Magnetic Properties of Transition-Metal Ions

01 / UNDERSTAND & APPLY

1. Identify the ion before counting spins

Magnetic behaviour connects an electron configuration to a measurable response to a field. For a transition-metal ion, first obtain oxidation state from the complete formula, then determine its d count. Only after that should the ligand geometry and splitting pattern be used to distribute electrons. Neutral-atom configurations are not automatically the configurations of the ions.

The outer ns electrons are removed before the (n−1)d electrons in the familiar first-row cations. Fe is commonly written [Ar]3d⁶4s², Fe²⁺ is 3d⁶ and Fe³⁺ is 3d⁵. Cu²⁺ is 3d⁹, whereas Zn²⁺ is 3d¹⁰. The unpaired-electron count is obtained from the occupied orbitals, not from oxidation state alone.

The shortcut d count=group number−oxidation state works for the usual transition-metal ions in this lesson. It does not say that a complex with six d electrons must have six unpaired electrons: each orbital accommodates at most two, and ligand splitting may favour pairing.

An isolated ion model is also different from a bulk metal or a magnetically coupled solid. Start with local electronic structure, then ask whether collective interactions or orbital contributions invalidate a simple spin-only prediction.

02 / UNDERSTAND & APPLY

2. Diamagnetism, paramagnetism and collective ordering

All electron clouds have a diamagnetic response: the induced magnetization opposes the applied field and produces a weak repulsion from stronger-field regions. For closed-shell molecules or ions, this is usually the dominant response. In the elementary isolated-ion classification, all electrons paired implies diamagnetic behaviour.

Unpaired electrons can provide permanent magnetic moments. Thermal motion randomizes their directions without a field; an applied field biases the population toward alignment, giving paramagnetism. A paramagnetic sample is attracted into a stronger field. It is not necessarily a permanent magnet after the field is removed.

Ferromagnetism is collective parallel ordering of magnetic moments below an ordering temperature, and can give strong spontaneous magnetization. Antiferromagnetic ordering has oppositely aligned sublattices with cancellation in the simplest case. Ferrimagnetism also has opposing sublattices but unequal moments leave a net magnetization. These are properties of interacting ensembles, not synonyms for having one unpaired electron.

A salt containing paramagnetic ions can display antiferromagnetic coupling, while a bulk metal can show band magnetism not captured by isolated free-ion counts. State whether the problem asks about a discrete complex, a free ion or a solid. Temperature, coupling and impurities can change the observed response.

03 / UNDERSTAND & APPLY

3. Derive and use the spin-only expression

Each unpaired electron contributes spin 1/2. If n unpaired electrons are aligned according to Hund’s rule within the relevant occupancy, their total spin is S=n/2. Taking the electron spin g factor approximately as 2 gives the spin-only effective moment 2√[S(S+1)] μB, equivalent to √[n(n+2)] μB.

The Bohr magneton μB is a unit of magnetic moment, approximately 9.274×10⁻²⁴ J T⁻¹. A reported value such as 4.90 BM means 4.90 Bohr magnetons; it is not a number of electrons or a field strength in tesla. The effective moment extracted from susceptibility is also not simply the maximum projected moment nμB.

For n=0,1,2,3,4,5 the spin-only values are 0,1.73,2.83,3.87,4.90,5.92 μB respectively. These landmarks help check arithmetic. A measured value near a landmark can support an electron assignment only when the spin-only approximation and sample model are suitable.

Orbital angular momentum, spin–orbit coupling, magnetic exchange, temperature-dependent populations and covalency can all produce deviations. The approximation often works reasonably for many first-row transition-metal ions with largely quenched orbital contributions, but it is not an exact universal experimental law.

04 / UNDERSTAND & APPLY

4. Octahedral high spin and low spin

In an octahedral field, the three t₂g orbitals lie below the two e_g orbitals, separated by Δₒ. Electrons occupy separate orbitals within a degenerate set before pairing. When lower orbitals are singly occupied, additional electrons may either pair there or enter the upper set; the competition is between splitting and pairing costs.

For d⁴–d⁷ octahedral ions, weak fields can favour high spin and strong fields low spin. In the simple model, Δₒ<P favours the high-spin alternative and Δₒ>P favours low spin. These terms compare alternative distributions at the same d count, not two different oxidation states.

For octahedral d⁶, high spin t₂g⁴e_g² has four unpaired electrons, whereas low spin t₂g⁶e_g⁰ has none. For d⁵, high spin t₂g³e_g² has five unpaired electrons and low spin t₂g⁵ has one. Draw each orbital to avoid equating the exponent with the number of unpaired electrons.

Octahedral d¹, d² and d³ fill the lower set without a pairing-versus-upper-level choice; d⁸, d⁹ and d¹⁰ also lack distinct ordinary high/low-spin alternatives in this elementary octahedral scheme. Geometry still matters: square-planar d⁸ can be all-paired, while a common tetrahedral d⁸ complex has two unpaired electrons.

05 / UNDERSTAND & APPLY

5. Geometry cannot be read from one number alone

Tetrahedral splitting is usually smaller than octahedral splitting for comparable metal and ligands. Most familiar tetrahedral first-row complexes are therefore high spin. Avoid the absolute assertion that no low-spin tetrahedral complex could ever exist; the school-level trend is broad but not a universal law of nature.

Square-planar d⁸ complexes commonly have a sufficiently high d(x²−y²) level that the eight electrons occupy the four lower orbitals in pairs. In contrast, tetrahedral Ni²⁺ is d⁸ with two unpaired electrons. Thus [Ni(CN)₄]²⁻ is a standard diamagnetic square-planar reference, and [NiCl₄]²⁻ a paramagnetic tetrahedral reference.

A measured moment can eliminate incompatible assignments under stated assumptions, but does not uniquely establish every bond angle or ligand arrangement. For example, several d counts can share one unpaired electron. Combine magnetic evidence with composition, ligand field, spectra and structural information.

Count ligand charge rather than assuming that a familiar ligand always produces one oxidation state. Then keep the geometry-specific orbital ordering explicit. Applying an octahedral t₂g/e_g diagram to a square-planar complex is a change of model, not a harmless notation choice.

06 / UNDERSTAND & APPLY

6. f-block moments and the orbital contribution

For lanthanoid ions, the 4f electrons are relatively shielded from the ligands by filled outer shells. The orbital contribution is often retained more strongly than in familiar 3d complexes. Spin–orbit coupling combines total orbital angular momentum L and total spin S into total angular momentum J. Consequently, counting unpaired electrons alone usually does not predict the measured moment.

For an isolated well-defined ground J multiplet in the free-ion approximation, μ_eff≈g_J√[J(J+1)] μB. The Landé factor is g_J=1+[J(J+1)+S(S+1)−L(L+1)]/[2J(J+1)]. This expression assumes a suitable angular-momentum coupling model and negligible thermal population of higher multiplets; it must not be applied mechanically when J=0 or when nearby states matter.

Gd³⁺ is 4f⁷ with S=7/2 and L=0 in its ground state. Then J=7/2, g≈2, and the result is √63≈7.94 μB, matching the spin-only estimate well. Ce³⁺ is 4f¹ with a ground-state orbital contribution; using L=3, S=1/2 and J=5/2 gives g_J=6/7 and a free-ion estimate about 2.54 μB, rather than 1.73 μB from one unpaired spin alone.

Sm³⁺ and Eu³⁺ illustrate additional limits because nearby levels and temperature-dependent effects matter. In particular, a J=0 ground state does not justify claiming zero observed susceptibility at every temperature. For actinoids, stronger spin–orbit effects, covalency and more complicated level structures further restrict simple free-ion or spin-only models.

The broad d- and f-block comparison is therefore causal: ligand-induced orbital quenching often makes spin-only useful for 3d complexes; shielded 4f electrons retain substantial orbital character; 5f systems can add stronger covalency and coupling complexity. It is a hierarchy of approximations, not a claim that one formula fits all metals.

07 / UNDERSTAND & APPLY

7. Colour begins with the absorbed photon

Ligands split metal d levels. If an allowed or weakly allowed transition between relevant states requires a visible photon, selective absorption can colour the complex. The absorbed colour is removed from the incident white-light mixture; the transmitted or reflected appearance is associated with the remaining light, often described approximately as the complementary colour.

Photon energy is E=hν=hc/λ. A larger energy gap corresponds to shorter absorbed wavelength. For a simple d¹ octahedral model, the main d–d absorption can directly estimate Δₒ. Multi-electron ions have electron–electron repulsion and multiple terms, so not every observed band equals a single Δₒ without further analysis.

In centrosymmetric complexes, d–d transitions are Laporte-forbidden in the simplest electric-dipole approximation, but vibrations and distortion can lend them intensity. Spin-changing transitions are also weak under the spin selection rule. Thus partially filled d orbitals do not guarantee intense visible colour. Tetrahedral complexes lack an inversion centre, which can make their d–d bands more intense.

Charge-transfer transitions move electron density substantially between ligand and metal and can be intense. MnO₄⁻ has Mn(VII), formally d⁰, yet is strongly coloured; its colour cannot be assigned to a d–d promotion from an occupied d orbital. Similarly, ligand-centred transitions may colour a compound even when the metal is d⁰ or d¹⁰.

Ligand identity, oxidation state, geometry and solvent can alter energy gaps and therefore colour. Colour alone is not a reliable unique identification of a metal, and a diamagnetic complex can still be coloured through allowed electronic transitions. Magnetism and optical absorption interrogate related but different aspects of electronic structure.

08 / UNDERSTAND & APPLY

8. Combine magnetic and optical evidence

Suppose a correctly assigned d⁶ complex is diamagnetic and has a visible absorption. An octahedral low-spin configuration is compatible with the magnetic evidence, and visible absorption is compatible with excitations from its filled lower levels into higher levels. “All paired” describes its ground state, not the absence of excited electronic states.

Suppose a sample has a moment larger than the elementary spin-only prediction. First check composition, oxidation state, geometry, purity and the convention used to extract the moment. Only then consider orbital contributions or exchange. A discrepancy is evidence that a model or assignment needs examination, not permission to invent a fractional unpaired electron.

In qualitative multiple-statement problems, distinguish “can” from “must.” A partially filled d subshell can support d–d transitions; their energy may be outside the visible range. Unpaired electrons can cause paramagnetism; a coupled solid may have a cancelling ground state. A ligand can shift colour without changing oxidation state.

09 / UNDERSTAND & APPLY

9. Revision decision path

For a discrete transition-metal complex: solve ligand-charge algebra, obtain dⁿ, select the actual geometry, fill its orbitals with the stated field assumptions, count unpaired electrons, and only then use the spin-only expression if justified. For f-block ions, explicitly ask whether orbital angular momentum must be retained.

For colour: name the candidate transition, relate the absorption energy to wavelength, and distinguish absorbed from observed light. Check charge-transfer and ligand-centred alternatives before concluding that d⁰ or d¹⁰ necessarily means colourless.

For solids: keep collective ordering separate from an isolated-ion moment. A complete explanation identifies both the microscopic source of moments and the interactions that align or cancel them.

• The number under the square root is n(n+2), not n².
• An effective moment is reported in μB or BM, not tesla.
• High/low spin depends on d count, geometry and splitting versus pairing.
• Spin-only is a model; f-block orbital contributions are substantive, not minor bookkeeping.