## Atoms

# Bohrâ€™s Second Postulate of Quantisation, The Line Spectra of the Hydrogen Atom

- Potential energy of \tt \overline{e} in n
^{th}orbit. \tt U_{n} = - \frac{1}{4 \pi \varepsilon_{0}} . \frac{ze^{2}}{r_{n}} - \tt U_{n} = - 27.2\ \frac{z^{2}}{n^{2}} ev
- When electron jumps from n
_{2}orbit to n_{1}orbit. hν = En_{2}− En_{1} - Frequency \tt v = \frac{2 \pi^{2} me^{4}}{(4 \pi \varepsilon_{0})^{2} n^{3}} \left[\frac{1}{n^{2}_{1}} - \frac{1}{n^{2}_{2}}\right]
- Frequency \tt v = RCZ^{2} \left(\frac{1}{n^{2}_{1}} - \frac{1}{n^{2}_{2}}\right) . \left\{R = \frac{me^{4}}{8 \pi \varepsilon_{0} h^{3}c}\right\}
- The wavelength of emitted radiation. \tt \frac{1}{\lambda} = z^{2} R \left(\frac{1}{n^{2}_{1}} - \frac{1}{n^{2}_{2}}\right)
- If electron jumps from n
_{2}to n_{1}. The number of spectral lines emitted \tt N = \frac{(n_{2}-n_{1}+1)(n_{2}-n_{1})}{2} - Energy level diagram for hydrogen atom.
- The wave number of Lyman Series.\tt \overline{V} = \frac{1}{\lambda} = R \left[\frac{1}{1^{2}} - \frac{1}{n^{2}_{2}}\right] (n
^{2}= 2, 3, 4, 5...∞) - For Lyman Series \tt \frac{1}{\lambda_{max}} = R \left[\frac{1}{1^{2}} - \frac{1}{2^{2}}\right]
- For Lyman Series λ
_{max}= 1212 Å - For Lyman Series \tt \frac{1}{\lambda_{min}}R \left[\frac{1}{1^{2}} - \frac{1}{\infty^{2}}\right]
- For Lyman Series λ
_{min}= 912 Å. - Lyman Series lies in ultraviolet region of electromagnetic spectrum.
- Lyman series is obtained in emission as well as in absorption spectrum.
- Wave number of Balmer series \tt \overline{v} = \frac{1}{\lambda} = R \left[\frac{1}{2^{2}} - \frac{1}{n^{2}_{2}}\right] {n
_{2}= 3,4,5 …} - For Balmer series \tt \frac{1}{\lambda_{max}} = R \left(\frac{1}{2^{2}} - \frac{1}{3^{2}}\right)
- For Balmer series λ
_{max}= 6563 Å - For Balmer series \tt \frac{1}{\lambda_{min}} = R \left(\frac{1}{2^{2}} - \frac{1}{\propto^{2}}\right)
- For Balmer series λ
_{min}= 3636 Å - Balmer series lies in visible region of Em radiation.
- Balmer series obtained only in Emission spectrum.
- The wave number of paschen series. \tt \overline{V} = \frac{1}{\lambda} = R \left[\frac{1}{3^{2}} - \frac{1}{n^{2}_{2}}\right] (n
_{2}= 4, 5, 6 ….. ∞) - For paschen series \tt \frac{1}{\lambda_{max}} = R \left[\frac{1}{3^{2}} - \frac{1}{4^{2}}\right]
- For paschen series λ
_{max}= 8202 Å - For Paschen series \tt \frac{1}{\lambda_{min}} = R \left(\frac{1}{3^{2}} - \frac{1}{\infty^{2}}\right)
- Paschen series lies in the infrared region of Em Spectrum.
- Wave number of Bracket series \tt \overline{V}= R \left[\frac{1}{4^{2}} - \frac{1}{n^{2}_{2}}\right] {n
_{2 }= 5, 6, ...∞ - For Bracket series \tt \frac{1}{\lambda_{max}} = R \left[\frac{1}{4^{2}} - \frac{1}{5^{2}}\right]
- For Bracket series λ
_{max}= 40477 Å - For Bracket series \tt \frac{1}{\lambda_{min}} = R \left[\frac{1}{4^{2}} - \frac{1}{\infty^{2}}\right]
- For Bracket series λ
_{min}= 14572 Å - Bracket series lies in infrared region of Em. Spectrum.
- Wave number of P fund series \tt \overline{V} R \left[\frac{1}{5^{2}} - \frac{1}{n^{2}_{2}}\right] {n
_{2}= 6, 7 …. 8} - For P fund series \tt \frac{1}{\lambda_{max}} = R \left[\frac{1}{5^{2}} - \frac{1}{4^{2}}\right]
- For P fund series λ
_{max}= 74563 Å - For P fund series \tt \frac{1}{\lambda_{min}} = R \left[\frac{1}{5^{2}} - \frac{1}{\infty^{2}}\right]
- For P fund series λ
_{min}= 22768 Å - Energy of Hydrogen like atoms. \tt E_{n} = - \frac{13.6}{n^{2}} ev
- E
_{2}= Energy of Second orbit \tt E_{2} = \frac{-13.6}{2^{2}} = - 3.4 ev - E
_{3}= Energy of Third orbit \tt E_{3} = \frac{-13.6}{3^{2}} = - 1.51 ev - E
_{4}= Energy of Fourth Orbit \tt E_{4} = \frac{-13.6}{4^{2}} = - 0.85 ev - Bohr’s theory is applicable only for hydrogen like single electron atom.
- Bohr’s theory could not explain fine features of the hydrogen spectrum.
- Bohr’s theory does not explain why only circular orbits are chosen.
- Excitation energy of an atom is defined as the energy required to jump from the ground state to any one of excited states.
- Ionisation energy is the energy required to knock out an electron completely out of atom.

### Bohr's Second Postulate of Quantisation View the Topic in this video From 06:41 To 13:24

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1. This is called the Bohr radius, represented by the symbol a_{0}. Thus, a_{0}=\frac{h^{2}\varepsilon_{0}}{\pi\ m e^{2}}

2. If the speed of the electron is much less than the speed of light, the momentum is mv_{n}. Thus, λ = h/mv_{n}. From Eq.(12.24), we have

2π r_{n} = n h/mv_{n} or m v_{n} r_{n} = nh/2π

3.For the Balmer series, the Rydberg constant R is readily identified to be R=\frac{me^{4}}{8\ \varepsilon_{0}^{2}\ h^{3}c}