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.

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

r1 = 4πε0  (h/2π)²/m_ee² ————- Eq-1

Where,

Eo = Permittivity of free space

h = Planck’s constant = 6.63 x 10^-34 Js

m_e= Mass of an electron = 9.1 x 10^-³¹kg

e = Charge of an electron = 1.9 x 10^–19 C

m_p= Mass of a proton = 1.67 x 10 ^–27 kg

r= Distance between the electron and the proton

Coulomb attraction between an electron and a proton is given as:

F_C = e²/4πε0r²————-Eq-2

Gravitational force of attraction between an electron and a proton is given as:

F_G =G m_p m_e /r²   ——————- Eq-3                     ,

^Fc, =-^H-                                         -(3)

Where, G = Gravitational constant = 6.67 x 10⁻¹¹ N m²/kg²

If the electrostatic (Coulomb) force and the gravitational force between an electron and a proton are equal, then we can write:

Therefore , F_G=F_C

=>    G m_p m_e /r² = e²/4πε0r²

Therefore ,

e²/4πε0 =G m_p m_e—————Eq-4

Putting the value of equation (4) in equation (1),we get:

r1= (h/2π)²/G m_p m²_e

= [6.63 x 10^-34/(2 x 3.14)]²/(6.67 x 10⁻¹¹) x (1.67 x 10 ^–27) x (9.1 x 10^-³¹)²

≈1.21 x 10²⁹
It is known that the universe is 156 billion light years wide or 1.5 x 10²⁷ 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.