Atoms Class 12 Notes (Modern Physics)

Atomic Models (Thomson to Rutherford)

• Most alpha particles pass straight through — most of the atom is empty.
• A few are deflected at large angles or bounce back — a small massive positive nucleus.
• Nuclear size ~10⁻¹⁵ m, atomic size ~10⁻¹⁰ m.

Bohr's Model of the Hydrogen Atom

• Electrons occupy only stationary orbits of quantised angular momentum:
• $mvr = \frac{nh}{2\pi}$
• .
• Radius of nth orbit:
• $r_n = n^2 \times 0.529\,\text{\AA}$
• .
• Energy of nth level:
• $E_n = -\frac{13.6}{n^2}\,\text{eV}$
• (ground state
• $n=1$
• is −13.6 eV).
• Photon absorbed/emitted on a transition between levels
• $n_i \to n_f$
• :
• $\Delta E = E_i - E_f = h\nu$
• .

Hydrogen Line Spectra

• $\text{Lyman series}:$
• transitions to
• $n_f = 1$
• (ultraviolet).
• $\text{Balmer series}:$
• transitions to
• $n_f = 2$
• (visible).
• $\text{Paschen, Brackett, Pfund}:$
• transitions to
• $n_f = 3, 4, 5$
• (infrared).

Energy of the Electron and Ionisation

• Kinetic energy of electron:
• $KE = \frac{1}{2}mv^2 = \frac{e^2}{8\pi\varepsilon_0 r_n}$
• .
• Potential energy:
• $PE = -\frac{e^2}{4\pi\varepsilon_0 r_n}$
• ; total
• $E = \text{KE} + \text{PE} = -\text{KE}$
• .
• Ionisation energy of hydrogen from the ground state is 13.6 eV; ionisation potential is 13.6 V.

de Broglie Wavelength of the Electron

An electron acts as a wave whose wavelength is set by its momentum. For an electron accelerated through a potential difference V: $\lambda = \frac{1.23}{\sqrt{V}}\,\text{nm}$. The allowed Bohr orbits correspond to integer numbers of wavelengths fitting around the orbit: $2\pi r = n\lambda$.

Solved Examples

Example: Find the wavelength of the second line of the Balmer series (transition n = 4 → n = 2) using R_H = 1.097 × 10⁷ m⁻¹.

Solution: $\frac{1}{\lambda} = 1.097\times10^7\left(\frac{1}{4}-\frac{1}{16}\right) = 1.097\times10^7\times\frac{3}{16}$ giving $\lambda \approx 486\,\text{nm}$ (Balmer H-β).