Periodic Trends

Moving from Sc (Z=21) to Zn (Z=30), atomic radius decreases only slightly — from approximately 162 pm to 122 pm — compared with the dramatic decrease across Period 2 s/p elements. Each additional proton is accompanied by an additional 3d electron that shields other 3d electrons moderately well (Slater shielding constant contribution ≈ 0.35 per same-subshell electron). The net increase in effective nuclear charge (Z_eff) per element is therefore smaller than in the p-block. Notably, there is a slight increase in radius at Cr and Cu due to half-filled and fully filled 3d sub-shells respectively, where exchange energy stabilisation subtly redistributes electron density.

The 4f orbitals are diffuse and penetrate the core poorly. Their shielding constant against nuclear charge is ≈ 0.85 per electron (compared with ≈ 1.00 for inner-shell electrons), so each step across the 14 lanthanoids adds 1 proton but only 0.85 units of shielding, giving a net increase in Z_eff of ≈ +0.15 per element. This cumulative effect contracts the 4f, 5s, 5p, and eventually 5d/6s radii by about 15 pm in total. Consequence: Zr (4d series, r ≈ 160 pm) and Hf (5d series, r ≈ 159 pm) are nearly identical in size, making their chemical separation extraordinarily difficult. Similarly, Au and Pt have unusually high densities and distinctive relativistic effects that are amplified by the lanthanoid contraction.

First ionisation energies (IE₁) across the 3d series increase from Sc (631 kJ mol⁻¹) to Zn (906 kJ mol⁻¹), but the trend is irregular. Cr and Cu have anomalously high IE₁ values because their ground-state configurations ([Ar]3d⁵4s¹ and [Ar]3d¹⁰4s¹) benefit from exchange energy stabilisation in the 3d sub-shell, making it harder to remove the 4s electron. The second and third ionisation energies are often similar in magnitude for adjacent metals, explaining why many transition metals form M²⁺ and M³⁺ ions with comparable stability. In the 5d series, relativistic contraction of the 6s orbital further raises ionisation energies, making Au and Pt noble and difficult to oxidise.

The key to oxidation state variability is the small energy gap between the 3d and 4s orbitals. In the free atom, 4s is filled first (lower energy), but in ions, 3d becomes lower in energy than 4s, so 4s electrons are lost first upon ionisation. For Mn ([Ar]3d⁵4s²): removing 2 electrons gives Mn²⁺ (+2); removing 3 gives Mn³⁺ (+3); in strongly oxidising environments, all 7 valence electrons can be involved in covalent bonds (as in MnO₄⁻, oxidation state +7). Higher oxidation states are stabilised by electronegative ligands (O²⁻, F⁻) via ionic or covalent bonding. Lower oxidation states are stabilised by π-acceptor ligands such as CO (as in Mn₂(CO)₁₀, Mn in 0 state). The maximum oxidation state across the first-row metals rises from +3 (Sc) to +7 (Mn) then falls as pairing energy increases and more d electrons resist ionisation.

Slater's rules allow calculation of Z_eff = Z − σ, where σ is the shielding constant. For a 3d electron shielded by other 3d electrons: each contributes 0.35; electrons in n−1 shells contribute 1.00; same-shell (n, same l) electrons contribute 0.35. For Cr (Z=24): 5 other 3d electrons × 0.35 = 1.75; 18 core electrons × 1.00 = 18.00; σ = 19.75; Z_eff = 24 − 19.75 = 4.25. For Cu (Z=29): 9 other 3d electrons × 0.35 = 3.15; 18 core × 1.00 = 18.00; σ = 21.15; Z_eff = 7.85. The larger Z_eff for Cu explains its smaller atomic radius (128 pm vs. Cr 128 pm — nearly equal, reflecting the d¹⁰ configuration).

• Z_{\text{eff}} — Effective nuclear charge (dimensionless (in units of elementary charge)): The net positive charge experienced by a valence electron after accounting for shielding by inner and same-shell electrons.
• Z — Atomic number (dimensionless): Total number of protons in the nucleus.
• \sigma — Shielding constant (dimensionless): Sum of all shielding contributions from other electrons, calculated using Slater's rules (0.35 per same-subshell electron, 0.85 per n−1 shell electron, 1.00 per deeper shell electron for s/p; 1.00 for all inner electrons for d/f).

Problem. Manganese (Mn, Z=25) commonly forms both MnO₄⁻ (permanganate) and MnO (manganese(II) oxide). (a) Identify the oxidation state of Mn in each compound. (b) Explain why Mn can exist in such a wide range of oxidation states. (c) Calculate Z_eff for a 3d electron in Mn using Slater's rules.

Answer. Mn is +7 in MnO₄⁻ and +2 in MnO. Z_eff for a 3d electron in Mn = 5.60. The closeness of 3d and 4s energies, stabilisation of d⁵ by exchange energy, and bonding with electronegative ligands account for the wide range of oxidation states.

• Electron Configuration of d-Block Elements — Understanding anomalous configurations like Cr ([Ar]3d⁵4s¹) and Cu ([Ar]3d¹⁰4s¹) is essential to explain the irregular trends in atomic radius and ionisation energy across the transition series.
• Colour in Transition Metal Compounds — The variable oxidation states of d-block elements directly determine d-orbital splitting energies and hence the wavelengths of light absorbed by transition metal complexes, producing characteristic colours.
• Coordination Compounds and Ligand Field Theory — Oxidation state variability in transition metals determines the number of d electrons available for bonding with ligands, directly affecting complex geometry, stability, and magnetic properties.
• Ionisation Energy and Periodic Trends — The principles of effective nuclear charge and shielding used to explain IE trends in the main group are the same tools applied to understand the irregular IE variations across the d-block, including relativistic effects in the 5d series.