Bohr’s Second Postulate of Quantisation, The Line Spectra of the Hydrogen Atom

  • Potential energy of \tt \overline{e} in nth 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 n2 orbit to n1 orbit. hν = En2 − En1
  • 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 n2 to n1. 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] (n2 = 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 λmax = 1212 Å
  • For Lyman Series \tt \frac{1}{\lambda_{min}}R \left[\frac{1}{1^{2}} - \frac{1}{\infty^{2}}\right]
  • For Lyman Series λmin = 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] {n2 = 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 λmax = 6563 Å
  • For Balmer series \tt \frac{1}{\lambda_{min}} = R \left(\frac{1}{2^{2}} - \frac{1}{\propto^{2}}\right)
  • For Balmer series λmin = 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] (n2 = 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 λmax = 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] {n2 = 5, 6, ...∞
  • For Bracket series \tt \frac{1}{\lambda_{max}} = R \left[\frac{1}{4^{2}} - \frac{1}{5^{2}}\right]
  • For Bracket series λmax = 40477 Å
  • For Bracket series \tt \frac{1}{\lambda_{min}} = R \left[\frac{1}{4^{2}} - \frac{1}{\infty^{2}}\right]
  • For Bracket series λmin = 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] {n2 = 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 λmax = 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 λmin = 22768 Å
  • Energy of Hydrogen like atoms. \tt E_{n} = - \frac{13.6}{n^{2}} ev
  • E2 = Energy of Second orbit \tt E_{2} = \frac{-13.6}{2^{2}} = - 3.4 ev
  • E3 = Energy of Third orbit \tt E_{3} = \frac{-13.6}{3^{2}} = - 1.51 ev
  • E4 = 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 a0. 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 mvn. Thus, λ = h/mvn. From Eq.(12.24), we have
2π rn = n h/mvn or m vn rn = 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}