Periodic table

The periodic table of the elements is a tabular arrangement of the chemical elements, ordered by increasing atomic number (Z), recurring chemical properties, and electronic structure. It is the central organizing framework of chemistry, revealing periodic trends across rows (periods) and columns (groups).

Although the periodic law was first formulated in the 19th century—well before the development of quantum mechanics—the modern understanding of the table rests on solutions of the Schrödinger equation and, more fundamentally, the Dirac equation. These equations describe how electrons occupy quantized orbitals around nuclei, and the resulting configurations explain the recurring patterns of stability, reactivity, and bonding encoded in the periodic table.

Visualization: Throughout the element pages, valence-only orbital grids (noble-gas cores collapsed) are rendered with the Circular Grid Model of Atoms, so group patterns (e.g., ns¹ vs. ns², progressive np filling) are visible at a glance.

The Periodic Table: Harmonics, Nodes, and the Octet

Atomic orbitals are standing-wave solutions of the Schrödinger equation in a central potential. The angular part is described by spherical harmonics $Y_{\ell }^{\,m}(\theta ,\phi )$ , which form a $(2\ell +1)$ -fold degenerate set for each angular momentum $\ell$ (with $m=-\ell ,\ldots ,+\ell$ ). Including spin pairing, each subshell holds $2(2\ell +1)$ electrons:

s ( $\ell =0$ ) → 2
p ( $\ell =1$ ) → 6
d ( $\ell =2$ ) → 10
f ( $\ell =3$ ) → 14

Higher $\ell$ means more angular nodes—richer interference patterns—and completely filling all angular modes in a shell/subshell yields especially symmetric, low-reorganization states. For main-group elements in periods 2–3, a filled valence set $ns^{2}\,np^{6}$ (the “octet”) closes all available p-harmonics, producing an energetic gap to the next available modes; chemically, such closed shells are reluctant to rearrange.

Energy ordering (Madelung’s rule: increasing $n+\ell$ , then $n$ ) together with electron–electron interactions determines which standing-wave sets fill first. Known exceptions (e.g., Cr, Cu, Mo, Ag, Pd, Au, Pt, etc.) arise where small subshell splittings, correlation, and relativistic effects favor slightly different occupations. The observed capacities and the prominence of closed shells follow directly from rotational symmetry (SO(3)) and the resulting degeneracies of these standing waves.

Notation

In what follows, valence configurations are written as $ns^{x}np^{y}$ (and analogously for d, f), where $n$ is the shell (principal quantum number), $s,p,d,f$ are subshells ( $\ell =0,1,2,3$ ), and superscripts give electron counts.

Organization of the Periodic Table

Horizontal rows: Periods

Elements in the same period have the same number of occupied electron shells, but gradually changing properties from metals (left) to nonmetals (right).

With only the n = 1 shell (1s orbital)

Period 1 – Hydrogen (H) and Helium (He); only the 1s orbital is occupied.

With s and p subshells

Period 2 – From Lithium (Li) to Neon (Ne); first period with a p-block.
Period 3 – From Sodium (Na) to Argon (Ar); parallels Period 2 with the n = 3 shell.

With s, p, and d subshells

Period 4 – From Potassium (K) to Krypton (Kr); introduces 3d transition metals.
Period 5 – From Rubidium (Rb) to Xenon (Xe); includes 4d transition metals.

With s, p, d, and f subshells

Period 6 – From Caesium (Cs) to Radon (Rn); includes 5d transition metals and the lanthanides (4f).
Period 7 – From Francium (Fr) to Oganesson (Og); includes 6d transition metals and the actinides (5f, largely radioactive/synthetic).

Vertical columns

s-block (Groups 1–2)

Alkali metals (Group 1) – Li, Na, K, Rb, Cs, Fr. Soft, highly reactive metals; valence ns¹. (Hydrogen is in Group 1 by position and 1s¹ configuration, but is not an alkali metal.)
Alkaline earth metals (Group 2) – Be, Mg, Ca, Sr, Ba, Ra. Reactive metals; ns²; basic oxides/hydroxides.

Notes:

The s-block begins in period 1 with H (1s¹) and He (1s²).
Hydrogen is not normally classified as an alkali metal; although it sits above Group 1 with 1s¹ and shares some alkali-like features (e.g., H⁺ in acids, saline hydrides), it is a light, diatomic nonmetal and also shows halogen-like H⁻ chemistry.
Helium shows a valence-electron analogy to the alkaline earth metals (ns² like Group 2), but it is not classified as one; its closed 1s shell and extreme inertness align it with the noble gases.
Group placement follows convention and chemical similarity: despite being 1s² (s-block by configuration), helium is placed in Group 18 (noble gases) because it has a completely filled shell and near-inert behavior.

p-block (Groups 13–18; starts at period 2)

Columns are called groups and collect elements with similar valence configurations and broadly similar chemistry. Groups are numbered 1–18 (IUPAC).

Boron group (Group 13) – B, Al, Ga, In, Tl, Nh. ns²np¹.
Carbon group (Group 14) – C, Si, Ge, Sn, Pb, Fl. ns²np².
Pnictogens (Group 15) – N, P, As, Sb, Bi, Mc. ns²np³.
Chalcogens (Group 16) – O, S, Se, Te, Po, Lv. ns²np⁴.
Halogens (Group 17) – F, Cl, Br, I, At, Ts. ns²np⁵.
Noble gases (Group 18) – He, Ne, Ar, Kr, Xe, Rn, Og. ns²np⁶ (He is 1s²); chemically inert, closed shells.

d-block: Transition metals (Groups 3–12; starts at period 4)

Main article: /Transition metals

Electronic structure: typically (n−1)d^1–10 ns^0–2.
Properties: variable oxidation states; colored complexes (d–d and ligand-field); catalysis; magnetism; alloying.
Trends: radii contract across a period; lanthanide contraction influences 5d/6d; high densities/melting points.
Biology: essential cofactors (e.g., Fe in heme; Cu/Zn enzymes; Mn in PSII).

Note: Some definitions exclude Group 12 (Zn, Cd, Hg, Cn) if requiring an incomplete d subshell in the ion.

f-block (periods 6–7)

Often shown offset below the main table:

Lanthanides (La–Lu) – 4f-block “rare earths”; magnets, optics, catalysts, phosphors.
Actinides (Ac–Lr) – 5f-block; radioactive; Th, U (fuel); transuranics (Pu, Am, …).

Trends of the Periodic Table

The table encodes recurring patterns:

Ionization energy and electronegativity generally increase across a period and decrease down a group.
Atomic radius generally decreases across a period and increases down a group.
Metallic character generally decreases across a period and increases down a group.
Electron affinity is often more irregular (notably in the 2nd period and for closed/half-closed subshells).
Reactivity trends: alkali metals become more reactive down the group; halogens typically less reactive.

Chemical Reactivity

Main article: /Chemical Reactivity

The chemical behavior of elements is governed primarily by the arrangement of their valence electrons. Atoms tend to achieve stable configurations by filling or emptying their outermost orbitals, which explains recurring similarities within groups of the periodic table. Elements with nearly full or nearly empty valence shells (such as halogens and alkali metals) are highly reactive, while those with complete shells (noble gases) are largely inert.

Across a period, reactivity patterns reflect the progressive filling of orbitals: alkali metals on the left readily lose a single electron, while halogens on the right strongly attract one. Down a group, atomic size and shielding modify these trends, making alkali metals more reactive but halogens less so. These patterns embody the principle of periodicity, where orbital structure dictates chemical properties.

As a teaching metaphor, the periodic table can be seen as a kind of quantum harmony: electrons fill standing-wave orbitals like notes on an instrument, with stable "chords" arising from closed shells. In this view, chemical reactions are the rearrangements of these quantum notes into more stable resonances.

Molecular Harmonies and Aromaticity

Main article: /Chemical Bonding

Chemical bonding describes the forces that hold atoms together in molecules and solids. At its core, bonding arises from the tendency of atoms to achieve stable electron configurations, often by filling or emptying their outermost orbitals.

Several major types of bonding are distinguished:

Ionic bonds – transfer of electrons between metals and nonmetals, producing oppositely charged ions (e.g., NaCl).
Covalent bonds – sharing of electron pairs between atoms, forming directional links that underlie most molecular structures (e.g., H₂O, CO₂).
Metallic bonds – delocalized electrons shared among many atoms, explaining conductivity and malleability in metals.
Weaker interactions – hydrogen bonds, van der Waals forces, and π–π stacking, which shape molecular geometry, biomolecular folding, and materials behavior.

Resonance and aromaticity provide additional stability in many molecules: delocalized electrons form standing-wave patterns across multiple atoms, producing coherent and highly stable electronic structures (e.g., benzene).

Together, these bonding modes explain the immense diversity of chemical matter, from simple salts and diatomic gases to complex biomolecules and advanced materials.