Coordination Compounds Formula Sheet

Last-minute revision sheet for Coordination Compounds. Every formula, key relation, and high-yield fact below is pulled straight from the chapter topics — scan it the night before the exam.

Core Formulas

Quantity	Formula	Notes
Magnetic moment	$\mu = \sqrt{n(n+2)}\ \text{BM}$	$n$ = number of unpaired electrons
Overall stability constant	$\beta_n = K_1 \times K_2 \times K_3 \times \dots \times K_n$	Product of stepwise constants
Dissociation (instability) constant	$K_d = \dfrac{1}{\beta_n}$	$K_d \times \beta_n = 1$
Octahedral CFSE	$\text{CFSE} = [(-0.4)\,n_{t_{2g}} + (+0.6)\,n_{e_g}]\,\Delta_0$	Add $+P$ per forced pairing
Tetrahedral CFSE	$\text{CFSE} = [(-0.6)\,n_{e} + (+0.4)\,n_{t_2}]\,\Delta_t$
Tetrahedral vs octahedral splitting	$\Delta_t \approx \dfrac{4}{9}\,\Delta_0$	$\Delta_t$ always small
Splitting energy ↔ wavelength	$\Delta_0 = h\nu = \dfrac{hc}{\lambda}$	Basis of colour (d–d transitions)

$$\boxed{\mu = \sqrt{n(n+2)}\ \text{BM}}$$$$\boxed{\beta_n = K_1 K_2 K_3 \dots K_n \qquad K_d = \frac{1}{\beta_n}}$$

Werner’s Theory & Counting Ions

Concept	Key relation
Primary valency	= Oxidation state of metal (ionizable, non-directional, satisfied by anions)
Secondary valency	= Coordination number (non-ionizable, directional, fixed geometry)
Ions from $[M(\text{ligands})_x]Y_n$	$= n + 1$ (1 cation + $n$ anions)
AgCl precipitated	= number of ionizable $\text{Cl}^-$ outside the coordination sphere

Conductivity / ion count for CoCl₃·xNH₃ series:

Compound	Werner formula	Ions	AgCl ppt
CoCl₃·6NH₃	$[\text{Co(NH}_3)_6]\text{Cl}_3$	4	3 mol
CoCl₃·5NH₃	$[\text{Co(NH}_3)_5\text{Cl}]\text{Cl}_2$	3	2 mol
CoCl₃·4NH₃	$[\text{Co(NH}_3)_4\text{Cl}_2]\text{Cl}$	2	1 mol
CoCl₃·3NH₃	$[\text{Co(NH}_3)_3\text{Cl}_3]$	0	0

Coordination number	Geometry (Werner)
2	Linear
4	Tetrahedral or Square planar
6	Octahedral

Nomenclature Quick Rules

• Order: cation before anion; within sphere → ligands (alphabetical, ignore di/tri prefixes) then metal + oxidation state in Roman numerals.
• Anionic complex: metal gets -ate suffix (often Latin: iron → ferrate, copper → cuprate, silver → argentate, gold → aurate, tin → stannate, lead → plumbate).
• Prefixes: simple ligands → di, tri, tetra, penta, hexa; complex ligands (en, etc.) → bis, tris, tetrakis.
• Bridging ligand: μ (mu). Ambidentate binding atom: κ (kappa).

Ligand	Name	Ligand	Name
H₂O	aqua	NH₃	ammine
CO	carbonyl	NO	nitrosyl
Cl⁻	chlorido	Br⁻	bromido
CN⁻	cyanido	OH⁻	hydroxido
O²⁻	oxido	SO₄²⁻	sulfato
NO₂⁻ (κN)	nitrito-κN	ONO⁻ (κO)	nitrito-κO
SCN⁻ (κS)	thiocyanato-κS	NCS⁻ (κN)	thiocyanato-κN

Isomerism

graph TD
    A[Isomerism] --> B[Structural]
    A --> C[Stereo]
    B --> B1[Ionization]
    B --> B2[Hydrate/Solvate]
    B --> B3[Linkage]
    B --> B4[Coordination]
    B --> B5[Ligand]
    C --> C1[Geometrical cis/trans, fac/mer]
    C --> C2[Optical Δ/Λ]

Type	Same	Different	Example
Ionization	Formula	Ions formed	$[\text{Co(NH}_3)_5\text{Br}]\text{SO}_4$ vs $[\text{Co(NH}_3)_5\text{SO}_4]\text{Br}$
Hydrate	Formula	H₂O in/out of sphere	$[\text{Cr(H}_2\text{O})_6]\text{Cl}_3$ vs $[\text{Cr(H}_2\text{O})_5\text{Cl}]\text{Cl}_2\cdot\text{H}_2\text{O}$
Linkage	Formula	Binding atom	$[\text{Co(NH}_3)_5\text{NO}_2]^{2+}$ vs $[\text{Co(NH}_3)_5\text{ONO}]^{2+}$
Coordination	Formula	Ligand distribution	$[\text{Co(NH}_3)_6][\text{Cr(CN)}_6]$ vs $[\text{Cr(NH}_3)_6][\text{Co(CN)}_6]$
Geometrical	Bonding	Spatial arrangement	cis vs trans
Optical	Bonding	Mirror images	Δ vs Λ

Number of geometrical isomers (count fast):

Square planar	#	Octahedral	#
MA₂B₂	2 (cis, trans)	MA₄B₂	2 (cis, trans)
MA₂BC	3	MA₃B₃	2 (fac, mer)
MABCD	3	MA₂B₂C₂	up to 5

Bonding — Valence Bond Theory (VBT)

CN	Hybridization	Geometry	Bond angle	Example
2	sp	Linear	180°	$[\text{Ag(NH}_3)_2]^+$
4	sp³	Tetrahedral	109.5°	$[\text{NiCl}_4]^{2-}$
4	dsp²	Square planar	90°	$[\text{Ni(CN)}_4]^{2-}$
5	sp³d	Trigonal bipyramidal	90°, 120°	$[\text{Fe(CO)}_5]$
6	sp³d²	Octahedral (outer)	90°	$[\text{CoF}_6]^{3-}$
6	d²sp³	Octahedral (inner)	90°	$[\text{Co(NH}_3)_6]^{3+}$

• Inner orbital (d²sp³): uses (n−1)d orbitals; strong field; pairing forced; usually diamagnetic; stronger bonds.
• Outer orbital (sp³d²): uses nd orbitals; weak field; no pairing; usually paramagnetic.
• Always remove s electrons before d when forming ions (e.g. Fe³⁺ = [Ar]3d⁵, not 3d⁵4s²).

Reference complexes (memorise the row outcomes):

Complex	d-electrons	Hybridization	Geometry	Unpaired e⁻	Magnetic
$[\text{Fe(CN)}_6]^{4-}$	d⁶	d²sp³	Octahedral	0	Diamagnetic
$[\text{FeF}_6]^{4-}$	d⁶	sp³d²	Octahedral	4	Paramagnetic
$[\text{Co(NH}_3)_6]^{3+}$	d⁶	d²sp³	Octahedral	0	Diamagnetic
$[\text{CoF}_6]^{3-}$	d⁶	sp³d²	Octahedral	4	Paramagnetic
$[\text{Ni(CN)}_4]^{2-}$	d⁸	dsp²	Square planar	0	Diamagnetic
$[\text{NiCl}_4]^{2-}$	d⁸	sp³	Tetrahedral	2	Paramagnetic
$[\text{Cu(NH}_3)_4]^{2+}$	d⁹	sp³	Tetrahedral/Square	1	Paramagnetic
$[\text{Ag(NH}_3)_2]^+$	d¹⁰	sp	Linear	0	Diamagnetic

Spectrochemical Series

$$\boxed{I^- < Br^- < SCN^- < Cl^- < S^{2-} < F^- < OH^- < C_2O_4^{2-} < H_2O < NCS^- < EDTA < NH_3 < en < NO_2^- < CN^- < CO}$$

Weak field (small Δ₀, high spin) ←————→ Strong field (large Δ₀, low spin)

Crystal Field Theory (CFT)

Octahedral splitting: d orbitals split into lower t₂g (d_xy, d_yz, d_xz) and higher e_g (d_x²–y², d_z²).

• e_g raised by +0.6Δ₀; t₂g lowered by −0.4Δ₀.
• Barycenter check: $2(+0.6\Delta_0) + 3(-0.4\Delta_0) = 0$.
• Tetrahedral: order reverses (e lower, t₂ higher) and $\Delta_t \approx \tfrac{4}{9}\Delta_0$ → always high spin.

High vs low spin: decided by Δ₀ vs pairing energy P.

Condition	Result
$\Delta_0 < P$ (weak field)	High spin (unpaired electrons maximised)
$\Delta_0 > P$ (strong field)	Low spin (t₂g filled first, pairing)

High/low spin distinction only matters for d⁴, d⁵, d⁶, d⁷ in octahedral fields.

Factors increasing Δ₀: stronger field ligand (up spectrochemical series); higher metal oxidation state ($\text{Fe}^{3+} > \text{Fe}^{2+}$); larger d orbitals ($3d < 4d < 5d$).

CFSE Table — Octahedral

dⁿ	High spin CFSE	Low spin CFSE
d⁰	0	0
d¹	−0.4Δ₀	−0.4Δ₀
d²	−0.8Δ₀	−0.8Δ₀
d³	−1.2Δ₀	−1.2Δ₀
d⁴	−0.6Δ₀	−1.6Δ₀ + P
d⁵	0	−2.0Δ₀ + 2P
d⁶	−0.4Δ₀	−2.4Δ₀ + 2P
d⁷	−0.8Δ₀	−1.8Δ₀ + P
d⁸	−1.2Δ₀	−1.2Δ₀
d⁹	−0.6Δ₀	−0.6Δ₀
d¹⁰	0	0

Colour

• Cause: d–d transitions (t₂g → e_g) absorb visible light; observed colour is the complementary of the absorbed colour.
• Colourless when d⁰ (no electron to excite, e.g. $[\text{Sc(H}_2\text{O})_6]^{3+}$) or d¹⁰ (no empty d orbital, e.g. $[\text{Zn(H}_2\text{O})_6]^{2+}$).
• Worked value: $[\text{Ti(H}_2\text{O})_6]^{3+}$, Δ₀ = 20,300 cm⁻¹ → band maximum λ ≈ 493 nm. The band is broad (Jahn–Teller split, ~430–580 nm), so green and yellow are absorbed and the transmitted red + blue give the purple colour. (A single 493 nm line alone is blue-green, whose exact complement is red/orange-red, not purple.)

Stability of Complexes

Factor	Effect on stability	Example
Higher charge / smaller size (charge density)	↑	$\text{Al}^{3+} > \text{Mg}^{2+} > \text{Na}^+$
Better Lewis base (ligand basicity)	↑	$\text{CN}^- > \text{NH}_3 > \text{H}_2\text{O}$
Hard–hard or soft–soft (HSAB) match	↑	Fe³⁺–F⁻; Ag⁺–CN⁻
Chelate effect	↑↑	en > 2 NH₃
Macrocyclic effect	↑↑↑	porphyrin
Higher CFSE	↑	d³, low-spin d⁶

Irving–Williams series (first-row M²⁺, same ligand):

$$\boxed{Mn^{2+} < Fe^{2+} < Co^{2+} < Ni^{2+} < Cu^{2+} > Zn^{2+}}$$

• Chelate effect is mainly entropic: e.g. $[\text{Ni(H}_2\text{O})_6]^{2+} + 6\,\text{NH}_3$ has ΔS ≈ 0 ($\log\beta_6 = 8.61$), but $+3\,\text{en}$ releases more particles → ΔS > 0 ($\log\beta_3 = 18.28$), ~5 billion times more stable.
• Thermodynamic stability (β) ≠ kinetic stability (labile/inert). Labile: d⁰, d¹⁰, high-spin d⁴–d⁷, large ions. Inert: low-spin d³ and d⁶ (Cr³⁺, Co³⁺), high charge density.
• $\Delta G^\circ = -RT \ln \beta$.
• Stability scale: large β (>10⁸) = very stable; 10⁴–10⁸ = moderate; <10⁴ = weak.

High-Yield Application Facts

Biological metal centres:

Complex	Metal	Ligand / function
Hemoglobin	Fe²⁺	porphyrin (CN = 6, octahedral); reversible O₂ binding
Chlorophyll	Mg²⁺	modified porphyrin; light absorption
Vitamin B₁₂	Co³⁺	corrin ring
Hemocyanin	Cu⁺	O₂ transport (blue blood)
Carboxypeptidase	Zn²⁺	protein-digesting enzyme

• CO poisoning: CO binds Fe²⁺ ~200× more strongly than O₂ (strong field, won’t release) → blocks O₂ transport.
• Cisplatin = cis-$[\text{Pt(NH}_3)_2\text{Cl}_2]$ (anticancer); trans isomer inactive — only cis cross-links adjacent guanines on DNA.
• Chelation therapy / MRI: EDTA chelates Pb²⁺ (β ≈ 10¹⁸); $[\text{Gd(DTPA)}]^{2-}$ is the safe MRI contrast agent (free Gd³⁺ toxic).

Qualitative analysis colours:

Ion	Reagent	Complex	Colour
Fe³⁺	SCN⁻	$[\text{Fe(SCN)}]^{2+}$	blood red
Cu²⁺	NH₃	$[\text{Cu(NH}_3)_4]^{2+}$	deep blue
Ni²⁺	DMG	$[\text{Ni(DMG)}_2]$	rose-red
Co²⁺	SCN⁻	$[\text{Co(SCN)}_4]^{2-}$	blue
Fe²⁺	K₄[Fe(CN)₆]	$\text{Fe}_4[\text{Fe(CN)}_6]_3$	Prussian blue

Key industrial/metallurgy reactions:

$$4\text{Au} + 8\text{CN}^- + \text{O}_2 + 2\text{H}_2\text{O} \rightarrow 4[\text{Au(CN)}_2]^- + 4\text{OH}^-$$$$2[\text{Au(CN)}_2]^- + \text{Zn} \rightarrow [\text{Zn(CN)}_4]^{2-} + 2\text{Au}$$$$\text{Ni}(s) + 4\text{CO}(g) \xrightarrow{50\text{–}60\,^\circ\text{C}} [\text{Ni(CO)}_4](g) \xrightarrow{200\,^\circ\text{C}} \text{Ni}(s) + 4\text{CO}(g) \quad (\text{Mond process})$$$$\text{AgBr}(s) + 2\text{S}_2\text{O}_3^{2-} \rightarrow [\text{Ag(S}_2\text{O}_3)_2]^{3-} + \text{Br}^- \quad (\text{photographic fixing})$$

Catalysts: Wilkinson’s $[\text{RhCl(PPh}_3)_3]$ (alkene hydrogenation); Ziegler–Natta TiCl₄ + Al(C₂H₅)₃ (polymerisation); Monsanto $[\text{Rh(CO)}_2\text{I}_2]^-$ (CH₃OH → CH₃COOH).

Quick-Recall Mnemonics

• Counting ions: $[M(\text{L})_x]Y_n$ → total ions = $n + 1$.
• Structural isomer types: Ionization, Hydrate, Linkage, Ligand, Coordination (+ stereo: Geometrical, Optical).
• trans = PLANE of symmetry → never optically active.
• Irving–Williams: Mn < Fe < Co < Ni < Cu > Zn.
• Stability ladder: Macrocyclic > Chelate > Monodentate.
• Labile vs inert: d⁰ and d¹⁰ are labile; low-spin d³ and d⁶ are inert.

Related Topics

• Werner’s Theory
• IUPAC Nomenclature
• Isomerism
• Bonding Theories
• Crystal Field Theory
• Stability of Coordination Compounds
• Applications