Estimate the number of moles of water in all the Earth's oceans. Assume water covers 75% of the Earth to an average depth of 3 km.

there are approximately n ≈ 10²² moles

Explanation:

Since the radius of the earth is approximately R=6378 km = 6.378*10⁶ m, then the surface S of the earth would be

S = 4*π*R²

since the water covers 75% of the Earth's surface, the surface covered by water Sw is

Sw=0.75*S

the volume for a surface Sw and a depth D = 3 km = 3000 m (approximating the volume through a rectangular shape) is

V=Sw*D

the mass of water under a volume V, assuming a density ρ = 1000 kg/m³ is

m=ρ*V

the number of moles n of water (molecular weight M = 18 g/mole = 1.8*10⁻² kg/mole) for a mass m is

n = m/M

then

n = m/M = ρ*V/M = ρ*Sw*D/M = 0.75*ρ*S*D/M = 3/4*ρ*4*π*R² * D/M = 3*π*ρ*R² * D/M

n=3*π*ρ*R² * D/M

replacing values

n=3*π*ρ*R² * D/M = 3*π*1000 kg/m³ * (6.378*10⁶ m) ² * 3000 m / (1.8*10⁻² kg/mole) = 3*π*6.378*3/1.8 * 10²⁰ = 100.18 * 10²⁰ ≈ 10²² moles

n ≈ 10²² moles