Question

Grains of fine California beach sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide, which has a density of 2.9 × 10³ kg/m³. What mass of sand grains would have a total surface area (the total area of all the individual spheres) equal to the surface area of a cube 0.8 m on an edge?

Answer 1

surface area of cube = 0.8^3 = 0.512 m^3

surface area of grain = 4 pi r^2

= 4 x pi x (50 x 10^-6)^2

= 3.14 x 10^-8 m^2

number of grains = (0.512) / (3.14 x 10^-8)

= 1.63 x 10^7 grains

volume of each grain = 4 x pi x r^3 / 3

= 4 x pi x (50 x 10^-6)^3 / 3

= 5.24 x 10^-13 m^3

total volume = (5.24 x 10^-13) (1.63 x 10^7)

= 8.53 x 10^-6 m^3

mass = volume x density = (8.53 x 10^-6) (2.9x 10^3)

= 0.0247 kg or 24.7 grams