How can I calculate the densities of a hydrogen nucleus and a hydrogen atom?

You use the formula for density and the published radii to calculate the densities. The values are ##6.70 × 10^14"kg/dm"^3## and ##2.7"kg/dm"^3##.

The formulas to use are

##"Density" = "mass"/"volume"## or ##ρ = m/V## and

##V = (4/3)πr^3##

Hydrogen nuclei

A hydrogen nucleus is a proton, and the mass of a proton is ##1.67 × 10^(-27)"kg"##.

The most recent value for the radius of a proton is ##8.41 × 10^(-16)"m"##.

##V = (4/3)πr^3 = (4/3)π(8.41 × 10^-16"m")^3 = 2.49 × 10^(-45)"m"^3##

##ρ = m/V = (1.67 × 10^(-27)"kg")/(2.49 × 10^(-45)"m"^3)= 6.70 × 10^17"kg/m"^3 = 6.70 × 10^14"kg/dm"^3##

H atoms

The mass of a hydrogen atom is ##1.67 × 10^(-27)"kg"##.

The radius of a hydrogen atom, calculated using self-consistent field functions, is ##"53 pm"## or ##53 × 10^(-12)"m"##.

##V = (4/3)πr^3 = (4/3)π(53 × 10^(-12)"m")^3 = 6.24 × 10^(-31)"m"^3##

##ρ = m/V = (1.67 × 10^(-27)"kg")/(6.24 × 10^(-31)"m"^3) = 2.7 × 10^3"kg/m"^3 = "2.7 kg/dm"^3##