Atomic Spectra and Bohr Model of the Hydrogen Atom

  • According to Bohr the electrons revolve around nucleus in discrete circular orbits.
  • The centripetal force is provided by the electrostatic attraction between the electron and nucleus.
  • The allowed orbits for electron are those in which the electron does not radiate energy called stationary orbits.
  • The angular momentum of the revolving electron is quantized.
  • Angular momentum L = mvr = \tt \frac{n h}{2 \pi} {n = 1, 2, 3 ……}
  • According to Bohr transition rule when electron jumps from higher energy orbit to lower energy orbit. The difference energy is emitted as photon energy.
  • Photon energy E = hν (or) E2 − E1 = hν (ν = frequency)
  • From Bohr theory for revolving electron \tt \frac{mv^{2}}{r} = \frac{1}{4 \pi \varepsilon_{0}} \frac{ze^{2}}{r^{2}}
  • Radius of nth Bohr orbit \tt r_{n} = \frac{n^{2} h^{2} \varepsilon_{0}}{\pi m ze^{2}} in hydrogen atom
  • \tt r_{n} = 0.53 \frac{n^{2}}{z} {nth orbit}
  • For hydrogen like atom \tt r_{n}\propto \frac{n^{2}}{z} (nth orbit)
  • Speed of electron in nth orbit \tt V_{n}= \frac{ze^{2}}{2 \varepsilon_{0} n h}
  • \tt V_{n} \propto \frac{Z}{n}
  • Speed of \tt \overline{e} in nth orbit \tt V = \frac{1}{137} \frac{c}{n}.
  • For first orbit of hydrogen \tt V = \frac{C}{137}.
  • Time period of revolution of electron in nth orbit \tt T_{n} = \frac{4 \varepsilon o^{2} n^{3} b^{3}}{mz^{2}e^{4}}
  • Time period \tt T_{n} \propto \frac{n^{3}}{z^{2}}
  • Total energy of electron in the nth orbit \tt E_{n} = - \frac{me^{4}}{8 h^{2} \varepsilon_{0}^{2}} . \frac{z^{2}}{n^{2}}
  • \tt E_{n} = - 13.6 \times \frac{z^{2}}{n^{2}} (ev)
  • Angular frequency of nth electron \tt \omega_{n} = \frac{8 \pi^{3} z^{2} e^{4} m}{u \pi \varepsilon_{0} n^{3} h^{3}}
  • Angular frequency of nth \tt \overline{e} \ \omega_{n} \propto \frac{z^{2} m}{n^{3}}
  • Angular frequency for first Bohr orbit of hydrogen atom \tt \omega_n=\frac{4.459 \times 10^{16} z^{2}}{n^{3}}
  • Electric current due to electron motion \tt I = \frac{4 \pi^{2} z^{2} e^{5} m}{(4 \pi \varepsilon_{0})^{2} n^{3} h^{3}}
  • Electric current due to electron motion \tt I \propto \frac{z^{2} m}{n^{3}}
  • Magnetic field at the centre due to revolving e, \tt B_{n} = \frac{\mu_{0} 8 \pi^{4} z^{3} e^{7} m^{2}}{(4 \pi \varepsilon_{0})^{3} n^{5} h^{5}}
  • \tt B_{n} \propto \frac{z^{3} m^{2}}{n^{5}}

Atomic Spectra View the Topic in this video From 00:07 To 11:24

View the Topic in this video From 00:05 To 06:40

Disclaimer: may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. makes no representations whatsoever concerning the content of these sites and the fact that has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by with respect to such sites, its services, the products displayed, its owners, or its providers.

1. The stationary orbits are those in which angular momentum of the electron is an integral multiple of \frac{h}{2\pi}\left(=\hbar\right)
i.e., mvr = n\left(\frac{h}{2\pi}\right), n being integer or the principle quantum number.

2. Radius of orbit in Bohr's Quantisation: r_{n}=\frac{n^{2}h^{2}\varepsilon_{0}}{\pi me^{2}Z}

3. Velocity of Electron in nth orbit: v=\frac{nh}{2\pi\ mr}

4. Kinetic Energy (K) of Electron in nth Orbit: \Rightarrow K=\frac{1}{2}mv^{2}=\frac{Ze^{2}}{8\pi \varepsilon_{0}r}

5. Total Energy (E) of Electron in nth orbit:
Total energy = K.E + P.E
\Rightarrow E=\frac{Ze^{2}}{8\pi\varepsilon_{0}r}-\frac{Ze^{2}}{4\pi\varepsilon_{0}r}\Rightarrow E=-\frac{Ze^{2}}{8\pi\varepsilon_{0}r}

6.Frequency of Emitted Radiation, wave number (\overline{\upsilon}) is given by \overline{\upsilon}=\frac{1}{\lambda}=Z^{2}R\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)

7. Lyman series : \frac{1}{\lambda}=R\ \frac{1}{1^{2}}-\frac{1}{n^{2}} n = 2,3,4....

8. Paschen series : \frac{1}{\lambda}=R\left(\frac{1}{3^{2}}-\frac{1}{n^{2}}\right) n = 4,5,6...

9. Brackett series : \frac{1}{\lambda}=R\left(\frac{1}{4^{2}}-\frac{1}{n^{2}}\right) n = 5,6,7...

10. Pfund series : \frac{1}{\lambda}=R\left(\frac{1}{5^{2}}-\frac{1}{n^{2}}\right) n = 6,7,8...

11. The Balmer formula Eq.(12.5) may be written in terms of frequency of the light, recalling that
c = vλ or \frac{1}{\lambda}=\frac{v}{c}

Thus, Eq. (12.5) becomes v=Rc\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right)