Spherical TiO₂ nanoparticles 20 nm in diameter (density 4.0 g/cm³) are all the same size and fully dispersed. Which value is closest to the surface area per gram of powder (the specific surface area)?

Answer: B. About 75 m²/g

Dividing the surface area of one sphere, πd², by its mass, ρπd³/6, gives a specific surface area per mass of S = 6/(ρd). Substituting ρ = 4.0 g/cm³ = 4.0×10⁶ g/m³ and d = 20×10⁻⁹ m gives S = 6/(4.0×10⁶ × 20×10⁻⁹) = 6/0.080 = 75 m²/g. About 150 m²/g puts the radius, 10 nm, in place of d, and about 37.5 m²/g puts the diameter, 20 nm, in place of r in the radius form S = 3/(ρr). The specific surface area is inversely proportional to particle size, so halving the size doubles it. In real powders the particles touch and agglomerate, so measured values are usually smaller than this calculated value.