Question

Derive an expression for radius of n^th Bohr orbit.

Answer 1

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, the centripetal force on the electron = the electrostatic force of attraction exerted on the electron by the nucleus.

∴ `(mv^2)/r = 1/(4 pi epsilon_0) * (Z e^2)/r^2`    ...(i)

where ε₀ is the permittivity of free space.

∴ v² = `(Z e^2)/(4 pi epsilon_0 m r)`    ...(2)

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

mvr = `(nh)/(2 pi)`    ...(3)

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

∴ v = `(n h)/(2 pi m  r)`

∴ v² = `(n^2 h^2)/(4 pi^2 m^2 r^2)`    ...(4)

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

`(Z e^2)/(4 pi epsilon_0 m r) = (n^2 h^2)/(4 pi^2 m^2 r^2)`

Since ε₀, h, Z, m, 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.