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Derive an expression for the radius of the nth Bohr orbit in an atom. Hence, show that the radius of the orbit is directly proportional to the square of the principal quantum number.

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Consider an electron revolving in the nth orbit around the nucleus of an atom with the atomic number Z. Let m and – e be the mass and charge of the electron, r the radius of the orbit and v the linear speed of the electron.

According to Bohr’s first postulate, centripetal force on the electron = electrostatic force of attraction exerted on the electron by the nucleus.

where ɛis the permittivity of free space.

According to Bohr's second postulate, the orbital angular momentum of the electron,

where h is Planck’s constant and n is the principal quantum number which takes integral values 1, 2, 3, …… etc.

Equating the right sides of Eqs. (2) and (4),

Since, ε0, h, Z, in and e are constants, it follows that r ∝ n2 , i.e., the radius of a Bohr orbit of the electron in an atom is directly proportional to the square of the principal quantum number.

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