DE-BROGLIE'S EXPLANATION OF BOHR'S SECOND POSTULATE OF QUANTIZATION OF ANGULAR MOMENTUM (BOHR'S QUANTUM CONDITION FROM de-BROGLIE HYPOTHESIS)
Bohr's quantum condition of angular momentum:
Question: Derive Bohr's quantum condition of angular momentum of an electron in an atom from de-Broglie hypothesis.
Solution:
Bohr could not explain as to why only certain orbits or energy levels were allowed for orbiting the electrons around the nucleus of atom. However, de-Broglie established a connection between the wave nature of electron and the stable orbits in the Bohr's model of atom. de-Broglie assumed that an electron orbit would be stable only if it contained an integral multiple of electron wavelength. The first orbit must contain one electron wavelength, the second orbit must contain two electron wavelengths, the third orbit must contain three electron wavelengths and so on.
Proof :
de-Broglie hypothesis is analogous to the standing waves formed on a vibrating string of certain length. He assumed the circumference of the orbit of an electron as the length of the string.
Thus, the circumference of the orbit of an electron is equal to the integral multiple of the de-Broglie wavelength of the electron moving in the orbit.
i.e.,
$2\pi r = n\lambda$, where n = 1, 2, 3, \dots etc.
But
$\lambda = \frac{h}{mv}$
(de-Broglie wavelength of moving electron)
\therefore
$2\pi r = \frac{nh}{mv}$
or
$mvr = \frac{nh}{2\pi}$
But mvr = L, the angular momentum,
so $L = n \frac{h}{2\pi}$
which is Bohr's postulate of the quantization of angular momentum.
Drawbacks or Limitations of Bohr's Atomic Model
Question: What are the drawbacks of Bohr's model of atom?
Solution:
Bohr's atomic theory lacks consistency and has its own contradictions. It is neither based on a pure quantum theory nor pure classical theory.
Bohr's atomic model has the following limitations:
1. This model could not explain the spectra of complex atoms having more than one electron. But it successfully explains the spectra of simple atoms (i.e., the atoms having only one electron). For example, it can explain the spectra of hydrogen atom and hydrogen like atoms ($He^+, Li^{++}$ etc).
2. Bohr's model of atom could not explain fine structure of the spectral lines of Balmer series. When the spectral lines of a Balmer series was observed under a powerful microscope, it was found that each spectral line consists of closely spaced lines.
3. Bohr's atomic model does not give any indication regarding the arrangement and distribution of electrons in an atom.
4. This model could not account for the wave nature of electrons.
5. It does not give any idea about the relative intensities of spectral lines in the spectrum.
6. This model could not explain the (Zeeman effect) i.e. splitting of spectral line into a number of spectral lines under the effect of magnetic field.
7. This model could not explain (Stark effect) i.e. the splitting of spectral line into a number of spectral lines under the effect of electric field.