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

The collection of different spectral lines obtained due to transition of an electron in hydrogen atom from upper energy levels to lower energy levels is called the Hydrogen Spectrum.

The spectrum consisting of bright lines on a dark background, emitted when an atomic gas is excited at low pressure by passing an electric current through it, is called the Emission Line Spectrum.

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

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

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

Bohr's model applies to one-electron species such as H, He⁺, Li²⁺, etc.

Postulates of Bohr's Model:

1. Electrons revolve in fixed circular paths called stationary states, orbits, shells, or energy levels. Each has a definite, fixed energy. Energy of the electron increases as it moves away from the nucleus.

2. Angular momentum of an electron is always an integral multiple of \[\frac{h}{2\pi}:\]
   \[mvr=n\cdot\frac{h}{2\pi}\]
   where m = mass, v = velocity, r = orbital radius, n = principal quantum number.

Energy is emitted or absorbed only when an electron jumps between energy levels — not while it is in a stationary orbit. When falling from higher (E₂) to lower (E₁) energy level:

\[\Delta E=E_2-E_1=h\nu=\frac{hc}{\lambda}\]

Dalton's atomic theory laid the foundation of modern chemistry with four core postulates:

1. All matter is made up of extremely small particles called atoms.
2. Atoms of the same element are identical to each other in mass and properties; atoms of different elements differ.
3. Atoms can neither be created nor destroyed — they are indestructible.
4. Atoms combine in fixed, simple whole-number ratios to form compound atoms (molecules).

Note: Modern discoveries have refined some postulates (e.g., isotopes show atoms of the same element can differ in mass), but the core framework remains foundational.

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

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.