Ncert Exercises & Solutions

12.12 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.

Hint: The centripetal force is responsible for the circular motion of an electron about the nucleus.
Step 1: Find the radius of the first Bohr orbit.

The radius of the first Bohr orbit is given by the relation,

r_{1} = \frac{4 π \in_{0} {(\frac{h}{2 π})}^{2}}{m_{e} e^{2}} . . . (1)

where,

\in_{0}

= Permittivity of free space

h = Planck's constant =

6.62 \times 10^{- 34} Js

m_{e}

= mass of an electron =

9.1 \times 10^{- 31} kg

e = Charge of an electron =

1.9 \times 10^{- 19} C

m_{p}

= Mas of a proton =

1.67 \times 10^{- 27} kg

r= Distance between the electron and the proton
Step 2: Find condition when the gravitational force is equivalent to the electrostatic force.
Coulomb attraction between an electron and a proton is given as:

F_{c} = \frac{e^{2}}{4 π \in_{0} r^{2}} . . . (2)

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

F_{G} = \frac{G m_{p} m_{e}}{r^{2}}

where G=Gravitational constant

= 6.67 \times 10^{- 11} N m^{2} / k g^{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}

\frac{G m_{p} m_{e}}{r^{2}} = \frac{e^{2}}{2 π \in_{0} r^{2}}

∴ \frac{e^{2}}{4 π \in_{0}} = {Gm}_{p} m_{e}

Step 3: Find the radius of the first Bohr orbit for this gravitational force.
Putting this value in equation (1), we get;

r_{1} = \frac{{(\frac{h}{2 π})}^{2}}{{Gm}_{p} {m_{e}}^{2}}

= \frac{{(\frac{6.62 \times 10^{- 34}}{2 \times 3.14})}^{2}}{6.67 \times 10^{- 11} \times 1.67 \times 10^{- 27} \times {(9.1 \times 10^{- 31})}^{2}} \approx 1.21 \times 10^{29} m

It is known that the universe is 156 billion light years wide or
1.5 x

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.

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