Question

Write the three postulates of Bohr model of hydrogen atom. Show that, in this model, the frequency of revolution of an electron in its \(n^{th}\) orbit is proportional to \(\left(\dfrac{1}{n^3}\right)\).

Hint:

State the three Bohr postulates. Then use r proportional to n squared and v proportional to 1 over n in f = v/(2 pi r).

Answer 1

Step 1: Postulates of the Bohr model:
1. An electron in the atom revolves around the nucleus in certain stable circular orbits without radiating energy. The Coulomb force supplies the centripetal force.
2. Only those orbits are allowed in which the angular momentum is an integral multiple of \( h/2\pi \): \( m v r = \dfrac{nh}{2\pi} \), with \( n = 1, 2, 3, \ldots \)
3. Radiation is emitted or absorbed only when the electron jumps between two orbits. The photon energy is \( h\nu = E_i - E_f \).

Step 2: Radius and speed in the nth orbit:
Coulomb force gives centripetal force: \( \dfrac{m v^2}{r} = \dfrac{e^2}{4\pi\varepsilon_0 r^2} \).
Using \( v = \dfrac{nh}{2\pi m r} \) in this gives
\[ r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2}, \qquad v_n = \frac{e^2}{2 \varepsilon_0 n h} \]
So \( r_n \propto n^2 \) and \( v_n \propto 1/n \).

Step 3: Frequency of revolution:
\[ f = \frac{v_n}{2\pi r_n} = \frac{e^2}{2\varepsilon_0 n h} \times \frac{\pi m e^2}{2\pi n^2 h^2 \varepsilon_0} = \frac{m e^4}{4 \varepsilon_0^2 h^3} \cdot \frac{1}{n^3} \]
All quantities except n are constants, so \( f \propto \dfrac{1}{n^3} \).

Final Answer:
The frequency of revolution is \( f = \dfrac{m e^4}{4 \varepsilon_0^2 h^3 n^3} \), so it is proportional to \( 1/n^3 \). \[ \boxed{f \propto \frac{1}{n^3}} \]