Atomic Radius

Key Ideas

• Definition: atomic radius ≈ half the distance between the nuclei of two adjacent identical atoms.
• Covalent radius: measured for nonmetals sharing a covalent bond (half the bond length).
• Metallic radius: measured for metals in a metallic crystal (half the internuclear distance).
• van der Waals radius: half the closest distance between two non-bonded atoms; always larger than the covalent radius.
• Across a period (→): radius decreases. Zeff increases, but the principal quantum number n stays the same, so the same shell is pulled tighter.
• Down a group (↓): radius increases. Electrons occupy higher-n shells that are farther from the nucleus and more shielded.
• Typical magnitude: atomic radii are on the order of tens to a few hundred picometers (pm); 1 pm = 10⁻¹² m.

Equations and Variables

There is no single equation that produces atomic radius, but the trend follows from effective nuclear charge:

• Zeff = Z − S (see the effective-nuclear-charge note).
  • Larger Zeff → stronger pull → smaller radius (within a shell).
• n (principal quantum number): larger n → electrons farther out → larger radius.

A useful memory anchor: radius ∝ n²/Zeff (the exact quantum-mechanical result for one-electron species; for many-electron atoms it is only a qualitative guide).

How It Works

1. Protons in the nucleus attract the electrons; the more protons (higher Z), the stronger the pull on a given shell.
2. Inner electrons shield outer electrons, reducing the net pull to Zeff.
3. Across a period, electrons are added to the same shell (n constant), so shielding stays roughly constant while Z — and therefore Zeff — increases.
4. The stronger Zeff contracts the electron cloud, so radius decreases left to right.
5. Down a group, each element adds an entire new shell (n increases by 1), which places the valence electrons farther from the nucleus.
6. The new inner shells also shield the valence electrons, so the radius increases down the group even though Z also increases.

Worked Example

Rank the following in order of increasing atomic radius: K, Ca, As, Br.

All four elements are in period 4, where radius decreases left to right. The order left → right in period 4 is K, Ca, As, Br, so:

• K (largest, far left) > Ca > As > Br (smallest, far right).

Increasing radius: Br < As < Ca < K.

Second check — a group trend: rank F, Cl, Br, I in order of increasing radius. These are the halogens in group 17; radius increases down the group, so: Increasing radius: F < Cl < Br < I (F ≈ 71 pm, Cl ≈ 99 pm, Br ≈ 114 pm, I ≈ 133 pm, covalent radii).

Common Confusions

• "More protons always means a bigger atom." — Wrong: more protons means a stronger pull. Across a period, more protons shrink the atom; only down a group (where new shells are added) does size grow.
• "Covalent and van der Waals radii are the same." — Wrong: the van der Waals radius (non-bonded contact) is larger than the covalent (bonded) radius.
• "Anions and cations follow the same trend as neutral atoms." — No: removing electrons makes a cation smaller than its parent atom; adding electrons makes an anion larger (see the ionic-radius note).
• "The radius increases smoothly and uniformly." — Mostly true for the main groups, but the d-block (transition metals) and the lanthanide contraction produce plateaus and small anomalies.