Bohr's Atomic Model

In order to explain the stability of an atom, Neils Bohr gave a new arrangement of electrons in the atom. According to Neils Bohr:

• The electrons revolve rapidly around the nucleus in fixed circular paths called energy levels or shells. The 'energy levels' or 'shells' or 'orbits' are represented in two ways: either by the numbers 1, 2, 3, 4, 5 and 6 or by letters K, L, M, N, O and P. The energy levels are counted from center outwards.
• Each energy level is associated with a fixed amount of energy. The shell nearest to the nucleus has minimum energy and the shell farthest from the nucleus has maximum energy.
• There is no change in the energy of electrons as long as they keep revolving with the same energy level. But,  an electron jumps from a lower energy level to a higher one, after absorbing some while some energy is emitted when an electron jump from higher energy level to lower one.

This model of the atom was able to explain the stability of the atom. It also explained the phenomenon of atomic spectra and ionization of gases. The important postulates of Bohr Theory are:

1. Electrons revolve round the nucleus with definite velocities in concentric circular orbits situated at definite distances from the nucleus. The energy of an electron in a certain orbit remains constant. As long as it remains in that orbit, it neither emits nor absorbs energy. These are termed stationary states or main energy states.
2. The angular momentum of an electron is quantized. Thus, the motion of an electron is restricted to those orbits where its angular momentum is an integral multiple of h/2π, where h is Planck’s constant.
3. The energy of an electron changes only when it moves from one orbit to another. An electronic transition from an inner orbit to outer orbit involves absorption of energy. Similarly, when an electron jumps from an outer orbit to inner orbit it releases energy, which is equal to the difference between the two energy levels.

If the energy of an electron in the outer orbit is E₂ and energy of electron in the inner orbit is E₁ then E₂ - E₁ = ΔE = hν.

Spectrum of Hydrogen Atom

Bohr atomic model states that the spectrum series arises when an electron jumps from an initial stationary orbit (n_i) to the final orbit (n_f). 

Now if the electron is in n^th orbit then

\[The\ electrostatic\ force\ of\ attraction\ =\frac{e^{2}}{4\pi \epsilon _{0}\ r_{n}^{2}}\] \[The\ centripetal\ force=\frac{mv^{2}}{r_{n}}\] \[\therefore \frac{e^{2}}{4\pi \epsilon _{0}\ r_{n}^{2}}\ =\frac{mv^{2}}{r_{n}}\Rightarrow \frac{e^{2}r_{n}}{4\pi \epsilon _{0}\ }=mv^{2} r_{n}^{2} \ \cdot \cdot \cdot (i)\] \[Form\ Bohr\ postulate\ mvr_{n}=n\frac{h}{2\pi }\Rightarrow mv^{2}r_{n}^{2}=\frac{n^{2}h^{2}}{m4\pi ^{2}}\cdot \cdot \cdot (ii)\] \[From eqn (i) and (ii) \frac{e^{2}r_{n}}{4\pi \epsilon _{0}\ }=\frac{n^{2}h^{2}}{m4\pi ^{2}} \Rightarrow r_{n}= \frac{n^{2}h^{2}\epsilon _{0}}{m\pi e^{2}}\]

The total energy E_n of the electron in n^th orbit,

\[E_{n}= -\frac{e^{2}}{8\pi \epsilon _{0}r_{n}}=-\frac{e^{2}}{8\pi \epsilon _{0}}\times \frac{m\pi e^{2}}{n^{2}h^{2}\epsilon _{0}}=-\frac{me^{4}}{8n^{2}h^{2}\epsilon _{0}^{2}}\]

Therefore frequency of the spectra for transition of the electron n_i orbit to n_f orbit is ν=(E_i-E_f)/h

\[\therefore \nu =(E_{i}-E_{f})/h=\frac{me^{4}}{8h^{3}\epsilon _{0}^{2}}\left ( \frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}} \right )\] \[\therefore wave\ number\ \frac{1}{\lambda } =\frac{me^{4}}{8ch^{3}\epsilon _{0}^{2}}\left ( \frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}} \right )\]

Name of the spectral lines are as follows:

nf	ni	Name of the spectral lines
1	2, 3, 4, 5, 6, 7, 8…	Lymen series
2	3, 4, 5, 6, 7, 8……	Balmer series
3	4, 5, 6, 7, 8………	Pashan series
4	5, 6, 7, 8…………	Brackett series
5	6, 7, 8…………….	Pfund series

Drawbacks of Bohr Model

1. Bohr model could not explain those atoms which have more than one electron like lithium, helium. This model was applicable only for those atoms which have one electron.
2. This model failed to explain Zeeman Effect and stark effect.
3. Bohr model could not explain the uncertainty principle of Heisenberg.
4. Bohr model was not related with classification and periodicity of elements.
5. By using Bohr atomic model, one can’t explain the intensity of spectrum line.