Question

Write the drawbacks of Rutherford’s atomic model. How did Bohr remove them? Show that different orbits in Bohr’s atom are not equally spaced.

Hint:

Bohr’s model successfully explained atomic stability and spectral lines by introducing quantized energy levels.

Answer 1

Drawbacks of Rutherford’s Atomic Model:

(i) According to classical electromagnetic theory, an accelerating charged particle emits radiation. Hence, an electron revolving around the nucleus should continuously lose energy, spiral into the nucleus, and make the atom unstable.

(ii) As electrons spiral inward, their angular velocity and frequency would continuously change, leading to the emission of a continuous spectrum. This contradicts the experimentally observed line spectra of atoms.

Bohr’s Explanation:

To overcome these drawbacks, Bohr proposed that electrons revolve in certain stable orbits without radiating energy. His main postulates are:

• (i) Electrons can revolve in specific stable orbits without emitting radiant energy.
• (ii) Only those orbits are permitted in which the angular momentum of the electron is an integer multiple of \( \frac{h}{2\pi} \), i.e. \[ mvr = n\frac{h}{2\pi} \]
• (iii) Radiation is emitted or absorbed only when an electron jumps between two permitted orbits. The energy of the photon emitted or absorbed equals the energy difference between the two orbits.

The radius of the \( n^{\text{th}} \) orbit is given by:

\[ r_n = \frac{n^2 h^2}{4\pi^2 m e^2} \quad \text{or} \quad r_n \propto n^2 \]

Alternatively:

The difference between the radii of two consecutive orbits is:

\[ r_{n+1} - r_n = k\left[(n+1)^2 - n^2\right] \]

Simplifying,

\[ r_{n+1} - r_n = k(2n + 1) \]

Since this difference depends on \( n \), it is not constant.