Bohr’s atomic model laid down various postulates for the arrangement of electrons in different orbits around the nucleus. According to Bohr’s atomic model, the angular momentum of the electron orbiting around the nucleus is quantized. He further added that electrons move only in those orbits where angular momentum of electron is an integral multiple of h/2?. This postulate regarding the quantisation of angular momentum of electron was later explained by Louis de Broglie. According to him, a moving electron in its circular orbit behaves like a particle wave.
De Broglie’s explanation to quantization of angular momentum of electron:
Behavior of particle waves can be viewed analogous to the waves travelling on a string. Particle waves can lead to standing waves held under resonant conditions. When a stationary string is plucked, a number of wavelengths are excited. On the other hand, we know that only those wavelengths survive which form a standing wave in the string, that is, which have nodes at the ends. Thus, in a string, standing waves are formed only when the total distance travelled by a wave is an integral number of wavelengths. Hence, for any electron moving in kth circular orbit of radius rk, the total distance is equal to the circumference of the orbit, 2πrk.
2πrk = kλ ———————- (1)
Where, λ is the de Broglie wavelength.
We know that de Broglie wavelength is given by,
λ = h/p,
Where, p is electron’s momentum
h = Planck’s constant
λ = h/mvk ————————– (2)
Where, mvk is the momentum of electron revolving in the kth orbit. Inserting the value of λ from equation 2 in equation 1 we get,
2πrk = kh/mvk
mvkrk = kh/2π
Hence, de Broglie hypothesis successfully proves Bohr’s second postulate stating the quantization of angular momentum of the orbiting electron. We can also conclude that the quantized electron orbits and energy states are due to the wave nature of the electron.
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