Developments Leading to Bohr’s Atomic Model

🚫 Problems with Rutherford’s Model

  • If electrons were stationary, the dense nucleus would pull them in due to electrostatic attraction (like a mini Thomson model).
  • It didn’t explain how electrons are distributed around the nucleus or their energies.

🔍 Key Developments for Bohr’s Model

Two breakthroughs helped Bohr improve Rutherford’s model:

  1. Dual nature of electromagnetic radiation (wave + particle properties).
  2. Atomic spectra experiments.

🌊 Wave Nature of Light

  • James Maxwell (1870) proposed: Accelerating charged particles create electromagnetic waves (Fig. 2.6).
  • Properties:
    • Electric & magnetic fields vibrate perpendicular to each other and to the wave’s direction.
    • Light waves don’t need a medium (unlike sound!) → travel in vacuum.
    • Different wavelengths = different radiation types (radio waves, microwaves, visible light, etc.).
  • Visible light is only a tiny part of the electromagnetic spectrum (Fig. 2.7):
    Range: Violet (400 nm, \(7.50 \times 10^{14}\) Hz) → Red (750 nm, \(4 \times 10^{14}\) Hz).

🔦 Particle Nature of Light: Planck’s Quantum Theory

  • Max Planck (1900) solved the “black-body radiation” puzzle:
    ✨ Energy is quantized → Atoms absorb/emit energy in chunks called quanta.
  • Energy of 1 quantum: \[ E = h\nu \] where \(h = 6.626 \times 10^{-34} \text{ J s}\) (Planck’s constant), \(\nu\) = frequency.
  • Like stairs: You can stand on Step 1, 2, or 3, but not between steps! Energy levels are discrete: \[ E = 0,\ h\nu,\ 2h\nu,\ 3h\nu,\ … \]

⚡ Photoelectric Effect (Einstein, 1905)

  • When light hits metals (e.g., potassium), electrons eject instantly (if light frequency > threshold \(\nu_0\)).
  • Key observations:
    • Number of ejected electrons ∝ light intensity.
    • Kinetic energy of electrons ∝ light frequency (not intensity!).
  • Einstein’s explanation using quanta:
    Light consists of particles (photons). Each photon energy = \(h\nu\).
    Equation for ejected electron’s kinetic energy: \[ \frac{1}{2}m_e v^2 = h\nu – h\nu_0 \] where \(\nu_0\) = threshold frequency, \(m_e\) = electron mass, \(v\) = electron velocity.

🌈 Atomic Spectra: Hydrogen’s Fingerprint

  • Hot gases emit light at specific wavelengths → line spectra (Fig. 2.10).
  • Hydrogen’s spectrum has series (named by discoverers):
    Series\(n_1\)\(n_2\)Region
    Lyman12,3,…Ultraviolet
    Balmer23,4,…Visible
    Paschen34,5,…Infrared
  • Rydberg’s formula for hydrogen’s wavenumber (\(\bar{\nu}\)): \[ \bar{\nu} = 109,677 \left( \frac{1}{n_1^2} – \frac{1}{n_2^2} \right) \text{ cm}^{-1} \] where \(n_1, n_2\) = integers, \(n_2 > n_1\).

🎯 Bohr’s Model for Hydrogen Atom (1913)

Bohr combined Planck’s quanta + atomic spectra to fix Rutherford’s model. Postulates:

  1. Electrons orbit nucleus in fixed circular paths (orbits) with constant energy.
  2. Electrons jump between orbits by absorbing/emitting energy: \[ \Delta E = E_2 – E_1 = h\nu \]
  3. Quantized angular momentum: \[ m_e v r = n \frac{h}{2\pi} \quad (n = 1,2,3,…) \] where \(m_e\) = electron mass, \(v\) = velocity, \(r\) = orbit radius.

💡 NEET Must-Know Concepts

  1. Dual nature of light: Wave (interference/diffraction) + Particle (photoelectric effect).
  2. Photoelectric equation: \(K.E. = h\nu – h\nu_0\) (Einstein’s explanation).
  3. Bohr’s postulates: Fixed orbits + Quantized angular momentum (\(m_e v r = n \frac{h}{2\pi}\)).
  4. Hydrogen spectrum series: Lyman (UV), Balmer (Visible), Paschen (IR).
  5. Rydberg formula: \(\bar{\nu} = 109,677 \left( \frac{1}{n_1^2} – \frac{1}{n_2^2} \right) \text{ cm}^{-1}\).

🧠 Practice Problems (with Solutions)

Problem 1: Calculate wavelength of All India Radio’s 1368 kHz broadcast.

\[ \lambda = \frac{c}{\nu} = \frac{3 \times 10^8 \text{ m/s}}{1368 \times 10^3 \text{ Hz}} = 219.3 \text{ m (Radio wave)} \]

Problem 2: Find energy of one mole of photons (\(\nu = 5 \times 10^{14}\) Hz).

Energy of 1 photon = \(h\nu = 3.313 \times 10^{-19}\) J
For 1 mole: \(3.313 \times 10^{-19} \times 6.022 \times 10^{23} = 199.51\) kJ/mol

Problem 3: Kinetic energy of electron ejected from metal (\(\nu_0 = 7.0 \times 10^{14}\) s\(^{-1}\), \(\nu = 1.0 \times 10^{15}\) s\(^{-1}\)).

\[ K.E. = h(\nu – \nu_0) = 6.626 \times 10^{-34} (3.0 \times 10^{14}) = 1.988 \times 10^{-19} \text{ J} \]