What it does
Nanoparticles are small enough that a large share of their atoms sit on the surface, which is why they are such good catalysts and why their properties change with size. This page turns a particle size into the numbers you need for a paper, a dose of material in an experiment or a comparison with gas adsorption:
- Surface to volume S/V = 6/d, the surface area and volume of one sphere, and the specific surface area 6/(ρd) in m²/g.
- Atoms per particle (volume ÷ volume per atom, M/(ρN_A)) and the share of atoms on the surface: those within one atom diameter of the surface, 1 − (1 − 2a/d)³.
- Magic-number clusters: closed-shell cuboctahedra (cut from an fcc crystal) and Mackay icosahedra with n shells have N = (10n³ + 15n² + 11n + 3)/3 atoms (13, 55, 147, 309, 561 …) and 10n² + 2 surface atoms. The page splits the surface into corner, edge and face atoms, which have different numbers of neighbours and so behave differently in catalysis, and lists the next sizes.
- Concentration: from a mass concentration (for example from ICP), the number of particles per mL, the molar concentration of particles (often nanomolar), the atoms per mL and the surface area per mL.
- BET-equivalent diameter 6/(ρ × SSA) from a measured specific surface area, and how it compares with your size.
How to use it
- Type the density and molar mass of the material (for a compound, of one formula unit) and the particle diameter. Leave the atom diameter blank to estimate it from the volume per atom, or type the nearest-neighbour distance.
- For clusters, pick the shape and the number of shells. The page suggests the n that matches your diameter.
- Type a mass concentration and a measured surface area if you have them. Download the results as an image, PDF or Word file.
Worked example
Gold spheres 20 nm across (ρ = 19.3 g/cm³, M = 196.97 g/mol) have S/V = 0.3 /nm and a specific surface area of 15.54 m²/g. Each atom takes 0.01695 nm³, so an atom diameter is about 0.3187 nm and each particle holds 247,171 atoms, of which 9.26 % (22,886) are on the surface. A particle weighs 8.084 × 10⁻¹⁷ g.
A cuboctahedron with 3 shells has 147 atoms, 92 on the surface (62.6 %): 12 corners, 48 edge atoms and 32 face atoms. A 20 nm particle matches about n = 31.
At 50 µg/mL of gold there are 6.185 × 10¹¹ particles per mL, a particle concentration of 1.027 nM. A powder of the same gold with a measured surface area of 12 m²/g has a BET-equivalent diameter of 25.91 nm, 1.3 times the 20 nm size.
Tips
- The surface-shell share depends on what you call an atom diameter; use the same choice when comparing sizes.
- Particle concentrations from mass assume every particle has the same size. For a spread of sizes, the mass sits mostly in the biggest particles, so the true number is higher than this estimate.
- A BET diameter much larger than the TEM size usually means the particles are stuck together.
FAQ
Why are 13, 55 and 147 “magic”? Those are the numbers that complete a full shell, so the cluster has no partly filled outer layer and is especially stable.
Do cuboctahedra and icosahedra really have the same counts? Yes, shell by shell, but the icosahedron has only (111)-like faces and 30 edges, while the cuboctahedron has (100) and (111) faces and 24 edges.
Is my input stored? No, all the sums happen on this page.
Where the numbers come from
- S/V = 6/d, specific surface area 6/(ρd), and the magic-number cluster counts N = (10n³ + 15n² + 11n + 3)/3 with 10n² + 2 surface atoms (Mackay 1962, Acta Cryst. 15, 916; Benfield 1992, J. Chem. Soc. Faraday Trans. 88, 1107) are standard closed forms. fibub's code is checked in the tests against clusters built atom by atom in NumPy.
- The example uses the density and molar mass of gold, rounded; the concentration and surface area are made-up round values.
- Nothing you type is sent or stored.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.