Chemical Periodicity: Atomic Structure, Periodic Properties and s, p, d, f Blocks

1.1The hydrogen atom: wave functions, quantum numbers, nodes and radial distribution

The Schrödinger equation for the hydrogen atom (and one-electron ions He⁺, Li²⁺) can be solved exactly in spherical polar coordinates. Each solution, an orbital, is a product of a radial part and an angular part:

• The radial part R depends on n and l; the angular part Y depends only on l and m. So all p orbitals (2p, 3p, 4p) have the same angular shape; only their size and radial nodes change with n.
• For an s orbital (l = 0) Y is a constant (1/√4π): an s orbital depends on r only and is spherically symmetric. Angular dependence starts with p orbitals.
• Born interpretation: ψ² (strictly |ψ|²) is the probability density. ψ itself has no physical meaning and can be positive or negative; the radial function of 2s changes sign at its node. An acceptable ψ is single-valued, finite, continuous and normalisable.

Quantum numbers

• A shell has n² orbitals and 2n² electrons. d orbitals exist only from n = 3 (five degenerate d orbitals for n = 3, 4, …); f orbitals from n = 4.
• In hydrogen all orbitals of one shell have the same energy. In many-electron atoms the energy depends on n + l (lower first; for equal n + l, lower n first): 4s (4) < 3d (5), and 3d (5) is higher than 3p (4).

Nodes

• Angular nodes are planes, except for d_z², whose two angular nodes are cones. So p orbitals have one nodal plane, four of the d orbitals have two planes and d_z² has none.
• Example: 3s (2, 0), 3p_x (1 radial, 1 plane), 3d_z² (0 radial, 2 conical). All orbitals with n = 3 together have 3 × 1 + 4 × 2 = 11 nodal planes.

Radial distribution function

The probability of finding the electron in a thin shell between r and r + dr is 4πr²R² dr. This radial distribution function (RDF) depends on n, l and r, not on the orientation (m). It is zero at the nucleus because of the r² factor, even for s orbitals.

• The number of peaks is n − l. For l = n − 1 (1s, 2p, 3d) there is a single peak at r = n²a₀: 1s at 52.9 pm (a₀), 2p at 4a₀, 3d at 9a₀.
• Most probable radius of the outer peak: 1s (1a₀) < 2s (5.2a₀) < 3s (13.1a₀); within a shell 3s > 3p (12a₀) > 3d (9a₀). More radial nodes push the main peak outward.
• Penetration: 2s has a small inner peak inside the 2p maximum, so a 2s electron spends some time close to the nucleus, is less shielded and feels a larger effective nuclear charge. Penetration within a shell: s > p > d > f. This is why 2s lies below 2p and 4s fills before 3d in many-electron atoms.

1.2Electronic configuration

• Aufbau principle: fill orbitals in order of increasing energy (n + l rule): 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p.
• Pauli exclusion principle: no two electrons have the same four quantum numbers; an orbital holds two electrons of opposite spin.
• Hund’s rule: in degenerate orbitals electrons enter singly with parallel spins before pairing. 1s²2s²2p_x²2p_y¹ for nitrogen breaks Hund’s rule (correct: 2p_x¹2p_y¹2p_z¹).

Exceptions worth remembering

• The group 10 metals all differ: Ni 3d⁸4s², Pd 4d¹⁰5s⁰, Pt 5d⁹6s¹.
• Ions: electrons leave the highest n first, so 4s before 3d and 6s before 5d and 4f. Fe²⁺ = [Ar]3d⁶; Ce³⁺ = [Xe]4f¹; Lu³⁺ = [Xe]4f¹⁴.
• Counting questions: write the full configuration and count. Fe has 12 p electrons (2p⁶ 3p⁶) out of 26 (46.15 %); Na has 6 occupied orbitals (1s, 2s, three 2p, 3s); an atom with K 2, L 8, M 5 electrons is P (15) with 9 p electrons.

1.3The modern periodic table: s, p, d and f blocks

• Modern periodic law (Moseley): properties are periodic functions of atomic number, not atomic weight (which failed for Ar–K, Co–Ni, Te–I).
• Period = highest n in the configuration. Block = subshell receiving the last electron. Group: s-block = number of ns electrons; d-block = ns + (n − 1)d electrons; p-block = 10 + ns + np electrons.
• Example: Hs (108) = [Rn]5f¹⁴6d⁶7s² → period 7, d-block, group 8 (with Fe, Ru, Os). An ion M⁴⁺ = [Xe]4f¹⁴5d¹⁰ comes from [Xe]4f¹⁴5d¹⁰6s²6p² = Pb, period 6, group 14.
• Names: group 1 alkali metals, 2 alkaline earth metals, 15 pnictogens, 16 chalcogens, 17 halogens, 18 noble gases; groups 3–12 = d-block (transition elements); 13–18 = p-block; f-block = inner transition elements (lanthanoids 4f, actinoids 5f).
• He (1s²) is placed with the noble gases although it has no p electrons: it is the only noble gas without an ns²np⁶ shell.
• Metallic character falls across a period and rises down a group. Metalloids: B, Si, Ge, As, Sb, Te. Bi is the only true metal of group 15; Tl and Sn, Pb are metals.

d-block (transition) elements

• Partly filled d subshells give variable oxidation states, coloured and paramagnetic compounds, complex formation and catalytic activity. d⁰ and d¹⁰ ions (Sc³⁺, Ti⁴⁺, Cu⁺, Zn²⁺) are colourless and diamagnetic. Spin-only moment μ = √[n(n + 2)] BM.
• Enthalpy of atomisation follows the number of unpaired d electrons available for metallic bonding: it peaks in the middle (V ≈ 515 kJ mol⁻¹ is the highest of the 3d series) and dips at Mn (half-filled, stable d⁵) and Zn (d¹⁰). Zn, Cd, Hg have the lowest melting and boiling points.
• Oxidation states: in a d-block group the higher states become more stable down the group (Cr(III) at the top, W(VI) at the bottom of group 6), the opposite of the p-block (inert pair effect).
• High oxidation state + small F gives covalent, volatile molecular halides: WF₆ is a gas (b.p. ≈ 17 °C), TiCl₄ a liquid.
• Lanthanide contraction: the 14 poorly-shielding 4f electrons between La and Hf shrink the 5d elements so that Zr ≈ Hf, Nb ≈ Ta, Mo ≈ W in size and chemistry.

f-block: lanthanoids and actinoids

• Lanthanoids (Ce–Lu, with La) are silvery-white, soft, highly electropositive and reactive metals (E° Ln³⁺/Ln ≈ −2.3 V): they tarnish in air and liberate H₂ from water.
• +3 is the characteristic state. Ce also gives a stable +4 (4f⁰; ceric = Ce(IV), a strong oxidant used in cerimetry), Eu and Yb give +2 (4f⁷, 4f¹⁴).
• La³⁺ (4f⁰) and Lu³⁺ (4f¹⁴) are colourless and diamagnetic. Ln³⁺ radii fall steadily: La³⁺ 103 > Eu³⁺ 95 > Y³⁺ 90 > Lu³⁺ 86 pm (Y³⁺ sits between Ho³⁺ and Er³⁺).
• Their complexing ability is weaker than that of d-block ions: the 4f orbitals are buried and take no part in bonding, so Ln³⁺ bonds are mainly ionic.
• La is counted both as the first lanthanoid (f-block, inner transition, rare earth) and as the first member of the 5d transition series. Rare earth metals = Sc, Y and La–Lu.

1.4Shielding and Slater’s rules

An outer electron does not feel the whole nuclear charge Z; inner electrons screen part of it. The effective nuclear charge is Z* = Z − S, and Slater’s rules estimate the screening constant S.

1. Write the configuration in groups: (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p) …
2. Electrons in groups to the right contribute 0.
3. For an s or p electron: each other electron in the same group 0.35 (0.30 within 1s); each electron in shell n − 1: 0.85; each electron in n − 2 and below: 1.00.
4. For a d or f electron: each other electron in the same group 0.35; every electron in groups to the left 1.00.

• Use the true configuration: Cr is 3d⁵4s¹, not 3d⁴4s² (which gives a wrong 4.95).
• Z* rises across a period (about +0.65 per step for the valence electron), which explains the fall in radius and the rise in ionisation enthalpy.