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#### CHAPTER 12 ATOMS

de Broglie’s explanation of Bohr’s second postulate of quantisation

The second postulate in the Bohr’s atomic model states that the angular momentum of the electron orbiting around the nucleus is quantised (that is, L= nh/2π; n = 1, 2, 3 …). Why should the angular momentum have only those values that are integral multiples of h/2π?

This can be explained using de Broglie’s hypothesis. According to his hypothesis, material particles, such as electrons, also have a wave nature. He argued that the electron in its circular orbit, as proposed by Bohr, must be seen as a particle wave. Just like the waves travelling on a string, particle waves too can lead to standing waves under resonant conditions. When a string is plucked, a vast number of wavelengths are excited. However only those wavelengths survive which have nodes at the ends and form the standing wave in the string. For an electron moving in nth circular orbit of radius rn, the total distance is the circumference of the orbit, 2πrn.

Standing waves are formed when the total distance travelled by a wave is one wavelength, two wavelengths, or any integral number of wavelengths. Waves with other wavelengths interfere with themselves upon reflection and their amplitudes quickly drop to zero. Thus,

Now λ = h/p, where p is the magnitude of the electron’s momentum. If the speed of the electron is much less than the speed of light, the momentum is mvn. Thus,

Combining these equations, we have,

This is the quantum condition proposed by Bohr for the angular momentum of the electron.

This equation is the basis of explaining the discrete orbits and energy levels in hydrogen atom. Thus de Broglie hypothesis provided an explanation for Bohr’s second postulate for the quantisation of angular momentum of the orbiting electron.