Here I use a microscope to determine the volume of a sand grain. Calculation is below.
***** Overview *****
The number of sand grains on a beach can be estimated by dividing the total volume of sand on the beach by the volume of one sand grain (plus air).
Sand grains = volume of sand / volume of one grain
***** Calculating Volume of Sand on the Beach *****
The volume of the beach can be calculated by multiplying the area of the cross section of the beach (a triangle) by the length of the beach. In the video I walked up the hypotenuse of the triangle, but we assume that the length of the base is similar.
length of beach: 20 000m
width of beach: 50m
height of beach 2m
Volume = ((50m*2m)/2)*20000m = 1 million cubic meters
***** Calculating the volume of one sand grain *****
We have to determine the radius of a sand grain first. To do this, we determine the projected area of a sand grain and then using the formula for the area of a circle, calculate the radius. We can then use the radius of the grain to determine the volume of the grain.
Mass of reference grain: 0.60g
Diameter of reference grain: 0.5mm (magnified)
Area of reference grain = 2 * pi * r^2 = 2 * 3.14 * (0.5mm)^2 = 7.85 * 10^-7 m^2
With the area of the reference grain and the mass, we can now calculate the area of all enlarged sand grains:
Mass of all magnified grains = 1.94g
Number of magnified grains = 47
Mass of one magnified grain = 1.94g / 47 = 0,041g
We can now calculate the area of one magnified grain, by direct proportion. Important: the area is not the area on paper, but the “real” area. Everything is magnified.
mass of reference grain / area of reference grain = mass of one magnified grain / area of magnified grain
0.60g / 7.85 * 10^-7 m^2 = 0,041g / X
X = projected area of grain = 5,36417E-08 m^2
We can now calculate the radius of one grain
A = pi * r^2
5,36417E-08 m^2 / pi = r^2
r = 0,000130703 m
This is 0.13mm, which is plausible
We use the formula for the volume of a sphere to calculate the volume of a grain:
V = 4/3 * pi * r^3
V = 9,34819E-12 m^3
We double the volume to account for air:
V (with air) = 1,86964E-11 m^3
***** Calculating the number of sand grains *****
Sand grains = volume of sand / volume of one grain
Sand grains = 10^6 m^3 / 1,86964E-11 m^3
Sand grains = 5,35 * 10^16
For ease of calculation, we estimate the number of sand grains to be about 10^16 (also because I overestimated the length of the beach somewhat). The order of magnitude is important, not the actual number.