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 in the form of electromagnetic waves. The energy of an accelerating electron should therefore continuously decrease, causing the electron to spiral inward and eventually fall into the nucleus. As a result, such an atom cannot be stable.
(ii) As the electrons spiral inward, their angular velocities and hence their frequencies would change continuously. This would lead to the emission of a continuous spectrum, which contradicts the line spectrum that is actually observed.
Bohr's explanation: Bohr postulated stable orbits in which electrons do not radiate energy. The postulates are as follows: \begin{itemize} \item (i) An electron in an atom can revolve in certain stable orbits without the emission of radiant energy.
\item (ii) The electron revolves around the nucleus in only those orbits for which the angular momentum is an integral multiple of \(\frac{h}{2\pi}\).
\item (iii) An electron might make a transition from one of its specified non-radiating orbits to another of lower energy. When this happens, a photon is emitted with energy equal to the difference in energy between the initial and final states. \end{itemize} The radius of the \(n^{th}\) 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 in the radius of consecutive orbits is: \[ r_{n+1} - r_n = k \left[ (n+1)^2 - n^2 \right] \] This simplifies to: \[ r_{n+1} - r_n = k (2n + 1) \] which depends on \(n\), meaning it is not a constant.