Question

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.

Answer 1

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 m²/kg²

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 × 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.