Class 12 Physics
CBSE and UP Board
Atoms
Chapter-12 Exercise 12.12
NCERT Solutions for Class 12 Physics Chapter 12 Question-12
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.
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Radius of the first Bohr orbit is given by the relation,
r1 = 4πε0 (h/2π)²/mee² ————- Eq-1
Where,
Eo = Permittivity of free space
h = Planck’s constant = 6.63 x 10-34 Js
me= Mass of an electron = 9.1 x 10-³¹kg
e = Charge of an electron = 1.9 x 10–19 C
mp= 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:
FC = e²/4πε0r²————-Eq-2
Gravitational force of attraction between an electron and a proton is given as:
FG =G mp me /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 , FG=FC
=> G mp me /r² = e²/4πε0r²
Therefore ,
e²/4πε0 =G mp me—————Eq-4
Putting the value of equation (4) in equation (1),we get:
r1= (h/2π)²/G mp m²e
= [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.