Read the following passage and answer the questions.
In Bohr’s model of the hydrogen atom, the electron moves only in certain allowed orbits. Bohr’s second postulate picks out these orbits. Of all his postulates, it looked the most puzzling, because it gave no reason for this choice. The French physicist Louis de Broglie later gave a reason. He had suggested that material particles, such as electrons, also have a wave nature. He argued that the electron in its circular orbit must be seen as a particle wave. Waves travelling on a stretched string can form standing waves when the conditions are right. In the same way, the particle wave of the electron can form a standing wave along its orbit. This idea explains why only certain orbits are allowed. It links the allowed orbits of the atom to the wave nature of the electron.
The electron of a hydrogen atom moves in the \(n = 3\) orbit, of radius \(4.77\times10^{-10}\ \mathrm{m}\). According to de Broglie’s explanation, the magnitude of its momentum is (Take \(h = 6.6\times10^{-34}\ \mathrm{J\,s}\).)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(6.6\times10^{-25}\ \mathrm{kg\,m\,s^{-1}}\)
In de Broglie’s picture, a whole number of wavelengths fits around the orbit. For the \( n \)th orbit, \( 2\pi r_n=n\lambda \). The momentum then follows from \( \lambda=h/p \).
Given:
\( n=3 \)
\( r_3=4.77\times10^{-10}\ \mathrm{m} \)
\( h=6.6\times10^{-34}\ \mathrm{J\,s} \)
de Broglie wavelength in this orbit:
\( \begin{aligned} \lambda &= \frac{2\pi r_3}{3} \\ &= \frac{2\times3.14\times4.77\times10^{-10}}{3} \\ &= 1.0\times10^{-9}\ \mathrm{m} \end{aligned} \)
Momentum of the electron:
\( \begin{aligned} p &= \frac{h}{\lambda} \\ &= \frac{6.6\times10^{-34}}{1.0\times10^{-9}} \\ &= 6.6\times10^{-25}\ \mathrm{kg\,m\,s^{-1}} \end{aligned} \)
From the above, \( p=6.6\times10^{-25}\ \mathrm{kg\,m\,s^{-1}} \). Hence, the magnitude of the momentum of the electron in the \( n=3 \) orbit is \( 6.6\times10^{-25}\ \mathrm{kg\,m\,s^{-1}} \).
NCERT (2026-27):
De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation – “… For an electron moving in nth circular orbit of radius rn, the total distance is the circumference of the orbit, 2πrn. Thus”
\( 2\pi r_n=n\lambda,\ n=1,2,3\ldots \)\( \qquad (12.12) \)
De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation – “… From Chapter 11, we have λ = h/p, where p is the magnitude of the electron’s momentum. …”