Question

Use de-Broglie's hypothesis to write the relation for the n^th radius of Bohr orbit in terms of Bohr's quantization condition of orbital angular momentum ?

Answer 1

According to Bohr’s postulates, 

\[mvr = \frac{nh}{2\pi}\]   ...... (1)

(where mvr = angular momentum of an electron and n is an integer).

Thus, the centripetal force,

\[\frac{m v^2}{r}\] (experienced by the electron) is due to the electrostatic attraction, 

\[\frac{kZ e^2}{r^2}\].

Where,
Z = Atomic number

Therefore, 

\[\frac{m v^2}{r} = \frac{kZ e^2}{r^2}\].

Substituting the value of v² from (1), we obtain:

\[\frac{m}{r}\frac{n^2 h^2}{4 \pi^2 m^2 r^2} = \frac{kZ e^2}{r^2}\]

\[\therefore r = \frac{n^2 h^2}{4 \pi^2 mkZ e^2}\]

The relation for the n^th radius of Bohr orbit in terms of Bohr's quantization condition of orbital angular momentum

\[= \frac{n^2 h^2}{4 \pi^2 mkZ e^2}\].