Revision: Atoms and Nuclei >> Atoms Physics Science (English Medium) Class 12 CBSE

Completely removing an electron from atom is called ionisation.

The atom emits a photon with energy equal to the difference between the two energy levels, a phenomenon known as luminescence.

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

\[v_n=\frac{nh}{2\pi mr_n}\]

\[r=\frac{n^2h^2}{4\pi^2mkZe^2}\]

\[v=\frac{2\pi kZe^2}{nh}\]

For hydrogen atom (Z = 1):

\[v=\frac{2\pi ke^2}{nh}=\alpha\frac{c}{n}\]

where α is the fine structure constant and \[\alpha=\frac{1}{137}.\]

Total Energy of Electron in n-th Orbit (General):

• \[E_n=\frac{-Z^2me^4}{8\varepsilon_0^2n^2h^2}\]

Total Energy (Alternate form):

• \[E_n=-\frac{2\pi^2mk^2Z^2e^4}{n^2h^2}\]

Total Energy for Hydrogen-like Atom (Simplified):

Radius of the n-th Bohr Orbit (General):

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

\[\mathrm{i.e.,}r_n\propto n^2\mathrm{and}r_n\propto\frac{1}{Z}\]

Radius of n-th orbit for Hydrogen-like atom:

\[r_n=0.53\left(\frac{n^2}{Z}\right)\mathrm{\r{A}}\]

\[\bar{v}=\frac{1}{\lambda}=RZ^2\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\mathrm{m}^{-1}\]

where \[R=1.097\times10^7\mathrm{m}^{-1}\] (Rydberg constant)

\[E_\text{ionisation}=13.6Z^2\mathrm{~eV}\]

\[E_n=-Rhc\left(\frac{1}{n^2}\right)\]

General Formula for all Spectral Series:

\[\frac{1}{\lambda}=R\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]\]

where \[R=1.097\times10^7\mathrm{m}^{-1},n_1=\text{final state},n_2=\text{initial state},n_2>n_1\]

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

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:

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

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

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

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

Since n_i and n_f are integers → Discrete line spectrum

• Most alpha particles passed through the gold foil without any deflection, proving the atom is mostly empty space.
• Around 0.14% of incident alpha particles are scattered by more than 1°.
• Around 1 in 8000 alpha particles are deflected by more than 90°.
• Large-angle deflections indicated a small, dense, positively charged nucleus at the centre of the atom.
• The gold foil used had a thickness of 2.1 × 10⁻⁷ m; the alpha particles had an energy of 5.5 MeV.
• Electrons have negligible mass and do not affect the trajectory of incident alpha particles.
• This experiment disproved Thomson's plum-pudding model and established the nuclear structure of the atom.

• The radius of Bohr's orbit is proportional to \[n^{2}\] and inversely proportional to Z.
• For hydrogen (Z = 1), the ground state (n = 1) radius is 0.53 Å, known as Bohr's radius.
• The velocity of an electron decreases as the orbital number (n) increases.
• For hydrogen, orbital speed of electron equals \[\alpha\frac{c}{n}\]​, where \[\alpha=\frac{1}{137}\]​.
• The total energy of an electron in any orbit is negative, indicating a bound state.
• For hydrogen-like atoms, the energy of an electron in the n-th orbit is \[-13.6\frac{Z^2}{n^2}\mathrm{~eV}.\] .

• For hydrogen (Z = 1): ground state energy = −13.6 eV; at n = ∞, energy = 0 eV.
• Energy levels for hydrogen: n=1: −13.6 eV, n=2: −3.4 eV, n=3: −1.511 eV, n=4: −0.850 eV, n=5: −0.544 eV.
• In normal conditions, electrons are in the ground state, occupying orbitals closest to the nucleus.
• Beyond ionisation potential, the electron is no longer bound — energy levels form a continuum (starts at 13.6 eV above ground in hydrogen).
• Electrons in orbitals close to the nucleus are stable (need more energy to remove); electrons farther away are less stable.
• If energy supplied ≥ ionisation energy, ionisation occurs.
• Spectral Series (from energy level transitions): Lyman, Balmer, Paschen, Bracket, Pfund series.

• Lyman series — n₁=1, n₂: 2→∞; converges toward 91–122 nm; UV region
• Balmer series — n₁=2, n₂: 3→∞; converges toward 365–657 nm; visible region (Hα = Red, Hβ = Blue-green, Hγ = Blue)
• Paschen series — n₁=3, n₂: 4→∞; converges toward 821–1876 nm; infrared (IR)
• Brackett series — n₁=4, n₂: 5→∞; converges toward 1459–4053 nm; IR region
• Pfund series — n₁=5, n₂: 6→∞; converges toward 2280–7462 nm; IR region
• Humphreys series — n₁=6, n₂: 7→∞; converges toward 3283 nm–∞; IR region
• Setting n₁=1 and n₂ from 2 to ∞ gives the Lyman series converging to 91 nm

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.

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. 

\[h\nu=E_i-E_f\]

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