Question

Derive an expression for the linear speed of an electron in a Bohr orbit. Hence, show that it is inversely proportional to the principal quantum number.

Answer 1

Consider an electron revolving in the nth Bohr 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 ε₀ is the permittivity of free space.

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

∴ r = \(\cfrac{nh}{2πmv}\)

Substituting this expression for r in Eqn (2), we get,

as Z, e, ε₀ and h are constants.

[Note : In this topic, unless stated otherwise, m = m_e , r = r_n, and v = v_n.]