🧪 Nanoparticle Surface-to-Volume Ratio & Melting-Point Depression Simulator

A sphere's surface-to-volume ratio is exactly 3/r — pure geometry, no approximation. For a coin-sized particle that ratio is negligible, but for a nanoparticle a few nanometers across it becomes enormous, and this simulator makes the consequence chain visible and quantitative rather than just asserted. Pick a real metal (gold, copper, aluminum or iron) and sweep its radius from a micrometer down to a single nanometer: the surface-atom fraction is computed from actual sphere-shell geometry, the melting-point depression uses the real Gibbs–Thomson relation with each metal's literature solid–liquid interfacial energy, molar volume and latent heat of fusion, and the surface atom count follows from the particle's true atomic packing. Watch all four coupled, physically grounded curves move together as the particle crosses from bulk-like behavior into the nanoscale regime where surface effects take over.