# What is the ratio of the density of the hydrogen nucleus to the density of the complete hydrogen atom?

## The nucleus of the hydrogen atom is a single proton, which has a radius of about 1.0 x 10-15 m. The single electron in a hydrogen atom normally orbits the nucleus at a distance of about 5.3 x 10-11 m.

May 25, 2016

The ratio is 1.5 × 10^14.

#### Explanation:

Density of proton

The mass of a proton is 1.67 × 10^"-27"color(white)(l) "kg".

The volume comes from the formula

color(blue)(|bar(ul(color(white)(a/a) V = 4/3πr^3color(white)(a/a)|)))" "

V = 4/3π × (1.0 × 10^"-15" color(white)(l)"m")^3 = 4.19 × 10^"-45"color(white)(l) "m"^3

ρ = m/V = (1.67 × 10^"-27"color(white)(l) "kg")/(4.19 × 10^"-45"color(white)(l) "m"^3) = 3.99 × 10^17color(white)(l) "kg/m"^3

Density of hydrogen atom

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

V = 4/3π × (5.3 × 10^"-11" color(white)(l)"m")^3 = 6.24 × 10^"-31"color(white)(l) "m"^3

ρ = m/V = (1.67 × 10^"-27"color(white)(l) "kg")/(6.24 × 10^"-31"color(white)(l) "m"^3) = 2.68 × 10^3 color(white)(l)"kg/m"^3

Ratio of densities

ρ_p/ρ_H = (3.99 × 10^17 color(red)(cancel(color(black)("kg/m"^3))))/(2.68 × 10^3 color(red)(cancel(color(black)("kg/m"^3)))) = 1.5 × 10^14