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.

#### Solution

Radius of the first Bohr orbit is given by the relation,

`r_1 = 4pi in_0 (h/(2pi))^2/(m_e e^2)` ....(1)

Where,

∈_{0} = Permittivity of free space

*h* = Planck’s constant = 6.63 × 10^{−34} Js

*m*_{e} = Mass of an electron = 9.1 × 10^{−31} kg

*e* = Charge of an electron = 1.9 × 10^{−19} C

*m*_{p} = Mass of a proton = 1.67 × 10^{−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/(4piin_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 = 6.67 × 10^{−11} N m^{2}/kg^{2}

If the electrostatic (Coulomb) 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/(4piin_0 r^2)`

`:. e^2/(4piin_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)`

`= (((6.63 xx 10^(-34))/(2xx3.14))^2)/(6.67 xx 10^(-11) xx 1.67 xx 10^(-27) xx (9.1 xx 10^(-31))^2) ~~ 1.21 xx 10^(29) m`

It is known that the universe is 156 billion light years wide or 1.5 × 10^{27 }m wide. Hence, we can conclude that the radius of the first Bohr orbit is much greater than the estimated size of the whole universe.