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NCERT Class XII Chapter
Atoms
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Question : 12 of 18
Marks: +1, -0
The gravitational attraction between electron and proton in a hydrogen atom is weaker than the coulomb attraction by a factor of about 10−4010^{-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.
Solution:  
The radius of the first orbit in Bohr’s model is calculated by considering the electrostatic force between electrons and proton in nucleus.
r0r_0 = n2h24π2meke2\frac{n^2 h^2}{4\pi^2 m_e k e^2} where k = 14πε0\frac{1}{4\pi \varepsilon_0}
If we consider the atom bound by gravitational force Gmpmer2\frac{G m_p m_e}{r^2} , then the term ke2k e^2 should be replaced by GmpmeG m_p m_e. The radius of the first Bohr orbit ina gravitational bound hydrogen atom will be r′ = n2h24π2Gmpme2\frac{n^2 h^2}{4\pi^2 G m_p m_e^2}
(1)2(6.6×10−34)24×9.87×.67×10−11×1.625×10−27×(9.1×10−31)2\frac{(1)^2 (6.6 \times 10^{-34})^2}{4 \times 9.87 \times .67 \times 10^{-11} \times 1.625 \times 10^{-27} \times (9.1 \times 10^{-31})^2}
r′ = 1.194 × 102910^{29} m ≈ 1.2 × 102910^{29} m.
This radius is much greater than the estimated size of the whole universe.
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