Atomic Radius

The atomic radius is one of the most fundamental properties of an element, describing the effective size of an atom. Because electrons occupy probabilistic orbitals rather than fixed shells, atoms do not have sharp, well-defined edges — so chemists define atomic radius operationally, by measuring the distances between bonded or non-bonded atoms and dividing by two. Three main definitions are used: the covalent radius (half the internuclear distance between two identical atoms joined by a single covalent bond), the metallic radius (half the internuclear distance between adjacent atoms in a metallic lattice), and the van der Waals radius (half the internuclear distance between two non-bonded atoms of the same element in the solid state). For example, the Cl–Cl bond length in Cl₂ is 198 pm, giving a covalent radius of 99 pm for chlorine. Atomic radii follow clear, predictable trends across the periodic table that arise directly from two competing factors: the nuclear charge (number of protons, Z) pulling electrons inward, and the shielding effect of inner-shell electrons reducing the effective nuclear charge (Z_eff) felt by outer electrons. Moving across a period from left to right, Z increases while electrons are added to the same principal quantum shell, so Z_eff rises and the electron cloud is pulled closer to the nucleus — atomic radius decreases. Moving down a group, each successive element adds a new principal quantum shell farther from the nucleus, and increased shielding from additional inner electrons reduces Z_eff, so atomic radius increases significantly. Understanding atomic radius is essential for explaining and predicting many other periodic trends: ionisation energy, electronegativity, ionic radius, and reactivity all depend on how tightly a nucleus holds its outer electrons. A larger atom with a lower Z_eff holds its valence electrons more loosely, making it easier to ionise and more likely to form metallic bonds, while a smaller atom with high Z_eff attracts shared electrons strongly and exhibits high electronegativity. Mastery of atomic radius trends therefore provides a foundation for understanding the entire landscape of chemical behaviour across the periodic table.

Key takeaways
  • Covalent Radius
  • Metallic Radius
  • Van der Waals Radius
  • Trend Across a Period (left to right)
  • Trend Down a Group
  • Effective Nuclear Charge and Slater's Rules

Covalent Radius

The covalent radius is defined as half the internuclear distance between two identical atoms bonded by a single covalent bond, measured by X-ray crystallography or spectroscopy. For example, the H–H bond length in H₂ is 74 pm, so the covalent radius of hydrogen is 37 pm. Covalent radii can be added approximately to predict bond lengths between different atoms: r_bond ≈ r_A + r_B. Multiple bonds are shorter than single bonds, so double-bond and triple-bond covalent radii are also tabulated (e.g., C single-bond radius: 77 pm; double-bond: 67 pm; triple-bond: 60 pm).

Metallic Radius

For metals, atoms are packed together in a crystal lattice and share delocalised electrons. The metallic radius is half the internuclear distance between adjacent metal atoms in the solid crystal structure. Metallic radii are generally slightly larger than covalent radii for the same element because delocalised bonding involves less electron-pair repulsion compression. For example, sodium has a metallic radius of 186 pm. Metallic radius is most relevant when comparing trends across the d-block (transition metals), where values change more slowly due to the d-orbital shielding effect.

Van der Waals Radius

Noble gases and non-bonded atoms in molecular solids are held together only by weak London dispersion forces. The van der Waals radius is half the shortest internuclear distance between two non-bonded atoms of the same element. It represents the effective size of the electron cloud when atoms simply touch without forming a chemical bond. Van der Waals radii are always larger than covalent radii: for example, chlorine has a covalent radius of 99 pm but a van der Waals radius of 175 pm. These radii are important in molecular modelling, drug design, and understanding steric effects.

Trend Across a Period (left to right)

Across Period 3: Na (186 pm) > Mg (160 pm) > Al (143 pm) > Si (117 pm) > P (110 pm) > S (104 pm) > Cl (99 pm) > Ar (188 pm, van der Waals). From Na to Cl, Z increases from 11 to 17, but all valence electrons are in the n = 3 shell. Because inner-shell shielding stays nearly constant (the same 10 core electrons), Z_eff rises from approximately +1 to +7. The stronger nuclear attraction contracts the outer electron cloud progressively. Argon's listed van der Waals radius appears larger because it is a different type of measurement — its covalent radius is not defined as it forms no bonds.

Trend Down a Group

Down Group 1: Li (152 pm) < Na (186 pm) < K (227 pm) < Rb (248 pm) < Cs (265 pm). Each step down adds a new principal quantum shell (n increases), placing valence electrons farther from the nucleus. Additionally, each new inner shell provides extra shielding, reducing Z_eff for the outermost electron. Although Z increases, the shielding increases proportionally — in fact, Slater's rules show that a complete inner shell shields the outer electron by approximately 0.85 per electron in the n−1 shell and 1.00 per electron in inner shells, making the increase in distance dominate. The result is a consistent and significant increase in atomic radius going down any group.

Effective Nuclear Charge and Slater's Rules

The effective nuclear charge is calculated as Z_eff = Z − σ, where σ (sigma) is the shielding constant estimated by Slater's rules. Electrons in the same shell shield each other by 0.35, electrons in the n−1 shell shield by 0.85, and electrons in deeper shells shield by 1.00. For a sodium valence electron (Z = 11, configuration [Ne]3s¹): σ = (8 × 0.85) + (2 × 1.00) = 6.80 + 2.00 = 8.80, so Z_eff = 11 − 8.80 = 2.20. For chlorine (Z = 17, [Ne]3s²3p⁵): σ = (6 × 0.35) + (8 × 0.85) + (2 × 1.00) = 2.10 + 6.80 + 2.00 = 10.90, so Z_eff = 17 − 10.90 = 6.10. The much higher Z_eff for Cl explains its much smaller radius compared to Na.

The Key Equation

Zeff=Z−σZ_{\text{eff}} = Z - \sigma
  • 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 electrons
  • Z — Atomic number (dimensionless): The total number of protons in the nucleus, equal to the actual nuclear charge in units of e
  • \sigma — Shielding constant (dimensionless): The total shielding experienced by a valence electron, calculated using Slater's rules by summing contributions from all other electrons

Worked Example

Problem. Using Slater's rules, calculate the effective nuclear charge experienced by a 3p valence electron in sulfur (Z = 16, electron configuration [Ne]3s²3p⁴). Predict whether sulfur or phosphorus (Z = 15, [Ne]3s²3p³) has the larger atomic radius and explain why.

Zeff(P)=4.80<Zeff(S)=5.45  ⟹  r(P)>r(S)Z_{\text{eff}}(\text{P}) = 4.80 < Z_{\text{eff}}(\text{S}) = 5.45 \implies r(\text{P}) > r(\text{S})

Answer. Phosphorus has the larger atomic radius (≈110 pm) compared to sulfur (≈104 pm) because phosphorus has a lower effective nuclear charge (Z_eff ≈ 4.80) than sulfur (Z_eff ≈ 5.45), meaning its valence electrons experience less nuclear attraction and occupy a more expanded electron cloud.

Figure
Diagram
Atomic Radius Trends Across Period 3 and Down Group 1Element (Period 3, left to right)Covalent Radius (pm)NaMgAlSiPSCl010020030018616014311711010499Group 1 Radii (pm) — trend DOWN groupLi 152Na 186K 227Rb 248Cs 265← Radius increases down the group← Radius decreases across period →(Higher Z_eff as proton number rises in same shell)
Origins · 1 minDmitri Mendeleev and how this was discovered
DM
Dmitri Mendeleev
1834–1907 · Russian

Mendeleev arranged the 63 elements known in his time into the first widely accepted periodic table (1869), ordering them by atomic mass and grouping elements with similar chemical properties. He noticed periodic repetition of properties — including the size of atoms — and courageously left gaps for undiscovered elements (gallium, germanium, scandium), predicting their properties with remarkable accuracy. His periodic law — that the properties of elements are a periodic function of their atomic mass (later understood to be atomic number) — laid the conceptual foundation for all modern discussion of periodic trends, including atomic radius. Mendeleev was so confident in the periodicity of elemental properties that he publicly predicted the existence, atomic mass, density, and chemical behaviour of three undiscovered elements. When gallium was discovered in 1875 fitting his predictions almost exactly, his periodic table gained universal acceptance.

  • Ionisation Energy — Ionisation energy is inversely related to atomic radius: smaller atoms with higher Z_eff require more energy to remove a valence electron. Understanding atomic radius trends directly explains ionisation energy trends across periods and groups.
  • Electronegativity — Electronegativity measures an atom's ability to attract bonding electrons. Smaller atomic radius correlates with higher electronegativity because the nucleus exerts a stronger pull on shared electrons over a shorter distance.
  • Ionic Radius — Ionic radii are derived from atomic radii: cations (positive ions) are smaller than their parent atoms due to electron loss increasing Z_eff, while anions are larger due to increased electron-electron repulsion. Atomic radius is the essential baseline.
  • Electron Configuration — Atomic radius trends arise directly from electron configurations — specifically the principal quantum number of valence electrons and the number of core shielding electrons. Understanding orbital filling is prerequisite to explaining radius trends.
  • Shielding Effect and Z_eff — The shielding effect and Slater's rules are the mechanistic explanation for atomic radius trends. Inner electrons reduce Z_eff felt by outer electrons, directly determining how tightly the atom's radius is contracted.