 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}}

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

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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)