Question

Write shortcomings of Bohr’s model. Show that Bohr’s second postulate of orbital quantisation can be explained on the basis of De-Broglie hypothesis of wave nature of electron.

Answer 1

Limitations of Bohr Model:

Bohr model has many limitations. Some are given here :

1. The Bohr model is applicable to hydrogenic atoms. It cannot be extended even to more two electron atoms such as helium. The analysis of atoms with more than one electron was attempted on the lines of Bohr’s model for hydrogenic atoms but did not meet with any success. Difficulty lies in the fact that each electron interacts not only with the positively charged nucleus but also with all other electrons.

2. It does not explain why only circular orbits should be chosen when elliptical orbits are also possible.

3. In the spectrum of hydrogen, certain spectral lines are not single lines but a group of closed lines with slightly different frequencies. Bohr’s theory could not explain these fine structures of the hydrogen spectrum.

4. Bohr’s theory does not tell anything about the relative intensities of the various spectral lines. Bohr’s theory predicts only the frequencies of these lines.

5. It does not explain the further splitting of spectral lines in a magnetic field (Zeeman effect) or in an electric field (Stark effect).

6. The formulation of Bohr model involves electrical force between positively charged nucleus and electron. It does not include the electrical forces between electrons which necessarily appear in multi-electron atoms.

Explanation of Bohr’s Second Postulate by de-Broglie Theory:

In 1927, Louis de-Broglie argued that the electron in its circular orbit, as proposed by Bohr’ must be seen as a particle wave. In analogy two waves travelling on a string particle wave can lead to standing waves under resonant condition. We know that 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 waves in the string. It means that in a string, standing waves are formed when the total distance travelled by a waves down the string and back 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. For an electron moving in n^th circular orbit of radius r_n . The total distance is the circumference of the orbit, 2πr_n Thus, 2πr_n = nλ,    n = 1, 2, 3, …. …………. (1)

Figure illustrates a standing particle wave on a circular orbit for n = 4 i.e., 2πr_n = 4λ, where λ is the De-Broglie wavelength of the electron moving in n^th orbit. As 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 mv_n. thus, λ = \(\frac{h}{m v_{n}}\) From equation (1), we have
2πr_n = \(\frac{n h}{m v_{n}}\) or mv_nr_n = \(\frac{n h}{2 \pi}\)

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. The quantised electron orbits and energy states are due to the wave nature of the electron and only resonant standing waves can persist.