Rutherford’s Model of Atom

Rutherford developed the planetary model of the atom which put all the protons in the nucleus and the electrons orbited around the nucleus like planets around the sun.

As per the Rutherford’s nuclear model of the atom, the electrostatic force of attraction (F_e) between the revolving electrons and the nucleus provides the requisite centripetal force (F_c) to keep them in their orbits. 

Thus, for a stable hydrogen atom

F_c=F_e

\[\Rightarrow \frac{mv^{2}}{r}=\frac{1}{4\pi \epsilon _{0}}\frac{e^{2}}{r{2}}\]

The kinetic energy of the electron,

\[K=\frac{1}{2}mv^{2}=\frac{e^{2}}{8\pi \epsilon _{0}r}\]

The electrostatic potential energy of the electron,

\[U=\int_{\infty }^{r }F_{e}dr=\frac{e^{2}}{4\pi \epsilon _{0}}\int_{\infty }^{r } \frac{dr}{r^{2}}=-\frac{e^{2}}{4\pi \epsilon _{0}r}\]

Thus the total energy E of the electron in a hydrogen atom is,

\[E=K+U=\frac{e^{2}}{8\pi \epsilon _{0}r}-\frac{e^{2}}{4\pi \epsilon _{0}r}=-\frac{e^{2}}{8\pi \epsilon _{0}r}\]

The total energy of the electron is negative. This implies the fact that the electron is bound to the nucleus. 

Drawbacks:

The Rutherford’s model suffers from the following drawbacks:

1. An electron revolving around the nucleus gets accelerated towards the nucleus. According to the electromagnetic theory, an accelerating charged particle must emit radiation, and lose energy. Because of this loss of energy, the electron would slow down, and will not be able to withstand the attraction of the nucleus. As a result, the electron should follow a spiral path, and ultimately fall into nucleus. But, this does not happen rather atoms are stable. This indicates that there is something wrong in the Rutherford’s model of atom.

2. The Rutherford’s model of atom does not say anything about the arrangement of electrons in an atom.

3. The Rutherford’s model of atom can’t explain line spectra of Hydrogen atoms.