mg²⁺(g): IE₂ = +1450
4. ½ O₂(g) → O(g): +249
5. O(g) + e⁻ → O⁻(g): EA₁ = −141
6. O⁻(g) + e⁻ → O²⁻(g): EA₂ = +744 (endothermic!)
7. Mg²⁺(g) + O²⁻(g) → MgO(s): U_lattice = ?

ΔH_f(MgO) = −602 kJ/mol. Closing: U = −602 − 148 − 738 − 1450 − 249 + 141 − 744 = −3790 kJ/mol

This enormous lattice energy is the only reason MgO is stable as an ionic solid despite the highly endothermic formation of O²⁻ from O(g). Notice how multiply-charged oxide chemistry depends on gigantic Madelung sums to compensate the unfavorable second electron affinity — a recurring theme for sulfides, oxides, nitrides, carbides.

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21. Selected named effects, equations, and unit conventions

• Mulliken population analysis: partition of MO electron density to atomic centers; basis-set-dependent.
• Tanabe–Sugano diagrams: plot of d-electron term energies versus Δ_o/B (Racah parameter) for each dⁿ configuration; predicts UV-Vis spectra.
• Racah parameters A, B, C: electron-repulsion parameters for free-ion d-electron terms; B for V³⁺ ~860 cm⁻¹, drops to ~700 in complexes (nephelauxetic effect, β = B(complex)/B(ion) < 1, soft-ligand-dependent).
• Nephelauxetic series: F⁻ > H₂O > NH₃ > ox²⁻ > Cl⁻ > CN⁻ > Br⁻ > I⁻ (decreasing β = expansion of d-orbital cloud).
• Group theory in inorganic spectroscopy: O_h has 48 operations, characters in C₄v, D₃h, D₄h used in IR/Raman selection rules. See group-theory-for-chemistry.
• SI primary conventions: lattice energy in kJ/mol; bond lengths in pm (legacy Å in older texts; 1 Å = 100 pm); cross-sections in barns (1 b = 10⁻²⁸ m²); magnetic moment in Bohr magnetons μ_B = 9.274 × 10⁻²⁴ J/T; activity in Bq (replacing curies in modern reporting).

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22. History and context

• Mendeleev 1869: periodic table.
• Werner 1893: coordination chemistry. Nobel 1913.
• Lewis 1916: electron-pair bonding. (Never won, despite being a perennial nominee.)
• Brønsted, Lowry 1923: proton acid–base.
• Lewis 1923: electron-pair acid–base.
• Bethe 1929: CFT.
• Slater 1930: shielding rules.
• Pauling 1932: electronegativity, hybridization. Nobel 1954, again Peace 1962.
• Mulliken 1934 / 1955: MO theory, populations. Nobel 1966.
• Jahn–Teller 1937: theorem.
• Seaborg 1940s–60s: actinide concept, transuranic elements. Nobel 1951.
• Wilkinson, Fischer 1952: ferrocene structure. Nobel 1973.
• Gillespie–Nyholm 1957: VSEPR.
• Pearson 1963: HSAB.
• Lipscomb 1976: borane bonding.
• Van Vleck 1977: magnetism + LFT.
• Cram, Lehn, Pedersen 1987: supramolecular host-guest chemistry (crown ethers, cryptands).
• Smalley, Curl, Kroto 1996: C₆₀ fullerene.
• Grubbs, Schrock, Chauvin 2005: olefin metathesis.
• Heck, Negishi, Suzuki 2010: cross-coupling.

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Adjacent

• organic-chemistry-foundations — sibling discipline; many organometallic and bioinorganic motifs sit at the boundary.
• analytical-chemistry-methods — XRD, ICP-MS, EXAFS, EPR, UV-Vis used to characterize inorganic systems.
• _index — solid-state structures, defects, ionic conductors, perovskite photovoltaics, rare-earth magnets.
• chemical-process-economics — Haber-Bosch, contact, Ostwald scaled to billions of dollars and gigatons.
• group-theory-for-chemistry — symmetry operations, character tables, MO and vibrational selection rules.
• _index — computational-chemistry stack (DFT, ab initio, plane-wave codes) that backs the modern interpretation of bonding, MO ordering, and relativistic effects.