Question

The stationary Bohr’s orbit can be readily explained on the basis of wave nature of electron if it is assumed that :
(A) Wave in any of the orbits is the stationary wave.
(B) The position of the maxima and minima of the wave does not change with time.
(C) The length of the circular orbit must be an integral multiple of the wavelength.
(D) Wave in any of the orbits is not a stationary wave.

Answer 1

Hint: To solve the given problem, we should have information about Bohr’s atomic model and the wave nature of electrons. Bohr’s atomic model assumes that the whole positive charge and entire mass of the atom is concentrated at the central core of the atom known as Nucleus which acquires very less space in the atom . It does not consider the wave nature of electrons and is applicable for hydrogen atoms or hydrogen-like ions.

Complete answer:
Step-1 :
Here,we have given stationary Bohr’s orbit which means it is revolving or following the same path and is more often observed as still. This means that the electron is neither radiating nor observing energy. So, option (A) is correct.

Step-2 :
In a wave, if the position of maxima and minima does not change with time, it is regarded as a still wave and can be considered as Bohr’s orbit as it is a stationary wave. So, option (B) is correct.

Step-3 :
In Bohr’s postulates, the electrons revolve around certain permitted orbits that have the angular momentum as an integral multiple of $ \dfrac{h}{2 \pi} $ . So, the length of the circular orbit will be an integral multiple of wavelength.
So, option (C) is also correct.

Note:
The waves associated with matter have wavelength known as de Broglie’s wavelength which gives us $ \lambda = \dfrac{h}{mv} $ and according to angular momentum, we have $ n \lambda = 2 \pi r $ considering the orbit as circular, which later gives us $ mvr = \dfrac{nh}{2 \pi}$ .