d- and f-Block Elements · Lesson 5 of 12
Magnetism and the Origin of Colour
“Count unpaired electrons, calculate spin-only moments and explain the scope and limits of d–d colour.”
• Distinguish diamagnetism, paramagnetism and collective ferromagnetism. • Calculate spin-only magnetic moments with stated assumptions. • Fill the split d orbitals of common hydrated ions. • Explain colour through light absorption and recognise charge-transfer and ligand effects.
Unpaired electrons respond to a magnetic field
A sample with unpaired electrons is generally attracted into an applied magnetic field: it is paramagnetic. A closed-shell sample has only a weak induced response opposing the field and is diamagnetic. Ferromagnets such as bulk iron show cooperative alignment of many moments and magnetic domains. Ferromagnetism is therefore a collective ordered phenomenon, not simply a solution ion with an exceptionally large spin-only value.
Electron spin and orbital angular momentum both contribute to magnetism. For many first-row complexes the orbital contribution is substantially quenched by the surrounding ligands, so a spin-only estimate is useful. It is approximate: Co²⁺ commonly deviates appreciably, and measured Fe²⁺ moments can also exceed the simple prediction. Later f-block moments often require spin–orbit and orbital contributions.
| Unpaired electrons n | Spin-only μ / BM |
|---|---|
| 0 | 0.00 |
| 1 | 1.73 |
| 2 | 2.83 |
| 3 | 3.87 |
| 4 | 4.90 |
| 5 | 5.92 |
Octahedral hydration splits the five d orbitals
A free ion has five d orbitals of equal energy before further interactions are considered. In an octahedral complex, six ligands approach along the coordinate axes. The two d orbitals directed most toward the ligands form the higher-energy e_g set; the three oriented between these axes form the lower-energy t₂g set. Electrons occupy these orbitals according to the competition between splitting and pairing energy. Water usually produces the high-spin arrangements used for the first-row aqua ions here.
| Hydrated ion | Formal d count | Octahedral occupation | Unpaired n | Spin-only / BM |
|---|---|---|---|---|
| Ti²⁺ | d² | t₂g² e_g⁰ | 2 | 2.83 |
| V²⁺ | d³ | t₂g³ e_g⁰ | 3 | 3.87 |
| Cr³⁺ | d³ | t₂g³ e_g⁰ | 3 | 3.87 |
| Mn²⁺ | d⁵ | t₂g³ e_g² | 5 | 5.92 |
| Fe²⁺ | d⁶ | t₂g⁴ e_g² | 4 | 4.90 |
| Fe³⁺ | d⁵ | t₂g³ e_g² | 5 | 5.92 |
| Co²⁺ | d⁷ | t₂g⁵ e_g² | 3 | 3.87 |
| Ni²⁺ | d⁸ | t₂g⁶ e_g² | 2 | 2.83 |
| Cu²⁺ | d⁹ | t₂g⁶ e_g³ | 1 | 1.73 |
For n=2, √8=2.828…, conventionally 2.83 BM; rounding it to 2.84 is a small numerical error. The t₂g² and t₂g³ entries place electrons singly in separate lower orbitals. A high-spin d⁵ arrangement has one electron in each of all five d orbitals. Beyond d⁵, extra electrons pair: d⁶ has four unpaired, d⁷ three, d⁸ two and d⁹ one. “d⁶ means six unpaired electrons” is impossible because only five d orbitals exist.
Problem
Calculate spin-only moments for divalent ions with Z=25 and Z=27.
- 1.Z=25 identifies Mn. Mn²⁺ is d⁵ and its aqua complex is high-spin: n=5. Hence μ=√(5×7)=√35=5.916≈5.92 BM.
- 2.Z=27 identifies Co. Co²⁺ is d⁷; its usual octahedral aqua occupation has three unpaired electrons. μ=√(3×5)=√15=3.873≈3.87 BM.
- 3.The values are model predictions. Actual Co²⁺ often has a larger moment because orbital contributions are not fully quenched.
Problem
Interpret approximate moments 2.2 BM for K₄[Mn(CN)₆], 5.3 BM for [Fe(H₂O)₆]²⁺ and 5.9 BM for K₂[MnCl₄].
- 1.In [Mn(CN)₆]⁴⁻, x−6=−4 gives Mn(II), d⁵. The small moment is consistent with low-spin d⁵, one unpaired electron, rather than high-spin d⁵. Its deviation from 1.73 BM reminds us the model is approximate.
- 2.In [Fe(H₂O)₆]²⁺, water is neutral, so Fe is +2 and d⁶. A moment near 5.3 BM is consistent with high-spin d⁶, four unpaired electrons, with non-spin-only contributions.
- 3.In [MnCl₄]²⁻, Mn is +2 and d⁵. A weak-field tetrahedral complex is high-spin with five unpaired electrons; √35≈5.92 BM agrees well.
- 4.Moment alone does not uniquely determine geometry. Charges, d counts, ligand information and other measurements must be combined.
Observed colour is the light that remains
A solution appears coloured when it absorbs some visible wavelengths more strongly than others. Ligands split metal d levels; absorption can promote an electron from a lower to an available higher d level when the photon energy matches the gap. The transmitted light is depleted in the absorbed colours and can appear complementary. The observed colour is not usually the colour of the photons absorbed. Ligand identity, geometry, oxidation state, concentration and hydrolysis can all change the spectrum.
| Species in a stated aqueous context | d count | Typical appearance |
|---|---|---|
| Sc³⁺, Ti⁴⁺ | d⁰ | Colourless simple aqua chemistry |
| Ti³⁺ | d¹ | Violet/purple |
| VO²⁺ (V(IV)) | d¹ | Blue |
| V³⁺ | d² | Green |
| V²⁺ | d³ | Violet |
| [Cr(H₂O)₆]³⁺ | d³ | Violet; other Cr(III) species can be green |
| Cr²⁺ | d⁴ | Blue |
| Mn²⁺ | d⁵ | Very pale pink |
| Fe²⁺ | d⁶ | Pale green |
| Fe³⁺ in many ordinary aqueous solutions | d⁵ | Yellow/brown affected by hydrolysis |
| Co²⁺ aqua | d⁷ | Pink |
| Ni²⁺ aqua | d⁸ | Green |
| Cu²⁺ aqua | d⁹ | Blue |
| Zn²⁺, Cu⁺ | d¹⁰ | No ordinary d–d colour |
d⁰ has no d electron to excite and d¹⁰ lacks a vacant d state for the simple within-d promotion, so these ions do not show ordinary d–d absorption. This does not mean all d⁰ or d¹⁰ compounds are colourless. MnO₄⁻ and CrO₄²⁻ have formal d⁰ metals and intense purple or yellow colours from charge-transfer transitions involving ligand and metal orbitals. Conversely, high-spin d⁵ Mn²⁺ is only weakly coloured because its d–d transitions are strongly restricted. Partial filling alone does not determine intensity or guarantee that absorption lies in the visible range.
Aqua colour tables describe particular species, not all salts of a metal. Changing Co²⁺ from a water-rich environment to a chloride-rich environment can change pink toward blue. A nominal Fe³⁺ solution commonly contains hydrolysed species; calling its yellow appearance solely a simple d–d effect omits this chemistry. These qualifications allow useful colour patterns without turning them into misleading universal tests.
| Ion | Spin-only / BM | Representative observed / BM |
|---|---|---|
| Sc³⁺ | 0 | 0 |
| Ti³⁺ | 1.73 | 1.75 |
| Ti²⁺ | 2.83 | 2.76 |
| V²⁺ | 3.87 | 3.86 |
| Cr²⁺ | 4.90 | 4.80 |
| Mn²⁺ | 5.92 | 5.96 |
| Fe²⁺ | 4.90 | 5.3–5.5 |
| Co²⁺ | 3.87 | 4.4–5.2 |
| Ni²⁺ | 2.83 | about 2.9–3.4 |
| Cu²⁺ | 1.73 | about 1.8–2.2 |
| Zn²⁺ | 0 | approximately 0 |
The observed entries illustrate particular hydrated-ion or solid-state measurements, not universal constants of an element. They show why a model is assessed against a stated chemical environment. Ti²⁺ is the d² titanium ion in this sequence; substituting the symbol Tl would incorrectly identify thallium. Rounded spin-only values in the table use direct square-root evaluation.
Quick Quiz
Quiz
What is the spin-only moment for two unpaired electrons?
Explanation: √[2(2+2)]=√8≈2.83 BM.
How many unpaired electrons does high-spin octahedral Fe²⁺ have?
Explanation: Fe²⁺ is d⁶ with t₂g⁴e_g²: one pair and four singly occupied orbitals.
Why can permanganate be purple despite a formal d⁰ metal?
Explanation: Colour can involve ligand-to-metal charge transfer rather than a d–d promotion.
Which statement about bulk ferromagnetic iron is correct?
Explanation: Domains and exchange coupling distinguish collective ferromagnetism.
What does a larger splitting gap imply for the absorbed photon?
Explanation: Since ΔE=hc/λ, energy and wavelength vary inversely.
Practice Problems
- For gaseous Mn³⁺, Cr³⁺, V³⁺ and Ti³⁺, count unpaired d electrons.
- A spin-only moment is 4.90 BM. Infer n and discuss whether geometry follows.
- Predict ordinary aqueous colour for Ti³⁺, V³⁺, Cu⁺, Sc³⁺, Mn²⁺, Fe³⁺ and Co²⁺.
- Explain why a strong-field ligand can change a d⁵ moment.
- Compute photon energy for absorption at 500 nm using h=6.626×10⁻³⁴ J s and c=2.998×10⁸ m s⁻¹.
- Why is a spin-only calculation for Ce³⁺ an estimate with a major limitation?
Their d counts are d⁴,d³,d²,d¹. In the free-ion Hund arrangement, these give 4,3,2,1 unpaired electrons. In aqueous solution octahedral d³ Cr³⁺ has especially favourable t₂g³ occupation, but an absolute stability ranking requires specified redox processes, not only gas-phase unpaired counts.
Key Takeaways
• Unpaired electrons generally produce paramagnetism; closed shells are diamagnetic. • Ferromagnetism involves collective order in a solid. • Spin-only moments are approximate and depend on unpaired electrons, not total d electrons. • Ligand-field splitting and pairing determine electron distribution in complexes. • Aqua-ion colours depend on absorption, speciation and ligand environment. • d⁰ and d¹⁰ suppress ordinary d–d colour but do not rule out charge-transfer colour. • Magnetic moments support structural interpretation when combined with other evidence.