When the diameter of spherical particles is reduced from 1 µm to 10 nm, by what factor does the surface area per unit volume (specific surface area) increase?

Answer: B. 100 times

A sphere has a surface area of 4πr² and a volume of (4/3)πr³, so the surface area per unit volume is 3/r, inversely proportional to the radius (diameter). When the diameter falls to 1/100, from 1 µm = 1000 nm to 10 nm, the specific surface area increases 100 times. Ten times takes the square root of the ratio, and 10,000 times confuses this with the ratio of the surface areas of single particles (the square of the diameter ratio). In particles about 10 nm in diameter, roughly a tenth of all atoms are at the surface, and the properties of the surface atoms strongly influence those of the particle as a whole. Nanoparticles are used as catalysts because of their large specific surface area.