The Atomic Radius Calculator estimates the atomic radius of any element from principal quantum number and effective nuclear charge using the Slater orbital approximation. Computes covalent and van der Waals radius relationships — used in quantum chemistry and teaching periodic trends.
216.4
pm
2.164
A
122
pm
42.46
216.4
pm
2.164
A
122
pm
42.46
Why does atomic radius decrease across a period but increase down a group? Why is the gap between Na and Cl so large? The answers lie in the balance between nuclear charge pulling electrons inward and the shielding of inner electrons softening that pull. The calculator for atomic radius applies the Slater orbital approximation to estimate atomic radius from first principles, making periodic trends quantitatively understandable rather than just qualitatively memorized.
The orbital radius r for an electron in the outermost shell can be estimated from the Bohr model scaled by effective nuclear charge:
r ≈ a₀ × n² / Z_eff
where a₀ = 0.529 Å is the Bohr radius, n is the principal quantum number of the outer shell, and Z_eff is the effective nuclear charge experienced by the outer electrons. Z_eff = Z − S, where Z is the actual atomic number and S is the shielding constant calculated from Slater's rules. For sodium (Z=11, n=3, Z_eff ≈ 2.2): r ≈ 0.529 × 9 / 2.2 = 2.16 Å. The actual covalent radius of Na is 1.66 Å — the Slater estimate is directionally correct and captures trends, though it overestimates absolute radii. Use this online calculator to compute estimated radii for any element.
Two competing effects govern atomic radius trends:
The effective nuclear charge calculator computes Z_eff using Slater's rules as the key intermediate variable.
Atomic radius is not a single well-defined quantity — it depends on the bonding context:
The ionization energy calculator and periodic table calculators provide complementary tools for periodic property analysis.
The lanthanide contraction is a striking anomaly: the atomic radii of hafnium (Hf, Period 6) and zirconium (Zr, Period 5) are nearly identical despite Hf being one full period lower. This occurs because the 14 lanthanide elements between them progressively fill the poorly-shielding 4f orbitals, increasing Z_eff substantially and shrinking the atom. The practical consequence: Hf and Zr are chemically nearly indistinguishable, making their separation in ore processing one of the most difficult challenges in inorganic chemistry.
The Bohr model radius for the most probable electron distance is:
r = a0 x n² / Z_eff
Where a0 = 52.9 pm (Bohr radius for hydrogen), n is the principal quantum number, and Z_eff is the effective nuclear charge. This formula captures the two main factors determining atomic size: higher n (more shells) increases radius, while higher Z_eff (more nuclear attraction) decreases it.
The bond length method uses the additivity approximation:
d(A-B) = r(A) + r(B)
Where d is the measured bond length (from X-ray crystallography or spectroscopy) and r values are covalent radii. If one radius is known, the other can be determined. This works well for single covalent bonds but overestimates for multiple bonds and ionic bonds.
The atomic volume is calculated assuming a spherical atom: V = (4/3) x pi x r^3. This helps compare atomic sizes and estimate packing efficiencies in crystal structures.
The Bohr model radius gives a rough estimate that captures periodic trends but may differ from tabulated covalent or van der Waals radii by 20-50%. Across a period, the estimated radius decreases because Z_eff increases while n stays constant. Down a group, radius increases because n increases faster than Z_eff. The bond-length-derived radius provides a more practical covalent radius value. The atomic volume provides perspective on how much space an atom occupies, useful for understanding density and packing in solids.
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Results
Sodium with n=3, Z_eff=2.2 gives a Bohr radius of 216 pm, compared to the experimental metallic radius of 186 pm and covalent radius of 166 pm. The Na-Cl bond length of 308 pm minus Na+ radius (186 pm as metallic, but ionic radius 102 pm more appropriate) illustrates the complexity of radius definitions.
Inputs
Results
Carbon with n=2, Z_eff=3.14 gives a Bohr radius of 67.4 pm, somewhat close to the covalent radius of 77 pm. Using the C-C single bond length of 154 pm and one C radius of 77 pm correctly gives the second C radius as 77 pm, demonstrating the additivity principle for homonuclear bonds.
Atoms do not have fixed boundaries, so radius depends on how it is measured. Covalent radius is half the bond length between two identical atoms. Van der Waals radius is the effective radius for non-bonded contacts. Metallic radius is half the nearest-neighbor distance in a metal crystal. Ionic radius depends on the charge and coordination number. Each definition serves different purposes.
Atomic radius decreases across a period from left to right. Each element adds one proton and one electron, but the electron goes into the same shell and shields poorly. The increasing effective nuclear charge pulls all electrons closer to the nucleus. For example, sodium (186 pm) to argon (71 pm) shows a dramatic decrease across period 3.
Atomic radius increases down a group because each successive element adds a new electron shell (higher n). Although Z_eff also increases down a group, the effect of the larger shell dominates. For example, lithium (152 pm) to cesium (265 pm) shows the steady increase in Group 1.
Cations are smaller than their parent atoms because removing electrons reduces electron-electron repulsion and the remaining electrons are held more tightly by the unchanged nuclear charge. Anions are larger because added electrons increase repulsion and the nuclear charge is spread over more electrons. For example, Na (186 pm) vs Na+ (102 pm), and Cl (99 pm) vs Cl- (181 pm).
The Bohr radius (a0 = 52.9 pm = 0.529 Angstrom) is the most probable distance between the nucleus and electron in a hydrogen atom in its ground state. It serves as the natural unit of length in atomic physics and is used as the scale factor in atomic radius calculations: r = a0 x n² / Z_eff.
The d-block contraction (also called scandide contraction) occurs because d electrons are poor shielders. As the 3d subshell fills across the first transition series, Z_eff increases significantly, causing atoms after the d-block to be smaller than expected. This makes Ga (135 pm) only slightly larger than Al (125 pm) despite being one period lower.
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