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.

• A. The radius of the first Bohr orbit and the estimated size of the whole universe is same.
• B. The radius of the first Bohr orbit is much greater than the estimated size of the whole universe.
• C. The radius of the first Bohr orbit is much smaller than the estimated size of the whole universe.
• D. None of the above.

Answer 1

Answer: (B)

The radius of the first Bohr orbit is much greater than the estimated size of the whole universe.

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

$r_1=\frac{4\pi {ϵ}_0(\frac{h}{2\pi }{)}^2}{m_e{e}^2}$ ......(1)

Where,

${ϵ}_0=$ Permittivity of free space

$h=$ Planck's constant

$m_e=$ Mass of an electron

$e=$ Charge of an electron

$m_p=$ Mass of a proton

$r=$ Distance between the electron and the proton

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

$F_C=\frac{e^2}{4\pi {ϵ}_0{r}^2}$ .....(2)

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

$F_G=\frac{G{m}_p{m}_c}{r^2}$ .....(3)

Where

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

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

$F_G=F_C$

$\frac{G{m}_p{m}_c}{r^2}=\frac{e^2}{4\pi {ϵ}_0{r}^2}$

$\therefore \frac{e^2}{4\pi {ϵ}_0}=G{m}_p{m}_c$ .....(4)

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

$r_1=\frac{(\frac{h}{2\pi }{)}^2}{G{m}_p{m}_c^2}$

$=\frac{(\frac{6.63\times {10}^{-34}}{2\times 3.14}{)}^2}{6.67\times {10}^{-11}\times 1.67\times {10}^{-27}\times (9.1\times {10}^{-31}{)}^2}\approx 1.21\times {10}^{29}m$

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