The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about `10^(-40)`. An alternative way of looking at this fact is to estimate the radius of the first Bohr orbit of a hydrogen atom if the electron and proton were bound by gravitational attraction. You will find the answer interesting.

Explanation

Radius of the first Bohr orbit is given by the relation
`r_(1)=(4piin_(0)(h/(2pi))^(2))/(m_(e)e^(2))` ….(i)
Where
`in_(0)` =permittivity of free space
h=Planck's constant `=6.63xx10^(-34)Js`
`m_(e)` = Mass of an electron `=9.1xx10^(-31)kg`
e= Charge of an electron `=1.9xx10^(-19)C`
`m_(p)` =Mass of a proton `=1.67xx10^(-27)kg`
r= Distance between the electron and the proton
Coulomb attraction between an electron and a proton is given as:
`F_(c)=(e^(2))/(4pi in_(0)r^(2))` ....(2)
Gravitational force of attraction between an electron and a proton is given as:
`F_(G)=(Gm_(p)m_(e))/r^(2)` ....(3)
Where, G= Gravitational constant `=6.67xx10^(-11)N m^(2)//kg^(2)`
If the electrostatic (couloumb) force and the gravitational force between an electron and a proton are equal, then we can write:
`:.F_(G)=F_(C)`
`(Gm_(p)m_(e))/r^(2)=e^(2)/(4pi in_(0)r^(2))`
`:.e^(2)/(4pi in_(0))=Gm_(p)m_(e)` .....(4)
Putting the value of equation (4) in equation (1) we get:
`r_(1)=(h/(2pi))^(2)/(Gm_(p)m_(e)^(2))`
`(6.63xx10^(-34)/(2xx3.14))^(2)/(6.67xx10^(-11)xx1.67xx10^(-27)xx(9.1xx10^(-31))^(2)) approx 1.21xx10^(29)m`
It is known that the universe is 156 billion light years wide or `1.5xx10^(27)` m wide Hence, we can conclude that the radius of the first Bohr orbit is much greater then the estimated size of the whole universe.