Overview: Atoms

\[F=\frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{r^2}\]

Where:

• Z = atomic number

• r = distance between α-particle and nucleus

\[d=\frac{1}{4\pi\varepsilon_0}\frac{2Ze^2}{K}\]

Centripetal Force = Electrostatic Force

\[\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2}=\frac{mv^2}{r}\]

\[r=\frac{e^2}{4\pi\varepsilon_0mv^2}\]

Kinetic Energy:

\[K=\frac{1}{2}mv^2\]

Potential Energy:

\[U=-\frac{e^2}{4\pi\varepsilon_0r}\]

Total Energy:

\[E=-\frac{e^2}{8\pi\varepsilon_0r}\]

Postulate 1:

An atom could revolve in certain stable orbits without the emission of radiant energy.

Postulate 2:

The electron revolves around the nucleus only in those orbits for which the angular momentum is some integral multiple of h/2π, where h is Planck’s constant (= 6.6 × 10^–34 J s). Thus, the angular momentum (L) of the orbiting electron is quantised. That is

\[L=\frac{nh}{2\pi}\]

Postulate 3:

An electron might make a transition from one of its specified non-radiating orbits to another of lower energy. 

\[h\nu=E_i-E_f\]

\[r_n=\frac{\varepsilon_0n^2h^2}{\pi me^2}\]

\[E_n=-\frac{13.6}{n^2}\mathrm{~eV}\]

\[h\nu=E_{n_i}-E_{n_f}\]

Since n_i and n_f are integers → Discrete line spectrum

Based on the experiment, Rutherford proposed that:

• An atom has a small, dense, positively charged nucleus at its centre.

• Almost all the mass of the atom is concentrated in the nucleus.

• Electrons revolve around the nucleus.

• Most of the atom is empty space.

The impact parameter is the perpendicular distance of the initial velocity vector of the a-particle from the centre of the nucleus.

When an excited gas emits radiation of specific discrete wavelengths, it produces bright lines on a dark background called an emission line spectrum.

When white light passes through a gas, some wavelengths are absorbed and appear as dark lines in the continuous spectrum, called the absorption spectrum

1. Applicable only to hydrogenic atoms

2. Cannot explain multi-electron atoms

3. Cannot explain the relative intensity of spectral lines

4. Does not include electron–electron interaction

1. An atom should be unstable

• Electron is accelerating
• Accelerating charge radiates energy
• Electron should spiral into the nucleus

2. Cannot explain the line spectrum of hydrogen